Introduction to General Relativity - G. T.hooft

INTRODUCTIONTOGENERALRELATIVITYG. t HooftCAPUTCOLLEGE1998Institute for Theoretical PhysicsUtrecht University,Princetonplein 5, 3584 CC Utrecht, the Netherlandsversion 30/1/98PROLOGUEGeneral relativity is a beautiful scheme for describing the gravitational eld and theequationsitobeys. Nowadaysthistheoryisoftenusedasaprototypeforother, moreintricate constructions to describe forces between elementary particles or other branches offundamental physics. This is why in an introduction to general relativity it is of importanceto separate as clearly as possible the various ingredients that together give shape to thisparadigm.After explaining the physical motivations we rst introduce curved coordinates, thenadd to this the notion of an ane connection eld and only as a later step add to that themetric eld. One then sees clearly how space and time get more and more structure, untilnally all we have to do is deduce Einsteins eld equations.Asforapplicationsofthetheory,theusualonessuchasthegravitational redshift,theSchwarzschildmetric, theperihelionshift andlightdeectionareprettystandard.They can be found in the cited literature if one wants any further details. I do pay someextraattentiontoanapplicationthat maywell becomeimportantinthenearfuture:gravitational radiation. The derivations given are often tedious, but they can be producedrather elegantly using standard Lagrangian methods from eld theory, which is what willbe demonstrated in these notes.LITERATUREC.W. Misner,K.S. Thorne and J.A. Wheeler,Gravitation, W.H. Freeman and Comp.,San Francisco 1973, ISBN 0-7167-0344-0.R. Adler, M. Bazin, M. Schier, Introduction to General Relativity, Mc.Graw-Hill 1965.R. M. Wald, General Relativity, Univ. of Chicago Press 1984.P.A.M. Dirac, General Theory of Relativity, Wiley Interscience 1975.S. Weinberg, GravitationandCosmology: PrinciplesandApplicationsof theGeneralTheory of Relativity, J. Wiley & Sons. year ???S.W. Hawking, G.F.R. Ellis, The large scale structure of space-time, Cambridge Univ.Press 1973.S. Chandrasekhar, The Mathematical Theory of Black Holes, Clarendon Press, OxfordUniv. Press, 1983Dr. A.D. Fokker, Relativiteitstheorie, P. Noordho, Groningen, 1929.1J.A. Wheeler, A Journey into Gravity and Spacetime, Scientic American Library, NewYork, 1990, distr. by W.H. Freeman & Co, New York.CONTENTSPrologue 1literature 11. Summary of the theory of Special Relativity. Notations. 32. The E otv os experiments and the equaivalence principle. 73. The constantly accelerated elevator. Rindler space. 94. Curved coordinates. 135. The ane connection. Riemann curvature. 196. The metric tensor. 257. The perturbative expansion and Einsteins law of gravity. 308. The action principle. 359. Spacial coordinates. 3910. Electromagnetism. 4311. The Schwarzschild solution. 4512. Mercury and light rays in the Schwarzschild metric. 5013. Generalizations of the Schwarzschild solution. 5514. The Robertson-Walker metric. 5815. Gravitational radiation. 6221. SUMMARY OF THE THEORY OF SPECIAL RELATIVITY. NOTATIONS.Special Relativityisthetheoryclaimingthatspaceandtimeexhibitaparticularsymmetry pattern. This statement contains two ingredients which we further explain:(i) There is a transformation law, and these transformations form a group.(ii) Consider a system in which a set of physical variables is described as being a correctsolutiontothelawsofphysics. Thenif allthesephysicalvariablesaretransformedappropriately according to the given transformation law, one obtains a new solutionto the laws of physics.A point-event is a point in space, given by its three coordinates x = (x, y, z), at a giveninstant tintime. Forshort, wewill call thisapointinspace-time, anditisafourcomponent vector,x=___x0x1x2x3___=___ctxyz___. (1.1)Herecisthevelocityof light. Clearly, space-timeisafourdimensional space. Thesevectors are often written asx, where is an index running from 0 to 3. It will howeverbeconvenienttouseaslightlydierentnotation, x, =1, . . . , 4, wherex4=ictandi =1. The intermittent use of superscript indices ({}) and subscript indices ({}) isof no signicance in this section, but will become important later.InSpecialRelativity,thetransformationgroupiswhatonecouldcallthevelocitytransformations, or Lorentz transformations. It is the set of linear transformations,(x)

=4

=1L x(1.2)subject to the extra condition that the quantity dened by2=4

=1(x)2= |x|2c2t2( 0) (1.3)remainsinvariant. Thisconditionimpliesthatthecoecients Lformanorthogonalmatrix:4

=1L L=;4

=1LL= .(1.4)3Becauseof the i inthedenitionof x4, thecoecients Li4andL4imust bepurelyimaginary. The quantitiesandare Kronecker delta symbols:==1 if = , and 0 otherwise. (1.5)One can enlarge the invariance group with the translations:(x)

=4

=1L x+a, (1.6)in which case it is referred to as the Poincare group.We introduce summation convention:If an index occurs exactly twice in a multiplication (at one side of the = sign) it will auto-matically be summed over from 1 to 4 even if we do not indicate explicitly the summationsymbol . Thus, Eqs (1.2)(1.4) can be written as:(x)

=L x, 2=xx=(x)2,L L=, LL= .(1.7)If we donot wantto sum overan indexthat occurs twice,or if we wantto sumoveranindexoccuringthreetimes,we putone of the indicesbetweenbrackets soas toindicatethat it does not participate in the summation convention. Greek indices, , . . . run from1 to 4; latin indicesi, j, . . . indicate spacelike components only and hence run from 1 to 3.A special element of the Lorentz group isL=_____1 0 0 00 1 0 0 0 0 cosh i sinh 0 0 i sinh cosh _____, (1.8)where is a parameter. Orx

=x ; y

=y ;z

=z cosh ct sinh ;t

= zc sinh +t cosh .(1.9)This is a transformation from one coordinate frame to another with velocityv/c=tanh (1.10)4with respect to each other.Units of length and time will henceforth be chosen such thatc=1 . (1.11)Note thatthe velocityvgiven in (1.10) will always belessthan thatoflight. Thelightvelocity itself is Lorentz-invariant. This indeed has been the requirement that lead to theintroduction of the Lorentz group.Many physical quantities are not invariant but covariant under Lorentz transforma-tions. For instance, energyEand momentump transform as a four-vector:p=___pxpypziE___; (p)

=L p. (1.12)Electro-magnetic elds transform as a tensor:F=_____ 0 B3B2iE1B30 B1iE2 B2B10 iE3 iE1iE2iE30_____; (F)

=LL F. (1.13)It is of importance to realize what this implies: although we have the well-known pos-tulate that an experimenter on a moving platform, when doing some experiment, will ndthe same outcomes as a colleague at rest, we must rearrange the results before comparingthem. Whatcouldlooklikeanelectriceldforoneobservercouldbeasuperpositionof anelectricandamagneticeldfor theother. Andsoon. Thisiswhat wemeanwith covariance asopposedto invariance. Muchmoresymmetrygroupscouldbefoundin Nature thanthe onesknown,if onlyweknewhowtorearrangethe phenomena. Thetransformation rule could be very complicated.We now have formulated the theory of Special Relativity in such a way that it has be-come very easy to check if some suspect Law of Nature actually obeys Lorentz invariance.Left- and right hand side of an equation must transform the same way, and this is guar-anteed if they are written as vectors or tensors with Lorentz indices always transformingas follows:(X......)

=LL. . . L L. . . X....... (1.14)5Note thatthistransformationruleis justas ifweweredealingwith productsofvectorsXY, etc. Quantities transformingasineq. (1.14)arecalledtensors. Duetotheorthogonality (1.4) ofLone can multiply and contract tensors covariantly, e.g.:X=YZ(1.15)is a tensor (a tensor with just one index is called a vector), ifYandZare tensors.The relativistically covariant form of Maxwells equations is:F= J; (1.16)F +F +F=0 ; (1.17)F=A A, (1.18)J=0. (1.19)Here stands for /x, and the current four-vector Jis dened as J(x) =_

j(x), ic(x)_, inunitswhere0and0havebeennormalizedtoone. Aspecial ten-sor is, which is dened by1234=1 ;== ;=0 ifany two ofits indices are equal.(1.20)This tensor is invariant under the set of homogeneous Lorentz tranformations, in fact forall Lorentz transformationsLwith det(L) = 1. One can rewrite Eq. (1.17) as F=0 . (1.21)A particle with mass m and electric charge q moves along a curve x(s), where s runs from to +, with(sx)2= 1 ; (1.22)m2sx=q F sx. (1.23)The tensorTemdened by1Tem=Tem=FF +14FF , (1.24)1N.B. SometimesTis dened in dierent units, so that extra factors4appear in the denominator.6describes the energy density, momentum density and mechanical tension of the eldsF.In particular the energy density isTem44= 12F24i +14FijFij=12(

E2+

B2) , (1.25)whereweremindthereaderthatLatinindices i, j, . . .onlytakethevalues1, 2and3.Energy and momentum conservation implies that, if at any given space-time pointx, weaddthecontributions of all elds andparticles toT(x), thenfor this total energy-momentum tensor,T=0 . (1.26)2. THE EOTVOS EXPERIMENTS AND THE EQUIVALENCE PRINCIPLE.Suppose that objects made of dierent kinds of material would react slightly dierentlytothepresenceof agravitational eldg, byhavingnot exactlythesameconstant ofproportionality between gravitational mass and inertial mass:

F(1)=M(1)inerta(1)=M(1)gravg ,

F(2)=M(2)inerta(2)=M(2)gravg ;a(2)=M(2)gravM(2)inertg =M(1)gravM(1)inertg=a(1).(2.1)These objects would show dierent accelerations a and this would lead to eects that canbedetectedveryaccurately. Inaspaceship, theaccelerationwouldbedeterminedbythematerialthespaceshipismadeof; anyotherkindofmaterialwouldbeaccelerateddierently, and the relative acceleration would be experienced as a weak residual gravita-tional force. On earth we can also do such experiments. Consider for example a rotatingplatform with a parabolic surface. A spherical object would be pulled to the center by theearths gravitational force but pushed to the brim by the centrifugal counter forces of thecircular motion. If these two forces just balance out, the object could nd stable positionsanywhere on the surface, but an object made of dierent material could still feel a residualforce.Actually the Earth itself is such a rotating platform, and this enabled the Hungarianbaron Roland von E otv os to check extremely accurately the equivalence between inertialmass and gravitational mass (the Equivalence Principle). The gravitational force on anobject on the Earths surface is

