This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleGrowth of Accreting Supermassive Black HoleSeeds and Neutrino Radiation
Correspondence should be addressed to Gagik Ter-Kazarian gago 50yahoocom
Received 1 May 2014 Revised 26 June 2014 Accepted 27 June 2014
Academic Editor Gary Wegner
Copyright copy 2015 Gagik Ter-KazarianThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In the framework of microscopic theory of black hole (MTBH) which explores the most important processes of rearrangementof vacuum state and spontaneous breaking of gravitation gauge symmetry at huge energies we have undertaken a large series ofnumerical simulations with the goal to trace an evolution of themass assembly history of 377 plausible accreting supermassive blackhole seeds in active galactic nuclei (AGNs) to the present time and examine the observable signatures today Given the redshiftsmasses and luminosities of these black holes at present time collected from the literature we compute the initial redshifts andmasses of the corresponding seed black holes For the present masses119872BH119872⊙
≃ 11 times 106 to 13 times 1010 of 377 black holes the
computed intermediate seedmasses are ranging from119872SeedBH 119872⊙
≃ 264 to 29times105 We also compute the fluxes of ultrahigh energy(UHE) neutrinos produced via simple or modified URCA processes in superdense protomatter nuclei The AGNs are favored aspromising pure UHE neutrino sources because the computed neutrino fluxes are highly beamed along the plane of accretion diskpeaked at high energies and collimated in smaller opening angle (120579 ≪ 1)
1 Introduction
With typical bolometric luminosities sim1045minus48 erg sminus1 theAGNs are amongst the most luminous emitters in the uni-verse particularly at high energies (gamma-rays) and radiowavelengths From its historical development up to currentinterests the efforts in theAGNphysics have evoked the studyof a major unsolved problem of how efficiently such hugeenergies observed can be generatedThis energy scale severelychallenges conventional source models The huge energyrelease from compact regions of AGN requires extremelyhigh efficiency (typically ge10 per cent) of conversion of restmass to other forms of energy This serves as the mainargument in favour of supermassive black holes with massesof millions to billions of times the mass of the Sun as centralengines ofmassive AGNsThe astrophysical black holes comein a wide range of masses from ge3119872
⊙for stellar mass black
holes [1] to sim1010119872⊙for supermassive black holes [2 3]
Demography of local galaxies suggests that most galaxiesharbour quiescent supermassive black holes in their nucleiat the present time and that the mass of the hosted blackhole is correlated with properties of the host bulge The
visible universe should therefore contain at least 100 billionsupermassive black holes A complex study of evolution ofAGNs requires an answer to the key questions such as howdidthe first black holes form how did massive black holes get tothe galaxy centers and how did they grow in accreting massnamely an understanding of the important phenomenon ofmass assembly history of accreting supermassive black holeseeds The observations support the idea that black holesgrow in tandem with their hosts throughout cosmic historystarting from the earliest times While the exact mechanismfor the formation of the first black holes is not currentlyknown there are several prevailing theories [4] Howevereach proposal towards formation and growth of initial seedblack holes has its own advantage and limitations in provingthe whole view of the issue In this report we review the massassembly history of 377 plausible accreting supermassiveblack hole seeds in AGNs and their neutrino radiation inthe framework of gravitation theory which explores themostimportant processes of rearrangement of vacuum state anda spontaneous breaking of gravitation gauge symmetry athuge energies We will proceed according to the followingstructureMost observational theoretical and computational
Hindawi Publishing CorporationJournal of AstrophysicsVolume 2015 Article ID 205367 30 pageshttpdxdoiorg1011552015205367
2 Journal of Astrophysics
aspects of the growth of black hole seeds are summarizedin Section 2 The other important phenomenon of ultrahighenergy cosmic rays in relevance to AGNs is discussed inSection 3 The objectives of suggested approach are outlinedin Section 4 In Section 5 we review the spherical accretionon superdense protomatter nuclei in use In Section 6 wediscuss the growth of the seed black hole at accretion andderive its intermediate mass initial redshift and neutrinopreradiation time (PRT) Section 7 is devoted to the neutrinoradiation produced in superdense protomatter nuclei Thesimulation results of the seed black hole intermediate massesPRTs seed redshifts and neutrino fluxes for 377 AGN blackholes are brought in Section 8 The concluding remarks aregiven in Section 9 We will refrain from providing lengthydetails of the proposed gravitation theory at huge energiesand neutrino flux computations For these the reader isinvited to visit the original papers and appendices of thepresent paper In the latter we also complete the spacetimedeformation theory in the model context of gravitation bynew investigation of building up the complex of distortion(DC) of spacetime continuum and showing how it restoresthe world-deformation tensor which still has been put in byhand Finally note that we regard the considered black holesonly as the potential neutrino sources The obtained resultshowever may suffer if not all live black holes at present residein final stage of their growth driven by the formation ofprotomatter disk at accretion and they radiate neutrino Weoften suppress the indices without notice Unless otherwisestated we take geometrized units throughout this paper
2 A Breakthrough in Observational andComputational Aspects on Growth of BlackHole Seeds
Significant progress has been made in the last few years inunderstanding how supermassive black holes form and growGiven the currentmasses of 106minus9119872
⊙ most black hole growth
happens in the AGN phase A significant fraction of the totalblack hole growth 60 [6] happens in the most luminousAGN quasars In an AGN phase which lasts sim108 years thecentral supermassive black hole can gain up to sim107minus8119872
⊙
so even the most massive galaxies will have only a few ofthese events over their lifetime Aforesaid gathers supportespecially from a breakthroughmade in recent observationaltheoretical and computational efforts in understanding ofevolution of black holes and their host galaxies particularlythrough self-regulated growth and feedback from accretion-powered outflows see for example [4 7ndash18] Whereas themultiwavelength methods are used to trace the growth ofseed BHs the prospects for future observations are reviewedThe observations provide strong support for the existenceof a correlation between supermassive black holes and theirhosts out to the highest redshifts The observations of thequasar luminosity function show that the most supermassiveblack holes get most of their mass at high redshift whileat low redshift only low mass black holes are still growing[19] This is observed in both optical [20] and hard X-ray luminosity functions [19 21] which indicates that this
result is independent of obscuration Natarajan [13] hasreported that the initial black hole seeds form at extremelyhigh redshifts from the direct collapse of pregalactic gasdiscs Populating dark matter halos with seeds formed inthis fashion and using a Monte-Carlo merger tree approachhe has predicted the black hole mass function at highredshifts and at the present time The most aspects of themodels that describe the growth and accretion history ofsupermassive black holes and evolution of this scenario havebeen presented in detail by [9 10] In these models at earlytimes the properties of the assembling black hole seeds aremore tightly coupled to properties of the dark matter haloas their growth is driven by the merger history of halosWhile a clear picture of the history of black hole growth isemerging significant uncertainties still remain [14] and inspite of recent advances [6 13] the origin of the seed blackholes remains an unsolved problem at present The NuSTARdeep high-energy observations will enable obtaining a nearlycomplete AGN survey including heavily obscured Compton-thick sources up to 119911 sim 15 [22] A similar mission ASTRO-H [23] will be launched by Japan in 2014These observationsin combination with observations at longer wavelengths willallow for the detection and identification of most growingsupermassive black holes at 119911 sim 1 The ultradeep X-ray andnear-infrared surveys covering at least sim1 deg2 are requiredto constrain the formation of the first black hole seeds Thiswill likely require the use of the next generation of space-based observatories such as the James Webb Space Telescopeand the International X-ray Observatory The superb spatialresolution and sensitivity of the Atacama Large MillimeterArray (ALMA) [24] will revolutionize our understanding ofgalaxy evolution Combining these new data with existingmultiwavelength information will finally allow astrophysi-cists to pave the way for later efforts by pioneering some ofthe census of supermassive black hole growth in use today
3 UHE Cosmic-Ray Particles
The galactic sources like supernova remnants (SNRs) ormicroquasars are thought to accelerate particles at least upto energies of 3 times 1015 eV The ultrahigh energy cosmic-ray (UHECR) particles with even higher energies have sincebeen detected (comprehensive reviews can be found in [25ndash29]) The accelerated protons or heavier nuclei up to energiesexceeding 1020 eV are firstly observed by [30] The cosmic-ray events with the highest energies so far detected haveenergies of 2 times 1011 GeV [31] and 3 times 1011 GeV [32] Theseenergies are 107 times higher than the most energetic man-made accelerator the LHC at CERN These highest energiesare believed to be reached in extragalactic sources like AGNsor gamma-ray bursts (GRBs) During propagation of suchenergetic particles through the universe the threshold forpion photoproduction on the microwave background is sim2times 1010 GeV and at sim3 times 1011 GeV the energy-loss distance isabout 20Mpc Propagation of cosmic rays over substantiallylarger distances gives rise to a cutoff in the spectrum at sim1011 GeV as was first shown by [33 34] the GZK cutoffThe recent confirmation [35 36] of GZK suppression in the
Journal of Astrophysics 3
cosmic-ray energy spectrum indicates that the cosmic rayswith energies above the GZK cutoff 119864GZK sim 40EeV mostlycome from relatively close (within the GZK radius 119903GZK sim100Mpc) extragalactic sources However despite the detailedmeasurements of the cosmic-ray spectrum the identificationof the sources of the cosmic-ray particles is still an openquestion as they are deflected in the galactic and extragalacticmagnetic fields and hence have lost all information abouttheir originwhen reaching Earth Only at the highest energiesbeyond sim10196 GeV cosmic-ray particles may retain enoughdirectional information to locate their sources The lattermust be powerful enough to sustain the energy density inextragalactic cosmic rays of about 3times10minus19 erg cmminus3 which isequivalent tosim8times 1044 ergMpcminus3 yrminus1Though it has not beenpossible up to now to identify the sources of galactic or extra-galactic cosmic rays general considerations allow limitingpotential source classes For example the existing data on thecosmic-ray spectrum and on the isotropic 100MeV gamma-ray background limit significantly the parameter space inwhich topological defects can generate the flux of the highestenergy cosmic rays and rule out models with the standardX-particle mass of 1016 GeV and higher [37] Eventually theneutrinos will serve as unique astronomical messengers andthey will significantly enhance and extend our knowledgeon galactic and extragalactic sources of the UHE universeIndeed except for oscillations induced by transit in a vacuumHiggs field neutrinos can penetrate cosmological distancesand their trajectories are not deflected by magnetic fieldsas they are neutral providing powerful probes of highenergy astrophysics in ways which no other particle canMoreover the flavor composition of neutrinos originating atastrophysical sources can serve as a probe of new physicsin the electroweak sector Therefore an appealing possibilityamong the various hypotheses of the origin of UHECR isso-called Z-burst scenario [38ndash51] This suggests that if ZeVastrophysical neutrino beam is sufficiently strong it canproduce a large fraction of observed UHECR particles within100Mpc by hitting local light relic neutrinos clustered in darkhalos and form UHECR through the hadronic Z (s-channelproduction) andW-bosons (t-channel production) decays byweak interactions The discovery of UHE neutrino sourceswould also clarify the productionmechanism of the GeV-TeVgamma rays observed on Earth [43 52 53] as TeV photonsare also produced in the up-scattering of photons in reactionsto accelerated electrons (inverse-Compton scattering) Thedirect link between TeV gamma-ray photons and neutrinosthrough the charged and neutral pion production which iswell known from particle physics allows for a quite robustprediction of the expected neutrino fluxes provided thatthe sources are transparent and the observed gamma raysoriginate from pion decay The weakest link in the Z-bursthypothesis is probably both unknown boosting mechanismof the primary neutrinos up to huge energies of hundredsZeV and their large flux required at the resonant energy 119864] ≃119872
2
119885(2119898]) ≃ 42 times 10
21 eV (eV119898]) well above the GZKcutoff Such a flux severely challenges conventional sourcemodels Any concomitant photon flux should not violateexisting upper limits [37 48 49 54] The obvious question is
then raised where in the Cosmos are these neutrinos comingfrom It turns out that currently at energies in excess of10
19 eV there are only two good candidate source classes forUHE neutrinos AGNs and GRBs The AGNs as significantpoint sources of neutrinos were analyzed in [50 55 56]While hard to detect neutrinos have the advantage of repre-senting aforesaid unique fingerprints of hadron interactionsand therefore of the sources of cosmic rays Two basicevent topologies can be distinguished track-like patterns ofdetected Cherenkov light (hits) which originate from muonsproduced in charged-current interactions of muon neutrinos(muon channel) spherical hit patterns which originate fromthe hadronic cascade at the vertex of neutrino interactionsor the electromagnetic cascade of electrons from charged-current interactions of electron neutrinos (cascade channel)If the charged-current interaction happens inside the detectoror in case of charged-current tau-neutrino interactions thesetwo topologies overlap which complicates the reconstruc-tion At the relevant energies the neutrino is approximatelycollinear with the muon and hence the muon channel isthe prime channel for the search for point-like sources ofcosmic neutrinos On the other hand cascades deposit allof their energy inside the detector and therefore allow fora much better energy reconstruction with a resolution of afew 10 Finally numerous reports are available at presentin literature on expected discovery potential and sensitivityof experiments to neutrino point-like sources Currentlyoperating high energy neutrino telescopes attempt to detectUHE neutrinos such as ANTARES [57 58] which is themostsensitive neutrino telescope in theNorthernHemisphere Ice-Cube [35 59ndash64] which is worldwide largest and hence mostsensitive neutrino telescope in the Southern HemisphereBAIKAL [65] as well as the CR extended experiments ofTheTelescope Array [66] Pierre Auger Observatory [67 68] andJEM-EUSO mission [69] The JEM-EUSO mission whichis planned to be launched by a H2B rocket around 2015-2016 is designed to explore the extremes in the universe andfundamental physics through the detection of the extremeenergy (119864 gt 10
20 eV) cosmic rays The possible originsof the soon-to-be famous 28 IceCube neutrino-PeV events[59ndash61] are the first hint for astrophysical neutrino signalAartsen et al have published an observation of two sim1 PeVneutrinos with a 119875 value 28120590 beyond the hypothesis thatthese events were atmospherically generated [59] The anal-ysis revealed an additional 26 neutrino candidates depositingldquoelectromagnetic equivalent energiesrdquo ranging from about30 TeV up to 250 TeV [61] New results were presented at theIceCube Particle Astrophysics Symposium (IPA 2013) [62ndash64] If cosmic neutrinos are primarily of extragalactic originthen the 100GeV gamma ray flux observed by Fermi-LATconstrains the normalization at PeV energies at injectionwhich in turn demands a neutrino spectral index Γ lt 21 [70]
4 MTBH Revisited Preliminaries
For the benefit of the reader a brief outline of the key ideasbehind the microscopic theory of black hole as a guidingprinciple is given in this section to make the rest of the
4 Journal of Astrophysics
paper understandable There is a general belief reinforced bystatements in textbooks that according to general relativity(GR) a long-standing standard phenomenological black holemodel (PBHM)mdashnamely the most general Kerr-Newmanblack hole model with parameters of mass (119872) angularmomentum (119869) and charge (119876) still has to be put in byhandmdashcan describe the growth of accreting supermassiveblack hole seed However such beliefs are suspect and shouldbe critically reexamined The PBHM cannot be currentlyaccepted as convincing model for addressing the afore-mentioned problems because in this framework the verysource of gravitational field of the black hole is a kind ofcurvature singularity at the center of the stationary blackhole A meaningless central singularity develops which ishidden behind the event horizon The theory breaks downinside the event horizon which is causally disconnected fromthe exterior world Either the Kruskal continuation of theSchwarzschild (119869 = 0 119876 = 0) metric or the Kerr (119876 = 0)metric or the Reissner-Nordstrom (119869 = 0) metric showsthat the static observers fail to exist inside the horizonAny object that collapses to form a black hole will go onto collapse to a singularity inside the black hole Therebyany timelike worldline must strike the central singularitywhich wholly absorbs the infalling matter Therefore theultimate fate of collapsing matter once it has crossed theblack hole surface is unknown This in turn disables anyaccumulation of matter in the central part and thus neitherthe growth of black holes nor the increase of their mass-energy density could occur at accretion of outside matteror by means of merger processes As a consequence themass and angular momentum of black holes will not changeover the lifetime of the universe But how can one be surethat some hitherto unknown source of pressure does notbecome important at huge energies and halt the collapse Tofill the voidwhich the standard PBHMpresents one plausibleidea to innovate the solution to alluded key problems wouldappear to be the framework of microscopic theory of blackhole This theory has been originally proposed by [71] andreferences therein and thoroughly discussed in [72ndash75]Here we recount some of the highlights of the MTBHwhich is the extension of PBHM and rather completes it byexploring the most important processes of rearrangementof vacuum state and a spontaneous breaking of gravitationgauge symmetry at huge energies [71 74 76] We will notbe concerned with the actual details of this framework butonly use it as a backdrop to validate the theory with someobservational tests For details the interested reader is invitedto consult the original papers Discussed gravitational theoryis consistent with GR up to the limit of neutron stars Butthis theory manifests its virtues applied to the physics ofinternal structure of galactic nuclei In the latter a significantchange of properties of spacetime continuum so-calledinner distortion (ID) arises simultaneously with the stronggravity at huge energies (see Appendix A) Consequently thematter undergoes phase transition of second kind whichsupplies a powerful pathway to form a stable superdenseprotomatter core (SPC) inside the event horizon Due to thisthe stable equilibrium holds in outward layers too and thusan accumulation of matter is allowed now around the SPC
The black hole models presented in phenomenological andmicroscopic frameworks have been schematically plotted inFigure 1 to guide the eye A crucial point of the MTBH isthat a central singularity cannot occur which is now replacedby finite though unbelievably extreme conditions held in theSPC where the static observers existed The SPC surroundedby the accretion disk presents the microscopic model ofAGNThe SPC accommodates the highest energy scale up tohundreds of ZeV in central protomatter core which accountsfor the spectral distribution of the resulting radiation ofgalactic nuclei External physics of accretion onto the blackhole in earlier part of its lifetime is identical to the processesin Schwarzschildrsquos model However a strong difference inthe model context between the phenomenological black holeand the SPC is arising in the second part of its lifetime(see Section 6) The seed black hole might grow up drivenby the accretion of outside matter when it was gettingmost of its mass An infalling matter with time forms theprotomatter disk around the protomatter core tapering offfaster at reaching out the thin edge of the event horizon Atthis metric singularity inevitably disappears (see appendices)and the neutrinos may escape through vista to outsideworld even after the neutrino trapping We study the growthof protomatter disk and derive the intermediate mass andinitial redshift of seed black hole and examine luminositiesneutrino surfaces for the disk In this framework we havecomputed the fluxes of UHE neutrinos [75] produced in themediumof the SPC via simple (quark and pionic reactions) ormodified URCA processes even after the neutrino trapping(G Gamow was inspired to name the process URCA afterthe name of a casino in Rio de Janeiro when M Schenbergremarked to him that ldquothe energy disappears in the nucleusof the supernova as quickly as the money disappeared at thatroulette tablerdquo) The ldquotrappingrdquo is due to the fact that asthe neutrinos are formed in protomatter core at superhighdensities they experience greater difficulty escaping from theprotomatter core before being dragged along with thematternamely the neutrinos are ldquotrappedrdquo comove with matterThe part of neutrinos annihilates to produce further thesecondary particles of expected ultrahigh energies In thismodel of course a key open question is to enlighten themechanisms that trigger the activity and how a large amountofmatter can be steadily funneled to the central regions to fuelthis activity In high luminosity AGNs the large-scale internalgravitational instabilities drive gas towards the nucleus whichtrigger big starbursts and the coeval compact cluster justformed It seemed they have some connection to the nuclearfueling through mass loss of young stars as well as their tidaldisruption and supernovae Note that we regard the UHECRparticles as a signature of existence of superdence protomattersources in the universe Since neutrino events are expected tobe of sufficient intensity our estimates can be used to guideinvestigations of neutrino detectors for the distant future
5 Spherical Accretion onto SPC
As alluded to above the MTBH framework supports the ideaof accreting supermassive black holes which link to AGNsIn order to compute the mass accretion rate in use it is
Journal of Astrophysics 5
AGN
ADAD
EH
infin
(a)
AGN
PDPC
ADAD
EHSPC
(b)
Figure 1 (a) The phenomenological model of AGN with the central stationary black hole The meaningless singularity occurs at the centerinside the black hole (b) The microscopic model of AGN with the central stable SPC In due course the neutrinos of huge energies mayescape through the vista to outside world Accepted notations EH = event horizon AD = accretion disk SPC = superdense protomatter corePC = protomatter core
necessary to study the accretion onto central supermassiveSPC The main features of spherical accretion can be brieflysummed up in the following three idealized models thatillustrate some of the associated physics [72]
51 Free Radial Infall We examine the motion of freelymoving test particle by exploring the external geometry of theSPC with the line element (A7) at 119909 = 0 Let us denote the 4-vector of velocity of test particle V120583 = 119889120583119889 120583 = (119905 )and consider it initially for simplest radial infall V2 = V3 =0 We determine the value of local velocity V lt 0 of theparticle for the moment of crossing the EH sphere as well asat reaching the surface of central stable SPC The equation ofgeodesics is derived from the variational principle 120575 int 119889119878 = 0which is the extremum of the distance along the wordline forthe Lagrangian at hand
2119871 = (1 minus 1199090)2 1199052
minus (1 + 1199090)2 119903
2
minus 2sin2 1205932 minus 2 120579
2
(1)
where 119905 equiv 119889 119889120582 is the 119905-component of 4-momentum and 120582is the affine parameter along the worldline We are using anaffine parametrization (by a rescaling 120582 rarr 120582(120582
1015840)) such that
119871 = const is constant along the curve A static observermakesmeasurements with local orthonormal tetrad
119890=10038161003816100381610038161 minus 1199090
1003816100381610038161003816
minus1
119890119905 119890
= (1 + 119909
0)minus1
119890119903
119890=
minus1119890120579 119890
= ( sin )
minus1
119890120579
(2)
The Euler-Lagrange equations for and can be derivedfrom the variational principle A local measurement of theparticlersquos energy made by a static observer in the equatorialplane gives the time component of the 4-momentum asmeasured in the observerrsquos local orthonormal frame This
is the projection of the 4-momentum along the time basisvector The Euler-Lagrange equations show that if we orientthe coordinate system as initially the particle is moving in theequatorial plane (ie = 1205872 120579 = 0) then the particle alwaysremains in this plane There are two constants of the motioncorresponding to the ignorable coordinates and namelythe 119864-ldquoenergy-at-infinityrdquo and the 119897-angular momentum Weconclude that the free radial infall of a particle from theinfinity up to the moment of crossing the EH sphere aswell as at reaching the surface of central body is absolutelythe same as in the Schwarzschild geometry of black hole(Figure 2(a)) We clear up a general picture of orbits justoutside the event horizon by considering the Euler-Lagrangeequation for radial component with ldquoeffective potentialrdquo Thecircular orbits are stable if119881 is concave up namely at gt 4where is the mass of SPC The binding energy per unitmass of a particle in the last stable circular orbit at = 4is bind = (119898 minus 119864) ≃ 1 minus (2732)
12 Namely this isthe fraction of rest-mass energy released when test particleoriginally at rest at infinity spirals slowly toward the SPC tothe innermost stable circular orbit and then plunges into itThereby one of the important parameters is the capture crosssection for particles falling in from infinity 120590capt = 120587119887
2
maxwhere 119887max is the maximum impact parameter of a particlethat is captured
52 Collisionless Accretion The distribution function for acollisionless gas is determined by the collisionless Boltzmannequation or Vlasov equation For the stationary and sphericalflow we obtain then
(119864 gt 0) = 16120587 (119866)2
120588infinVminus1infin119888minus2 (3)
6 Journal of Astrophysics
AGN
SPC
x0 = 1x0 = 2
x0 = 0
t = infin
vr lt 0 vr = 0
r = infin
EH
(a)
1minus 1 1+x0
10
15
25
ntimes10
minus40(g
cmminus
3 )
(b)
Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of
BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere
(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect
the metric singularity vanishes and the particles well pass EH sphere
where the particle density 120588infin
is assumed to be uniform atfar from the SPC and the particle speed is V
infin≪ 1 During
the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899
infin)1199090rarr1
asymp
minus(ln 00)2 V
infin up to the ID threshold value
119889()
minus13=
04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909
0= 1) and in the sequel form the protomatter
disk around the protomatter core
53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903
119904outside the
event horizon The locally measured particle velocity readsV = (1 minus
00119864
2) where 119864 = 119864
infin119898 = (
00(1 minus 119906
2))12 and
119864infin
is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877
1198922 the proper velocity equals the speed of light
|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903
119904 it is expected that the particles be
nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)
as they were nonrelativistic at infinity (119886infin≪ 1) Considering
the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate
= 212058711989811989911990411990352
119904(ln
00)1015840
119904 (4)
where prime (1015840)119904denotes differentiation with respect to at
the point 119903119904 The gas compression can be estimated as
119899infin
asymp11990352
119904
21199032[(ln
00)1015840
119904
1 + 119903119903()]
12
(5)
The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109
6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole
The key objectives of the MTBH framework are then anincrease of the mass 119872Seed
BH gravitational radius 119877Seed119892
andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed
rarr
BH may most usefully be the increase of gravitational radiusor mass of black hole
Δ119877119892= 119877
BH119892minus 119877
Seed119892=2119866
1198882119872
119889=2119866
1198882120588119889119881119889
Δ119872BH = 119872BH minus119872SeedBH = 119872
SeedBHΔ119877
119892
119877Seed119892
(6)
where 119872119889 120588
119889 and 119881
119889 respectively are the total mass
density and volume of protomatter disk At the value BH119892
of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black
Journal of Astrophysics 7
PC PD120579
1205880
120593Z
Z0Z1
EHBH EHSeed
1205881120588
d120576d = d2Rg asymp 120579
RSPCRBHg
RSeedg
Rd
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
aspects of the growth of black hole seeds are summarizedin Section 2 The other important phenomenon of ultrahighenergy cosmic rays in relevance to AGNs is discussed inSection 3 The objectives of suggested approach are outlinedin Section 4 In Section 5 we review the spherical accretionon superdense protomatter nuclei in use In Section 6 wediscuss the growth of the seed black hole at accretion andderive its intermediate mass initial redshift and neutrinopreradiation time (PRT) Section 7 is devoted to the neutrinoradiation produced in superdense protomatter nuclei Thesimulation results of the seed black hole intermediate massesPRTs seed redshifts and neutrino fluxes for 377 AGN blackholes are brought in Section 8 The concluding remarks aregiven in Section 9 We will refrain from providing lengthydetails of the proposed gravitation theory at huge energiesand neutrino flux computations For these the reader isinvited to visit the original papers and appendices of thepresent paper In the latter we also complete the spacetimedeformation theory in the model context of gravitation bynew investigation of building up the complex of distortion(DC) of spacetime continuum and showing how it restoresthe world-deformation tensor which still has been put in byhand Finally note that we regard the considered black holesonly as the potential neutrino sources The obtained resultshowever may suffer if not all live black holes at present residein final stage of their growth driven by the formation ofprotomatter disk at accretion and they radiate neutrino Weoften suppress the indices without notice Unless otherwisestated we take geometrized units throughout this paper
2 A Breakthrough in Observational andComputational Aspects on Growth of BlackHole Seeds
Significant progress has been made in the last few years inunderstanding how supermassive black holes form and growGiven the currentmasses of 106minus9119872
⊙ most black hole growth
happens in the AGN phase A significant fraction of the totalblack hole growth 60 [6] happens in the most luminousAGN quasars In an AGN phase which lasts sim108 years thecentral supermassive black hole can gain up to sim107minus8119872
⊙
so even the most massive galaxies will have only a few ofthese events over their lifetime Aforesaid gathers supportespecially from a breakthroughmade in recent observationaltheoretical and computational efforts in understanding ofevolution of black holes and their host galaxies particularlythrough self-regulated growth and feedback from accretion-powered outflows see for example [4 7ndash18] Whereas themultiwavelength methods are used to trace the growth ofseed BHs the prospects for future observations are reviewedThe observations provide strong support for the existenceof a correlation between supermassive black holes and theirhosts out to the highest redshifts The observations of thequasar luminosity function show that the most supermassiveblack holes get most of their mass at high redshift whileat low redshift only low mass black holes are still growing[19] This is observed in both optical [20] and hard X-ray luminosity functions [19 21] which indicates that this
result is independent of obscuration Natarajan [13] hasreported that the initial black hole seeds form at extremelyhigh redshifts from the direct collapse of pregalactic gasdiscs Populating dark matter halos with seeds formed inthis fashion and using a Monte-Carlo merger tree approachhe has predicted the black hole mass function at highredshifts and at the present time The most aspects of themodels that describe the growth and accretion history ofsupermassive black holes and evolution of this scenario havebeen presented in detail by [9 10] In these models at earlytimes the properties of the assembling black hole seeds aremore tightly coupled to properties of the dark matter haloas their growth is driven by the merger history of halosWhile a clear picture of the history of black hole growth isemerging significant uncertainties still remain [14] and inspite of recent advances [6 13] the origin of the seed blackholes remains an unsolved problem at present The NuSTARdeep high-energy observations will enable obtaining a nearlycomplete AGN survey including heavily obscured Compton-thick sources up to 119911 sim 15 [22] A similar mission ASTRO-H [23] will be launched by Japan in 2014These observationsin combination with observations at longer wavelengths willallow for the detection and identification of most growingsupermassive black holes at 119911 sim 1 The ultradeep X-ray andnear-infrared surveys covering at least sim1 deg2 are requiredto constrain the formation of the first black hole seeds Thiswill likely require the use of the next generation of space-based observatories such as the James Webb Space Telescopeand the International X-ray Observatory The superb spatialresolution and sensitivity of the Atacama Large MillimeterArray (ALMA) [24] will revolutionize our understanding ofgalaxy evolution Combining these new data with existingmultiwavelength information will finally allow astrophysi-cists to pave the way for later efforts by pioneering some ofthe census of supermassive black hole growth in use today
3 UHE Cosmic-Ray Particles
The galactic sources like supernova remnants (SNRs) ormicroquasars are thought to accelerate particles at least upto energies of 3 times 1015 eV The ultrahigh energy cosmic-ray (UHECR) particles with even higher energies have sincebeen detected (comprehensive reviews can be found in [25ndash29]) The accelerated protons or heavier nuclei up to energiesexceeding 1020 eV are firstly observed by [30] The cosmic-ray events with the highest energies so far detected haveenergies of 2 times 1011 GeV [31] and 3 times 1011 GeV [32] Theseenergies are 107 times higher than the most energetic man-made accelerator the LHC at CERN These highest energiesare believed to be reached in extragalactic sources like AGNsor gamma-ray bursts (GRBs) During propagation of suchenergetic particles through the universe the threshold forpion photoproduction on the microwave background is sim2times 1010 GeV and at sim3 times 1011 GeV the energy-loss distance isabout 20Mpc Propagation of cosmic rays over substantiallylarger distances gives rise to a cutoff in the spectrum at sim1011 GeV as was first shown by [33 34] the GZK cutoffThe recent confirmation [35 36] of GZK suppression in the
Journal of Astrophysics 3
cosmic-ray energy spectrum indicates that the cosmic rayswith energies above the GZK cutoff 119864GZK sim 40EeV mostlycome from relatively close (within the GZK radius 119903GZK sim100Mpc) extragalactic sources However despite the detailedmeasurements of the cosmic-ray spectrum the identificationof the sources of the cosmic-ray particles is still an openquestion as they are deflected in the galactic and extragalacticmagnetic fields and hence have lost all information abouttheir originwhen reaching Earth Only at the highest energiesbeyond sim10196 GeV cosmic-ray particles may retain enoughdirectional information to locate their sources The lattermust be powerful enough to sustain the energy density inextragalactic cosmic rays of about 3times10minus19 erg cmminus3 which isequivalent tosim8times 1044 ergMpcminus3 yrminus1Though it has not beenpossible up to now to identify the sources of galactic or extra-galactic cosmic rays general considerations allow limitingpotential source classes For example the existing data on thecosmic-ray spectrum and on the isotropic 100MeV gamma-ray background limit significantly the parameter space inwhich topological defects can generate the flux of the highestenergy cosmic rays and rule out models with the standardX-particle mass of 1016 GeV and higher [37] Eventually theneutrinos will serve as unique astronomical messengers andthey will significantly enhance and extend our knowledgeon galactic and extragalactic sources of the UHE universeIndeed except for oscillations induced by transit in a vacuumHiggs field neutrinos can penetrate cosmological distancesand their trajectories are not deflected by magnetic fieldsas they are neutral providing powerful probes of highenergy astrophysics in ways which no other particle canMoreover the flavor composition of neutrinos originating atastrophysical sources can serve as a probe of new physicsin the electroweak sector Therefore an appealing possibilityamong the various hypotheses of the origin of UHECR isso-called Z-burst scenario [38ndash51] This suggests that if ZeVastrophysical neutrino beam is sufficiently strong it canproduce a large fraction of observed UHECR particles within100Mpc by hitting local light relic neutrinos clustered in darkhalos and form UHECR through the hadronic Z (s-channelproduction) andW-bosons (t-channel production) decays byweak interactions The discovery of UHE neutrino sourceswould also clarify the productionmechanism of the GeV-TeVgamma rays observed on Earth [43 52 53] as TeV photonsare also produced in the up-scattering of photons in reactionsto accelerated electrons (inverse-Compton scattering) Thedirect link between TeV gamma-ray photons and neutrinosthrough the charged and neutral pion production which iswell known from particle physics allows for a quite robustprediction of the expected neutrino fluxes provided thatthe sources are transparent and the observed gamma raysoriginate from pion decay The weakest link in the Z-bursthypothesis is probably both unknown boosting mechanismof the primary neutrinos up to huge energies of hundredsZeV and their large flux required at the resonant energy 119864] ≃119872
2
119885(2119898]) ≃ 42 times 10
21 eV (eV119898]) well above the GZKcutoff Such a flux severely challenges conventional sourcemodels Any concomitant photon flux should not violateexisting upper limits [37 48 49 54] The obvious question is
then raised where in the Cosmos are these neutrinos comingfrom It turns out that currently at energies in excess of10
19 eV there are only two good candidate source classes forUHE neutrinos AGNs and GRBs The AGNs as significantpoint sources of neutrinos were analyzed in [50 55 56]While hard to detect neutrinos have the advantage of repre-senting aforesaid unique fingerprints of hadron interactionsand therefore of the sources of cosmic rays Two basicevent topologies can be distinguished track-like patterns ofdetected Cherenkov light (hits) which originate from muonsproduced in charged-current interactions of muon neutrinos(muon channel) spherical hit patterns which originate fromthe hadronic cascade at the vertex of neutrino interactionsor the electromagnetic cascade of electrons from charged-current interactions of electron neutrinos (cascade channel)If the charged-current interaction happens inside the detectoror in case of charged-current tau-neutrino interactions thesetwo topologies overlap which complicates the reconstruc-tion At the relevant energies the neutrino is approximatelycollinear with the muon and hence the muon channel isthe prime channel for the search for point-like sources ofcosmic neutrinos On the other hand cascades deposit allof their energy inside the detector and therefore allow fora much better energy reconstruction with a resolution of afew 10 Finally numerous reports are available at presentin literature on expected discovery potential and sensitivityof experiments to neutrino point-like sources Currentlyoperating high energy neutrino telescopes attempt to detectUHE neutrinos such as ANTARES [57 58] which is themostsensitive neutrino telescope in theNorthernHemisphere Ice-Cube [35 59ndash64] which is worldwide largest and hence mostsensitive neutrino telescope in the Southern HemisphereBAIKAL [65] as well as the CR extended experiments ofTheTelescope Array [66] Pierre Auger Observatory [67 68] andJEM-EUSO mission [69] The JEM-EUSO mission whichis planned to be launched by a H2B rocket around 2015-2016 is designed to explore the extremes in the universe andfundamental physics through the detection of the extremeenergy (119864 gt 10
20 eV) cosmic rays The possible originsof the soon-to-be famous 28 IceCube neutrino-PeV events[59ndash61] are the first hint for astrophysical neutrino signalAartsen et al have published an observation of two sim1 PeVneutrinos with a 119875 value 28120590 beyond the hypothesis thatthese events were atmospherically generated [59] The anal-ysis revealed an additional 26 neutrino candidates depositingldquoelectromagnetic equivalent energiesrdquo ranging from about30 TeV up to 250 TeV [61] New results were presented at theIceCube Particle Astrophysics Symposium (IPA 2013) [62ndash64] If cosmic neutrinos are primarily of extragalactic originthen the 100GeV gamma ray flux observed by Fermi-LATconstrains the normalization at PeV energies at injectionwhich in turn demands a neutrino spectral index Γ lt 21 [70]
4 MTBH Revisited Preliminaries
For the benefit of the reader a brief outline of the key ideasbehind the microscopic theory of black hole as a guidingprinciple is given in this section to make the rest of the
4 Journal of Astrophysics
paper understandable There is a general belief reinforced bystatements in textbooks that according to general relativity(GR) a long-standing standard phenomenological black holemodel (PBHM)mdashnamely the most general Kerr-Newmanblack hole model with parameters of mass (119872) angularmomentum (119869) and charge (119876) still has to be put in byhandmdashcan describe the growth of accreting supermassiveblack hole seed However such beliefs are suspect and shouldbe critically reexamined The PBHM cannot be currentlyaccepted as convincing model for addressing the afore-mentioned problems because in this framework the verysource of gravitational field of the black hole is a kind ofcurvature singularity at the center of the stationary blackhole A meaningless central singularity develops which ishidden behind the event horizon The theory breaks downinside the event horizon which is causally disconnected fromthe exterior world Either the Kruskal continuation of theSchwarzschild (119869 = 0 119876 = 0) metric or the Kerr (119876 = 0)metric or the Reissner-Nordstrom (119869 = 0) metric showsthat the static observers fail to exist inside the horizonAny object that collapses to form a black hole will go onto collapse to a singularity inside the black hole Therebyany timelike worldline must strike the central singularitywhich wholly absorbs the infalling matter Therefore theultimate fate of collapsing matter once it has crossed theblack hole surface is unknown This in turn disables anyaccumulation of matter in the central part and thus neitherthe growth of black holes nor the increase of their mass-energy density could occur at accretion of outside matteror by means of merger processes As a consequence themass and angular momentum of black holes will not changeover the lifetime of the universe But how can one be surethat some hitherto unknown source of pressure does notbecome important at huge energies and halt the collapse Tofill the voidwhich the standard PBHMpresents one plausibleidea to innovate the solution to alluded key problems wouldappear to be the framework of microscopic theory of blackhole This theory has been originally proposed by [71] andreferences therein and thoroughly discussed in [72ndash75]Here we recount some of the highlights of the MTBHwhich is the extension of PBHM and rather completes it byexploring the most important processes of rearrangementof vacuum state and a spontaneous breaking of gravitationgauge symmetry at huge energies [71 74 76] We will notbe concerned with the actual details of this framework butonly use it as a backdrop to validate the theory with someobservational tests For details the interested reader is invitedto consult the original papers Discussed gravitational theoryis consistent with GR up to the limit of neutron stars Butthis theory manifests its virtues applied to the physics ofinternal structure of galactic nuclei In the latter a significantchange of properties of spacetime continuum so-calledinner distortion (ID) arises simultaneously with the stronggravity at huge energies (see Appendix A) Consequently thematter undergoes phase transition of second kind whichsupplies a powerful pathway to form a stable superdenseprotomatter core (SPC) inside the event horizon Due to thisthe stable equilibrium holds in outward layers too and thusan accumulation of matter is allowed now around the SPC