Fg= GNMMgravrr3, (2.2)7whereGNis Newtons constant of gravity, andMis the Earths mass. The centrifugalforce is

F= Minert2raxis, (2.3)where is the Earths angular velocity andraxis=r ( r)2(2.4)is the distance from the Earths rotational axis. The combined force an object (i) feels onthesurfaceis

F(i)=

F(i)g+

F(i). Iffortwoobjects, (1)and(2), theseforces,

F(1)and

F(2), are not exactly parallel, one could measure=

F(1)

F(2)|F(1)||F(2)|_M(1)inertM(1)gravM(2)inertM(2)grav_(r )( r)rGNM(2.5)where we assumed that the gravitational force is much stronger than the centrifugal one.Actually, for the Earth we have:GNM2r3300 . (2.6)From (2.5) we see that the misalignment is given by (1/300) cos sin _M(1)inertM(1)gravM(2)inertM(2)grav_, (2.7)where is the latitude of the laboratory in Hungary, fortunately suciently far from boththe North Pole and the Equator.E otv os found no such eect, reaching an accuracy of one part in 107for the equivalenceprinciple. By observingthattheEarth alsorevolves aroundtheSunone canrepeattheexperiment using the Suns gravitational eld. The advantage one then has is that the eectone searches for uctuates dayly.This was R.H. Dickes experiment, in which he establishedanaccuracyofonepartin1011. ThereareplanstolounchadedicatedsatellitenamedSTEP (Satellite Test of the Equivalence Principle), to check the equivalence principle withan accuracy of one part in 1017. One expects that there will be no observable deviation. Inany case it will be important to formulate a theory of the gravitational force in which theequivalence principle is postulated to hold exactly. Since Special Relativity is also a theoryfrom which never deviations have been detected it is natural to ask for our theory of thegravitational forcealsotoobeythepostulatesof specialrelativity. Thetheoryresultingfrom combining these two demands is the topic of these lectures.83. THE CONSTANTLY ACCELERATED ELEVATOR. RINDLER SPACE.Theequivalenceprincipleimpliesanewsymmetryandassociatedinvariance. Therealization of this symmetry and its subsequent exploitation will enable us to give a uniqueformulation of this gravity theory. This solution was rst discovered by Einstein in 1915.We will now describe the modern ways to construct it.Consideranidealizedelevator, thatcanmakeanykindsof vertical movements,includingafreefall. Whenitmakesafreefall, all objectsinsideitwill beacceleratedequally, accordingtotheEquivalencePrinciple. Thismeansthatduringthetimetheelevator makes a free fall, its inhabitants will not experience any gravitational eld at all;they are weightless.Conversely, we can consider a similar elevator in outer space, far away from any star orplanet. Now give it a constant acceleration upward. All inhabitants will feel the pressurefrom the oor, just as if they were living in the gravitational eld of the Earth or any otherplanet. Thus, we can construct an articial gravitational eld. Let us consider such anarticial gravitational eld more closely. Suppose we want this articial gravitational eldto be constant in space and time. The inhabitant will feel a constant acceleration.An essential ingredient in relativity theory is the notion of a coordinate grid. So letus introduce a coordinate grid , = 1, . . . , 4, inside the elevator, such that points on itswalls are given by iconstant,i = 1, 2, 3. An observer in outer space uses a Cartesian grid(inertial frame)xthere. The motion of the elevator is described by the functionsx().Let the origin of the coordinates be a point in the middle of the oor of the elevator, andlet it coincide with the origin of the x coordinates. Now consider the line = (0, 0, 0, i).What is the corresponding curvex(

0, )? If the acceleration is in thezdirection it willhave the formx()= _0, 0, z(), it()_. (3.1)Time runs constantly for the inside observer. Hence_x_2=(z)2(t)2= 1 . (3.2)The acceleration is g, which is the spacelike components of2x2=g. (3.3)At= 0 we can also take the velocity of the elevator to be zero, hencex= (

0,i), (at= 0) . (3.4)9Atthatmomenttandcoincide,andif wewantthatthe accelerationgis constantwealso want at= 0 thatg = 0, henceg=(

0,iF)=Fxat = 0 , (3.5)where for the time beingFis an unknown constant.Now this equation is Lorentz covariant. So not only at= 0 but also at all times weshould haveg=Fx. (3.6)Eqs. (3.3) and (3.6) giveg=F(x+A) , (3.7)x()= Bcosh(g) +Csinh(g) A, (3.8)F, A, Band Care constants. Dene F= g2. Then, from (3.1), (3.2) and the boundaryconditions:(g)2=F= g2, B=1g___0010___, C=1g___000i___, A=B, (3.9)and since at= 0 the acceleration is purely spacelike we nd that the parameterg is theabsolute value of the acceleration.Wenoticethatthepositionoftheelevatorooratinhabitanttime isobtainedfromthepositionat=0byaLorentzboostaroundthepoint= A. Thismustimply that the entire elevator is Lorentz-boosted. The boost is given by (1.8) with = g .This observation gives us immediately the coordinates of all other points of the elevator.Suppose that at= 0,x(

, 0)=(

, 0) (3.10)Then at othervalues,x(

,i)=_____12cosh(g )_3+1g_1gi sinh(g )_3+1g______. (3.11)10a03,x3 = const.3 = const.x0past horizonfuture horizonFig. 1. Rindler Space. The curved solid line represents the oor of the elevator,3= 0. A signal emitted from point acan never be received by an inhabitant ofRindler Space, who lives in the quadrant at the right.The 3, 4 components of the coordinates, imbedded in the x coordinates, are picturedinFig. 1. Thedescriptionofaquadrantofspace-timeintermsofthecoordinatesiscalledRindlerspace. FromEq. (3.11)itshouldbeclearthatanobserverinsidetheelevator feels no eects that depend explicitly on his time coordinate, since a transitionfrom to

isnothingbutaLorentztransformation. Wealsonoticesomeimportanteects:(i) We see that the equal lines converge at the left. It follows that the local clock speed,which is given by = _(x/)2, varies with hight3:=1 +g 3, (3.12)(ii) The gravitational eld strength felt locally is 2g(), which is inversely proportionaltothe distance tothe pointx= A. Soeven though oureldis constantin thetransverse direction and with time, it decreases with hight.(iii) The region of space-time described by the observer in the elevator is only part of all ofspace-time (the quadrant at the right in Fig. 1, where x3+1/g> |x0|). The boundarylines are called (past and future) horizons.All these are typically relativistic eects. In the non-relativistic limit (g 0) Eq. (3.11)simply becomes:x3=3+12g2; x4=i = 4. (3.13)Accordingtotheequivalenceprincipletherelativisticeectswediscoveredhereshouldalsobefeaturesofgravitational eldsgeneratedbymatter. Letusinspectthemonebyone.11Observation (i) suggests that clocks will run slower if they are deep down a gravita-tional eld. Indeed one may suspect that Eq. (3.12) generalizes into=1 +V (x) , (3.14)whereV (x) is the gravitational potential. Indeedthis will turn out to be true,providedthat the gravitational eld is stationary. This eect is called the gravitational red shift.(ii) is also a relativistic eect. It could have been predicted by the following argument.The energy density of a gravitational eld is negative. Since the energy of two massesM1andM2atadistancerapartisE= GNM1M2/rwecancalculate theenergydensityof a eldgasT44= (1/8GN)g2. Sincewe had normalizedc = 1this is also its massdensity. Butthenthismassdensityinturnshouldgenerateagravitationaleld! Thiswould imply2

g?= 4GNT44= 12g2, (3.15)sothatindeedtheeldstrengthshoulddecreasewithheight. Howeverthisreasoningisapparentlytoosimplistic, sinceoureldobeysadierentialequationasEq. (3.15)butwithout the coecient12.The possible emergenceof horizons, our observation (iii), will turn out to be a veryimportant newfeatureof gravitational elds. Undernormal circumstancesof coursetheeldsaresoweakthatnohorizonwill beseen, butgravitational collapsemayproducehorizons. If this happens there will be regions in space-time from which no signals can beobserved. In Fig. 1 we see that signals from a radio station at the pointawill never reachan observer in Rindler space.Themostimportantconclusiontobedrawnfromthischapteristhatinordertodescribe a gravitational eld one may have to perform a transformation from the coordi-natesthat were used inside the elevator where one feels the gravitational eld, towardscoordinatesxthat describe empty space-time, in which freely falling objects move alongstraight lines. Now we know that in an empty space without gravitational elds the clockspeeds, andthe lengthsof rulers,aredescribedbyadistancefunctionas givenin Eq.(1.3). We can rewrite it asd2=gdxdx; g=diag(1, 1, 1, 1) , (3.16)We wrote here dand dxto indicate that we look at the innitesimal distance betweentwo points close together in space-time. In terms of the coordinates appropriate for the2Temporarilywedonot showtheminussignusuallyinsertedtoindicatethat theeldispointeddownward.12elevator we have for innitesimal displacements d,dx3= cosh(g )d3+_1 +g 3_sinh(g )d ,dx4=i sinh(g )d3+i_1 +g 3_cosh(g )d .(3.17)implyingd2= _1 +g 3_2d2+ (d

)2. (3.18)If we write this asd2=g(x)dd= (d

)2+ (1 +g 3)2(d4)2, (3.19)thenweseethat all eects that gravitational elds haveonrulers andclocks canbedescribed in terms of a space (and time) dependent eldg(x). Only in the gravitationaleld of a Rindler space can one nd coordinates xsuch that in terms of these the functiongtakes the simple form of Eq. (3.16). We will see that g(x) is all we need to describethe gravitational eld completely.Spaces in which the innitesimal distance d is described by a space(time) dependentfunctiong(x) are called curved or Riemann spaces. Space-time is a Riemann space. Wewill now investigate such spaces more systematically.4. CURVED COORDINATES.Eq. (3.11)isaspecial caseof acoordinatetransformationrelevantforinspectingthe Equivalence Principle for gravitational elds. It is not a Lorentz transformation sinceitisnotlinearin. WeseeinFig. 1thatthecoordinatesarecurved. Theemptyspace coordinates could be called straight because in terms of them all particles move instraight lines. However, such a straight coordinate frame will only exist if the gravitationaleld has the same Rindler form everywhere, whereas in the vicinity of stars and planets istakes much more complicated forms.But in the latter case we can also usethe equivalence Principle: the laws of gravityshouldbeformulatedsuchawaythatanycoordinateframethatuniquelydescribesthepoints in our four-dimensional space-time can be used in principle. None of these frameswill besuperiortoanyofthe otherssincein anyoftheseframesonewill feelsomesortof gravitational eld3. Let us start with just one choice of coordinates x=(t,x,y,z).Fromthischapteronwardsitwill nolongerbeuseful tokeepthefactor i inthetime3Therewillbesomelimitationsinthesenseofcontinuityanddierentiabilityaswewillsee.13component because it doesnt simplify things. It has become convention to denex0=tand drop thex4which wasit. So now runs from 0 to 3. It will be of importance nowthat the indices for the coordinates be indicated as super scripts,.Let there now be some one-to-one mapping onto another set of coordinatesu,ux; x=x(u) . (4.1)Quantities depending on these coordinates will simply be called elds. A scalar eldis a quantity that depends onx but does not undergo further transformations, so that inthe new coordinate frame (we distinguish the functions of the new coordinatesu from thefunctions ofx by using the tilde, )=(u)=_x(u)_. (4.2)Now dene the gradient (and note that we use a sub script index)(x)=x(x)xconstant, for = . (4.3)Rememberthatthepartial derivativeisdenedbyusinganinnitesimal displacementdx,(x + dx)=(x) +dx+O(dx2) . (4.4)We derive(u + du)=(u) +xudu+O(du2)=(u) + (u)du. (4.5)Therefore in the new coordinate frame the gradient is(u)=x, _x(u)_, (4.6)where we use the notationx,def=ux(u)u=constant, (4.7)so the comma denotes partial derivation.Noticethatinall theseequationssuperscript indicesandsubscript indicesalwayskeep their position and they are used in such a way that in the summation convention onesubscript and one superscript occur:

(. . .)(. . .)14Of course one can transform back from thex to theu coordinates:(x)=u, _u(x)_. (4.8)Indeed,u,x,=, (4.9)(the matrixu,is the inverseofx, ) A specialcase would be if the matrixx,wouldbeanelementoftheLorentzgroup. TheLorentzgroupisjustasubgroupofthemuchlarger set of coordinate transformations considered here. We see that(x) transforms asa vector. All eldsA(x) that transform just like the gradients(x), that is,A(u)=x, A_x(u)_, (4.10)will be called covariant vector elds, co-vector for short, even if they cannot be written asthe gradient of a scalar eld.Notethattheproductofascalareldandaco-vectorAtransformsagainasaco-vector:B=A;B(u)=(u) A(u)=_x(u)_x,A_x(u)_=x, B_x(u)_.(4.11)Now consider the direct productB=A(1)A(2). It transforms as follows:B(u)=x,x, B_x(u)_. (4.12)A collection of eld components that can be characterised with a certain number of indices,,. . . and that transforms according to (4.12) is called a covariant tensor.Warning: In a tensorsuch asBone may not sumover repeated indices to obtaina scalar eld. This is because the matricesx,in general do not obey the orthogonalityconditions (1.4) of the Lorentz transformations L. One is not advised to sum over two re-peated subscript indices. Nevertheless we would like to formulate things such as MaxwellsequationsinGeneralRelativity,andthereofcourseinnerproductsofvectorsdooccur.To enable us to do this we introduce another type of vectors: the so-called contra-variantvectors and tensors. Sincea contravariant vectortransforms dierently froma covariantvector we have to indicate his somehow. This we do by putting its indices upstairs: F(x).The transformation rule for such a superscript index is postulated to beF(u)=u,F_x(u)_, (4.13)15as opposed to the rules (4.10), (4.12) for subscript indices; and contravariant tensors F...transform as productsF(1)F(2)F(3). . .. (4.14)Wewill alsosee mixedtensors havingbothupper (superscript) andlower (subscript)indices. They transform as the corresponding products.Exercise: check that the transformation rules (4.10) and (4.13) form groups, i.e. thetransformationx u yields the same tensor as the sequencex v u. Make useof the fact that partial dierentiation obeysxu=xvvu. (4.15)Summation over repeated indices is admitted if one of the indices is a superscript and oneis a subscript:F(u) A(u)=u,F_x(u)_x,A_x(u)_, (4.16)and since the matrixu, is the inverse ofx, (according to 4.9), we haveu,x,=, (4.17)so that the productFA indeed transforms as a scalar:F(u) A(u)=F_x(u)_A_x(u)_. (4.18)Note that sincethesummation conventionmakes ussumoverrepeatedindiceswith thesame name, we must ensure in formulae such as (4.16) that indices not summed over areeach given a dierent name.We recognise that in Eqs.(4.4) and (4.5) the innitesimal displacement of a coordinatetransforms as a contravariant vector. This is why coordinates are given superscript indices.Eq. (4.17) also tells us that the Kronecker delta symbol (provided it has one subscript andone superscript index) is an invariant tensor: it has the same form in all coordinate grids.Gradients of tensorsThegradientofascalareldtransformsasacovariantvector. Aregradientsofcovariant vectors and tensors again covariant tensors?Unfortunately no. Let us from nowonindicatepartial dierentiation/xsimplyas . Sometimeswewill useanevenshorter notation:x==,. (4.19)16From (4.10) we nd A(u)=uA(u)=u_xu A_x(u)__=xuxuxA_x(u)_+2xuuA_x(u)_=x,x,A_x(u)_+x,, A_x(u)_.(4.20)The last term here deviates from the postulated tensor transformation rule (4.12).Now notice thatx,,=x,,, (4.21)whichalwaysholds for ordinarypartial dierentiations. Fromthis it followsthat theantisymmetric part ofA is a covariant tensor:F=AA ;F(u)=x,x,F_x(u)_.(4.22)Thisisanessential ingredientinthemathematicaltheoryofdierentialforms. Wecancontinue this way: ifA = AthenF=A +A +A(4.23)is a fully antisymmetric covariant tensor.Next, consider afullyantisymmetrictensor ghavingas manyindices as thedimensionality of space-time (lets keep space-time four-dimensional). Then one can writeg= , (4.24)(see the denition of in Eq. (1.20)) since the antisymmetry condition xes the values ofall coecients of gapart from one common factor. Although carries no indices itwill turn out not to transform as a scalar eld. Instead, we nd: (u)=det(x,)_x(u)_. (4.25)A quantity transforming this way will be called a density.The determinant in (4.25) can act as the Jacobian of a transformation in an integral.If(x) is some scalar eld (or the inner product of tensors with matching superscript andsubscript indices) then the integral17_(x)(x)d4x (4.26)is independent of the choice of coordinates, because_d4x . . . =_d4u det(x/u) . . . . (4.27)This can also be seen from the denition (4.24):_ g du du du du=_g dx dx dx dx.(4.28)Two important properties of tensors are:1) The decomposition theorem.Everytensor X......canbewrittenasanitesumof productsof covariantandcontravariant vectors:X......=N