The black hole models presented in phenomenological andmicroscopic frameworks have been schematically plotted inFigure 1 to guide the eye A crucial point of the MTBH isthat a central singularity cannot occur which is now replacedby finite though unbelievably extreme conditions held in theSPC where the static observers existed The SPC surroundedby the accretion disk presents the microscopic model ofAGNThe SPC accommodates the highest energy scale up tohundreds of ZeV in central protomatter core which accountsfor the spectral distribution of the resulting radiation ofgalactic nuclei External physics of accretion onto the blackhole in earlier part of its lifetime is identical to the processesin Schwarzschildrsquos model However a strong difference inthe model context between the phenomenological black holeand the SPC is arising in the second part of its lifetime(see Section 6) The seed black hole might grow up drivenby the accretion of outside matter when it was gettingmost of its mass An infalling matter with time forms theprotomatter disk around the protomatter core tapering offfaster at reaching out the thin edge of the event horizon Atthis metric singularity inevitably disappears (see appendices)and the neutrinos may escape through vista to outsideworld even after the neutrino trapping We study the growthof protomatter disk and derive the intermediate mass andinitial redshift of seed black hole and examine luminositiesneutrino surfaces for the disk In this framework we havecomputed the fluxes of UHE neutrinos [75] produced in themediumof the SPC via simple (quark and pionic reactions) ormodified URCA processes even after the neutrino trapping(G Gamow was inspired to name the process URCA afterthe name of a casino in Rio de Janeiro when M Schenbergremarked to him that ldquothe energy disappears in the nucleusof the supernova as quickly as the money disappeared at thatroulette tablerdquo) The ldquotrappingrdquo is due to the fact that asthe neutrinos are formed in protomatter core at superhighdensities they experience greater difficulty escaping from theprotomatter core before being dragged along with thematternamely the neutrinos are ldquotrappedrdquo comove with matterThe part of neutrinos annihilates to produce further thesecondary particles of expected ultrahigh energies In thismodel of course a key open question is to enlighten themechanisms that trigger the activity and how a large amountofmatter can be steadily funneled to the central regions to fuelthis activity In high luminosity AGNs the large-scale internalgravitational instabilities drive gas towards the nucleus whichtrigger big starbursts and the coeval compact cluster justformed It seemed they have some connection to the nuclearfueling through mass loss of young stars as well as their tidaldisruption and supernovae Note that we regard the UHECRparticles as a signature of existence of superdence protomattersources in the universe Since neutrino events are expected tobe of sufficient intensity our estimates can be used to guideinvestigations of neutrino detectors for the distant future
5 Spherical Accretion onto SPC
As alluded to above the MTBH framework supports the ideaof accreting supermassive black holes which link to AGNsIn order to compute the mass accretion rate in use it is
Journal of Astrophysics 5
AGN
ADAD
EH
infin
(a)
AGN
PDPC
ADAD
EHSPC
(b)
Figure 1 (a) The phenomenological model of AGN with the central stationary black hole The meaningless singularity occurs at the centerinside the black hole (b) The microscopic model of AGN with the central stable SPC In due course the neutrinos of huge energies mayescape through the vista to outside world Accepted notations EH = event horizon AD = accretion disk SPC = superdense protomatter corePC = protomatter core
necessary to study the accretion onto central supermassiveSPC The main features of spherical accretion can be brieflysummed up in the following three idealized models thatillustrate some of the associated physics [72]
51 Free Radial Infall We examine the motion of freelymoving test particle by exploring the external geometry of theSPC with the line element (A7) at 119909 = 0 Let us denote the 4-vector of velocity of test particle V120583 = 119889120583119889 120583 = (119905 )and consider it initially for simplest radial infall V2 = V3 =0 We determine the value of local velocity V lt 0 of theparticle for the moment of crossing the EH sphere as well asat reaching the surface of central stable SPC The equation ofgeodesics is derived from the variational principle 120575 int 119889119878 = 0which is the extremum of the distance along the wordline forthe Lagrangian at hand
2119871 = (1 minus 1199090)2 1199052
minus (1 + 1199090)2 119903
2
minus 2sin2 1205932 minus 2 120579
2
(1)
where 119905 equiv 119889 119889120582 is the 119905-component of 4-momentum and 120582is the affine parameter along the worldline We are using anaffine parametrization (by a rescaling 120582 rarr 120582(120582
1015840)) such that
119871 = const is constant along the curve A static observermakesmeasurements with local orthonormal tetrad
119890=10038161003816100381610038161 minus 1199090
1003816100381610038161003816
minus1
119890119905 119890
= (1 + 119909
0)minus1
119890119903
119890=
minus1119890120579 119890
= ( sin )
minus1
119890120579
(2)
The Euler-Lagrange equations for and can be derivedfrom the variational principle A local measurement of theparticlersquos energy made by a static observer in the equatorialplane gives the time component of the 4-momentum asmeasured in the observerrsquos local orthonormal frame This
is the projection of the 4-momentum along the time basisvector The Euler-Lagrange equations show that if we orientthe coordinate system as initially the particle is moving in theequatorial plane (ie = 1205872 120579 = 0) then the particle alwaysremains in this plane There are two constants of the motioncorresponding to the ignorable coordinates and namelythe 119864-ldquoenergy-at-infinityrdquo and the 119897-angular momentum Weconclude that the free radial infall of a particle from theinfinity up to the moment of crossing the EH sphere aswell as at reaching the surface of central body is absolutelythe same as in the Schwarzschild geometry of black hole(Figure 2(a)) We clear up a general picture of orbits justoutside the event horizon by considering the Euler-Lagrangeequation for radial component with ldquoeffective potentialrdquo Thecircular orbits are stable if119881 is concave up namely at gt 4where is the mass of SPC The binding energy per unitmass of a particle in the last stable circular orbit at = 4is bind = (119898 minus 119864) ≃ 1 minus (2732)
12 Namely this isthe fraction of rest-mass energy released when test particleoriginally at rest at infinity spirals slowly toward the SPC tothe innermost stable circular orbit and then plunges into itThereby one of the important parameters is the capture crosssection for particles falling in from infinity 120590capt = 120587119887
2
maxwhere 119887max is the maximum impact parameter of a particlethat is captured
52 Collisionless Accretion The distribution function for acollisionless gas is determined by the collisionless Boltzmannequation or Vlasov equation For the stationary and sphericalflow we obtain then
(119864 gt 0) = 16120587 (119866)2
120588infinVminus1infin119888minus2 (3)
6 Journal of Astrophysics
AGN
SPC
x0 = 1x0 = 2
x0 = 0
t = infin
vr lt 0 vr = 0
r = infin
EH
(a)
1minus 1 1+x0
10
15
25
ntimes10
minus40(g
cmminus
3 )
(b)
Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of
BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere
(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect
the metric singularity vanishes and the particles well pass EH sphere
where the particle density 120588infin
is assumed to be uniform atfar from the SPC and the particle speed is V
infin≪ 1 During
the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899
infin)1199090rarr1
asymp
minus(ln 00)2 V
infin up to the ID threshold value
119889()
minus13=
04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909
0= 1) and in the sequel form the protomatter
disk around the protomatter core
53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903
119904outside the
event horizon The locally measured particle velocity readsV = (1 minus
00119864
2) where 119864 = 119864
infin119898 = (
00(1 minus 119906
2))12 and
119864infin
is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877
1198922 the proper velocity equals the speed of light
|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903
119904 it is expected that the particles be
nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)
as they were nonrelativistic at infinity (119886infin≪ 1) Considering
the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate
= 212058711989811989911990411990352
119904(ln
00)1015840
119904 (4)
where prime (1015840)119904denotes differentiation with respect to at
the point 119903119904 The gas compression can be estimated as
119899infin
asymp11990352
119904
21199032[(ln
00)1015840
119904
1 + 119903119903()]
12
(5)
The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109
6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole
The key objectives of the MTBH framework are then anincrease of the mass 119872Seed
BH gravitational radius 119877Seed119892
andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed
rarr
BH may most usefully be the increase of gravitational radiusor mass of black hole
Δ119877119892= 119877
BH119892minus 119877
Seed119892=2119866
1198882119872
119889=2119866
1198882120588119889119881119889
Δ119872BH = 119872BH minus119872SeedBH = 119872
SeedBHΔ119877
119892
119877Seed119892
(6)
where 119872119889 120588
119889 and 119881
119889 respectively are the total mass
density and volume of protomatter disk At the value BH119892
of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black
Journal of Astrophysics 7
PC PD120579
1205880
120593Z
Z0Z1
EHBH EHSeed
1205881120588
d120576d = d2Rg asymp 120579
RSPCRBHg
RSeedg
Rd
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
cosmic-ray energy spectrum indicates that the cosmic rayswith energies above the GZK cutoff 119864GZK sim 40EeV mostlycome from relatively close (within the GZK radius 119903GZK sim100Mpc) extragalactic sources However despite the detailedmeasurements of the cosmic-ray spectrum the identificationof the sources of the cosmic-ray particles is still an openquestion as they are deflected in the galactic and extragalacticmagnetic fields and hence have lost all information abouttheir originwhen reaching Earth Only at the highest energiesbeyond sim10196 GeV cosmic-ray particles may retain enoughdirectional information to locate their sources The lattermust be powerful enough to sustain the energy density inextragalactic cosmic rays of about 3times10minus19 erg cmminus3 which isequivalent tosim8times 1044 ergMpcminus3 yrminus1Though it has not beenpossible up to now to identify the sources of galactic or extra-galactic cosmic rays general considerations allow limitingpotential source classes For example the existing data on thecosmic-ray spectrum and on the isotropic 100MeV gamma-ray background limit significantly the parameter space inwhich topological defects can generate the flux of the highestenergy cosmic rays and rule out models with the standardX-particle mass of 1016 GeV and higher [37] Eventually theneutrinos will serve as unique astronomical messengers andthey will significantly enhance and extend our knowledgeon galactic and extragalactic sources of the UHE universeIndeed except for oscillations induced by transit in a vacuumHiggs field neutrinos can penetrate cosmological distancesand their trajectories are not deflected by magnetic fieldsas they are neutral providing powerful probes of highenergy astrophysics in ways which no other particle canMoreover the flavor composition of neutrinos originating atastrophysical sources can serve as a probe of new physicsin the electroweak sector Therefore an appealing possibilityamong the various hypotheses of the origin of UHECR isso-called Z-burst scenario [38ndash51] This suggests that if ZeVastrophysical neutrino beam is sufficiently strong it canproduce a large fraction of observed UHECR particles within100Mpc by hitting local light relic neutrinos clustered in darkhalos and form UHECR through the hadronic Z (s-channelproduction) andW-bosons (t-channel production) decays byweak interactions The discovery of UHE neutrino sourceswould also clarify the productionmechanism of the GeV-TeVgamma rays observed on Earth [43 52 53] as TeV photonsare also produced in the up-scattering of photons in reactionsto accelerated electrons (inverse-Compton scattering) Thedirect link between TeV gamma-ray photons and neutrinosthrough the charged and neutral pion production which iswell known from particle physics allows for a quite robustprediction of the expected neutrino fluxes provided thatthe sources are transparent and the observed gamma raysoriginate from pion decay The weakest link in the Z-bursthypothesis is probably both unknown boosting mechanismof the primary neutrinos up to huge energies of hundredsZeV and their large flux required at the resonant energy 119864] ≃119872
2
119885(2119898]) ≃ 42 times 10
21 eV (eV119898]) well above the GZKcutoff Such a flux severely challenges conventional sourcemodels Any concomitant photon flux should not violateexisting upper limits [37 48 49 54] The obvious question is
then raised where in the Cosmos are these neutrinos comingfrom It turns out that currently at energies in excess of10
19 eV there are only two good candidate source classes forUHE neutrinos AGNs and GRBs The AGNs as significantpoint sources of neutrinos were analyzed in [50 55 56]While hard to detect neutrinos have the advantage of repre-senting aforesaid unique fingerprints of hadron interactionsand therefore of the sources of cosmic rays Two basicevent topologies can be distinguished track-like patterns ofdetected Cherenkov light (hits) which originate from muonsproduced in charged-current interactions of muon neutrinos(muon channel) spherical hit patterns which originate fromthe hadronic cascade at the vertex of neutrino interactionsor the electromagnetic cascade of electrons from charged-current interactions of electron neutrinos (cascade channel)If the charged-current interaction happens inside the detectoror in case of charged-current tau-neutrino interactions thesetwo topologies overlap which complicates the reconstruc-tion At the relevant energies the neutrino is approximatelycollinear with the muon and hence the muon channel isthe prime channel for the search for point-like sources ofcosmic neutrinos On the other hand cascades deposit allof their energy inside the detector and therefore allow fora much better energy reconstruction with a resolution of afew 10 Finally numerous reports are available at presentin literature on expected discovery potential and sensitivityof experiments to neutrino point-like sources Currentlyoperating high energy neutrino telescopes attempt to detectUHE neutrinos such as ANTARES [57 58] which is themostsensitive neutrino telescope in theNorthernHemisphere Ice-Cube [35 59ndash64] which is worldwide largest and hence mostsensitive neutrino telescope in the Southern HemisphereBAIKAL [65] as well as the CR extended experiments ofTheTelescope Array [66] Pierre Auger Observatory [67 68] andJEM-EUSO mission [69] The JEM-EUSO mission whichis planned to be launched by a H2B rocket around 2015-2016 is designed to explore the extremes in the universe andfundamental physics through the detection of the extremeenergy (119864 gt 10
20 eV) cosmic rays The possible originsof the soon-to-be famous 28 IceCube neutrino-PeV events[59ndash61] are the first hint for astrophysical neutrino signalAartsen et al have published an observation of two sim1 PeVneutrinos with a 119875 value 28120590 beyond the hypothesis thatthese events were atmospherically generated [59] The anal-ysis revealed an additional 26 neutrino candidates depositingldquoelectromagnetic equivalent energiesrdquo ranging from about30 TeV up to 250 TeV [61] New results were presented at theIceCube Particle Astrophysics Symposium (IPA 2013) [62ndash64] If cosmic neutrinos are primarily of extragalactic originthen the 100GeV gamma ray flux observed by Fermi-LATconstrains the normalization at PeV energies at injectionwhich in turn demands a neutrino spectral index Γ lt 21 [70]
4 MTBH Revisited Preliminaries
For the benefit of the reader a brief outline of the key ideasbehind the microscopic theory of black hole as a guidingprinciple is given in this section to make the rest of the
4 Journal of Astrophysics
paper understandable There is a general belief reinforced bystatements in textbooks that according to general relativity(GR) a long-standing standard phenomenological black holemodel (PBHM)mdashnamely the most general Kerr-Newmanblack hole model with parameters of mass (119872) angularmomentum (119869) and charge (119876) still has to be put in byhandmdashcan describe the growth of accreting supermassiveblack hole seed However such beliefs are suspect and shouldbe critically reexamined The PBHM cannot be currentlyaccepted as convincing model for addressing the afore-mentioned problems because in this framework the verysource of gravitational field of the black hole is a kind ofcurvature singularity at the center of the stationary blackhole A meaningless central singularity develops which ishidden behind the event horizon The theory breaks downinside the event horizon which is causally disconnected fromthe exterior world Either the Kruskal continuation of theSchwarzschild (119869 = 0 119876 = 0) metric or the Kerr (119876 = 0)metric or the Reissner-Nordstrom (119869 = 0) metric showsthat the static observers fail to exist inside the horizonAny object that collapses to form a black hole will go onto collapse to a singularity inside the black hole Therebyany timelike worldline must strike the central singularitywhich wholly absorbs the infalling matter Therefore theultimate fate of collapsing matter once it has crossed theblack hole surface is unknown This in turn disables anyaccumulation of matter in the central part and thus neitherthe growth of black holes nor the increase of their mass-energy density could occur at accretion of outside matteror by means of merger processes As a consequence themass and angular momentum of black holes will not changeover the lifetime of the universe But how can one be surethat some hitherto unknown source of pressure does notbecome important at huge energies and halt the collapse Tofill the voidwhich the standard PBHMpresents one plausibleidea to innovate the solution to alluded key problems wouldappear to be the framework of microscopic theory of blackhole This theory has been originally proposed by [71] andreferences therein and thoroughly discussed in [72ndash75]Here we recount some of the highlights of the MTBHwhich is the extension of PBHM and rather completes it byexploring the most important processes of rearrangementof vacuum state and a spontaneous breaking of gravitationgauge symmetry at huge energies [71 74 76] We will notbe concerned with the actual details of this framework butonly use it as a backdrop to validate the theory with someobservational tests For details the interested reader is invitedto consult the original papers Discussed gravitational theoryis consistent with GR up to the limit of neutron stars Butthis theory manifests its virtues applied to the physics ofinternal structure of galactic nuclei In the latter a significantchange of properties of spacetime continuum so-calledinner distortion (ID) arises simultaneously with the stronggravity at huge energies (see Appendix A) Consequently thematter undergoes phase transition of second kind whichsupplies a powerful pathway to form a stable superdenseprotomatter core (SPC) inside the event horizon Due to thisthe stable equilibrium holds in outward layers too and thusan accumulation of matter is allowed now around the SPC
The black hole models presented in phenomenological andmicroscopic frameworks have been schematically plotted inFigure 1 to guide the eye A crucial point of the MTBH isthat a central singularity cannot occur which is now replacedby finite though unbelievably extreme conditions held in theSPC where the static observers existed The SPC surroundedby the accretion disk presents the microscopic model ofAGNThe SPC accommodates the highest energy scale up tohundreds of ZeV in central protomatter core which accountsfor the spectral distribution of the resulting radiation ofgalactic nuclei External physics of accretion onto the blackhole in earlier part of its lifetime is identical to the processesin Schwarzschildrsquos model However a strong difference inthe model context between the phenomenological black holeand the SPC is arising in the second part of its lifetime(see Section 6) The seed black hole might grow up drivenby the accretion of outside matter when it was gettingmost of its mass An infalling matter with time forms theprotomatter disk around the protomatter core tapering offfaster at reaching out the thin edge of the event horizon Atthis metric singularity inevitably disappears (see appendices)and the neutrinos may escape through vista to outsideworld even after the neutrino trapping We study the growthof protomatter disk and derive the intermediate mass andinitial redshift of seed black hole and examine luminositiesneutrino surfaces for the disk In this framework we havecomputed the fluxes of UHE neutrinos [75] produced in themediumof the SPC via simple (quark and pionic reactions) ormodified URCA processes even after the neutrino trapping(G Gamow was inspired to name the process URCA afterthe name of a casino in Rio de Janeiro when M Schenbergremarked to him that ldquothe energy disappears in the nucleusof the supernova as quickly as the money disappeared at thatroulette tablerdquo) The ldquotrappingrdquo is due to the fact that asthe neutrinos are formed in protomatter core at superhighdensities they experience greater difficulty escaping from theprotomatter core before being dragged along with thematternamely the neutrinos are ldquotrappedrdquo comove with matterThe part of neutrinos annihilates to produce further thesecondary particles of expected ultrahigh energies In thismodel of course a key open question is to enlighten themechanisms that trigger the activity and how a large amountofmatter can be steadily funneled to the central regions to fuelthis activity In high luminosity AGNs the large-scale internalgravitational instabilities drive gas towards the nucleus whichtrigger big starbursts and the coeval compact cluster justformed It seemed they have some connection to the nuclearfueling through mass loss of young stars as well as their tidaldisruption and supernovae Note that we regard the UHECRparticles as a signature of existence of superdence protomattersources in the universe Since neutrino events are expected tobe of sufficient intensity our estimates can be used to guideinvestigations of neutrino detectors for the distant future
5 Spherical Accretion onto SPC
As alluded to above the MTBH framework supports the ideaof accreting supermassive black holes which link to AGNsIn order to compute the mass accretion rate in use it is
Journal of Astrophysics 5
AGN
ADAD
EH
infin
(a)
AGN
PDPC
ADAD
EHSPC
(b)
Figure 1 (a) The phenomenological model of AGN with the central stationary black hole The meaningless singularity occurs at the centerinside the black hole (b) The microscopic model of AGN with the central stable SPC In due course the neutrinos of huge energies mayescape through the vista to outside world Accepted notations EH = event horizon AD = accretion disk SPC = superdense protomatter corePC = protomatter core
necessary to study the accretion onto central supermassiveSPC The main features of spherical accretion can be brieflysummed up in the following three idealized models thatillustrate some of the associated physics [72]
51 Free Radial Infall We examine the motion of freelymoving test particle by exploring the external geometry of theSPC with the line element (A7) at 119909 = 0 Let us denote the 4-vector of velocity of test particle V120583 = 119889120583119889 120583 = (119905 )and consider it initially for simplest radial infall V2 = V3 =0 We determine the value of local velocity V lt 0 of theparticle for the moment of crossing the EH sphere as well asat reaching the surface of central stable SPC The equation ofgeodesics is derived from the variational principle 120575 int 119889119878 = 0which is the extremum of the distance along the wordline forthe Lagrangian at hand
2119871 = (1 minus 1199090)2 1199052
minus (1 + 1199090)2 119903
2
minus 2sin2 1205932 minus 2 120579
2
(1)
where 119905 equiv 119889 119889120582 is the 119905-component of 4-momentum and 120582is the affine parameter along the worldline We are using anaffine parametrization (by a rescaling 120582 rarr 120582(120582
1015840)) such that
119871 = const is constant along the curve A static observermakesmeasurements with local orthonormal tetrad
119890=10038161003816100381610038161 minus 1199090
1003816100381610038161003816
minus1
119890119905 119890
= (1 + 119909
0)minus1
119890119903
119890=
minus1119890120579 119890
= ( sin )
minus1
119890120579
(2)
The Euler-Lagrange equations for and can be derivedfrom the variational principle A local measurement of theparticlersquos energy made by a static observer in the equatorialplane gives the time component of the 4-momentum asmeasured in the observerrsquos local orthonormal frame This
is the projection of the 4-momentum along the time basisvector The Euler-Lagrange equations show that if we orientthe coordinate system as initially the particle is moving in theequatorial plane (ie = 1205872 120579 = 0) then the particle alwaysremains in this plane There are two constants of the motioncorresponding to the ignorable coordinates and namelythe 119864-ldquoenergy-at-infinityrdquo and the 119897-angular momentum Weconclude that the free radial infall of a particle from theinfinity up to the moment of crossing the EH sphere aswell as at reaching the surface of central body is absolutelythe same as in the Schwarzschild geometry of black hole(Figure 2(a)) We clear up a general picture of orbits justoutside the event horizon by considering the Euler-Lagrangeequation for radial component with ldquoeffective potentialrdquo Thecircular orbits are stable if119881 is concave up namely at gt 4where is the mass of SPC The binding energy per unitmass of a particle in the last stable circular orbit at = 4is bind = (119898 minus 119864) ≃ 1 minus (2732)
12 Namely this isthe fraction of rest-mass energy released when test particleoriginally at rest at infinity spirals slowly toward the SPC tothe innermost stable circular orbit and then plunges into itThereby one of the important parameters is the capture crosssection for particles falling in from infinity 120590capt = 120587119887
2
maxwhere 119887max is the maximum impact parameter of a particlethat is captured
52 Collisionless Accretion The distribution function for acollisionless gas is determined by the collisionless Boltzmannequation or Vlasov equation For the stationary and sphericalflow we obtain then
(119864 gt 0) = 16120587 (119866)2
120588infinVminus1infin119888minus2 (3)
6 Journal of Astrophysics
AGN
SPC
x0 = 1x0 = 2
x0 = 0
t = infin
vr lt 0 vr = 0
r = infin
EH
(a)
1minus 1 1+x0
10
15
25
ntimes10
minus40(g
cmminus
3 )
(b)
Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of
BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere
(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect
the metric singularity vanishes and the particles well pass EH sphere
where the particle density 120588infin
is assumed to be uniform atfar from the SPC and the particle speed is V
infin≪ 1 During
the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899
infin)1199090rarr1
asymp
minus(ln 00)2 V
infin up to the ID threshold value
119889()
minus13=
04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909
0= 1) and in the sequel form the protomatter
disk around the protomatter core
53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903
119904outside the
event horizon The locally measured particle velocity readsV = (1 minus
00119864
2) where 119864 = 119864
infin119898 = (
00(1 minus 119906
2))12 and
119864infin
is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877
1198922 the proper velocity equals the speed of light
|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903
119904 it is expected that the particles be
nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)
as they were nonrelativistic at infinity (119886infin≪ 1) Considering
the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate
= 212058711989811989911990411990352
119904(ln
00)1015840
119904 (4)
where prime (1015840)119904denotes differentiation with respect to at
the point 119903119904 The gas compression can be estimated as
119899infin
asymp11990352
119904
21199032[(ln
00)1015840
119904
1 + 119903119903()]
12
(5)
The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109
6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole
The key objectives of the MTBH framework are then anincrease of the mass 119872Seed
BH gravitational radius 119877Seed119892
andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed
rarr
BH may most usefully be the increase of gravitational radiusor mass of black hole
Δ119877119892= 119877
BH119892minus 119877
Seed119892=2119866
1198882119872
119889=2119866
1198882120588119889119881119889
Δ119872BH = 119872BH minus119872SeedBH = 119872
SeedBHΔ119877
119892
119877Seed119892
(6)
where 119872119889 120588
119889 and 119881
119889 respectively are the total mass
density and volume of protomatter disk At the value BH119892
of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black
Journal of Astrophysics 7
PC PD120579
1205880
120593Z
Z0Z1
EHBH EHSeed
1205881120588
d120576d = d2Rg asymp 120579
RSPCRBHg
RSeedg
Rd
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
paper understandable There is a general belief reinforced bystatements in textbooks that according to general relativity(GR) a long-standing standard phenomenological black holemodel (PBHM)mdashnamely the most general Kerr-Newmanblack hole model with parameters of mass (119872) angularmomentum (119869) and charge (119876) still has to be put in byhandmdashcan describe the growth of accreting supermassiveblack hole seed However such beliefs are suspect and shouldbe critically reexamined The PBHM cannot be currentlyaccepted as convincing model for addressing the afore-mentioned problems because in this framework the verysource of gravitational field of the black hole is a kind ofcurvature singularity at the center of the stationary blackhole A meaningless central singularity develops which ishidden behind the event horizon The theory breaks downinside the event horizon which is causally disconnected fromthe exterior world Either the Kruskal continuation of theSchwarzschild (119869 = 0 119876 = 0) metric or the Kerr (119876 = 0)metric or the Reissner-Nordstrom (119869 = 0) metric showsthat the static observers fail to exist inside the horizonAny object that collapses to form a black hole will go onto collapse to a singularity inside the black hole Therebyany timelike worldline must strike the central singularitywhich wholly absorbs the infalling matter Therefore theultimate fate of collapsing matter once it has crossed theblack hole surface is unknown This in turn disables anyaccumulation of matter in the central part and thus neitherthe growth of black holes nor the increase of their mass-energy density could occur at accretion of outside matteror by means of merger processes As a consequence themass and angular momentum of black holes will not changeover the lifetime of the universe But how can one be surethat some hitherto unknown source of pressure does notbecome important at huge energies and halt the collapse Tofill the voidwhich the standard PBHMpresents one plausibleidea to innovate the solution to alluded key problems wouldappear to be the framework of microscopic theory of blackhole This theory has been originally proposed by [71] andreferences therein and thoroughly discussed in [72ndash75]Here we recount some of the highlights of the MTBHwhich is the extension of PBHM and rather completes it byexploring the most important processes of rearrangementof vacuum state and a spontaneous breaking of gravitationgauge symmetry at huge energies [71 74 76] We will notbe concerned with the actual details of this framework butonly use it as a backdrop to validate the theory with someobservational tests For details the interested reader is invitedto consult the original papers Discussed gravitational theoryis consistent with GR up to the limit of neutron stars Butthis theory manifests its virtues applied to the physics ofinternal structure of galactic nuclei In the latter a significantchange of properties of spacetime continuum so-calledinner distortion (ID) arises simultaneously with the stronggravity at huge energies (see Appendix A) Consequently thematter undergoes phase transition of second kind whichsupplies a powerful pathway to form a stable superdenseprotomatter core (SPC) inside the event horizon Due to thisthe stable equilibrium holds in outward layers too and thusan accumulation of matter is allowed now around the SPC
The black hole models presented in phenomenological andmicroscopic frameworks have been schematically plotted inFigure 1 to guide the eye A crucial point of the MTBH isthat a central singularity cannot occur which is now replacedby finite though unbelievably extreme conditions held in theSPC where the static observers existed The SPC surroundedby the accretion disk presents the microscopic model ofAGNThe SPC accommodates the highest energy scale up tohundreds of ZeV in central protomatter core which accountsfor the spectral distribution of the resulting radiation ofgalactic nuclei External physics of accretion onto the blackhole in earlier part of its lifetime is identical to the processesin Schwarzschildrsquos model However a strong difference inthe model context between the phenomenological black holeand the SPC is arising in the second part of its lifetime(see Section 6) The seed black hole might grow up drivenby the accretion of outside matter when it was gettingmost of its mass An infalling matter with time forms theprotomatter disk around the protomatter core tapering offfaster at reaching out the thin edge of the event horizon Atthis metric singularity inevitably disappears (see appendices)and the neutrinos may escape through vista to outsideworld even after the neutrino trapping We study the growthof protomatter disk and derive the intermediate mass andinitial redshift of seed black hole and examine luminositiesneutrino surfaces for the disk In this framework we havecomputed the fluxes of UHE neutrinos [75] produced in themediumof the SPC via simple (quark and pionic reactions) ormodified URCA processes even after the neutrino trapping(G Gamow was inspired to name the process URCA afterthe name of a casino in Rio de Janeiro when M Schenbergremarked to him that ldquothe energy disappears in the nucleusof the supernova as quickly as the money disappeared at thatroulette tablerdquo) The ldquotrappingrdquo is due to the fact that asthe neutrinos are formed in protomatter core at superhighdensities they experience greater difficulty escaping from theprotomatter core before being dragged along with thematternamely the neutrinos are ldquotrappedrdquo comove with matterThe part of neutrinos annihilates to produce further thesecondary particles of expected ultrahigh energies In thismodel of course a key open question is to enlighten themechanisms that trigger the activity and how a large amountofmatter can be steadily funneled to the central regions to fuelthis activity In high luminosity AGNs the large-scale internalgravitational instabilities drive gas towards the nucleus whichtrigger big starbursts and the coeval compact cluster justformed It seemed they have some connection to the nuclearfueling through mass loss of young stars as well as their tidaldisruption and supernovae Note that we regard the UHECRparticles as a signature of existence of superdence protomattersources in the universe Since neutrino events are expected tobe of sufficient intensity our estimates can be used to guideinvestigations of neutrino detectors for the distant future
5 Spherical Accretion onto SPC
As alluded to above the MTBH framework supports the ideaof accreting supermassive black holes which link to AGNsIn order to compute the mass accretion rate in use it is
Journal of Astrophysics 5
AGN
ADAD
EH
infin
(a)
AGN
PDPC
ADAD
EHSPC
(b)
Figure 1 (a) The phenomenological model of AGN with the central stationary black hole The meaningless singularity occurs at the centerinside the black hole (b) The microscopic model of AGN with the central stable SPC In due course the neutrinos of huge energies mayescape through the vista to outside world Accepted notations EH = event horizon AD = accretion disk SPC = superdense protomatter corePC = protomatter core
necessary to study the accretion onto central supermassiveSPC The main features of spherical accretion can be brieflysummed up in the following three idealized models thatillustrate some of the associated physics [72]
51 Free Radial Infall We examine the motion of freelymoving test particle by exploring the external geometry of theSPC with the line element (A7) at 119909 = 0 Let us denote the 4-vector of velocity of test particle V120583 = 119889120583119889 120583 = (119905 )and consider it initially for simplest radial infall V2 = V3 =0 We determine the value of local velocity V lt 0 of theparticle for the moment of crossing the EH sphere as well asat reaching the surface of central stable SPC The equation ofgeodesics is derived from the variational principle 120575 int 119889119878 = 0which is the extremum of the distance along the wordline forthe Lagrangian at hand
2119871 = (1 minus 1199090)2 1199052
minus (1 + 1199090)2 119903
2
minus 2sin2 1205932 minus 2 120579
2
(1)
where 119905 equiv 119889 119889120582 is the 119905-component of 4-momentum and 120582is the affine parameter along the worldline We are using anaffine parametrization (by a rescaling 120582 rarr 120582(120582
1015840)) such that
119871 = const is constant along the curve A static observermakesmeasurements with local orthonormal tetrad
119890=10038161003816100381610038161 minus 1199090
1003816100381610038161003816
minus1
119890119905 119890
= (1 + 119909
0)minus1
119890119903
119890=
minus1119890120579 119890
= ( sin )
minus1
119890120579
(2)
The Euler-Lagrange equations for and can be derivedfrom the variational principle A local measurement of theparticlersquos energy made by a static observer in the equatorialplane gives the time component of the 4-momentum asmeasured in the observerrsquos local orthonormal frame This
is the projection of the 4-momentum along the time basisvector The Euler-Lagrange equations show that if we orientthe coordinate system as initially the particle is moving in theequatorial plane (ie = 1205872 120579 = 0) then the particle alwaysremains in this plane There are two constants of the motioncorresponding to the ignorable coordinates and namelythe 119864-ldquoenergy-at-infinityrdquo and the 119897-angular momentum Weconclude that the free radial infall of a particle from theinfinity up to the moment of crossing the EH sphere aswell as at reaching the surface of central body is absolutelythe same as in the Schwarzschild geometry of black hole(Figure 2(a)) We clear up a general picture of orbits justoutside the event horizon by considering the Euler-Lagrangeequation for radial component with ldquoeffective potentialrdquo Thecircular orbits are stable if119881 is concave up namely at gt 4where is the mass of SPC The binding energy per unitmass of a particle in the last stable circular orbit at = 4is bind = (119898 minus 119864) ≃ 1 minus (2732)
12 Namely this isthe fraction of rest-mass energy released when test particleoriginally at rest at infinity spirals slowly toward the SPC tothe innermost stable circular orbit and then plunges into itThereby one of the important parameters is the capture crosssection for particles falling in from infinity 120590capt = 120587119887
2
maxwhere 119887max is the maximum impact parameter of a particlethat is captured
52 Collisionless Accretion The distribution function for acollisionless gas is determined by the collisionless Boltzmannequation or Vlasov equation For the stationary and sphericalflow we obtain then
(119864 gt 0) = 16120587 (119866)2
120588infinVminus1infin119888minus2 (3)
6 Journal of Astrophysics
AGN
SPC
x0 = 1x0 = 2
x0 = 0
t = infin
vr lt 0 vr = 0
r = infin
EH
(a)
1minus 1 1+x0
10
15
25
ntimes10
minus40(g
cmminus
3 )
(b)
Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of
BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere
(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect
the metric singularity vanishes and the particles well pass EH sphere
where the particle density 120588infin
is assumed to be uniform atfar from the SPC and the particle speed is V
infin≪ 1 During
the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899
infin)1199090rarr1
asymp
minus(ln 00)2 V
infin up to the ID threshold value
119889()
minus13=
04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909
0= 1) and in the sequel form the protomatter
disk around the protomatter core
53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903
119904outside the
event horizon The locally measured particle velocity readsV = (1 minus
00119864
2) where 119864 = 119864
infin119898 = (
00(1 minus 119906
2))12 and
119864infin
is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877
1198922 the proper velocity equals the speed of light
|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903
119904 it is expected that the particles be
nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)
as they were nonrelativistic at infinity (119886infin≪ 1) Considering
the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate
= 212058711989811989911990411990352
119904(ln
00)1015840
119904 (4)
where prime (1015840)119904denotes differentiation with respect to at
the point 119903119904 The gas compression can be estimated as
119899infin
asymp11990352
119904
21199032[(ln
00)1015840
119904
1 + 119903119903()]
12
(5)
The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109
6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole
The key objectives of the MTBH framework are then anincrease of the mass 119872Seed
BH gravitational radius 119877Seed119892
andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed
rarr
BH may most usefully be the increase of gravitational radiusor mass of black hole
Δ119877119892= 119877
BH119892minus 119877
Seed119892=2119866
1198882119872
119889=2119866
1198882120588119889119881119889
Δ119872BH = 119872BH minus119872SeedBH = 119872
SeedBHΔ119877
119892
119877Seed119892
(6)
where 119872119889 120588
119889 and 119881
119889 respectively are the total mass
density and volume of protomatter disk At the value BH119892
of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black
Journal of Astrophysics 7
PC PD120579
1205880
120593Z
Z0Z1
EHBH EHSeed
1205881120588
d120576d = d2Rg asymp 120579
RSPCRBHg
RSeedg
Rd
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Figure 1 (a) The phenomenological model of AGN with the central stationary black hole The meaningless singularity occurs at the centerinside the black hole (b) The microscopic model of AGN with the central stable SPC In due course the neutrinos of huge energies mayescape through the vista to outside world Accepted notations EH = event horizon AD = accretion disk SPC = superdense protomatter corePC = protomatter core
necessary to study the accretion onto central supermassiveSPC The main features of spherical accretion can be brieflysummed up in the following three idealized models thatillustrate some of the associated physics [72]
51 Free Radial Infall We examine the motion of freelymoving test particle by exploring the external geometry of theSPC with the line element (A7) at 119909 = 0 Let us denote the 4-vector of velocity of test particle V120583 = 119889120583119889 120583 = (119905 )and consider it initially for simplest radial infall V2 = V3 =0 We determine the value of local velocity V lt 0 of theparticle for the moment of crossing the EH sphere as well asat reaching the surface of central stable SPC The equation ofgeodesics is derived from the variational principle 120575 int 119889119878 = 0which is the extremum of the distance along the wordline forthe Lagrangian at hand
2119871 = (1 minus 1199090)2 1199052
minus (1 + 1199090)2 119903
2
minus 2sin2 1205932 minus 2 120579
2
(1)
where 119905 equiv 119889 119889120582 is the 119905-component of 4-momentum and 120582is the affine parameter along the worldline We are using anaffine parametrization (by a rescaling 120582 rarr 120582(120582
1015840)) such that
119871 = const is constant along the curve A static observermakesmeasurements with local orthonormal tetrad
119890=10038161003816100381610038161 minus 1199090
1003816100381610038161003816
minus1
119890119905 119890
= (1 + 119909
0)minus1
119890119903
119890=
minus1119890120579 119890
= ( sin )
minus1
119890120579
(2)
The Euler-Lagrange equations for and can be derivedfrom the variational principle A local measurement of theparticlersquos energy made by a static observer in the equatorialplane gives the time component of the 4-momentum asmeasured in the observerrsquos local orthonormal frame This
is the projection of the 4-momentum along the time basisvector The Euler-Lagrange equations show that if we orientthe coordinate system as initially the particle is moving in theequatorial plane (ie = 1205872 120579 = 0) then the particle alwaysremains in this plane There are two constants of the motioncorresponding to the ignorable coordinates and namelythe 119864-ldquoenergy-at-infinityrdquo and the 119897-angular momentum Weconclude that the free radial infall of a particle from theinfinity up to the moment of crossing the EH sphere aswell as at reaching the surface of central body is absolutelythe same as in the Schwarzschild geometry of black hole(Figure 2(a)) We clear up a general picture of orbits justoutside the event horizon by considering the Euler-Lagrangeequation for radial component with ldquoeffective potentialrdquo Thecircular orbits are stable if119881 is concave up namely at gt 4where is the mass of SPC The binding energy per unitmass of a particle in the last stable circular orbit at = 4is bind = (119898 minus 119864) ≃ 1 minus (2732)
12 Namely this isthe fraction of rest-mass energy released when test particleoriginally at rest at infinity spirals slowly toward the SPC tothe innermost stable circular orbit and then plunges into itThereby one of the important parameters is the capture crosssection for particles falling in from infinity 120590capt = 120587119887