t=1A(t)B(t). . .P(t)Q(t). . . . (4.29)Thenumberofterms, N,doesnothave tobelarger thanthe numberof componentsofthe tensor. By choosing in one coordinate frame the vectorsA, B,. . . each such that theyare nonvanishing for only one value of the index the proof can easily be given.2) The quotient theorem.Let there be given an arbitrary set of components X............. Let it be known that forall tensorsA...... (with a given, xed number of superscript and/or subscript indices)the quantityB......=X............A......transforms as a tensor. Then it follows thatXitself also transforms as a tensor.The proof can be given byinduction. First one choosesA to have justone index. Thenin one coordinate frame we choose it to have just one nonvanishing component. One thenuses(4.9) or(4.17). If Ahasseveralindicesonedecomposesitusingthedecompositiontheorem.Whathasbeenachievedinthischapteristhatwelearnedtoworkwithtensorsincurved coordinate frames. They can be dierentiated and integrated. But before we canconstruct physically interesting theories in curved spaces two more obstacles will have tobe overcome:18(i) Thusfarwehave only been able todierentiate antisymmetrically,otherwise the re-sulting gradients do not transform as tensors.(ii) Therestill aretwotypesof indices. Summationisonlypermittedif oneindexisasuperscript andoneis asubscript index. This is toomuchof alimitationforconstructing covariant formulations of the existing laws of nature, such as the Maxwelllaws. We will deal with these obstacles one by one.5. THE AFFINE CONNECTION. RIEMANN CURVATURE.Thespacedescribedinthepreviouschapterdoesnotyethaveenoughstructuretoformulate all known physical laws in it. For a good understanding of the structure now tobe added we rst must dene the notion of ane connection. Only in the next chapterwe will dene distances in time and space.(x)(x ) xSxFig. 2. Two contravariant vectors close to each other on a curveS.Let (x) be a contravariant vector eld, and let x() be the space-time trajectory Sof an observer. We now assume that the observer has a way to establish whether(x) isconstant or varies as his eigentimegoes by. Let us indicate the observed time derivativeby a dot:=dd _x()_. (5.1)The observer will have used a coordinate framex where he stays at the origin O of three-space. What will equation (5.1) be like in some other coordinate frameu?(x)=x, _u(x)_;x,def=dd _x()_=x,dd_u_x()__+x,,dud (u) .(5.2)Thus, if we wish to dene a quantitythat transforms as a contravector then in a generalcoordinate frame this is to be written as_u()_def=dd _u()_+ dud_u()_. (5.3)19Here, isaneweld, andnearthepointuthelocalobservercanuseapreferencecoordinate framex such thatu,x,, = . (5.4)In his preferencecoordinate frame, will vanish,but onlyonhiscurveS ! In generalitwill not be possible to nd a coordinate frame such that vanishes everywhere. Eq. (5.3)denestheparalel displacementofacontravariantvectoralongacurveS. Todothisaneweld was introduced,(u),called aneconnectioneldbyLevi-Civita. It is aeld, but not a tensor eld, since it transforms as_u(x)_=u,_x,x,(x) +x,,_. (5.5)Exercise:Prove (5.5) and show that two successive transformations of this type againproduces a transformation of the form (5.5).We now observe that Eq. (5.4) implies=, (5.6)and sincex,,=x,,, (5.7)thissymmetrywillalsoholdinanyothercoordinateframe. Now, inprinciple, onecanconsider spaces with a paralel displacement according to (5.3) where does not obey (5.6).In this case there are no local inertial frames where in some given point x one has = 0.This is called torsion. We will not pursue this, apart from noting that the antisymmetricpart of would be an ordinary tensor eld, which could always be added to our modelsat a later stage. So we limit ourselves now to the case that Eq. (5.6) always holds.A geodesic is a curvex() that obeysd2d2x() + dxddxd=0 . (5.8)Since dx/d is a contravariant vector this is a special case of Eq. (5.3) and the equationfor the curve will look the same in all coordinate frames.N.B.Ifonechoosesanarbitrary,dierentparametrizationofthecurve(5.8), usingaparameter thatisanarbitrarydierentiablefunctionof , oneobtainsadierentequation,d2d 2x( ) +( )dd x( ) + dxd dxd =0 . (5.8a)20where( )canbeanyfunctionof . Apparentlytheshapeofthecurveincoordinatespace does not depend on the function( ).Exercise: check Eq. (5.8a).Curves described by Eq.(5.8) could be dened to be the space-time trajectories of particlesmovingin agravitational eld. Indeed,ineverypointxthereexists acoordinate framesuchthat vanishesthere,sothatthe trajectory goes straight(the coordinate frameofthe freely falling elevator). In an accelerated elevator, the trajectories look curved, and anobserver inside the elevator can attribute this curvature to a gravitational eld.Thegravitationaleldisherebyidentiedasananeconnectioneld. Inthelit-eratureonealsondstheChristoelsymbol {}whichmeansthesamething. Theconvention used here is that of Hawking and Ellis.SincenowwehaveaeldthattransformsaccordingtoEq. (5.5)wecanuseittoeliminatetheoendinglastterminEq. (4.20). Wedeneacovariant derivativeof aco-vector eld:DA=AA . (5.9)This quantityDAneatly transforms as a tensor:D A(u)=x,x,DA(x) . (5.10)Notice thatDADA=AA, (5.11)so that Eq. (4.22) is kept unchanged.Similarly one can now dene the covariant derivative of a contravariant vector:DA=A+ A. (5.12)(notice the dierences with (5.9)!)It is not dicult now to dene covariant derivatives ofall other tensors:DX......=X......+X......+X....... . .X...... X....... . . .(5.13)Expressions (5.12) and (5.13) also transform as tensors.We also easily verify a product rule. Let the tensor Z be the product of two tensorsXandY :Z............=X......Y....... (5.14)21Then one has (in a notation where we temporarily suppress the indices)DZ=(DX)Y+X(DY ) . (5.15)Furthermore, if one sums over repeated indices (one subscript and one superscript, we willcall this a contraction of indices):(DX)......=D(X......) , (5.16)so that we can just as well omit the brackets in (5.16). Eqs. (5.15) and (5.16) can easily beproven to hold in any pointx,by choosing the referenceframe where vanishes at thatpointx.The covariant derivative of a scalar eld is the ordinary derivative:D=, (5.17)but this does not hold for a density function (see Eq. 4.24),D= . (5.18)D is a density times a covector. This one derives from (4.24) and=6 . (5.19)Thus we have found that if one introduces in a space or space-time a eld thattransforms accordingtoEq. (5.5), calledaneconnection, thenonecandene: 1)geodesiccurvessuchasthetrajectoriesof freelyfallingparticles, and2)thecovariantderivative of any vector and tensor eld. But what we do not yet have is (i) a unique def-inition of distance between points and (ii) a way to identify co vectors with contra vectors.Summation over repeated indices only makes sense if one of them is a superscript and theother is a subscript index.CurvatureNowagainconsideracurveSasinFig. 2, butcloseit(Fig. 3). Letushaveacontravector eld(x) with_x()_=0 ; (5.20)We take the curve to be very small so that we can write(x)=+,x+O(x2) . (5.21)22Fig. 3. Paralel displacement along a closed curve in a curved space.Willthiscontravectorreturntoitsoriginalvalueifwefollowitwhilegoingaroundthecurveonefullloop? Accordingto(5.3)itcertainlywilliftheconnectioneldvanishes: = 0. But if there is a strong gravity eld there might be a deviation. We nd:_d =0 ;=_ddd _x()_= _dxd_x()_d= _d_ + ,x_dxd_+,x_.(5.22)where we chose the function x() to be very small, so that terms O(x2) could be neglected.We have_d dxd=0 andD 0, ,(5.23)so that Eq. (5.22) becomes=12__xdxdd_R+higher orders inx. (5.24)Since_xdxdd +_xdxdd =0 , (5.25)only the antisymmetric part ofR matters. We chooseR= R(5.26)(the factor12in (5.24) is conventionally chosen this way). Thus we nd:R= + . (5.27)Wenowclaimthatthisquantitymusttransformasatruetensor. Thisshouldbesurprising since itself is not a tensor, and since there are ordinary derivatives in stead23of covariant derivatives. Theargumentgoes as follows. In Eq. (5.24) the l.h.s.,is atrue contravector, and also the quantityS=_xdxdd , (5.28)transforms as a tensor. Now we can chooseany way we want and also the surface ele-mentsSmay be chosen freely. Therefore we may use the quotient theorem (expandedto cover the case of antisymmetric tensors) to conclude that in that case the set of coe-cients R must also transform as a genuine tensor. Of course we can check explicitly byusing(5.5)thatthecombination(5.27)indeedtransformsasatensor, showingthattheinhomogeneous terms cancel out.Rtells us something about the extent to which this space is curved. It is calledthe Riemann curvature tensor. From (5.27) we deriveR +R +R=0 , (5.29)andDR +DR +DR=0 . (5.30)The latter equation, called Bianchi identity, can be derived most easily by noting that forevery point x a coordinate frame exists such that at that point x one has = 0 (thoughits derivative cannot be tuned to zero). One then only needs to take into account thoseterms of Eq. (5.27) that are linear in.Partial derivatives have the property that the order may be interchanged,=. This is no longer true for covariant derivatives. For any covector eld A(x) we ndDDADDA= RA, (5.31)and for any contravector eldA:DDADDA=RA, (5.32)which we can verify directly from the denition of R. These equations also show clearlywhy the Riemann curvature transforms as a true tensor; (5.31) and (5.32) hold for allAandAand the l.h.s. transform as tensors.An important theorem is that the Riemann tensor completely species the extent towhichspaceorspace-timeis curved, ifthis space-timeis simplyconnected. Toseethis,assumethat R=0everywhere. Considerthenapoint xandacoordinateframe24suchthat(x)=0. Thenfromthefactthat(5.27)vanisheswededucethatintheneighborhood of this point one can nd a quantityXsuch that(x

)=X(x

)+O(x x

)2. (5.33)Due to the symmetry (5.6) we haveX = Xand this in turn tells us that there isa quantityysuch that(x

)= y+ O(x x

)2. (5.34)If we useyas a new coordinate frame near the pointx then according to (5.5) the aneconnectionwillvanishnearthispoint. Thiswayonecanconstructaspecialcoordinateframe in the entire space such that the connection vanishes in the entire space (providedit is simply connected). Thus we see that if the Riemann curvature vanishes a coordinateframe can be constructed in terms of which all geodesics are straight lines and all covariantderivatives are ordinary derivatives. This is a at space.Warning: there is no universal agreement in the literature about sign conventions inthe denitions of d2, ,R, Tand the eldgof the next chapter. This shouldbenoimpedimentagainststudyingotherliterature. Onefrequentlyhastoadjustsignsand pre-factors.6. THE METRIC TENSOR.In a space with ane connection we have geodesics, but no clocks and rulers. Thesewe will introduce now. In Chapter 3 we saw that in at space one has a matrixg=___1 0 0 00 1 0 00 0 1 00 0 0 1___, (6.1)so that for the Lorentz invariant distance we can write2=t2+x2=gxx. (6.2)(timewill bethezerothcoordinate, whichis agreedupontobetheconventionif allcoordinates are chosen to stay real numbers). For a particle running along a timelike curveC = {x()} the increase in eigentimeTisT =_CdT, with dT2= gdxddxd d2def=gdxdx.(6.3)25Thisexpressionis coordinate independent. Weobservethatgisaco-tensorwithtwo subscript indices, symmetric under interchange of these. In curved coordinates we getg=g=g(x) . (6.4)This is the metric tensor eld. Only far away from stars and planets we can nd coordinatessuch that it will coincide with (6.1) everywhere. In general it will deviate from this slightly,butusuallynotverymuch. Inparticular wewill demandthat upondiagonalization onewill always nd three positive and one negative eigenvalue. This property can be shown tobe unchanged under coordinate transformations. The inverse ofgwhich we will simplyrefer to asgis uniquely dened bygg=. (6.5)This inverse is also symmetric under interchange of its indices.It now turns out that the introduction of such a two-index cotensor eld gives space-time more structure than the three-index ane connection of the previous chapter. Firstof all, the tensor ginduces one special choice for the ane connection eld. One simplydemands that the covariant derivative ofgvanishes:Dg=0 . (6.6)This indeed would have been a natural choice in Rindler space, since inside a freely fallingelevator one feels at space-times, i.e. bothgconstant and = 0. From (6.6) we see:g=g + g. (6.7)Write=g, (6.8)=. (6.9)Then one nds from (6.7)12_g +gg_= , (6.10)=g . (6.11)These equations now dene an ane connection eld. Indeed Eq. (6.6) follows from (6.10),(6.11). SinceD==0 , (6.12)26we also have for the inverse ofgDg=0 , (6.13)which follows from (6.5) in combination with the product rule (5.15).Butthemetrictensorgnotonlygivesusananeconnectioneld, itnowalsoenables us to replace subscript indices by superscript indices and back. For every covectorA(x) we dene a contravectorA(x) byA(x)=g(x)A(x) ; A=gA. (6.14)Veryimportantiswhatisimpliedbytheproductrule(5.15), togetherwith(6.6)and(6.13):DA=gDA ,DA=gDA.(6.15)Itfollows thatraisingorloweringindicesbymultiplication withgorgcanbe donebefore or after covariant dierentiation.The metric tensor also generates a density function:=_det(g) . (6.16)IttransformsaccordingtoEq. (4.25). Thiscanbeunderstoodbyobservingthatinacoordinate frame with in some pointxg(x)=diag(a, b, c, d) , (6.17)the volume element is given by abcd .The space of the previous chapter is called an ane space. In the present chapterwe have a subclass of the ane spaces called a metric space or Riemann space; indeed wecancallitaRiemannspace-time. Thepresenceofatime coordinateis betrayedbytheone negative eigenvalue ofg.27The geodesicsConsidertwoarbitrarypointsXandY inourmetricspace. ForeverycurveC={x()} that hasXandYas its end points,x(0)=X; x(1)=Y, (6, 18)we consider the integral=_Cds , (6.19)with eitherds2=gdxdx, (6.20)when the curve is spacelike, ords2= gdxdx, (6.21)whereeverthecurveistimelike. Forsimplicitywechoosethecurvetobespacelike, Eq.(6.20). The timelike case goes exactly analogously.Consider now an innitesimal displacement of the curve, keeping however X and Yintheir places:x