2
maxwhere 119887max is the maximum impact parameter of a particlethat is captured
52 Collisionless Accretion The distribution function for acollisionless gas is determined by the collisionless Boltzmannequation or Vlasov equation For the stationary and sphericalflow we obtain then
(119864 gt 0) = 16120587 (119866)2
120588infinVminus1infin119888minus2 (3)
6 Journal of Astrophysics
AGN
SPC
x0 = 1x0 = 2
x0 = 0
t = infin
vr lt 0 vr = 0
r = infin
EH
(a)
1minus 1 1+x0
10
15
25
ntimes10
minus40(g
cmminus
3 )
(b)
Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of
BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere
(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect
the metric singularity vanishes and the particles well pass EH sphere
where the particle density 120588infin
is assumed to be uniform atfar from the SPC and the particle speed is V
infin≪ 1 During
the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899
infin)1199090rarr1
asymp
minus(ln 00)2 V
infin up to the ID threshold value
119889()
minus13=
04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909
0= 1) and in the sequel form the protomatter
disk around the protomatter core
53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903
119904outside the
event horizon The locally measured particle velocity readsV = (1 minus
00119864
2) where 119864 = 119864
infin119898 = (
00(1 minus 119906
2))12 and
119864infin
is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877
1198922 the proper velocity equals the speed of light
|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903
119904 it is expected that the particles be
nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)
as they were nonrelativistic at infinity (119886infin≪ 1) Considering
the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate
= 212058711989811989911990411990352
119904(ln
00)1015840
119904 (4)
where prime (1015840)119904denotes differentiation with respect to at
the point 119903119904 The gas compression can be estimated as
119899infin
asymp11990352
119904
21199032[(ln
00)1015840
119904
1 + 119903119903()]
12
(5)
The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109
6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole
The key objectives of the MTBH framework are then anincrease of the mass 119872Seed
BH gravitational radius 119877Seed119892
andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed
rarr
BH may most usefully be the increase of gravitational radiusor mass of black hole
Δ119877119892= 119877
BH119892minus 119877
Seed119892=2119866
1198882119872
119889=2119866
1198882120588119889119881119889
Δ119872BH = 119872BH minus119872SeedBH = 119872
SeedBHΔ119877
119892
119877Seed119892
(6)
where 119872119889 120588
119889 and 119881
119889 respectively are the total mass
density and volume of protomatter disk At the value BH119892
of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black
Journal of Astrophysics 7
PC PD120579
1205880
120593Z
Z0Z1
EHBH EHSeed
1205881120588
d120576d = d2Rg asymp 120579
RSPCRBHg
RSeedg
Rd
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Figure 2 (a) The free radial infall of a particle from the infinity to EH sphere (1199090= 1) which is similar to the Schwarzschild geometry of
BH Crossing the EH sphere a particle continues infall reaching finally the surface (1199090= 2) of the stable SPC (b) Approaching the EH sphere
(1199090= 1) the particle concentration increases asymptotically until the threshold value of protomatter Then due to the action of cutoff effect
the metric singularity vanishes and the particles well pass EH sphere
where the particle density 120588infin
is assumed to be uniform atfar from the SPC and the particle speed is V
infin≪ 1 During
the accretion process the particles approaching the EHbecome relativistic Approaching event horizon the particleconcentration increases asymptotically as (()119899
infin)1199090rarr1
asymp
minus(ln 00)2 V
infin up to the ID threshold value
119889()
minus13=
04 fm (Figure 2(b)) Due to the action of cutoff effect themetric singularity then vanishes and the particles well passEH sphere (119909
0= 1) and in the sequel form the protomatter
disk around the protomatter core
53 Hydrodynamic Accretion For real dynamical conditionsfound in considered superdense medium it is expected thatthemean free path for collisionswill bemuch shorter than thecharacteristic length scale that is the accretion of ambientgas onto a stationary nonrotating compact SPC will behydrodynamical in nature For any equation of state obeyingthe causality constraint the sound speed implies 1198862 lt 1 andthe flowmust pass through a critical sonic point 119903
119904outside the
event horizon The locally measured particle velocity readsV = (1 minus
00119864
2) where 119864 = 119864
infin119898 = (
00(1 minus 119906
2))12 and
119864infin
is the energy at infinity of individual particle of the mass119898Thus the proper flow velocity V = 119906 rarr 0 and is subsonicAt = 119877
1198922 the proper velocity equals the speed of light
|V| = 119906 = 1 gt 119886 and the flow is supersonic This conditionis independent of the magnitude of 119906 and is not sufficient byitself to guarantee that the flow passes through a critical pointoutside EH For large ge 119903
119904 it is expected that the particles be
nonrelativistic with 119886 le 119886119904≪ 1 (ie 119879 ≪ 1198981198882119870 = 1013119870)
as they were nonrelativistic at infinity (119886infin≪ 1) Considering
the equation of accretion onto superdense protomatter corewhich is an analogue of Bondi equations for spherical steady-state adiabatic accretion onto the SPC we determine a massaccretion rate
= 212058711989811989911990411990352
119904(ln
00)1015840
119904 (4)
where prime (1015840)119904denotes differentiation with respect to at
the point 119903119904 The gas compression can be estimated as
119899infin
asymp11990352
119904
21199032[(ln
00)1015840
119904
1 + 119903119903()]
12
(5)
The approximate equality between the sound speed and themean particle speed implies that the hydrodynamic accretionrate is larger than the collisionless accretion rate by the largefactor asymp109
6 The Intermediate Mass PRT and InitialRedshift of Seed Black Hole
The key objectives of the MTBH framework are then anincrease of the mass 119872Seed
BH gravitational radius 119877Seed119892
andof the seed black hole BHSeed at accretion of outside matterThereby an infalling matter forms protomatter disk aroundprotomatter core tapering off faster at reaching the thin edgeof event horizon So a practical measure of growth BHSeed
rarr
BH may most usefully be the increase of gravitational radiusor mass of black hole
Δ119877119892= 119877
BH119892minus 119877
Seed119892=2119866
1198882119872
119889=2119866
1198882120588119889119881119889
Δ119872BH = 119872BH minus119872SeedBH = 119872
SeedBHΔ119877
119892
119877Seed119892
(6)
where 119872119889 120588
119889 and 119881
119889 respectively are the total mass
density and volume of protomatter disk At the value BH119892
of gravitational radius when protomatter disk has finallyreached the event horizon of grown-up supermassive black
Journal of Astrophysics 7
PC PD120579
1205880
120593Z
Z0Z1
EHBH EHSeed
1205881120588
d120576d = d2Rg asymp 120579
RSPCRBHg
RSeedg
Rd
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Figure 3 A schematic cross section of the growth of supermassiveblack hole driven by the formation of protomatter disk at accretionwhen protomatter disk has finally reached the event horizon ofgrown-up supermassive black hole
hole the volume 119889can be calculated in polar coordinates
(120588 119911 120593) from Figure 3
119889= int
BH119892
1205880
119889120588int
2120587
0
120588119889120601int
1199111(120588)
minus1199111(120588)
119889119911
minus int
119877119889
1205880
119889120588int
2120587
0
120588119889120601int
1199110(120588)
minus1199110(120588)
119889119911
(119877119889≪BH119892
)
≃radic2120587
3119877119889(
BH119892)
2
(7)
where 1199111(120588) ≃ 119911
0minus 119911
0(120588 minus 120588
0)(
BH119892minus 120588
0) 119911
0(120588) = radic119877
2
119889minus 1205882
and in approximation119877119889≪
BH119892
we set 1199110(120588
0) ≃ 120588
0≃ 119877
119889radic2
61 The Intermediate Mass of Seed Black Hole From the firstline of (6) by virtue of (7) we obtain
BH119892= 119896(1 plusmn radic1 minus
2
119896119877Seed119892) (8)
where 2119896 = 873 [km]119877119889120588119889119872
⊙ The (8) is valid at (2
119896)119877Seed119892le 1 namely
119877⊙
119877119889
ge 209[km]119877⊙
120588119889
120588⊙
119877Seed119892
119877⊙
(9)
For the values 120588119889= 26times10
16[g cm]minus3 (see below) and119877Seed
119892≃
295 [km](103 to 106) inequality (9) is reduced to 119877⊙119877
119889ge
234 times 108(1 to 103) or [cm]119877
119889ge 034(10
minus2 to 10) Thiscondition is always satisfied because for considered 377 blackholes with the masses 119872BH119872⊙
≃ 11 times 106 to 13 times 1010
we approximately have 119877119889119903OV ≃ 10
minus10 to 10minus7 [71] Notethat Woo and Urry [5] collect and compare all the AGNBHmass and luminosity estimates from the literature Accordingto (6) the intermediate mass of seed black hole reads
119872SeedBH119872
⊙
≃119872BH119872
⊙
(1 minus 168 times 10minus6 119877119889
[cm]119872BH119872
⊙
) (10)
62 PRT The PRT is referred to as a lapse of time 119879BH fromthe birth of black hole till neutrino radiation the earlierpart of the lifetime That is 119879BH = 119872119889
where is theaccretion rate In approximation at hand 119877
119889≪ 119877
119892 the PRT
reads
119879BH = 120588119889119881119889
≃ 933 sdot 10
15[g cmminus3
]
1198771198891198772
119892
(11)
In case of collisionless accretion (3) and (11) give
119879BH ≃ 26 sdot 1016 119877119889
cm10
minus24 g cmminus3
120588infin
Vinfin
10 km 119904minus1yr (12)
In case of hydrodynamic accretion (4) and (11) yield
119879BH ≃ 88 sdot 10381198771198891198772
119892cmminus3
11989911990411990352
119904 (ln11989200)1015840
119904
(13)
Note that the spherical accretion onto black hole in generalis not necessarily an efficient mechanism for converting rest-mass energy into radiation Accretion onto black hole maybe far from spherical accretion because the accreted gaspossesses angular momentum In this case the gas will bethrown into circular orbits about the black hole when cen-trifugal forces will become significant before the gas plungesthrough the event horizon Assuming a typical mass-energyconversion efficiency of about 120598 sim 10 in approximation119877119889≪ 119877
119892 according to (12) and (13) the resulting relationship
of typical PRT versus bolometric luminosity becomes
119879BH ≃ 032119877119889
119903OV(119872BH119872
⊙
)
210
39119882
119871bol[yr] (14)
We supplement this by computing neutrino fluxes in the nextsection
63 Redshift of Seed Black Hole Interpreting the redshiftas a cosmological Doppler effect and that the Hubble lawcould most easily be understood in terms of expansion of theuniverse we are interested in the purely academic questionof principle to ask what could be the initial redshift 119911Seed ofseed black hole if the mass the luminosity and the redshift119911 of black hole at present time are known To follow thehistory of seed black hole to the present time let us placeourselves at the origin of coordinates 119903 = 0 (according tothe Cosmological Principle this is mere convention) andconsider a light traveling to us along the minus119903 direction withangular variables fixed If the light has left a seed black holelocated at 119903
119904 120579
119904 and 120593
119904 at time 119905
119904 and it has to reach us at
a time 1199050 then a power series for the redshift as a function
of the time of flight is 119911Seed = 1198670(1199050minus 119905
119904) + sdot sdot sdot where 119905
0
is the present moment and 1198670is Hubblersquos constant Similar
expression 119911 = 1198670(1199050minus119905
1)+sdot sdot sdot can be written for the current
black hole located at 1199031 120579
1 and 120593
1 at time 119905
1 where 119905
1=
119905119904+ 119879BH as seed black hole is an object at early times Hence
in the first-order approximation byHubblersquos constant wemayobtain the following relation between the redshifts of seed
8 Journal of Astrophysics
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
and present black holes 119911Seed ≃ 119911+1198670119879BHThis relation is in
agreement with the scenario of a general recession of distantgalaxies away from us in all directions the furthest naturallybeing those moving the fastest This relation incorporatingwith (14) for the value 119867
0= 70 [km][sMpc] favored today
yields
119911Seed≃ 119911 + 2292 times 10
28 119877119889
119903OV(119872BH119872
⊙
)
2119882
119871bol (15)
7 UHE Neutrino Fluxes
The flux can be written in terms of luminosity as 119869]120576 =]1205764120587119863
2
119871(119911)(1 + 119911) where 119911 is the redshift and 119863
119871(119911) is the
luminosity distance depending on the cosmological modelThe (1+119911)minus1 is due to the fact that each neutrino with energy1015840
] if observed near the place and time of emission 1199051015840 will bered-shifted to energy ] =
1015840
]119877(1199051)119877(1199050) = 1015840
](1 + 119911)minus1 of
the neutrino observed at time 119905 after its long journey to uswhere 119877(119905) is the cosmic scale factor Computing the UHEneutrino fluxes in the framework of MTBH we choose thecosmological model favored today with a flat universe filledwith matterΩ
119872= 120588
119872120588
119888and vacuum energy densitiesΩ
119881=
120588119881120588
119888 therebyΩ
119881+Ω
119872= 1 where the critical energy density
120588119888= 3119867
2
0(8120587119866
119873) is defined through the Hubble parameter
1198670[77]
119863119871(119911) =
(1 + 119911) 119888
1198670radicΩ
119872
int
1+119911
1
119889119909
radicΩ119881Ω
119872+ 1199093
= 24 times 1028119868 (119911) cm
(16)
Here 119868(119911) = (1+119911) int1+1199111119889119909radic23 + 1199093 we set the values119867
0=
70 kmsMpc Ω119881= 07 andΩ
119872= 03
71 URCA Reactions The neutrino luminosity of SPC ofgiven mass by modified URCA reactions with no muonsreads [75]
URCA]120576 = 38 times 10
50120576119889(119872
⊙
)
175
[erg sminus1] (17)
where 120576119889= 1198892 119877
119892and 119889 is the thickness of the protomatter
disk at the edge of even horizon The resulting total UHEneutrino flux of cooling of the SPC can be obtained as
119869URCA]120576 ≃ 522 times 10
minus8
times120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(18)
where the neutrino is radiated in a cone with the beamingangle 120579 sim 120576
seen the nucleon modified URCA reactions can contributeefficiently only to extragalactic objects with enough smallredshift 119911 ≪ 1
72 Pionic Reactions The pionic reactions occurring in thesuperdense protomatter medium of SPC allow both thedistorted energy and momentum to be conserved This is theanalogue of the simple URCA processes
120587minus+ 119899 997888rarr 119899 + 119890
minus+ ]
119890 120587
minus+ 119899 997888rarr 119899 + 120583
minus+ ]
120583(19)
and the two inverse processes As in the modified URCAreactions the total rate for all four processes is essentially fourtimes the rate of each reaction alone The muons are alreadypresent when pions appear The neutrino luminosity of theSPC of given mass by pionic reactions reads [75]
120587
]120576 = 578 times 1058120576119889(119872
⊙
)
175
[erg sminus1] (20)
Then the UHE neutrino total flux is
119869120587
]120576 ≃ 791120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(21)
The resulting total energy-loss rate will then be dramaticallylarger due to the pionic reactions (19) rather than themodified URCA processes
73 Quark Reactions In the superdense protomattermediumthe distorted quark Fermi energies are far below the charmedc- t- and b-quark production thresholds Therefore onlyup- down- and strange quarks are presentThe120573 equilibriumis maintained by reactions like
119889 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119889 + ]
119890 (22)
119904 997888rarr 119906 + 119890minus+ ]
119890 119906 + 119890
minus997888rarr 119904 + ]
119890 (23)
which are 120573 decay and its inverse These reactions constitutesimple URCA processes in which there is a net loss of a ]
119897]119897
pair at nonzero temperatures In this application a sufficientaccuracy is obtained by assuming 120573-equilibrium and thatthe neutrinos are not retained in the medium of Λ-likeprotomatter The quark reactions (22) and (23) proceed atequal rates in 120573 equilibrium where the participating quarksmust reside close to their Fermi surface Hence the totalenergy of flux due to simple URCA processes is rather twicethan that of (22) or (23) alone For example the spectral fluxesof theUHEantineutrinos andneutrinos for different redshiftsfrom quark reactions are plotted respectively in Figures 4and 5 [75] The total flux of UHE neutrino can be written as
119869119902
]120576 ≃ 7068120576119889
1198682 (119911) (1 + 119911)(119872
⊙
)
175
[erg cmminus2 sminus1 srminus1]
(24)
8 Simulation
For simulation we use the data of AGNBH mass andluminosity estimates for 377 black holes presented by [5]These masses are mostly based on the virial assumption forthe broad emission lines with the broad-line region size
Journal of Astrophysics 9
5E6
3E7
2E7
25E7
1E7
15E7
0
y2 = E100ZeV y2 = E100ZeV
z = 001
z = 007
z = 002
z = 07
z = 05
z = 01
z = 003
z = 005
10000
20000
30000
40000
50000
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1 srminus
1 )
(dJq 120576dy2)120576 d
(041
ergc
mminus
2sminus
1srminus
1 )
00
4 8 12 16 200 4 8 12 16
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Figure 4 The spectral fluxes of UHE antineutrinos for different redshifts from quark reactions
5E7
2E8
2E5
1E8
15E8
16E5
12E5
28E5
24E5
00 4 8 12 16
y1 = E100ZeV y1 = E100ZeV
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
(dJq
120576dy1)120576 d
(01
ergc
mminus
2sminus
1 srminus
1 )
00
4 8 12 16 20
z = 001
z = 002
z = 003
z = 005
40000
80000
z = 007
z = 07
z = 05
z = 01
Figure 5 The spectral fluxes of UHE neutrinos for different redshifts from quark reactions
determined from either reverberation mapping or opticalluminosity Additional black hole mass estimates based onproperties of the host galaxy bulges either using the observedstellar velocity dispersion or using the fundamental planerelation Since the aim is to have more than a thousand ofrealizations each individual run is simplified with a useof previous algorithm of the SPC-configurations [71] as aworking model given in Appendix G Computing the cor-responding PRTs seed black hole intermediate masses andtotal neutrino fluxes a main idea comes to solving an inverseproblem Namely by the numerous reiterating integrations ofthe state equations of SPC-configurationswe determine those
required central values of particle concentration (0) and ID-field119909(0) for which the integrated totalmass of configurationhas to be equal to the black hole mass 119872BH given fromobservations Along with all integral characteristics theradius119877
119889is also computed which is further used in (10) (14)
(15) (18) (21) and (24) for calculating119872SeedBH 119879BH 119911
Seed and119869119894
]120576 respectivelyThe results are summed up in Tables 1 2 3 4and 5 Figure 6 gives the intermediate seed masses119872Seed
BH 119872⊙
versus the present masses 119872BH119872⊙of 337 black holes on
logarithmic scales For the present masses119872BH119872⊙≃ 11 times
106 to 13 times 1010 the computed intermediate seed masses
10 Journal of Astrophysics
6 7 8 9 10 11
log (MBHM⊙)
1
2
3
4
5
6
log
(MSe
edBH
M⊙)
Figure 6 The 119872SeedBH 119872⊙
-119872BH119872⊙relation on logarithmic scales
of 337 black holes from [5] The solid line is the best fit to data ofsamples
are ranging from 119872SeedBH 119872⊙
≃ 264 to 29 times 105 Thecomputed neutrino fluxes are ranging from (1) (quarkreactions)mdash119869119902]120576120576119889 [erg cm
minus2 sminus1 srminus1] ≃ 829times10minus16 to 318times10
minus4 with the average 119869119902]120576 ≃ 553times10minus10120576119889[erg cmminus2 sminus1 srminus1]
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
]120576 with the average119869URCA]120576 ≃ 241 times 10
minus20120576119889[erg cmminus2 sminus1 srminus1] In accordance
the AGNs are favored as promising pure neutrino sourcesbecause the computed neutrino fluxes are highly beamedalong the plane of accretion disk and peaked at high energiesand collimated in smaller opening angle 120579 sim 120576
119889= 1198892 119903
119892≪ 1
To render our discussion here a bit more transparent and toobtain some feeling for the parameter 120576
119889we may estimate
120576119889≃ 169times10
minus10 just for example only for the suppermassiveblack hole of typical mass sim109119872
⊙(2 119877
119892= 59times10
14 cm) andso 119889 sim 1 km But the key problem of fixing the parameter120576119889more accurately from experiment would be an important
topic for another investigation elsewhere
9 Conclusions
The growth of accreting supermassive black hole seeds andtheir neutrino radiation are found to be common phenom-ena in the AGNs In this report we further expose theassertions made in the framework of microscopic theoryof black hole via reviewing the mass assembly history of377 plausible accreting supermassive black hole seeds Afterthe numerous reiterating integrations of the state equationsof SPC-configurations we compute their intermediate seedmasses 119872Seed
BH PRTs initial redshifts 119911Seed and neutrinofluxes All the results are presented in Tables 1ndash5 Figure 6gives the intermediate seed masses 119872Seed
BH 119872⊙versus the
present masses 119872BH119872⊙of 337 black holes on logarithmic
scales In accordance the AGNs are favored as promisingpure UHE neutrino sources Such neutrinos may reveal clueson the puzzle of origin of UHE cosmic rays We regardthe considered black holes only as the potential neutrinosources The obtained results however may suffer and thatwould be underestimated if not all 377 live black holes in the
119872BH119872⊙≃ 11 times 10
6 to 13 times 1010 mass range at presentreside in final stage of their growth when the protomatterdisk driven by accretion has reached the event horizon
Appendices
A Outline of the Key Points of ProposedGravitation Theory at Huge Energies
Theproposed gravitation theory explores themost importantprocesses of rearrangement of vacuum state and a spon-taneous breaking of gravitation gauge symmetry at hugeenergies From its historical development the efforts in gaugetreatment of gravity mainly focus on the quantum gravityand microphysics with the recent interest for example inthe theory of the quantum superstring or in the very earlyuniverse in the inflationary model The papers on the gaugetreatment of gravity provide a unified picture of gravitymodified models based on several Lie groups Howevercurrently no single theory has been uniquely accepted as theconvincing gauge theory of gravitation which could lead to aconsistent quantum theory of gravity They have evoked thepossibility that the treatment of spacetimemight involve non-Riemannian features on the scale of the Planck length Thisnecessitates the study of dynamical theories involving post-Riemannian geometries It is well known that the notions ofspace and connections should be separated see for example[78ndash81] The curvature and torsion are in fact properties ofa connection and many different connections are allowedto exist in the same spacetime Therefore when consideringseveral connections with different curvature and torsion onetakes spacetime simply as a manifold and connections asadditional structures From this view point in a recent paper[82] we tackle the problem of spacetime deformation Thistheory generalizes and in particular cases fully recovers theresults of the conventional theory Conceptually and tech-niquewise this theory is versatile and powerful and manifestsits practical and technical virtue in the fact that through anontrivial choice of explicit form of the world-deformationtensor which we have at our disposal in general we havea way to deform the spacetime which displayed differentconnections which may reveal different post-Riemannianspacetime structures as corollary All the fundamental grav-itational structures in factmdashthe metric as much as thecoframes and connectionsmdashacquire a spacetime deformationinduced theoretical interpretation There is another line ofreasoningwhich supports the side of thismethodWe addressthe theory of teleparallel gravity and construct a consistentEinstein-Cartan (EC) theory with the dynamical torsionWe show that the equations of the standard EC theory inwhich the equation defining torsion is the algebraic typeand in fact no propagation of torsion is allowed can beequivalently replaced by the set of modified Einstein-Cartanequations in which the torsion in general is dynamicalMoreover the special physical constraint imposed upon thespacetime deformations yields the short-range propagatingspin-spin interaction For the self-contained arguments inAppendix A1 and Appendices B and C we complete the
Journal of Astrophysics 11
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Table 1 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from spatially resolved kinematics Columns(1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
Table 2 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from reverberation mapping Columns (1)name (2) redshift (3)AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in specialunit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log ofthe neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Table 3 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from optimal luminosity Columns (1) name(2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomatter core in special unit119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass in solar masses (8) log of theneutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where 119869119894 equiv log(119869119894V120576120576119889 erg cm
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
spacetime deformation theory [82] by new investigation ofbuilding up the distortion-complex of spacetime continuumand showing how it restores the world-deformation tensorwhich still has been put in by hand We extend neces-sary geometrical ideas of spacetime deformation in conciseform without going into the subtleties as applied to thegravitation theory which underlies the MTBH framework Ihave attempted to maintain a balance between being overlydetailed and overly schematic Therefore the text in theappendices should resemble a ldquohybridrdquo of a new investigationand some issues of proposed gravitation theory
A1 A First Glance at Spacetime Deformation Consider asmooth deformation map Ω 119872
4rarr M
4 written
in terms of the world-deformation tensor (Ω) the general(M
4) and flat (119872
4) smooth differential 4D-manifolds The
following notational conventions will be used throughoutthe appendices All magnitudes related to the space M
4
will be denoted by an over ldquordquo We use the Greek alphabet(120583 ] 120588 = 0 1 2 3) to denote the holonomic worldindices related to M
4and the second half of Latin alphabet
(119897 119898 119896 = 0 1 2 3) to denote the world indices related to119872
4The tensorΩ can be written in the formΩ = (Ω119898
119897=
119898
120583120583
119897) where the DC-members are the invertible distortion
matrix (119898
120583) and the tensor (120583
119897equiv 120597
119897120583 and 120597
119897= 120597120597119909
119897)The principle foundation of theworld-deformation tensor (Ω)comprises the following two steps (1) the basis vectors 119890
119898at
given point (119901 isin 1198724) undergo the distortion transformations
by means of and (2) the diffeomorphism 120583(119909) 1198724rarr
4is constructed by seeking new holonomic coordinates
120583(119909) as the solutions of the first-order partial differential
equations Namely
120583=
119897
120583119890119897
120583120583
119897= Ω
119898
119897119890119898 (A1)
where the conditions of integrability 120597119896120595120583
119897= 120597
119897120595120583
119896 and
nondegeneracy 120595 = 0 necessarily hold [83 84] Forreasons that will become clear in the sequel next we writethe norm 119889 equiv 119894 (see Appendix B) of the infinitesimal
displacement 119889120583 on the M4in terms of the spacetime
structures of1198724
119894 = = 120583otimes
120583
= Ω119898
119897119890119898otimes 120599
119897isin M
4 (A2)
A deformation Ω 1198724rarr M
4comprises the following
two 4D deformations∘
Ω 1198724rarr 119881
4and Ω 119881
4rarr
4 where 119881
4is the semi-Riemannian space and
∘
Ω andΩ are the corresponding world deformation tensors Thekey points of the theory of spacetime deformation areoutlined further in Appendix B Finally to complete thistheory we need to determine and figured in (A1) Inthe standard theory of gravitation they can be determinedfrom the standard field equations by means of the generallinear frames (C10) However it should be emphasized thatthe standard Riemannian space interacting quantum fieldtheory cannot be a satisfactory ground for addressing themost important processes of rearrangement of vacuum stateand gauge symmetry breaking in gravity at huge energiesThe difficulties arise there because Riemannian geometryin general does not admit a group of isometries and itis impossible to define energy-momentum as Noether localcurrents related to exact symmetries This in turn posedsevere problem of nonuniqueness of the physical vacuum andthe associated Fock space A definition of positive frequencymodes cannot in general be unambiguously fixed in the pastand future which leads to |in⟩ = |out⟩ because the state |in⟩is unstable against decay intomany particle |out⟩ states due tointeraction processes allowed by lack of Poincare invarianceA nontrivial Bogolubov transformation between past andfuture positive frequency modes implies that particles arecreated from the vacuum and this is one of the reasons for|in⟩ = |out⟩
A2 General Gauge Principle Keeping in mind the aforesaidwe develop the alternative framework of the general gaugeprinciple (GGP) which is the distortion gauge induced fiber-bundle formulation of gravitation As this principle was inuse as a guide in constructing our theory we briefly discussits general implications in Appendix DThe interested reader
Journal of Astrophysics 15
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Table 4 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from observed stellar velocity dispersionsColumns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radius of protomattercore in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediate mass insolar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
is invited to consult the original paper [74] for detailsIn this we restrict ourselves to consider only the simplestspacetime deformation map Ω 119872
4rarr 119881
4(Ω
120583
] equiv
120575120583
] ) This theory accounts for the gravitation gauge group119866119881generated by the hidden local internal symmetry 119880loc
We assume that a distortion massless gauge field 119886(119909) (equiv119886119899(119909)) has to act on the external spacetime groups This
field takes values in the Lie algebra of the abelian group119880
loc We pursue a principle goal of building up the world-deformation tensor Ω(119865) = (119886)(119886) where 119865 is the
18 Journal of Astrophysics
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Table 5 Seed black hole intermediate masses preradiation times redshifts and neutrino fluxes from fundamental plane-derived velocitydispersions Columns (1) name (2) redshift (3) AGN type SY2 Seyfert 2 (4) log of the bolometric luminosity (ergsminus1) (5) log of the radiusof protomatter core in special unit 119903OV = 1368 km (6) log of the black hole mass in solar masses (7) log of the seed black hole intermediatemass in solar masses (8) log of the neutrino preradiation time (yrs) (9) redshift of seed black hole (10) 119869119894=119902 (11) 119869119894=URCA and (12) 119869119894=120587 where119869119894equiv log(119869119894V120576120576119889 erg cm
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
differential form of gauge field 119865 = (12)119865119899119898120599119899and 120599
119898 Weconnect the structure group 119866
119881 further to the nonlinear
realization of the Lie group119866119863of distortion of extended space
1198726(rarr
6) (E1) underlying the119872
4This extension appears
to be indispensable for such a realization In using the 6Dlanguage we will be able to make a necessary reduction tothe conventional 4D space The laws guiding this redactionare given in Appendix EThe nonlinear realization techniqueor the method of phenomenological Lagrangians [85ndash91]provides a way to determine the transformation properties offields defined on the quotient space In accordance we treatthe distortion group 119866
119863and its stationary subgroup 119867 =
119878119874(3) respectively as the dynamical group and its algebraicsubgroup The fundamental field is distortion gauge field(a) and thus all the fundamental gravitational structures infactmdashthe metric as much as the coframes and connectionsmdashacquire a distortion-gauge induced theoretical interpretationWe study the geometrical structure of the space of parametersin terms of Cartanrsquos calculus of exterior forms and derivethe Maurer-Cartan structure equations where the distortionfields (a) are treated as the Goldstone fields
A3 A Rearrangement of Vacuum State Addressing therearrangement of vacuum state in realization of the group119866119881we implement the abelian local group [74]
119880loc= 119880 (1)
119884times 119880 (1) equiv 119880 (1)
119884times diag [119878119880 (2)] (A3)
on the space 1198726(spanned by the coordinates 120578) with
the group elements of exp[119894(1198842)120579119884(120578)] of 119880(1)
119884and
exp[11989411987931205793(120578)] of119880(1)This group leads to the renormalizable
theory because gauge invariance gives a conservation ofcharge and it also ensures the cancelation of quantumcorrections that would otherwise result in infinitely largeamplitudes This has two generators the third component1198793 of isospin related to the Pauli spin matrix 2 and
hypercharge 119884 implying 119876119889= 119879
3+ 1198842 where 119876119889 is
the distortion charge operator assigning the number minus1 toparticles but +1 to antiparticles The group (A3) entails twoneutral gauge bosons of 119880(1) or that coupled to 1198793 and of119880(1)
119884 or that coupled to the hypercharge 119884 Spontaneous
symmetry breaking can be achieved by introducing theneutral complex scalar Higgs field Minimization of thevacuum energy fixes the nonvanishing vacuum expectationvalue (VEV) which spontaneously breaks the theory leavingthe 119880(1)
119889subgroup intact that is leaving one Goldstone
boson Consequently the left Goldstone boson is gaugedaway from the scalar sector but it essentially reappears inthe gauge sector providing the longitudinally polarized spinstate of one of gauge bosons which acquires mass throughits coupling to Higgs scalar Thus the two neutral gaugebosons were mixed to form two physical orthogonal statesof the massless component of distortion field (119886) (119872
119886= 0)
which is responsible for gravitational interactions and itsmassive component (119886) (119872
119886= 0) which is responsible for
the ID-regime Hence a substantial change of the propertiesof the spacetime continuum besides the curvature may ariseat huge energiesThis theory is renormalizable because gaugeinvariance gives conservation of charge and also ensures the
20 Journal of Astrophysics
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
cancelation of quantum corrections that would otherwiseresult in infinitely large amplitudes Without careful thoughtwe expect that in this framework the renormalizability of thetheory will not be spoiled in curved space-time too becausethe infinities arise from ultraviolet properties of Feynmanintegrals in momentum space which in coordinate space areshort distance properties and locally (over short distances)all the curved spaces look like maximally symmetric (flat)space
A4 Model Building Field Equations The field equationsfollow at once from the total gauge invariant Lagrangianin terms of Euler-Lagrange variations respectively on bothcurved and flat spaces The Lagrangian of distortion gaugefield (119886) defined on the flat space is undegenerated Killingform on the Lie algebra of the group 119880loc in adjoint repre-sentation which yields the equation of distortion field (F1)We are interested in the case of a spherical-symmetric grav-itational field 119886
0(119903) in presence of one-dimensional space-
like ID-field 119886 (F6) In the case at hand one has the groupof motions 119878119874(3) with 2D space-like orbits 1198782 where thestandard coordinates are and The stationary subgroup of119878119874(3) acts isotropically upon the tangent space at the point ofsphere 1198782 of radius So the bundle119901 119881
4rarr
2 has the fiber1198782= 119901
minus1() isin 119881
4 with a trivial connection on it where 2
is the quotient-space 1198814119878119874(3) Considering the equilibrium
configurations of degenerate barionic matter we assume anabsence of transversal stresses and the transference of massesin 119881
4
1198791
1= 119879
2
2= 119879
3
3= minus () 119879
0
0= minus () (A4)
where () and () ( isin 3
) are taken to denote theinternal pressure and macroscopic density of energy definedin proper frame of reference that is being usedThe equationsof gravitation (119886
0) and ID (119886) fields can be given in Feynman
gauge [71] as
Δ1198860=1
2
00
12059700
1205971198860
()
minus [33
12059733
1205971198860
+ 11
12059711
1205971198860
+ 22
12059722
1205971198860
] ()
(Δ minus 120582minus2
119886) 119886 =
1
2
00
12059700
120597119886 ()
minus [33
12059733
120597119886+
11
12059711
120597119886+
22
12059722
120597119886] ()
times 120579 (120582119886minus
minus13)
(A5)
where is the concentration of particles and 120582119886= ℎ119898
119886119888 ≃
04 fm is the Compton lenghth of the ID-field (but substantialID-effects occur far below it) and a diffeomorphism (119903) 119872
4rarr 119881
4is given as 119903 = minus 119877
1198924 A distortion of the
basis in the ID regime in turn yields the transformationsof Poincare generators of translations Given an explicit formof distorted basis vectors (F7) it is straightforward to derivethe laws of phase transition for individual particle found inthe ID-region (119909
where 119864 and119898 and 119875 and are ordinary and distortedenergy momentum andmass at rest Hence thematter foundin the ID-region (119886 = 0) of space-time continuum hasundergone phase transition of II-kind that is each particlegoes off from the mass shellmdasha shift of mass and energy-momentum spectra occurs upwards along the energy scaleThe matter in this state is called protomatter with the ther-modynamics differing strongly from the thermodynamics ofordinary compressed matter The resulting deformed metricon 119881
4in holonomic coordinate basis takes the form
00= (1 minus 119909
0)2
+ 1199092
120583] = 0 (120583 = ])
33= minus [(1 + 119909
0)2
+ 1199092]
11= minus
2
22= minus
2sin2
(A7)
As a workingmodel we assume the SPC-configurations givenin Appendix G which are composed of spherical-symmetricdistribution of matter in many-phase stratified states Thisis quick to estimate the main characteristics of the equi-librium degenerate barionic configurations and will guideus toward first look at some of the associated physics Thesimulations confirm in brief the following scenario [71] theenergy density and internal pressure have sharply increasedin protomatter core of SPC-configuration (with respect tocorresponding central values of neutron star) proportionalto gravitational forces of compression This counteracts thecollapse and equilibrium holds even for the masses sim109119872
⊙
This feature can be seen for example from Figure 7 wherethe state equation of the II-class SPCII configuration with thequark protomatter core is plotted
B A Hard Look at Spacetime Deformation
The holonomic metric on M4can be recast in the form
= 120583]
120583
otimes ]= (
120583 ])
120583
otimes ] with components
120583] = (120583 ]) in dual holonomic base
120583
equiv 119889120583 In order
to relate local Lorentz symmetry to more general deformed
Journal of Astrophysics 21
log(PP
OV)
log(120588120588OV)
30
20
10
0
minus10
0 10 20 30
AGN
branch
branch
Neutron star
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Figure 7 The state equation of SPCII on logarithmic scales where119875 and 120588 are the internal pressure and density given in special units119875OV = 6469 times 10
36[erg cmminus3
] and 120588OV = 7194 times 1015[g cmminus3
]respectively
spacetime there is however a need to introduce the solderingtools which are the linear frames and forms in tangent fiber-bundles to the external smooth differential manifold whosecomponents are so-called tetrad (vierbein) fields The M
4
has at each point a tangent space
4 spanned by the
anholonomic orthonormal frame field as a shorthand forthe collection of the 4-tuplet (
0
3) where
119886=
119886
120583120583
We use the first half of Latin alphabet (119886 119887 119888 = 0 1 2 3) todenote the anholonomic indices related to the tangent spaceThe frame field then defines a dual vector of differential
forms = (0
3
) as a shorthand for the collection of the
119887
= 119887
120583119889
120583 whose values at every point form the dual
basis such that 119886rfloor
119887
= 120575119887
119886 where rfloor denotes the interior
product namely this is a 119862infin-bilinear map rfloor Ω1rarr Ω
0
withΩ119901 denoting the 119862infin-modulo of differential 119901-forms onM
4 In components we have
119886
120583119887
120583= 120575
119887
119886 On the manifold
M4 the tautological tensor field 119894 of type (1 1) can be
definedwhich assigns to each tangent space the identity lineartransformation Thus for any point isin M
4and any vector
isin M
4 one has 119894() = In terms of the frame field the
119886
give the expression for 119894 as 119894 = = 0otimes
0
+ sdot sdot sdot 3otimes
3
in the sense that both sides yield when applied to anytangent vector in the domain of definition of the frame fieldOne can also consider general transformations of the lineargroup 119866119871(4 119877) taking any base into any other set of fourlinearly independent fieldsThenotation
119886
119887
will be usedbelow for general linear frames Let us introduce so-called
first deformation matrices (120587(119909)119898119896and 119886
119897()) isin 119866119871(4 )
for all as follows
119898
120583=
120583
119896120587119898
119896
120583
119897=
120583
119896120587119896
119897
120583
119896120583
119898= 120575
119896
119898
119886
119898=
119886
120583
119898
120583
119886
119897=
119886
120583120583
119897
(B1)
where 120583]119896
120583119904
]= 120578
119896119904 120578
119896119904is themetric on119872
4 A deformation
tensorΩ119898
119897= 120587
119898
119896120587119896
119897 yields local tetrad deformations
119886=
119886
119898119890119898
119886
= 119886
119897120599119897
119890119896= 120587
119898
119896119890119898 120599
119896
= 120587119896
119897120599119897
(B2)
and 119894 = 119886otimes
119886
= 119890119896otimes120599
119896
isin M4The first deformationmatrices
120587 and in general give rise to the right cosets of the Lorentzgroup that is they are the elements of the quotient group119866119871(4 )119878119874(3 1) If we deform the cotetrad according to(B2) we have two choices to recast metric as follows eitherwriting the deformation of the metric in the space of tetradsor deforming the tetrad field
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119897119887
119898120599119897otimes 120599
119898
= 120574119897119898120599119897otimes 120599
119898
(B3)
where the second deformation matrix 120574119897119898 reads 120574
119897119898=
119900119886119887119886
119897119887
119898 The deformed metric splits as
120583] = Υ
2120578120583] + 120583] (B4)
provided that Υ = 119886119886= 120587
119896
119896and
120583] = (120574119886119897 minus Υ
2119900119886119897)
119886
120583119897
V
= (120574119896119904minus Υ
2120578119896119904)
119896
120583119904
V
(B5)
The anholonomic orthonormal frame field relates tothe tangent space metric 119900
119886119887= diag(+ minus minusminus) by 119900
119886119887=
(119886
119887) =
120583]119886120583119887
] which has the converse 120583] = 119900119886119887
119886
120583119887
]because
119886
120583119886
] = 120575120583
] With this provision we build up aworld-deformation tensorΩ yielding a deformation of the flat space119872
4 The 120574
119897119898can be decomposed in terms of symmetric
(119886119897)
and antisymmetric [119886119897]
parts of the matrix 119886119897= 119900
119886119888119888
119897(or
resp in terms of 120587(119896119897)
and 120587[119896119897]
where 120587119896119897= 120578
119896119904120587119904
119897) as
120574119886119897= Υ
2
119900119886119897+ 2ΥΘ
119886119897+ 119900
119888119889Θ
119888
119886Θ
119889
119897
+ 119900119888119889(Θ
119888
119886119889
119897+
119888
119886Θ
119889
119897) + 119900
119888119889119888
119886119889
119897
(B6)
where
119886119897= Υ119900
119886119897+ Θ
119886119897+
119886119897 (B7)
Υ = 119886
119886 Θ
119886119897is the traceless symmetric part and
119886119897is
the skew symmetric part of the first deformation matrix
22 Journal of Astrophysics
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
The anholonomy objects defined on the tangent space
4
read
119886
= 119889119886
=1
2119886
119887119888119887
and 119888
(B8)
where the anholonomy coefficients 119886119887119888 which represent the
curls of the base members are
119888
119886119887= minus
119888
([119886
119887]) =
119886
120583119887
](
120583119888
] minus ]119888
120583)
= minus119888
120583[
119886(
119887
120583) minus
119887(
119886
120583)]
= 2120587119888
119897119898
120583(120587
minus1119898
[119886120583120587minus1119897
119887])
(B9)
In particular case of constant metric in the tetradic space thedeformed connection can be written as
Γ119886
119887119888=1
2(
119886
119887119888minus 119900
1198861198861015840
1199001198871198871015840
1198871015840
1198861015840119888minus 119900
1198861198861015840
1199001198881198881015840
1198881015840
1198861015840119887) (B10)
All magnitudes related to the 1198814will be denoted by an over
ldquo ∘rdquo According to (A1) we have∘
Ω
119898
119897=∘
119863
119898
120583
∘
120595120583
119897and Ω
120583
] =
120583
120588120588
] provided
∘
119890120583=∘
119863
119897
120583119890119897
∘
119890120583
∘
120595120583
119897=∘
Ω
119898
119897119890119898
120588=
120583
120588
∘
119890120583
120588120588
] = Ω120583
V∘
119890120583
(B11)
In analogy with (B1) the following relations hold∘