()=x() +() , innitesimal,(0)=(1)=0,(6.22)then what is the innitesimal change in ?=_ds ;2dsds=(g)dxdx+ 2gdxd+O(d2)=(g)dxdx+ 2gdxdd d .(6.23)Now we make a restriction for the original curve:dsd=1 , (6.24)whichonecanalwaysrealisebychoosinganappropriateparametrizationof thecurve.(6.23) then reads=_d_12g,dxddxd+gdxddd_. (6.25)28We can take care of the d/d term by partial integration; usingddg=g,dxd, (6.26)we get=_d__12g,dxddxdg,dxddxdgd2xd2_+dd_gdxd__.= _d ()g_d2xd2+ dxddxd_.(6.27)The pure derivative term vanishes since we require to vanish at the end points, Eq.(6.22).We used symmetry under interchange of the indices and in the rst line and the deni-tions (6.10) and (6.11) for . Now, strictly following standard procedure in mathematicalphysics, we can demand that vanishes for all choices of the innitesimal function ()obeyingthe boundarycondition. Weobtain exactly the equation forgeodesics,(5.8). Ifwe hadnt imposed Eq. (6.24) we would have obtained (5.8a).We have spacelike geodesics (with Eq. 6.20) and timelike geodesics (with Eq. 6.21).One can show that for timelike geodesics is a relative maximum. For spacelike geodesicsit is on a saddle point. Only in spaces with a positive denitegthe lengthof the pathis a minimum for the geodesic.CurvatureAs for the Riemann curvature tensor dened in the previous chapter, we can now raiseand lower all its indices:R=gR , (6.28)and we can check if there are any further symmetries, apart from (5.26), (5.29) and (5.30).By writing down the full expressions for the curvature in terms ofgone ndsR= R=R . (6.29)By contracting two indices one obtains the Ricci tensor:R=R , (6.30)It now obeysR=R, (6.31)29We can contract further to obtain the Ricci scalar,R=gR=R. (6.32)The Bianchi identity (5.30) implies for the Ricci tensor:DR 12DR=0 . (6.33)We also writeG=R 12Rg, DG=0 . (6.34)The formalism developed in this chapter can be used to describe any kind of curvedspace or space-time. Every choice for the metric g(under certain constraints concerningitseigenvalues)canbeconsidered. Weobtainthetrajectoriesgeodesicsofparticlesmovingingravitational elds. Howeverso-farwehavenotdiscussedtheequationsthatdetermine the gravity eld congurations given some conguration of stars and planets inspace and time. This will be done in the next chapters.7. THE PERTURBATIVE EXPANSION AND EINSTEINS LAW OF GRAVITY.We have a law of gravity if we have some prescription to pin down the values of thecurvature tensor Rnear a given matter distribution in space and time. To obtain suchaprescription we wanttomake useof the given factthat Newtons law of gravity holdswhenever the non-relativistic approximation is justied. This will be the case in any regionof space and time that is suciently small so that a coordinate frame can be devised therethat is approximtely at. Thegravitational eldsare then sucientlyweakand then atthatspotwenotonlyknowfairlywell howtodescribethelawsofmatter, butwealsoknow how these weak gravitational elds are determined by the matter distribution there.In our small region of space-time we writeg(x)= +h , (7.1)where=___1 0 0 00 1 0 00 0 1 00 0 0 1___, (7.2)andhis a small perturbation. We nd (see (6.10):=12_h +hh_; (7.3)g= h+hh. . . . (7.4)30In this latter expression the indices were raised and lowered usingandinstead ofthegandg. This is a revised index- and summation convention that we only applyon expressions containingh.= +O(h2) . (7.5)The curvature tensor isR= +O(h2) , (7.6)and the Ricci tensorR= +O(h2)=12_2h +h +h h_+O(h2) .(7.7)The Ricci scalar isR= 2h +h +O(h2) . (7.8)A slowly moving particle hasdxd(1, 0, 0, 0) , (7.9)so that the geodesic equation (5.8) becomesd2d2xi()= i00. (7.10)Apparently,i= i00istoidentiedwith thegravitational eld. Nowinastationarysystem one may ignore time derivatives0. Therefore Eq. (7.3) for the gravitational eldreduces toi= i00=12ih00, (7.11)so that one may identify 12h00 as the gravitational potential. This conrms the suspicionexpressed in Chapter 3 that the local clock speed, which is =g00 1 12h00 , can beidentied with the gravitational potential, Eq. (3.18) (apart from an additive constant, ofcourse).NowletTbe the energy-momentum-stress-tensor; T44= T00is the mass-energydensity and since in our coordinate frame the distinction between covariant derivative andordinary deivatives is negligible, Eq. (1.26) for energy-momentum conservation readsDT=0 (7.12)31In other coordinate frames this deviates from ordinary energy-momentum conservation justbecause the gravitational elds can carry away energy and momentum;theTwe workwith presently will be only the contribution from stars and planets, not their gravitationalelds. Now Newtons equations for slowly moving matter implyi= i00= iV (x)=12ih00 ;ii= 4GNT44=4GNT00 ;

2h00=8GNT00(7.13)This wenowwishtorewriteinawaythat is invariant under general coordinatetransformations. This is averyimportant stepinthetheory. Insteadof havingonecomponent of theTdepend on certain partial derivatives of the connection elds wewantarelationbetweencovarianttensors. Theenergymomentumdensityformatter,T, satisfyingEq. (7.12),isclearlyacovarianttensor. Theonlycovarianttensorsonecan build from the expressionsin Eq. (7.13) are the Ricci tensorRand the scalarR.The two independent components that are scalars onder spacelike rotations areR00= 12

2h00 ; (7.14)and R=ijhij + 2(h00hii) . (7.15)Now these equations strongly suggest a relationship between the tensors Tand R,butwenowhavetobecareful. Eq. (7.15)cannotbeusedsinceitisnotaprioriclearwhetherwecanneglectthespacelikecomponentsof hij(wecannot). Themostgeneraltensor relation one can expect of this type would beR=AT +BgT, (7.16)where A and B are constants yet to be determined. Here the trace of the energy momentumtensor is, in the non-relativistic approximationT= T00 +Tii . (7.17)so the 00 component can be written asR00= 12