119863
119898
120583=∘
119890120583
119896 ∘
120587119898
119896
∘
120595120583
119897=∘
119890120583
119896
∘
120587119896
119897
∘
119890120583
119896 ∘
119890120583
119898= 120575
119896
119898
∘
120587119886
119898
=∘
119890119886
120583 ∘
119863
119898
120583
∘
120587119886
119897=∘
119890119886
120583
∘
120595120583
119897
(B12)
where∘
Ω
119898
119897=∘
120587119898
120588
∘
120587120588
119897and Ω
120583
] = 120583
120588120588
] We also have∘
119892120583]∘
119890119896
120583 ∘
119890119904
]= 120578
119896119904and
120583
120588= 119890]
120583]120588
120588
] = 119890120588
120583120583
]
119890]120583119890]120588= 120575
120583
120588
119886
120583= 119890
119886
120588
120583
120588
119886
] = 119890119886
120588120588
]
(B13)
The norm 119889 ∘119904 equiv 119894∘
119889 of the displacement 119889 ∘119909120583 on 1198814can be
written in terms of the spacetime structures of1198724as
119894
∘
119889 =∘
119890
∘
120599 =∘
Ω
119898
119897119890119898otimes 120599
119897isin 119881
4 (B14)
The holonomic metric can be recast in the form∘
119892 =∘
119892120583]∘
120599
120583
otimes
∘
120599
]=∘
119892 (∘
119890120583∘
119890])∘
120599
120583
otimes
∘
120599
] (B15)
The anholonomy objects defined on the tangent space∘
119879 ∘1199091198814
read∘
119862
119886
= 119889
∘
120599
119886
=1
2
∘
119862
119886
119887119888
∘
120599
119887
and
∘
120599
119888
(B16)
where the anholonomy coefficients∘
119862119886
119887119888 which represent the
curls of the base members are∘
119862
119888
119887119888= minus
∘
120599
119888
([∘
119890119886∘
119890119887])
=∘
119890119886
120583 ∘
119890119887
](
∘
120597120583
∘
119890119888
] minus∘
120597]∘
119890119888
120583)
= minus∘
119890119888
120583[∘
119890119886(∘
119890119887
120583
) minus∘
119890119887(∘
119890119886
120583
)]
(B17)
The (anholonomic) Levi-Civita (or Christoffel) connectioncan be written as
∘
Γ119886119887=∘
119890[119886rfloor119889
∘
120599119887]minus1
2(∘
119890119886rfloor∘
119890119887rfloor119889
∘
120599119888) and
∘
120599
119888
(B18)
where∘
120599119888is understood as the down indexed 1-form
∘
120599119888=
119900119888119887
∘
120599
119887
The norm 119894 (A2) can then be written in terms of thespacetime structures of 119881
4and119872
4as
119894 = = 120588otimes
120588
= 119886otimes
119886
= Ω120583
]∘
119890120583otimes
∘
120599
]
= Ω119886
119887119890119886119887
= Ω119898
119897119890119898otimes 120599
119897isin M
4
(B19)
provided
Ω119886
119887=
119886
119888119888
119887= Ω
120583
]∘
119890119886
120583
∘
119890119887
]
120588=
]120588
∘
119890]
120588
= 120583
120588∘
120599
120583
119888=
119888
119886 ∘
119890119886
119888
= 119888
119887
∘
120599
119887
(B20)
Under a local tetrad deformation (B20) a general spinconnection transforms according to
119886
119887120583=
119888
119886 ∘
120596119888
119889120583119889
119887+
119888
119886120583119888
119887= 120587
119897
119886120583120587119897
119887 (B21)
We have then two choices to recast metric as follows
= 119900119886119887119886
otimes 119887
= 119900119886119887119886
119888119887
119889
∘
120599
119888
otimes
∘
120599
119889
= 119888119889
∘
120599
119888
otimes
∘
120599
119889
(B22)
In the first case the contribution of the Christoffel symbolsconstructed by the metric
119886119887= 119900
119888119889119888
119886119889
119887reads
Γ119886
119887119888=1
2(∘
119862
119886
119887119888minus
1198861198861015840
1198871198871015840
∘
119862
1198871015840
1198861015840119888minus
1198861198861015840
1198881198881015840
∘
119862
1198881015840
1198861015840119887)
+1
21198861198861015840(∘
119890119888rfloor119889
1198871198861015840 minus∘
119890119887rfloor119889
1198881198861015840 minus∘
1198901198861015840rfloor119889
119887119888)
(B23)
As before the second deformation matrix 119886119887 can be
decomposed in terms of symmetric (119886119887)
and antisymmetric[119886119887]
parts of the matrix 119886119887= 119900
119886119888119888
119887 So
119886119887= Υ119900
119886119887+ Θ
119886119887+
119886119887 (B24)
where Υ = 119886119886 Θ
119886119887is the traceless symmetric part and
119886119887
is the skew symmetric part of the first deformationmatrix Inanalogy with (B4) the deformed metric can then be split as
120583] () = Υ
2
()∘
119892120583] + 120583] () (B25)
Journal of Astrophysics 23
where
120583] () = [119886119887 minus Υ
2
119900119886119887]∘
119890119886
120583
∘
119890119887
] (B26)
The inverse deformed metric reads
120583]() = 119900
119888119889minus1119886
119888minus1119887
119889
∘
119890119886
120583 ∘
119890119887
] (B27)
where minus1119886119888119888
119887=
119888
119887minus1119886
119888= 120575
119886
119887 The (anholonomic) Levi-
Civita (or Christoffel) connection is
Γ119886119887=
[119886rfloor119889
119887]minus1
2(
119886rfloor
119887rfloor119889
119888) and
119888
(B28)
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
where 119888is understood as the down indexed 1-form
119888=
119900119888119887119887
Hence the usual Levi-Civita connection is related tothe original connection by the relation
Γ120583
120588120590= Γ
120583
120588120590+ Π
120583
120588120590 (B29)
provided
Π120583
120588120590= 2
120583]] (120588nabla120590)Υ minus
120588120590119892120583]nabla]Υ
+1
2120583](nabla
120588]120590 + nabla120590120588] minus nabla]120588120590)
(B30)
where nabla is the covariant derivative The contravariantdeformed metric ]120588 is defined as the inverse of
120583] suchthat
120583]]120588= 120575
120588
120583 Hence the connection deformation Π120583
120588120590
acts like a force that deviates the test particles from thegeodesic motion in the space 119881
4 A metric-affine space
(4 Γ) is defined to have a metric and a linear connection
that need not be dependent on each other In general the lift-ing of the constraints of metric-compatibility and symmetryyields the new geometrical property of the spacetime whichare the nonmetricity 1-form
119886119887and the affine torsion 2-form
119886
representing a translational misfit (for a comprehensivediscussion see [92ndash95])These together with the curvature 2-form
119886
119887 symbolically can be presented as [96 97]
(119886119887
119886
119886
119887
) sim D (119886119887
119886
Γ119886
119887
) (B31)
where D is the covariant exterior derivative If the nonmetric-ity tensor
120582120583] = minusD120582120583] equiv minus120583] 120582 does not vanish
the general formula for the affine connection written in thespacetime components is
Γ120588
120583] =∘
Γ
120588
120583] + 120588
120583] minus 120588
120583] +1
2
120588
(120583]) (B32)
where∘
Γ
120588
120583] is the Riemann part and 120588
120583] = 2(120583])120588
+ 120588
120583]is the non-Riemann part the affine contortion tensor Thetorsion 120588
120583] = (12)120588
120583] = Γ120588
[120583]] given with respectto a holonomic frame 119889
120588
= 0 is the third-rank tensorantisymmetric in the first two indices with 24 independentcomponents In a presence of curvature and torsion the
coupling prescription of a general field carrying an arbitraryrepresentation of the Lorentz group will be
120583997888rarr D
120583=
120583minus119894
2(
119886119887
120583minus
119886119887
120583) 119869
119886119887 (B33)
with 119869119886119887
denoting the corresponding Lorentz generatorThe Riemann-Cartan manifold 119880
4 is a particular case of
the general metric-affine manifold M4 restricted by the
metricity condition 120582120583] = 0 when a nonsymmetric linear
connection is said to be metric compatible The Lorentz anddiffeomorphism invariant scalar curvature becomes eithera function of 119886
120583only or
120583]
() equiv 119886
120583119887
]120583]
119886119887
() = ( Γ)
equiv 120588]120583
120588120583] (Γ)
(B34)
C Determination of and in StandardTheory of Gravitation
Let 119886119887 = 119886119887
120583and 119889
120583 be the 1-forms of correspondingconnections assuming values in the Lorentz Lie algebra Theaction for gravitational field can be written in the form
119892=∘
119878 + 119876 (C1)
where the integral
∘
119878 = minus1
4aeligint⋆
∘
119877 = minus1
4aeligint⋆
∘
119877119888119889and
119888
and 119889
= minus1
2aeligint∘
119877radicminus119889Ω
(C2)
is the usual Einstein action with the coupling constant relat-ing to the Newton gravitational constant aelig = 8120587119866
119873119888
4 119878119876is
the phenomenological action of the spin-torsion interactionand ⋆ denotes the Hodge dual This is a 119862infin-linear map ⋆ Ω
119901rarr Ω
119899minus119901 which acts on the wedge product monomials ofthe basis 1-forms as ⋆(
1198861 sdotsdotsdot119886119901
) = 1205761198861 sdotsdotsdot119886119899
119886119901+1 sdotsdotsdot119886119899 Here we used
the abbreviated notations for the wedge product monomials1198861 sdotsdotsdot119886119901
= 1198861
and 1198862
and sdot sdot sdot and 119886119901 defined on the 119880
4space the
119886119894(119894 = 119901 + 1 119899) are understood as the down indexed
1-forms 119886119894= 119900
119886119894119887119887
and 1205761198861 sdotsdotsdot119886119899 is the total antisymmetricpseudotensor The variation of the connection 1-form 119886119887yields
120575119876=1
aeligint⋆T
119886119887and 120575
119886119887 (C3)
where
⋆ T119886119887=1
2⋆ (
119886and
119887) =
119888
and 119889
120576119888119889119886119887
=1
2
119888
120583] and 119889
120572120576119886119887119888119889120583]120572
(C4)
24 Journal of Astrophysics
and also
119886
= 119886
= 119889119886
+ 119886
119887and
119887
(C5)
The variation of the action describing themacroscopicmattersources
119898with respect to the coframe 120599119886 and connection 1-
form 119886119887 reads
120575119898= int120575
119898
= int(minus ⋆ 119886and 120575
119886
+1
2⋆ Σ
119886119887and 120575
119886119887)
(C6)
where⋆119886is the dual 3-form relating to the canonical energy-
momentum tensor 120583
119886 by
⋆119886=1
3120583
119886120576120583]120572120573
]120572120573(C7)
and ⋆Σ119886119887= minus ⋆ Σ
119887119886is the dual 3-form corresponding to the
canonical spin tensor which is identical with the dynamicalspin tensor
119886119887119888 namely
⋆Σ119886119887=
120583
119886119887120576120583]120572120573
]120572120573 (C8)
The variation of the total action = 119892+
119898 with respect to
the 119886 119886119887 and Φ gives the following field equations
(1)1
2
∘
119877119888119889and
119888
= aelig119889= 0
(2) ⋆ T119886119887= minus1
2aelig ⋆ Σ
119886119887
(3)120575
119898
120575Φ
= 0120575
119898
120575Φ
= 0
(C9)
In the sequel the DC-members and can readily bedetermined as follows
119897
119886= 120578
119897119898⟨
119886 119890
119898⟩
119886
119897= 120578
119897119898119886
(120599minus1)119898
(C10)
D The GGP in More Detail
Note that an invariance of the Lagrangian 119871Φ
under theinfinite-parameter group of general covariance (A5) in 119881
4
implies an invariance of 119871Φunder the 119866
119881group and vice
versa if and only if the generalized local gauge transforma-tions of the fields Φ() and their covariant derivative nabla
120583Φ()
are introduced by finite local119880119881isin 119866
119881gauge transformations
Φ1015840
() = 119880119881() Φ ()
[120583() nabla
120583Φ ()]
1015840
= 119880119881() [
120583() nabla
120583Φ ()]
(D1)
Here nabla120583denotes the covariant derivative agreeing with the
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
Lorentz group One has for example to set 120583() rarr 120583()
for the fields of spin (119895 = 0 1) for vector field [119869119886119887]119897
119896=
120575119897
119886120578119887119896minus 120575
119897
119887120578119886119896 but 120583() =
119886
120583()120574
119886 and 119869119886119887= minus(14)[120574
119886 120574
119887]
for the spinor field (119895 = 12) where 120574119886 are the DiracmatricesGiven the principal fiber bundle (119881
4 119866
119881 ) with the
structure group 119866119881 the local coordinates isin are =
( 119880119881) where isin 119881
4and 119880
119881isin 119866
119881 the total bundle space
is a smooth manifold and the surjection is a smoothmap rarr 119881
4 A set of open coverings U
119894 of 119881
4with
isin U119894 sub 119881
4satisfy ⋃
120572U
120572= 119881
4 The collection of matter
fields of arbitrary spins Φ() take values in standard fiber over
minus1(U
119894) = U
119894times
The fibration is given as ⋃
minus1() =
The local gauge will be the diffeomorphism map 119894
U119894times
1198814119866119881rarr
minus1(U
119894) isin since minus1
119894maps minus1(U
119894) onto
the direct (Cartesian) product U119894times
1198814119866119881 Here times
1198814represents
the fiber product of elements defined over space 1198814such that
(119894( 119880
119881)) = and
119894( 119880
119881) =
119894( (119894119889)
119866119881)119880
119881=
119894()119880
119881
for all isin U119894 where (119894119889)
119866119881is the identity element of the
group 119866119881 The fiber minus1 at isin 119881
4is diffeomorphic to
where is the fiber space such that minus1() equiv asymp The
action of the structure group119866119881on defines an isomorphism
of the Lie algebra g of 119866119881onto the Lie algebra of vertical
vector fields on tangent to the fiber at each isin calledfundamental To involve a drastic revision of the role of gaugefields in the physical concept of the spacetime deformationwe generalize the standard gauge scheme by exploring a newspecial type of distortion gauge field (119886) which is assumed toact on the external spacetime groups Then we also considerthe principle fiber bundle 119875(119872
4 119880
loc 119904) with the base space
1198724 the structure group119880loc and the surjection 119904Thematter
fields Φ(119909) take values in the standard fiber which is theHilbert vector space where a linear representation 119880(119909) ofgroup 119880loc is given This space can be regarded as the Liealgebra of the group 119880loc upon which the Lie algebra actsaccording to the law of the adjoint representation 119886 harrad 119886Φ rarr [119886Φ]
The GGP accounts for the gravitation gauge group 119866119881
generated by the hidden local internal symmetry 119880loc Thephysical system of the fields Φ() defined on 119881
4must be
invariant under the finite local gauge transformations119880119881(D1)
of the Lie group of gravitation119866119881(see Scheme 1) where 119877
120595(119886)
is the matrix of unitary map
119877120595(119886) Φ 997888rarr Φ
119878 (119886) 119877120595(119886) (120574
119896119863
119896Φ) 997888rarr (
]() nabla]Φ)
(D2)
Journal of Astrophysics 25
Φ998400( x) =UVΦ(x)
R998400120595(x x)
UlocΦ998400(x) =U
locΦ(x)
UV = R998400120595U
locRminus1120595
Φ(x)
Φ(x)
R120595(x x)
Scheme 1 The GGP
Here 119878(119865) is the gauge invariant scalar function 119878(119865) equiv(14)
minus1(119865) = (14)
119897
120583
120583
119897119863
119896= 120597
119896minus 119894aelig 119886
119896 In an illustration
of the point at issue the (D2) explicitly may read
Φ120583sdotsdotsdot120575
() = 120583
119897sdot sdot sdot
120575
119898119877 (119886)Φ
119897sdotsdotsdot119898(119909)
equiv (119877120595)120583sdotsdotsdot120575
119897sdotsdotsdot119898Φ
119897sdotsdotsdot119898(119909)
(D3)
and also
]() nabla]Φ
120583sdotsdotsdot120575
()
= 119878 (119865) 120583
119897sdot sdot sdot
120575
119898119877 (119886) 120574
119896119863
119896Φ
119897sdotsdotsdot119898(119909)
(D4)
In case of zero curvature one has 120595120583119897= 119863
120583
119897= 119890
120583
119897=
(120597119909120583120597119883
119897) 119863 = 0 where 119883119897 are the inertial coordinates
In this the conventional gauge theory given on the 1198724
is restored in both curvilinear and inertial coordinatesAlthough the distortion gauge field (119886
119860) is a vector field
only the gravitational attraction is presented in the proposedtheory of gravitation
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
120572(120582 = plusmn 120572 = 1 2 3) are linearly
independent unit basis vectors at the point (p) of interest ofthe given three-dimensional space 1198773
120582 The unit vectors 119874
120582
and 120590120572imply
⟨119874120582 119874
120591⟩ =
lowast120575120582120591 ⟨120590
120572 120590
120573⟩ = 120575
120572120573 (E2)
where 120575120572120573
is the Kronecker symbol and lowast
120575120582120591= 1 minus 120575
120582120591
Three spatial 119890120572= 120585 times 120590
120572and three temporal 119890
0120572= 120585
0times 120590
120572
components are the basis vectors respectively in spaces 1198773
and 1198793 where119874plusmn= (1radic2)(120585
0plusmn 120585) 1205852
0= minus120585
2= 1 ⟨120585
0 120585⟩ = 0
The 3D space 1198773plusmnis spanned by the coordinates 120578
(plusmn120572) In using
this language it is important to consider a reduction to thespace119872
4which can be achieved in the following way
(1) In case of free flat space 1198726 the subspace 1198793 is
isotropic And in so far it contributes in line elementjust only by the square of the moduli 119905 = |x0| x0 isin 1198793then the reduction 119872
6rarr 119872
4= 119877
3oplus 119879
1 can bereadily achieved if we use 119905 = |x0| for conventionaltime
(2) In case of curved space the reduction 1198816rarr 119881
4can
be achieved if we use the projection ( 1198900) of the tem-
poral component ( 1198900120572) of basis six-vector 119890( 119890
120572 119890
0120572) on
the given universal direction ( 1198900120572rarr 119890
0) By this we
choose the time coordinate Actually the Lagrangianof physical fields defined on 119877
6is a function of scalars
such that 119860(120582120572)119861(120582120572)= 119860
120572119861120572+ 119860
01205721198610120572 then upon
the reduction of temporal components of six-vectors119860
01205721198610120572= 119860
0120572⟨ 119890
0120572 119890
0120573⟩119861
0120573= 119860
0⟨ 119890
0 119890
0⟩119861
0= 119860
01198610
we may fulfill a reduction to 1198814
A distortion of the basis (E2) comprises the following twosteps We at first consider distortion transformations of theingredient unit vectors 119874
120591under the distortion gauge field
(119886)
(+120572)(119886) = Q
120591
(+120572)(119886) 119874
120591= 119874
++ aelig119886
(+120572)119874minus
(minus120572)(119886) = Q
120591
(minus120572)(119886) 119874
120591= 119874
minus+ aelig119886
(minus120572)119874+
(E3)
where Q (=Q120591
(120582120572)(119886)) is an element of the group 119876 This
induces the distortion transformations of the ingredient unitvectors 120590
120573 which in turn undergo the rotations
(120582120572)(120579) =
R120573
(120582120572)(120579)120590
120573 whereR(120579) isin 119878119874(3) is the element of the group
of rotations of planes involving two arbitrary axes around theorthogonal third axis in the given ingredient space1198773
120582 In fact
distortion transformations of basis vectors (119874) and (120590) arenot independent but rather are governed by the spontaneousbreaking of the distortion symmetry (for more details see[74]) To avoid a further proliferation of indices hereafter wewill use uppercase Latin (119860) in indexing (120582120572) and so forthThe infinitesimal transformations then read
119876 do not complete the group119867 to the dynamical group 119866119863
and therefore they cannot be interpreted as the generatorsof the quotien space 119866
119863119867 and the distortion fields 119886
119860
cannot be identified directly with the Goldstone fields arisingin spontaneous breaking of the distortion symmetry 119866
119863
These objections however can be circumvented becauseas it is shown by [74] the distortion group 119866
119863= 119876 times
119878119874(3) can be mapped in a one-to-one manner onto thegroup 119866
119863= 119878119874(3) times 119878119874(3) which is isomorphic to the
chiral group 119878119880(2) times 119878119880(2) in case of which the method ofphenomenological Lagrangians is well known In aftermathwe arrive at the key relation
tan 120579119860= minusaelig119886
119860 (E6)
26 Journal of Astrophysics
Given the distortion field 119886119860 the relation (E6) uniquely
determines six angles 120579119860of rotations around each of six (119860)
axes In pursuing our goal further we are necessarily led toextending a whole framework of GGP now for the base 12Dsmooth differentiable manifold
11987212= 119872
6oplus119872
6 (E7)
Here the1198726is related to the spacetime continuum (E1) but
the 1198726is displayed as a space of inner degrees of freedom
The
119890(120582120583120572)
= 119874120582120583otimes 120590
120572(120582 120583 = 1 2 120572 = 1 2 3) (E8)
are basis vectors at the point 119901(120577) of11987212
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
The matrices 119862(119886) generate the group of distortion transfor-mations of the bi-pseudo-vectors
119862120591](120582120583120572)
(119886) = 120575120591
120582120575]120583+ aelig119886
(120582120583120572)
lowast120575120591
120582
lowast
120575]120583 (E13)
but 119877(119886) isin 119878119874(3)120582120583mdashthe group of ordinary rotations of
the planes involving two arbitrary bases of the spaces 1198773120582120583
around the orthogonal third axes The angles of rotationsare determined according to (E6) but now for the extendedindices 119860 = (120582 120583 120572) and so forth
F Field Equations at Spherical Symmetry
The extended field equations followed at once in terms ofEuler-Lagrange variations respectively on the spaces 119872
12
and 12[74] In accordance the equation of distortion gauge
field 119886119860= (119886
(120582120572) 119886
(120591120573)) reads
120597119861120597119861119886119860minus (1 minus 120577
minus1
0) 120597
119860120597119861119886119861
= 119869119860= minus1
2radic119892120597119892
119861119862
120597119886119860
119879119861119862
(F1)
where 119879119861119862
is the energy-momentum tensor and 1205770is the
gauge fixing parameter To render our discussion here moretransparent below we clarify the relation between gravita-tional and coupling constants To assist in obtaining actualsolutions from the field equations wemay consider theweak-field limit and will envisage that the right-hand side of (F1)should be in the form
minus1
2(4120587119866
119873)radic119892 (119909)
120597119892119861119862(119909)
120597119909119860
119861119862 (F2)
Hence we may assign to Newtonrsquos gravitational constant 119866119873
the value
119866119873=aelig2
4120587 (F3)
The curvature of manifold 1198726rarr 119872
6is the familiar
distortion induced by the extended field components
119886(11120572)
= 119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= 119886(22120572)
equiv1
radic2
119886(minus120572)
(F4)
The other regime of ID presents at
119886(11120572)
= minus119886(21120572)
equiv1
radic2
119886(+120572)
119886(12120572)
= minus119886(22120572)
equiv1
radic2
119886(minus120572)
(F5)
To obtain a feeling for this point we may consider physicalsystems which are static as well as spherically symmetricalWe are interested in the case of a spherical-symmetric gravi-tational field 119886
0(119903) in presence of one-dimensional space-like
ID-field 119886
119886(113)
= 119886(223)
= 119886(+3)=1
2(minus119886
0+ 119886)
119886(123)
= 119886(213)
= 119886(minus3)=1
2(minus119886
0minus 119886)
119886(1205821205831)
= 119886(1205821205832)
= 0 120582 120583 = 1 2
(F6)
One can then easily determine the basis vectors (119890120582120572 119890
120591120573)
where tan 120579(plusmn3)= aelig(minus119886
0plusmn 119886) Passing back from the
6to
1198814 the basis vectors read
0= 119890
0(1 minus 119909
0) + 119890
3119909
3= 119890
3(1 + 119909
0) minus 119890
03119909
Journal of Astrophysics 27
1=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
1
+ (sin 120579(+3)+ sin 120579
(minus3)) 119890
2
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
01
+ (sin 120579(+3)minus sin 120579
(minus3)) 119890
02
2=1
2(cos 120579
(+3)+ cos 120579
(minus3)) 119890
2
minus (sin 120579(+3)+ sin 120579
(minus3)) 119890
1
+ (cos 120579(+3)minus cos 120579
(minus3)) 119890
02
minus (sin 120579(+3)minus sin 120579
(minus3)) 119890
01
(F7)where 119909
0equiv aelig119886
0 119909 equiv aelig119886
G SPC-Configurations
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
The equations describing the equilibrium SPC include thegravitational and ID field equations (A2) the hydrostaticequilibrium equation and the state equation specified foreach domain of many layered configurations The resultingstable SPC is formed which consists of the protomattercore and the outer layers of ordinary matter A layering ofconfigurations is a consequence of the onset of differentregimes in equation of state In the density range 120588 lt
454 times 1012 g cmminus3 one uses for both configurations the
simple semiempirical formula of state equation given byHarrison and Wheeler see for example [98] Above thedensity 120588 gt 454 times 1012 g cmminus3 for the simplicity the I-class SPCI configuration is thought to be composed of regularn-p-e (neutron-proton-electron) gas (in absence of ID) inintermediate density domain 454 times 1012 g cmminus3
le 120588 lt
120588119889and of the n-p-e protomatter in presence of ID at 120588 gt
120588119889 For the II-class SPCII configuration above the density120588119891119897= 409 times 10
14 g cmminus3 one considers an onset of meltingdown of hadrons when nuclear matter consequently turnsto quark matter found in string flip-flop regime In domain120588119891119897le 120588 lt 120588
119889 to which the distances 04 fm lt 119903
119873119873le
16 fm correspond one has the regular (ID is absent) stringflip-flop regime This is a kind of tunneling effect when thestrings joining the quarks stretch themselves violating energyconservation and after touching each other they switch on tothe other configuration [71] The basic technique adopted forcalculation of transition matrix element is the instantontechnique (semiclassical treatment) During the quantumtransition from a state 120595
1of energy
1to another one 120595
2
of energy 2 the lowering of energy of system takes place
and the quark matter acquires Δ correction to the classicalstring energy such that the flip-flop energy lowers the energyof quark matter consequently by lowering the critical densityor critical Fermi momentum If one for example looks forthe string flip-flop transition amplitude of simple system of
119902119902119902119902 described by the Hamiltonian and invariant action then one has
[d]eminusS⟩⟨ =|eminusHT| ⟨int⟩ (G1)
where 119879 is an imaginary time interval and [119889] is theintegration over all the possible string motionThe action isproportional to the area of the surface swept by the stringsin the finite region of ID-region of 119877
4The strings are initially
in the -configuration and finally in the -configurationThe maximal contribution to the path integral comes fromthe surface 120590
0of the minimum surface area ldquoinstantonrdquo A
computation of the transition amplitude is straightforward bysumming over all the small vibrations around 120590
0 In domain
120588119889le 120588 lt 120588
119886119904 one has the string flip-flop regime in presence
of ID at distances 025 fm lt 119903119873119873le 04 fmThat is the system
is made of quark protomatter in complete 120573-equilibriumwith rearrangement of string connections joining them Infinal domain 120588 gt 120588
119886119904 the system is made of quarks
in one bag in complete 120573-equilibrium at presence of IDThe quarks are under the weak interactions and gluonsincluding the effects of QCD-perturbative interactions TheQCD vacuum has a complicated structure which is inti-mately connected to the gluon-gluon interaction In mostapplications sufficient accuracy is obtained by assumingthat all the quarks are almost massless inside a bag Thelatter is regarded as noninteracting Fermi gas found in theID-region of the space-time continuum at short distances119903119873119873le 025 fm Each configuration is defined by the two free
parameters of central values of particle concentration (0)and dimensionless potential of space-like ID-field 119909(0) Theinterior gravitational potential 119909int
0(119903) matches the exterior
one 119909ext0(119903) at the surface of the configuration The central
value of the gravitational potential 1199090(0) can be found by
reiterating integrations when the sewing condition of theinterior and exterior potentials holds The key question ofstability of SPC was studied in [72] In the relativistic casethe total mass-energy of SPC is the extremum in equilibriumfor all configurations with the same total number of baryonsWhile the extrema of and 119873 occur at the same point ina one-parameter equilibrium sequence one can look for theextremum of = 1198882 minus
119861119873 on equal footing Minimizing
the energy will give the equilibrium configuration and thesecond derivative of will give stability information Recallthat for spherical configurations of matter instantaneouslyat rest small radial deviations from equilibrium are governedby a Sturm-Liouville linear eigenvalue equation [98] with theimposition of suitable boundary conditions on normalmodeswith time dependence 120585119894( 119905) = 120585119894()119890119894120596119905 A necessary andsufficient condition for stability is that the potential energybe positive defined for all initial data of 120585119894( 0) namelyin first order approximation when one does not take intoaccount the rotation and magnetic field if the square offrequency of normal mode of small perturbations is positiveA relativity tends to destabilize configurations Howevernumerical integrations of the stability equations of SPC [72]give for the pressure-averaged value of the adiabatic indexΓ1= (120597 ln 120597 ln )
119904the following values Γ
1asymp 2216 for
28 Journal of Astrophysics
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
the SPCI and Γ1 asymp 24 for SPCII configurations This clearlyproves the stability of resulting SPC Note that the SPC isalways found inside the event horizon sphere and thereforeit could be observed only in presence of accreting matter
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The very helpful and knowledgable comments from theanonymous referee which have essentially clarified the paperare much appreciated
References
[1] J A Orosz ldquoInventory of black hole binariesrdquo in Proceedings ofthe International Astronomical Union Symposium (IAU rsquo03) Kvan der Hucht A Herrero and C Esteban Eds vol 212 2003
[2] D Lynden-Bell ldquoGalactic nuclei as collapsed old quasarsrdquoNature vol 223 no 5207 pp 690ndash694 1969
[3] T R Lauer S M Faber D Richstone et al ldquoThe massesof nuclear black holes in luminous elliptical galaxies andimplications for the space density of the most massive blackholesrdquo Astrophysical Journal Letters vol 662 no 2 I pp 808ndash834 2007
[4] M Volonteri ldquoFormation of supermassive black holesrdquo Astron-omy and Astrophysics Review vol 18 no 3 pp 279ndash315 2010
[5] J Woo and C M Urry ldquoActive galactic nucleus black holemasses and bolometric luminositiesrdquo Astrophysical JournalLetters vol 579 no 2 pp 530ndash544 2002
[6] E Treister P Natarajan D B Sanders C Megan Urry KSchawinski and J Kartaltepe ldquoMajor galaxy mergers and thegrowth of supermassive black holes in quasarsrdquo Science vol 328no 5978 pp 600ndash602 2010
[7] A J Davis and P Natarajan ldquoAngular momentum and cluster-ing properties of early dark matter haloesrdquo Monthly Notices ofthe Royal Astronomical Society vol 393 no 4 pp 1498ndash15022009
[8] MVestergaard ldquoEarly growth and efficient accretion ofmassiveblack holes at high redshiftrdquo Astrophysical Journal Letters vol601 no 2 I pp 676ndash691 2004
[9] M Volonteri G Lodato and P Natarajan ldquoThe evolutionof massive black hole seedsrdquo Monthly Notices of the RoyalAstronomical Society vol 383 no 3 pp 1079ndash1088 2008
[10] M Volonteri and P Natarajan ldquoJourney to theMBH-120590 relationthe fate of low-mass black holes in the universerdquo MonthlyNotices of the Royal Astronomical Society vol 400 no 4 pp1911ndash1918 2009
[11] F Shankar D H Weinberg and J Miralda-Escude ldquoSelf-consistent models of the AGN and black hole populationsduty cycles accretion rates and the mean radiative efficiencyrdquoAstrophysical Journal Letters vol 690 no 1 pp 20ndash41 2009
[12] B C Kelly M Vestergaard X Fan P Hopkins L Hernquistand A Siemiginowska ldquoConstraints on black hole growthquasar lifetimes and Eddington ratio distributions from theSDSS broad-line quasar black holemass functionrdquoAstrophysicalJournal Letters vol 719 no 2 pp 1315ndash1334 2010
[13] P Natarajan ldquoThe formation and evolution of massive blackhole seeds in the early Universerdquo in Fluid Flows to BlackHoles ATribute to S Chandrasekhar on His Birth Centenary D J SaikiaEd pp 191ndash206 World Scientific 2011
[14] E Treister and C M Urry ldquoThe cosmic history of blackhole growth from deep multiwavelength surveysrdquo Advances inAstronomy vol 2012 Article ID 516193 21 pages 2012
[15] E Treister K Schawinski M Volonteri P Natarajan andE Gawiser ldquoBlack hole growth in the early Universe is self-regulated and largely hidden from viewrdquo Nature vol 474 no7351 pp 356ndash358 2011
[16] C J Willott L Albert D Arzoumanian et al ldquoEddington-limited accretion and the black hole mass function at redshift6rdquo Astronomical Journal vol 140 no 2 pp 546ndash560 2010
[17] V Bromm and A Loeb ldquoFormation of the first supermassiveblack holesrdquoAstrophysical Journal Letters vol 596 no 1 pp 34ndash46 2003
[18] B Devecchi and M Volonteri ldquoFormation of the first nuclearclusters and massive black holes at high redshiftrdquo AstrophysicalJournal Letters vol 694 no 1 pp 302ndash313 2009
[19] A J Barger L L Cowie R F Mushotzky et al ldquoThe cosmicevolution of hard X-ray-selected active galactic nucleirdquo Astro-nomical Journal vol 129 no 2 pp 578ndash609 2005
[20] S M Croom G T Richards T Shanks et al ldquoThe 2dF-SDSSLRG and QSO survey the QSO luminosity function at 04 lt119911 lt 26rdquoMNRAS vol 399 no 4 pp 1755ndash1772 2009
[21] Y Ueda M Akiyama K Ohta and T Miyaji ldquoCosmologicalevolution of the hard X-ray active galactic nucleus luminosityfunction and the origin of the hard X-ray backgroundrdquo TheAstrophysical Journal Letters vol 598 no 2 I pp 886ndash908 2003
[22] D R Ballantyne A R Draper K K Madsen J R Rigby and ETreister ldquoLifting the veil on obscured accretion active galacticnuclei number counts and survey strategies for imaging hardX-ray missionsrdquo Astrophysical Journal vol 736 article 56 no 12011
[23] T Takahashi K Mitsuda R Kelley et al ldquoThe ASTRO-Hmissionrdquo in Space Telescopes and Instrumentation 2010 Ultra-violet to Gamma Ray 77320Z M Arnaud S S Murray andT Takahashi Eds vol 7732 of Proceedings of SPIE San DiegoCalif USA June 2010
[24] L Yao E R Seaquist N Kuno and L Dunne ldquoCO moleculargas in infrared-luminous galaxiesrdquoAstrophysical Journal Lettersvol 588 no 2 I pp 771ndash791 2003
[25] A Castellina and F Donato ldquoAstrophysics of galactic chargedcosmic raysrdquo in Planets Stars and Stellar Systems T D Oswaltand G Gilmore Eds vol 5 of Galactic Structure and StellarPopulations pp 725ndash788 2011
[26] A Letessier-Selvon and T Stanev ldquoUltrahigh energy cosmicraysrdquoReviews ofModern Physics vol 83 no 3 pp 907ndash942 2011
[27] G Sigl ldquoHigh energy neutrinos and cosmic raysrdquo httparxivorgabs12020466
[28] K Kotera D Allard and A V Olinto ldquoCosmogenic neutrinosparameter space and detectabilty from PeV to ZeVrdquo Journal ofCosmology and Astroparticle Physics vol 2010 article 013 2010
[29] D Semikoz ldquoHigh-energy astroparticle physicsrdquo CERN YellowReport CERN-2010-001 2010
[30] J Linsley ldquoPrimary cosmic-rays of energy 1017 to 1020 ev theenergy spectrum and arrival directionsrdquo in Proceedings of the8th International Cosmic Ray Conference vol 4 p 77 1963
[31] NHayashida KHondaMHonda et al ldquoObservation of a veryenergetic cosmic ray well beyond the predicted 27 K cutoff in
Journal of Astrophysics 29
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
the primary energy spectrumrdquo Physical Review Letters vol 73no 26 pp 3491ndash3494 1994
[32] D J Bird S C Corbato H Y Dai et al ldquoDetection of a cosmicray with measured energy well beyond the expected spectralcutoff due to cosmic microwave radiationrdquo Astrophysical Jour-nal Letters vol 441 no 1 pp 144ndash150 1995
[33] K Greisen ldquoEnd to the cosmic-ray spectrumrdquo Physical ReviewLetters vol 16 no 17 pp 748ndash750 1966
[34] G T Zatsepin and V A Kuzrsquomin ldquoUpper limit of the spectrumof cosmic raysrdquo Journal of Experimental andTheoretical PhysicsLetters vol 4 pp 78ndash80 1966
[35] R Abbasi Y Abdou T Abu-Zayyad et al ldquoTime-integratedsearches for point-like sources of neutrinos with the 40-stringIceCube detectorrdquo The Astrophysical Journal vol 732 no 1article 18 2011
[36] J Abraham ldquoObservation of the suppression of the flux ofcosmic rays above 4m1019 eVrdquo Physical Review Letters vol 101no 6 Article ID 061101 2008
[37] R J Protheroe and T Stanev ldquoLimits onmodels of the ultrahighenergy cosmic rays based on topological defectsrdquo PhysicalReview Letters vol 77 no 18 pp 3708ndash3711 1996
[38] D Fargion ldquoUltrahigh energy neutrino scattering onto reliclight neutrinos in galactic halo as a possible source of highestextragalactic cosmic raysrdquo in Proceedings of the 25th Inter-national Cosmic Ray Conference (Held July-August 1997 inDurban South Africa) M S Potgieter C Raubenheimer andD J van der Walt Eds vol 7 pp 153ndash156 PotchefstroomUniversity Transvaal South Africa 1997
[39] D Fargion B Mele and A Salis ldquoUltra-high-energy neutrinoscattering onto relic light neutrinos in the galactic halo as apossible source of the highest energy extragalactic cosmic raysrdquoThe Astrophysical Journal vol 517 no 2 pp 725ndash733 1999
[40] D Fargion B Mele and A Salis ldquoUltrahigh energy neutrinoscattering onto relic light neutrinos in galactic halo as a possiblesource of highest energy extragalactic cosmicrdquo AstrophysicalJournal vol 517 pp 725ndash733 1999
[41] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[42] T J Weiler ldquoCosmic-ray neutrino annihilation on relic neutri-nos revisited amechanism for generating air showers above theGreisen-Zatsepin-Kuzmin cutoffrdquo Astroparticle Physics vol 11no 3 pp 303ndash316 1999
[43] V K Dubrovich D Fargion and M Y Khlopov ldquoPrimordialbound systems of superheavy particles as the source of ultra-high energy cosmic raysrdquo Nuclear Physics BmdashProceedings Sup-plements vol 136 no 1ndash3 pp 362ndash367 2004
[44] A Datta D Fargion and B Mele ldquoSupersymmetrymdashneutrinounboundrdquo Journal of High Energy Physics 0509007 2005
[45] S Yoshida G Sigl and S Lee ldquoExtremely high energy neutri-nos neutrino hot dark matter and the highest energy cosmicraysrdquo Physical Review Letters vol 81 no 25 pp 5505ndash55081998
[46] A Ringwald ldquoPossible detection of relic neutrinos and theirmassrdquo in Proceedings of the 27th International Cosmic Ray Con-ference R Schlickeiser Ed Invited Rapporteur and HighlightPapers p 262 Hamburg Germany August 2001
[47] Z Fodor S D Katz and A Ringwald ldquoRelic neutrino massesand the highest energy cosmic-raysrdquo Journal of High EnergyPhysics 2002
[48] O E Kalashev V A Kuzmin D V Semikoz and G SiglldquoUltrahigh-energy neutrino fluxes and their constraintsrdquo Phys-ical Review D vol 66 no 6 Article ID 063004 2002
[49] O E Kalashev V A Kuzmin D V Semikoz andG Sigl ldquoUltra-high energy cosmic rays from neutrino emitting accelerationsourcesrdquo Physical Review D vol 65 no 10 Article ID 1030032002
[50] A Y Neronov and D V Semikoz ldquoWhich blazars are neutrinoloudrdquo Physical Review D vol 66 no 12 Article ID 1230032002
[51] P Jain and S Panda ldquoUltra high energy cosmic rays from earlydecaying primordial black holesrdquo in Proceedings of the 29thInternational Cosmic Ray Conference B Sripathi Acharya Edvol 9 pp 33ndash36 Tata Institute of Fundamental Research PuneIndia 2005
[52] T K Gaisser F Halzen and T Stanev ldquoParticle astrophysicswith high energy neutrinosrdquo Physics Report vol 258 no 3 pp173ndash236 1995
[53] J Alvarez-Muniz and F Halzen ldquoPossible high-energy neutri-nos from the cosmic accelerator RX J17137-3946rdquoAstrophysicalJournal Letters vol 576 no 1 pp L33ndashL36 2002
[54] P S Coppi and F A Aharonian ldquoConstraints on the very highenergy emissivity of the universe from the diffuse GeV gamma-ray backgroundrdquoAstrophysical Journal Letters vol 487 no 1 ppL9ndashL12 1997
[56] F Halzen and E Zas ldquoNeutrino fluxes from active galaxies amodel-independent estimaterdquoAstrophysical Journal Letters vol488 no 2 pp 669ndash674 1997
[57] S Adrian-Martinez I Al Samarai A Albert et al ldquoMeasure-ment of atmospheric neutrino oscillations with the ANTARESneutrino tele-scoperdquo Physics Letters B vol 714 no 2ndash5 pp 224ndash230 2012
[58] S Adrian-Martınez J A Aguilar I Al Samarai et al ldquoFirstsearch for point sources of high energy cosmic neutrinos withthe ANTARES neutrino telescoperdquo The Astrophysical JournalLetters vol 743 no 1 article L14 2011
[59] M G Aartsen R Abbasi Y Abdou et al ldquoMeasurementof atmospheric neutrino oscillations with Ice-Cuberdquo PhysicalReview Letters vol 111 no 8 Article ID 081801 2013
[60] M G Aartsen R Abbasi Y Abdou et al ldquoFirst observationof PeV-energy neutrinos with IceCuberdquo Physical Review Lettersvol 111 Article ID 021103 2013
[61] M G Aartsen ldquoEvidence for high-energy extraterrestrial neu-trinos at the IceCube detectorrdquo Science vol 342 no 6161 ArticleID 1242856 2013
[62] C Kopper and IceCube Collabotration ldquoObservation of PeVneutrinos in IceCuberdquo in Proceedings of the IceCube ParticleAstrophysics Symposium (IPA 13) Madison Wis USA May2013 httpwipacwiscedumeetingshomeIPA2013
[63] N Kurahashi-Neilson ldquoSpatial clustering analysis of the veryhigh energy neutrinos in icecuberdquo in Proceedings of the IceCubeParticle Astrophysics Symposium (IPA rsquo13) IceCube Collabora-tion Madison Wis USA May 2013
[64] N Whitehorn and IceCube Collabotration ldquoResults fromIceCuberdquo in Proceedings of the IceCube Particle AstrophysicsSymposium (IPA 13) Madison Wis USA May 2013
[65] A V Avrorin V M Aynutdinov V A Balkanov et al ldquoSearchfor high-energy neutrinos in the Baikal neutrino experimentrdquoAstronomy Letters vol 35 no 10 pp 651ndash662 2009
30 Journal of Astrophysics
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983
[66] T Abu-Zayyad R AidaM Allen et al ldquoThe cosmic-ray energyspectrum observed with the surface detector of the tele-scopearray experimentrdquo The Astrophysical Journal Letters vol 768no 1 5 pages 2013
[67] P Abreu M Aglietta M Ahlers et al ldquoLarge-scale distributionof arrival directions of cosmic rays detected above 1018 eV at thepierre auger observatoryrdquoTheAstrophysical Journal SupplementSeries vol 203 no 2 p 20 2012
[68] ldquoA search for point sources of EeV neutronsrdquoThe AstrophysicalJournal vol 760 no 2 Article ID 148 11 pages 2012
[69] T Ebisuzaki Y Takahashi F Kajino et al ldquoThe JEM-EUSOmission to explore the extreme Universerdquo in Proceedings of the7th Tours Symposium on Nuclear Physics and Astrophysics vol1238 pp 369ndash376 Kobe Japan November 2009
[70] K Murase M Ahlers and B C Lacki ldquoTesting the hadronu-clear origin of PeV neutrinos observed with IceCuberdquo PhysicalReview D vol 88 Article ID 121301(R) 2013
[71] G Ter-Kazarian ldquoProtomatter and EHE CRrdquo Journal of thePhysical Society of Japan B vol 70 pp 84ndash98 2001
[72] G Ter-Kazarian S Shidhani and L Sargsyan ldquoNeutrino radi-ation of the AGN black holesrdquo Astrophysics and Space Sciencevol 310 no 1-2 pp 93ndash110 2007
[73] G Ter-Kazarian and L Sargsyan ldquoSignature of plausible accret-ing supermassive black holes in Mrk 261262 and Mrk 266rdquoAdvances in Astronomy vol 2013 Article ID 710906 12 pages2013
[74] G T Ter-Kazarian ldquoGravitation and inertia a rearrangement ofvacuum in gravityrdquo Astrophysics and Space Science vol 327 no1 pp 91ndash109 2010
[75] G Ter-Kazarian ldquoUltra-high energy neutrino fluxes fromsupermassive AGN black holesrdquo Astrophysics amp Space Sciencevol 349 pp 919ndash938 2014
[76] G T Ter-Kazarian ldquoGravitation gauge grouprdquo Il NuovoCimento vol 112 no 6 pp 825ndash838 1997
[77] A Neronov D Semikoz F Aharonian andO Kalashev ldquoLarge-scale extragalactic jets powered by very-high-energy gammaraysrdquo Physical Review Letters vol 89 no 5 pp 1ndash4 2002
[78] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports C vol 66no 6 pp 213ndash393 1980
[79] S Kobayashi and K Nomizu Foundations of Differential Geom-etry Interscience Publishers New York NY USA 1963
[80] J Plebanski ldquoForms and riemannian geometryrdquo in Proceedingsof the International School of Cosmology and Gravitation EriceItaly 1972
[81] A Trautman Differential Geometry for Physicists vol 2 ofMonographs and Textbooks in Physical Science BibliopolisNaples Fla USA 1984
[82] G Ter-Kazarian ldquoTwo-step spacetime deformation induceddynamical torsionrdquo Classical and Quantum Gravity vol 28 no5 Article ID 055003 2011
[83] L S Pontryagin Continous Groups Nauka Moscow Russia1984
[84] B A Dubrovin Contemporary Geometry Nauka MoscowRussia 1986
[85] S Coleman J Wess and B Zumino ldquoStructure of phenomeno-logical lagrangians Irdquo Physical Review vol 177 no 5 pp 2239ndash2247 1969
[86] C G Callan S Coleman J Wess and B Zumino ldquoStructure ofphenomenological lagrangians IIrdquo Physical Review vol 177 no5 pp 2247ndash2250 1969
[87] S Weinberg Brandeis Lectures MIT Press Cambridge MassUSA 1970
[88] A Salam and J Strathdee ldquoNonlinear realizations II Confor-mal symmetryrdquo Physical Review vol 184 pp 1760ndash1768 1969
[89] C J Isham A Salam and J Strathdee ldquoNonlinear realizationsof space-time symmetries Scalar and tensor gravityrdquo Annals ofPhysics vol 62 pp 98ndash119 1971
[90] D V Volkov ldquoPhenomenological lagrangiansrdquo Soviet Journal ofParticles and Nuclei vol 4 pp 1ndash17 1973
[91] V I Ogievetsky ldquoNonlinear realizations of internal and space-time symmetriesrdquo in Proceedings of 10th Winter School ofTheoretical Physics in Karpacz vol 1 p 117 Wroclaw Poland1974
[92] V de Sabbata and M Gasperini ldquoIntroduction to gravitationrdquoin Unified Field Theories of more than Four Dimensions Vde Sabbata and E Schmutzer Eds p 152 World ScientificSingapore 1985
[93] V de Sabbata and M Gasperini ldquoOn the Maxwell equations ina Riemann-Cartan spacerdquo Physics Letters A vol 77 no 5 pp300ndash302 1980
[94] V de Sabbata and C Sivaram Spin and Torsion in GravitationWorld Scientific Singapore 1994
[95] N J Poplawski ldquoSpacetime and fieldsrdquo httparxivorgabs09110334
[96] A Trautman ldquoOn the structure of the Einstein-Cartan equa-tionsrdquo in Differential Geometry vol 12 of Symposia Mathemat-ica pp 139ndash162 Academic Press London UK 1973
[97] J-P Francoise G L Naber and S T Tsou Eds Encyclopedia ofMathematical Physics Elsevier Oxford UK 2006
[98] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars A Wiley-Intercience Publication Wiley-Intercience New York NY USA 1983