2h00=(A+B)T00BTii, (7.18)tobecomparedwith(7.13). Itisof importancetorealisethatintheNewtonianlimittheTiiterm(thepressurep)vanishes, notonlybecausethepressureofordinary(non-relativistic) matter is very small, but also because it averages out to zero as a source: inthe stationary case we have0=Ti=jTji, (7.19)ddx1_T11dx2dx3= _dx2dx3_2T21 +3T31_=0 , (7.20)32and therefore, if our source is surrounded by a vacuum, we must have_T11dx2dx3=0 _d3xT11=0 ,and similarly,_d3xT22=_d3xT33=0 .(7.21)We must conclude that all one can deduce from (7.18) and (7.13) isA+B= 4GN . (7.22)Fortunately we have another piece of information. The trace of (7.16) isR = (A+ 4B)T. The quantityGin Eq. (6.34) is thenG=AT (12A+B)Tg , (7.23)and since we have both the Bianchi identity (6.34) and the energy conservation law (7.12)we getDG=0 ; DT=0 ; therefore (12A+B)(T)=0 . (7.24)NowT,thetraceofthe energy-momentumtensor,isdominated by T00. Thiswill ingeneral not be space-time independent. So our theory would be inconsistent unlessB= 12A ; A= 8GN , (7.25)using (7.22).We conclude that the only tensor equation consistent with Newtons equationin a locally at coordinate frame isR 12Rg= 8GNT , (7.26)where the sign of the energy-momentum tensor is dened by ( is the energy density)T44= T00=T00= . (7.27)This is Einsteins celebrated law of gravitation. From the equivalence principle it followsthat if this law holds in a locally at coordinate frame it should hold in any other frameas well.Since both left and right of Eq. (7.26) are symmetric under interchange of the indiceswe have here 10 equations. We know however that both sides obey the conservation lawDG=0 . (7.28)33Theseare4equationsthatareautomaticallysatised. Thisleaves6non-trivial equa-tions. They should determine the 10 components of the metric tensorg, so one expectsaremainingfreedomof 4equations. Indeedthecoordinatetransformationsareasyetundetermined, and there are 4 coordinates. Counting degrees of freedom this way suggeststhat Einsteins gravity equations should indeed determine the space-time metric uniquely(apart from coordinate transformations) and could replace Newtons gravity law. Howeverone has to be extremely careful with arguments of this sort. In the next chapter we showthat the equations are associated with an action principle, and this is a much better way toget some feeling for the internal self-consistency of the equations. Fundamental dicultiesare not completely resolved, in particular regarding the stability of the solutions.Note that (7.26) implies8GNT=R;R= 8GN_T 12Tg_.(7.29)therefore in parts of space-time where no matter is present one hasR=0 , (7.30)but the complete Riemann tensorRwill not vanish.The Weyltensor is dened by subtracting fromRa part in such a way that allcontractions of any pair of indices gives zero:C=R+12_gR +gR +13Rg g ( )_. (7.31)This construction is such that C has the same symmetry properties (5.26), (5.29) and(6.29) and furthermoreC=0 . (7.32)If one carefully counts the number of independent components one nds in a given pointx thatRhas 20 degrees of freedom, andRandCeach 10.The cosmological constantWe have seen that Eq. (7.26) can be derived uniquely; there is no room for correctionterms if we insistthat both the equivalence principle and the Newtonian limit are valid.But if we allow for a small deviation from Newtons law then another term can be imagined.Apart from (7.28) we also haveDg=0 , (7.33)34and therefore one might replace (7.26) byR 12Rg + g= 8GN T , (7.34)where is a constant of Nature, with a very small numerical value, called the cosmologicalconstant. The extra term may also be regarded as a renormalization:Tg , (7.35)implying some residual energy and pressure in the vacuum. Einstein rst introduced suchaterminordertoobtain interestingsolutions,butlaterregretted this. Inanycasearesidual gravitationaleldemanatingfromthevacuumhasneverbeendetected. Iftheterm exists it is very mysterious why the associated constant should be so close to zero.In modern eld theories it is dicult to understand why the energy and momentum densityof the vacuumstate (which just happens to be the state with lowest energy content) aretuned to zero. So we do not know why = 0, exactly or approximately, with or withoutEinsteins regrets.8. THE ACTION PRINCIPLE.We saw that a particles trajectory in a space-time with a gravitational eld is deter-mined by the geodesic equation (5.8), but also by postulating that the quantity=_ds, with (ds)2= gdxdx, (8.1)is stationary under innitesimal displacementsx() x() +x() :=0 . (8.2)This is an example of an action principle,beingthe action forthe particles motion inits orbit. Theadvantageof thisaction principleisits simplicityaswellasthefactthatthe expressions are manifestly covariant so that we see immediately that they will give thesameresultsinanycoordinateframe. Furthermoretheexistenceofsolutionsof(8.2)isvery plausible in particular if the expression for this action is bounded. For example, formost timelike curves is an absolute maximum.Now letgdef=det(g) . (8.3)Then consider in some volume Vof 4 dimensional space-time the so-called Einstein-Hilbertaction:I =_Vg Rd4x, (8.4)35whereR is the Ricci scalar (6.32). We saw in chapters 4 and 6 that with this factor gtheintegral (8.4)isinvariantundercoordinatetransformations, butifwekeepV nitethenof coursetheboundaryshouldbekeptunaected. Considernowaninnitesimalvariation of the metric tensorg: g=g +g , (8.5)such thatgand its rst derivatives vanish on the boundary ofV . The variation in theRicci tensorRto lowest order ingis given byR=R +12_D2g +DDg+DDg DDg_, (8.6)where we used thatgandRandRall transform as true tensors so that all thosecoecientsthatresultfromexpandingR(seeEq. 5.27)mustcombinewiththederivatives of gin such a way that they form covariant derivatives, such as DDg.Once we realise this we can derive (8.6) easily by choosing a coordinate frame where in agiven pointx the ane connection vanishes.Exercise: derive Eq. (8.6).Furthermore we have g=gg, (8.7)so withR = g Rwe haveR=RRg+_DDgD2g_. (8.8)Finally g=g(1 +g) ; (8.9)_ g= g (1 +12g) . (8.10)and so we nd for the variation of the integralIas a consequence of the variation (8.5):I =I +_Vg_R+12Rg_g +_Vg_DD gD2_g. (8.11)However,g DX=_g X_, (8.12)and therefore the secondhalf in (8.11) is an integral overa pure derivative and since wedemandedthatg(and itsderivatives)vanishattheboundarythesecondhalfofEq.(8.11) vanishes. So we ndI = _Vg Gg , (8.13)36withGasdenedin(6.34). Notethatinthesederivationswemixedsuperscriptandsubscriptindices. Only in (8.12) it is essentialthatXis a contra-vector since we insistin having an ordinary rather than a covariant derivative in order to be able to do partialintegration. Here we see that partial integration using covariant derivatives works out neprovided we have the factor g inside the integral as indicated.We read o from Eq. (8.13) that Einsteins equations for the vacuum, G= 0, areequivalent with demanding thatI =0 , (8.14)forall smoothvariations g(x). Inthepreviouschapteraconnectionwassuggestedbetween the gauge freedom in choosing the coordinates on the one hand and the conserva-tion law (Bianchi identity) forGon the other. We can now expatiate on this. For anysystem, even if it does not obey Einsteins equations, I will be invariant under innitesimalcoordinate transformations: x=x+u, g(x)= xx xx g( x) ;g( x)=g(x) +ug(x) +O(u2) ; xx= +u, +O(u2) ,(8.15)so that g(x)=g +ug +gu, +gu, +O(u2) . (8.16)This combination precisely produces the covariant derivatives ofu. Again the reason isthat all other tensors in the equation are true tensors so that non-covariant derivatives areoutlawed. And so we nd that the variation ingis g=g +Du +Du. (8.17)This leavesIalways invariant:I = 2_g GDu= 0 ; (8.18)for anyu(x). By partial integration one nds that the equationg uDG=0 (8.19)isautomaticallyobeydforall u(x). ThisiswhytheBianchiidentityDG=0, Eq.(6.34) is always automatically obeyed.37Theactionprinciplecanbeexpandedforthecasethatmatterispresent. Takeforinstance scalar elds(x). In ordinary at space-time these obey the Klein-Gordon equa-tion:(2m2)=0 . (8.20)In a gravitational eld this will have to be replaced by the covariant expression(D2m2)=(gDD m2)=0 . (8.21)It is not dicult to verify that this equation also follows by demanding thatJ=0J=12_g d4x(D2m2)=_g d4x_12(D)212m22_,(8.22)forall innitesimal variationsin(Notethat(8.21)followsfrom(8.22)viapartialintegrations which are allowed for covariant derivatives in the presence of the g term).Now consider the sumS=116GNI +J=_Vg d4x_R16GN12(D)212m22_, (8.23)and remember that(D)2=g. (8.24)Then variation in will yield the Klein-Gordon equation (8.21) for as usual. Variationingnow givesS=_Vg d4x_G16GN+12DD 14_(D)2+m22_g_g . (8.25)So we haveG= 8GNT, (8.26)if we writeT= DD +12_(D)2+m22_g . (8.27)NowsinceJisinvariantundercoordinatetransformations, Eqs. (8.15), itmustobeyacontinuity equation just as (8.18), (8.19):DT=0 , (8.28)whereas we also haveT44=12(

D)2+12m22+12(D0)2= H(x) , (8.29)38whichcan be identiedas the energydensityforthe eld. Thusthe {i0}componentsof (8.28) must represent the energy ow, which is the momentum density, and this impliesthat thisThas to coincide exactly with the ordinary energy-momentum density for thescalar eld. In conclusion, demanding (8.25) to vanish also for all innitesimal variationsingindeed gives us the correct Einstein equation (8.26).Finally, there is room for a cosmological term in the action:S=_Vg_R216GN12(D)212m22_. (8.30)This example with the scalar eld can immediately be extended to other kinds of mattersuchas otherelds,eldswith furtherinteractionterms (suchas4),andelectromag-netism, and even liquids and free point particles. Every time, all we need is the classicalactionSwhich we rewrite in a covariant way: Smatter = _ g Lmatter, to which we thenadd the Einstein-Hilbert action:S=_Vg_R 216GN+Lmatter_. (8.31)Of course we will often omit the term. Unless stated otherwise the integral symbol willstand short for _ d4x.9. SPECIAL COORDINATES.In the preceding chapters no restrictions were made concerning the choice of coordinateframe. Everychoiceisequivalenttoanyotherchoice(providedthemappingisone-to-one and dierentiable). Complete invariance was ensured. However,when one wishes tocalculate in detail the properties of some particular solution such as space-time surroundinga point particle or the history of the universe, one is forced to make a choice. Since we havea four-fold freedom for the use of coordinates we can in general formulate four equationsand then try to choose our coordinates such a way that these equations are obeyed. Suchequations are called gauge conditions. Of course one should choose the gauge conditionssuch a way that one can easily see how to obey them, and demonstrate that coordinatesobeying these equations exist. We discuss some examples.1) The temporal gauge. Chooseg00= 1 ; (9.1)g0i=0, (i = 1, 2, 3) . (9.2)39Atrstsightitseemseasytoshowthatonecanalwaysobeythese. Ifinanarbitrarycoordinate frame the equations (9.1) and (9.2) are not obeyed one writes g00=g00 + 2D0u0= 1 , (9.3) g0i=g0i +Diu0 +D0ui=0 . (9.4)u0(x, t) can be solved from eq. (9.3) by integrating (9.3) in the time direction, after whichwecannduibyintegrating(9.4)withrespecttotime. NowitistruethatEqs. (9.3)and (9.4) only correspond to coordinate transformations when u is innitesimal (see 8.17),butitseemseasytoobey(9.1)and(9.2)byiteration. Yetthereisadanger. Inthesecoordinatesthereisnogravitationaleld(onlyspace, notspace-time, iscurved), henceall linesof theformx(t)=constant areactuallygeodesicsasonecaneasilycheck(inEq. (5.8), i00 = 0 ). Therefore these are freely falling coordinates, but of course freelyfalling objects in general will go into orbits and hence either wander away from or collideagainst each other, at which instances these coordinates generate singularities.2) The gauge:g=0 . (9.5)This gauge has the advantage of being Lorentz invariant. The equations for innitesimalubecome g=g +Du +Du=0 . (9.6)(Note that ordinary and covariant derivatives must now be distinguished carefully) In aniterative procedure we rst solve foru. Letact on (9.6):22u +g=higher orders, (9.7)after which2u= g (u) +higher orders. (9.8)These are dAlembert equations of which the solutions are less singular than those of Eqs.(9.3) and (9.4).3) A smarter choice is the harmonic or De Donder gauge:g=0 . (9.9)Coordinates obeying this condition are called harmonic coordinates, for the following rea-son. Consider a scalar eldVobeyingD2V =0 , (9.10)or g_V V_=0 . (9.11)40Now let us choose four coordinatesx1,...,4that obey this equation. Note that these thenare not covariant equations because the index ofxis not participating:g_xx_=0 . (9.12)Now of course, in the gauge (9.9),x=0 ; x= . (9.13)Hence, inthesecoordinates,theequations(9.12)imply(9.9). Eq. (9.10)canbesolvedquite generally (it helps a lot that the equation is linear!)Forg= +h(9.14)with innitesimalhthis gauge diers slightly from gauge # 2:f=h 12h=0 , (9.15)and for innitesimaluwe havef=f +2u +uu=f +2u=0 (apart from higher orders)(9.16)so (of course) we get directly a dAlembert equation for u. Observe also that the equation(9.10) is themasslessKlein-Gordonequation thatextremizes theactionJofEq. (8.22)whenm = 0. In this gauge the innitesimal expression forRis simplyR= 122h , (9.17)which simplies practical calculations.TheactionprincipleforEinsteinsequationscanbeextendedsuchthatthegaugeconditionalsofollowsfromvaryingthesameactionastheonethatgeneratestheeldequations. This can be done various ways. Suppose the gauge condition is phrased asf_{g}, x_=0 , (9.18)and that it has been shown that a coordinate choice that obeys (9.18) always exists. Thenone adds to the invariant action (8.23), which we now callSinv.:Sgauge=_g (x)f(g, x)d4x, (9.19)Stotal=Sinv +Sgauge, (9.20)41where (x) is a new dynamical variable, called a Lagrange multiplier.Variation +immediately yields (9.18) as Euler-Lagrange equation. However, we can also consider as avariation the gauge transformation g(x)= x, x, g_ x(x)_. (9.21)ThenSinv=0 , (9.22)Sgauge=_f?=0 . (9.23)Now we must assume that there exists a gauge transformation that producesf(x)= (x x(1)) , (9.24)foranychoiceofthepointx(1)andtheindex. Thisispreciselytheassumptionthatunder any circumstance a gauge transformation exists that can tume f to zero. Then theEuler-Lagrange equation tells us thatSgauge = (x(1)) (x(1))=0 . (9.25)All other variations of gthat are not coordinate transformations then produce the usualequations as described in the previous chapter.A technical detail: often Eq. (9.24) cannot be realized by gauge transformations thatvanish everywhere on the boundary. This can be seen to imply that the solution = 0 willnotbeguaranteedbytheEuler-Lagrangeequations,butratherthattheyareconsistentwiththem, providedthat =0ischosenasaboundarycondition. Inthiscasetheequations generated by the action (9.20) may generate solutions with = 0 that have tobe discarded.Ifweallow(9.24)not toholdontheboundary,thenthecondition=0ontheboundarystillimplies(9.25).4210. ELECTROMAGNETISMWe write the Lagrangian for the Maxwell equations asL= 14FF +JA, (10.1)withF=A A ; (10.2)This means that for any variationA A +A, (10.3)the actionS=_Ld4x, (10.4)should be stationary when the Maxwell equations are obeyed. We see indeed that, ifAvanishes on the boundary,S=_ _F A +JA_d4x=_d4xA_F +J_,(10.5)using partial integration. Therefore (in our simplied units)F= J . (10.6)Describingnowtheinteractionsof theMaxwell eldwiththegravitational eldiseasy. We rst have to makeScovariant:SMax=_d4xg_14ggFF +gJA_, (10.7a)F=A A(unchanged) , (10.7b)andS=_g_R 216GN_+SMax. (10.8)Indices may be raised or lowered with the usual conventions.NotethatconventionsusedheredierfromotherssuchasJackson, Classical Electrodynamicsbyfactorssuchas4. Thereadermayhavetoadapttheexpressionsheretohisorherownnotation.43The energy-momentum tensor can be read o from (10.8) by varying with respect tog(and multiplying by 2):T= FF+_14FFJA_g ; (10.9)here J(with the superscript index) was kept as an external xed source. We have, in atspace-time, the energy density= T00=12(

E2+

B2) JA, (10.10)as usual.We also see that:1) TheinteractionoftheMaxwelleldwith gravitation is unique,thereis nofreedomto add an as yet unknown term.2) The Maxwell eld is a source of gravitational elds via its energy-momentum tensor,as was to be expected.3) The homogeneous equation in Maxwells laws, which follows from Eq. (10.7b),F +F +F=0 , (10.11)remains unchanged.4) VaryingA, we nd that the inhomogeneous equation becomesDF=gDF= J , (10.12)and hence recieves a contribution from the gravitational eld and the potentialg.Exercise:show, both with formal arguments and explicitly, that Eq. (10.11) does notchange if we replace the derivatives by covariant derivatives.Exercise: show that Eq. (10.12) can also be written as(g F)= g J, (10.13)and that(g J)=0 . (10.14)Thus g Jis the real conserved current, and Eq. (10.13) implies thatg acts as thedielectric constant of the vacuum.4411. THE SCHWARZSCHILD SOLUTION.Einsteins equation, (7.26), should be exactly valid. Therefore it is interesting to searchforexactsolutions. The simplestandmost importantone is emptyspacesurroundingastatic star or planet. There, one hasT=0 . (11.1)If the planet does not rotate very fast,the eects of this rotation (which do exist!) maybe ignored. Then there is spherical symmetry. Take spherical coordinates,(x0,x1,x2,x3)=(t,r,,) . (11.2)Spherical symmetry then impliesg02=g03=g12=g13=g23=0 , (11.3)and time-reversal symmetryg01=0 . (11.4)The metric tensor is then specied by writing down the length ds of the innitesimal lineelement:ds2= Adt2+Bdr2+Cr2d2+Dr2sin2 d2, (11.5)whereA,B,C and D can only depend on r. At large distance from the source we expect:r ; A,B,C,D 1 . (11.6)Furthermore spherical symmetry dictatesC=D. (11.7)Our freedom to choose the coordinates can be used to choose a newr coordinate: r= _C(r) r, so that Cr2= r2. (11.8)We then haveBdr2=B_C +r2CdCdr_2d r2def=Bd r2. (11.9)In the new coordinate one has (henceforth omitting the tilde ):ds2= Adt2+Bdr2+r2(d2+ sin2 d2) , (11.10)45whereA,B 1 asr . The signature of this metric must be (, +, +, +), so thatA > 0 and B> 0 . (11.11)Now for generalA andBwe must nd the ane connection they generate. Thereisamethodthatsavesusspaceinwriting(butdoesnotsaveusfromhavingtodothecalculations), because many of its coecients will be zero. If we know all geodesics x+ x x=0 , (11.12)then they uniquely determine all coecients. The variational principle for a geodesic is0=_ds=__gdxddxdd , (11.13)where is an arbitrary parametrization of the curve. In chapter 6 we saw that the originalcurve is chosen to have=s . (11.14)The square root is then one, and Eq. (6.23) then corresponds to12_gdxdsdxdsds=0 . (11.15)We write_ _At2+B r2+r2 2+r2sin2 2_dsdef=_Fds=0 . (11.16)The dot stands for dierentiation with respect tos.(11.16) generates the Lagrange equationddsF x=Fx. (11.17)For = 0 this isdds(2At)=0 , (11.18)ort +1A_Ar r_t=0 . (11.19)Comparing (11.12) we see that all 0vanish except010=001=A

/2A (11.20)46(the accent, , stands for dierentiation with respect to r; the 2 comes from symmetrizationof the subscript indices 0 and 1. For = 1 Eq. (11.17) implies r +B

2B r2+A

2Bt2rB2rB sin2 2=0 , (11.21)so that all 1are zero except100=A

/2B; 111=B

/2B;122= r/B; 133= (r/B) sin2, .(11.22)For = 2 and 3 we nd similarly:221=212=1/r ; 233= sin cos ;323=332=cot ; 313=331=1/r .(11.23)Furthermore we haveg=r2| sin|AB. (11.24)and from Eq. (5.18)=(g)/g= logg . (11.25)Therefore1=A

/2A+B

/2B + 2/r ,2=cot .(11.26)The equationR=0 , (11.27)now becomes (see 5.27)R= (logg),, + , + (logg),=0 . (11.28)Explicitly:R00=100,12100001 + 100(logg),1=(A

/2B)

A

2/2AB + (A

/2B)_A

2A +B

2B+ 2r_=12B_A

A

B

2BA

22A+ 2A

r_=0 ,(11.29)andR11= (logg),1,1 + 111,1010010111111221221331331 + 111(logg),1=0(11.30)47This produces12A_A

+A

B

2B+A

22A+ 2AB

rB_=0 . (11.31)Combining (11.29) and (11.31) we obtain2rB(AB)

=0 . (11.32)ThereforeAB = constant. Since atr we haveA andB 1 we concludeB=1/A. (11.33)In the direction one hasR22= logg),2,2 + 122,12122221323323 + 122(logg),1=0 .(11.34)This becomesR22= cot _ rB_

+2B cot2 rB_2r+ (AB)

2AB_=0 . (11.35)Using (11.32) one obtains(r/B)

=1 . (11.36)Upon integration,r/B=r 2M , (11.37)A=1 2Mr; B=_1 2Mr_1. (11.38)Here 2Mis an integration constant. We found the solution even though we did not yet useall equationsR= 0 available to us (and only a linear combination ofR00andR11wasused). It is not hard to convince oneself that indeedall equationsR= 0 are satised,rst by substituting (11.38) in (11.29) or (11.31), and then spherical symmetry with (11.35)will alsoensurethatR33= 0. Thereason whytheequations are over-determinedis theBianchi identity:DG=0 . (11.39)It will always be obeyed automatically, and implies that if most components ofGhavebeen set equal to zero the remainder will be forced to be zero too.The solution we found is the Schwarzschild solution (Schwarzschild, 1916):ds2= _1 2Mr_dt2+dr21 2Mr+r2_d2+ sin2 d2_. (11.40)48In (11.37) we inserted 2Mas an arbitrary integration constant. We see that far from theorigin,g00=1 2Mr1 + 2V (x) . (11.41)So the gravitational potentialV (x) goes to M/r, as near an object with massm, ifM=GN m (c = 1) . (11.42)Often we will normalize mass units such thatGN= 1.TheSchwarzschildsolutionissingularat r =2M, butthiscanbeseentobeanartefact of our coordinate choice. By studying the geodesics in this region one can discoverdierent coordinate frames in terms of which no singularity is seen. We here give the resultof such a procedure. Introduce new coordinates (Kruskal coordinates)(t, r, , ) (x, y, , ) , (11.43)dened by_r2M 1_er/2M=xy , (11.44a)et/2M=x/y , (11.44b)so thatdxx+ dyy=dr2M(1 2M/r) ;dxx dyy=dt2M.(11.45)The Schwarzschild line element is now given byds2=16M2_1 2Mr_dxdyxy+r2d2=32M3rer/2Mdxdy +r2d2(11.46)withd2def= d2+ sin2 d2. (11.47)The singularity atr = 2Mdisappeared. Remark that Eqs. (11.44) possess two solutions(x, y) for everyr, t. This implies that the completely extended vacuum solution (= solu-tionwithnomatterpresentasasourceofgravitational elds)consistsoftwouniversesconnected to each other at the center. Apart from a rotation over 45 the relation betweenKruskal coordinatesx, y and Schwarzschild coordinatesr, t close to the pointr = 2Mcan49be seen to be exactly as the one between the at space coordinates x3,x0and the Rindlercoordinates3,as discussed in chapter 3.The pointsr = 0 however remain singular in the Schwarzschild solution. The regularregion of the universe has the linexy= 1 (11.48)asitsboundary. Theregionx>0, y>0will beidentiedwiththeordinaryworldextending far from our source. The second universe, the region of space-time withx< 0and y< 0 has the same metric as the rst one. It is connected to the rst one by somethingone could call a wormhole. The physical signicance of this extended region however isvery limited, because:1) ordinary stars and planets contain matter (T = 0) within a certain radius r > 2M,so that for them the validity of the Schwarzschild solution stops there.2) Even if further gravitational contraction produces a black hole one nds that therewill still be imploding matter around (T = 0) that will cut o the second universecompletely from the rst.3) evenif therewerenoimplodingmatterpresenttheseconduniversecouldonlybereached by moving faster than the local speed of light.Exercise:Check these statements by drawing an xy diagram and indicating where thetwouniversesareandhowmatterandspacetravellerscanmoveabout. Showthatalsosignals cannot be exchanged between the two universes.If one draws an imploding star in the xy diagram one notices that the future horizonmay be physically relevant. One then has the so-called black hole solution.12. MERCURY AND LIGHT RAYS IN THE SCHWARZSCHILD METRIC.Historically the orbital motion of the planet Mercury in the Suns gravitational eldhasplayedanimportantroleasatestforthevalidityof General Relativity(althoughEinstein would have lounched his theory also if such tests had not been available)To describe this motion we have the variation equation (11.16) for the functionst(),r(),() and(), whereparametrizes the space-time trajectory. Writing r = dr/d,etc. we have_ __1 2Mr_t2+_1 2Mr_1 r2+r2_2+ sin2 2__d =0 , (12.1)50in which we put ds2/d2= 1 because the trajectory is timelike.The equations of motionfollow as Lagrange equations:dd (r2 )=r2sin cos 2; (12.2)dd (r2sin2 )=0 ; (12.3)dd__1 2Mr_t_=0 . (12.4)We did not yet write the equation for r. Instead of that it is more convenient to divideEq. (11.40) by ds2:1=_1 2Mr_t2_1 2Mr_1 r2r2_2+ sin2 2_. (12.5)Now even in the completely relativistic metric of the Schwarzschild solution all orbitswill be in at planes through the origin, since spherical symmetry allows us to choose asour initial condition=/2 ;=0 . (12.6)and then this will remain valid throughout because of Eq. (12.2). Eqs. (12.3) and (12.4)tell us:r2 =J=constant. (12.7)and_1 2Mr_t=E=constant. (12.8)Eq. (12.5) then becomes1=_1 2Mr_1E2_1 2Mr_1 r2J2/r2. (12.9)Just as in the Kepler problem it is convenient to treatr as a function of. t has alreadybeen eliminated. We now also eliminate s. Let us, for the remainder of this chapter, writedierentiation with respect to with an accent:r

= r/ . (12.10)From (12.7) and (12.9) one derives:1 2M/r=E2J2r

2/r4J2_1 2Mr_/r2. (12.11)51Notice that we can interpretEas energy andJas angular momentum. Write, just as inthe Kepler problem:r=1/u, r

= u

/u2; (12.12)1 2Mu=E2J2u2J2u2(1 2Mu) . (12.13)From this we nddud=__2Mu 1__u2+1J2_+E2/J2. (12.14)The formal solution is 0=_uu0du_E21J2+ 2MuJ2u2+ 2Mu3_12. (12.15)Exercise: showthatintheNewtonianlimittheu3termcanbeneglectedandthencompute the integral.The relativistic perihelion shift will be the extent to which the complete integral fromumin to umax (two roots of the third degree polynomial), multiplied by two, diers from 2.SunPlanetEarth:Venus:Mercury:Percentury:43".038".33".8Fig.4. Perihelion shift of a planet in its orbit around a central star.A neat way to obtain the perihelion shift is by dierentiating Eq. (12.13) once morewith respect to:2MJ2u

2u

u

2uu

+ 6Mu2u

=0 . (12.16)52Now of courseu

=0 (12.17)can be a solution (the circular orbit). Ifu

= 0 we divide byu

:u

+u=MJ2+ 3Mu2. (12.18)The last term is the relativistic correction. Suppose it is small. Then we have a well-knownproblem in mathematical physics:u

+u=A+u2. (12.19)One could expand u as a perturbative expansion in powers of , but we wish an expansionthat converges for all values of the independentvariable. Note that Eq. (12.13) allowsfor every value ofu only two possible values foru

so that the solution has to be periodicin. The unperturbedperiod is 2. Butwith theu2termpresentwe donotknowtheperiod exactly. Assume that it can be written as2_1 + +O(2)_. (12.20)Writeu=A+Bcos_(1 )+u1() +O(2) , (12.21)u

= B(1 2) cos_(1 )+u

1() +O(2) ; (12.22)u2=_A2+ 2ABcos_(1 )+B2cos2_(1 )_+O(2) . (12.23)We nd foru1:u

1 +u1=(2B + 2AB) cos +B2cos2 +A2, (12.24)where now the O() terms were omitted since they do not play any further role. This is justthe equation for a forced pendulum. If we do not want that the pendulum oscillates withan ever increasing period (u1muststay small for all values of) then the external forceisnotallowedtohaveaFouriercomponentwiththesameperiodicityasthependulumitself. Nowthetermwithcos in(12.24) isexactlyintheresonanceunlesswechoose=A. Then one hasu

1 +u1=12B2(cos2 + 1) +A2, (12.25)u1=12B2_1 1221 cos 2_+A2, (12.26)Notehereandinthefollowingthatthesolutionofanequationoftheformu

+u=

iAi cos i isu=

iAi cos i/(12i) +C1 cos +C2 sin. Thisissingularwhen 1.53which is exactly periodic. Apparently one has to choose the period to be 2(1 +A) if theorbitistobeperiodic in. Wendthataftereverypassagethroughtheperihelionitsposition is shifted by=2A=23M2J2, (12.27)(plus higher order corrections) in the direction of the planet itself (see Fig. 4).Nowwewishtocomputethetrajectoryof alightray. Itisalsoageodesic. Nowhowever ds = 0. In this limit we still have (12.1) (12.4), but now we setds/d =0 ,so that Eq. (12.5) becomes0=_1 2Mr_t2_1 2Mr_1 r2r2_2+ sin2 2_. (12.28)Since now the parameteris determined up to an arbitrary multiplicative constant, onlythe ratioJ/Ewill be relevant. Call thisj. Then Eq. (12.15) becomes=0 +_uu0du_j2u2+ 2Mu3_12. (12.29)As the left hand side of Eq. (12.13) must now be replaced by zero, Eq. (12.18) becomesu

+u=3Mu2. (12.30)An expansion in powers of M is now permitted (because the angle is now conned withinan interval a little larger than):u=Acos +v , (12.31)v

+v=3MA2cos2=32MA2(1 + cos 2) , (12.32)v=32MA2_1 13 cos 2_=MA2(2 cos2) . (12.33)So we have for smallM1r=u=Acos +MA2(2 cos2) . (12.34)The angles at which the ray enters and exits are determined by1/r=0, cos =1 1 + 8M2A22MA. (12.35)54SinceMis a small expansion parameter and | cos | 1 we must choose the minus sign:cos 2MA= 2M/r0, (12.36) _2+ 2M/r0_, (12.37)wherer0is the smallest distance of the light ray to the central source. In total the angleof deection between in- and outgoing ray is in lowest order:=4M/r0. (12.38)In conventional units this equation reads=4GNm

r0c2. (12.39)m

is the mass of the central star.Exercise:show that this is twice what one would expect if a light ray could be regardedas a non-relativistic particle in a hyperbolic orbit around the star.Exercise: show that expression (12.27) in ordinary units reads as=6GNm

a(1 2)c2, (12.40)wherea is the major axis of the orbit, its excentricity andc the velocity of light.13. GENERALIZATIONS OF THE SCHWARZSCHILD SOLUTION.a). The Reissner-Nordstrom solution.Spherical symmetrycanstill beusedasastartingpointfortheconstructionof asolution of the combined Einstein-Maxwell equations for the elds surrounding a planetwith electric chargeQ and massm. Just as Eq. (11.10) we chooseds2= Adt2+Bdr2+r2(d2+ sin2 d2) , (13.1)but now also a static electric eld:Er=E(r) ; E=E=0 ;

B=0 . (13.2)This implies thatF01= F10=E(r) and all othercomponentsof Fare zero. Letusassume that the sourceJof this eld is inside the planetand we are only interested inthe solution outside the planet. So there we haveJ=0 . (13.3)55If we move the indices upstairs we getF10=E(r)/AB, (13.4)and usingg= ABr2sin , (13.5)we nd that according to (10.13)r_E(r)r2AB_=0 . (13.6)Thus the inhomogeneous Maxwell law tells us thatE(r)=QAB4r2, (13.7)whereQ is an integration constant,to beidentiedwith electric chargesinceatr bothA andBtend to 1.The homogeneous Maxwell law (10.11) is automatically obeyed because there is a eldA0(potential eld) withEr= rA0. (13.8)The eld (13.7) contributes toT:T00= E2/2B= AQ2/322r4; (13.9)T11=E2/2A=BQ2/322r4; (13.10)T22= E2r2/2AB= Q2/322r2(13.11)T33=T22 sin2= Q2sin2 /322r2. (13.12)We ndT=gT=0 ; R=0 , (13.13)a general property of the free Maxwell eld. In this case we have (GN= 1)R= 8 T . (13.14)Herewith the equations (11.29) (11.31) becomeA

A

B

2BA

22A+ 2A

r=ABQ2/2r4,A

+A

B

2B+A

22A+ 2AB

rB= ABQ2/2r4.(13.15)56We nd that Eq. (11.32) still holds so that here alsoB=1/A. (13.16)Eq. (11.36) is now replaced by(r/B)

1= Q2/4r2. (13.17)This gives upon integrationr/B=r 2M +Q2/4r . (13.18)So now we have instead of Eq. (11.38),A=1 2Mr+Q24r2; B=1/A. (13.19)This is the Reissner-Nordstrom solution (1916, 1918).If we chooseQ2/4< M2there are two horizons, the roots of the equationA = 0:r=r=M _M2Q2/4 . (13.20)Againthesesingularitiesareartefactsof ourcoordinatechoiceandcanberemovedbygeneralizations of the Kruskal coordinates. Now one nds that there would be an innitesequenceof ghost universesconnectedtoours, if thehorizonshadnt beenblockedbyimploding matter. See Hawking and Ellis for a much more detailed description.b) The Kerr solutionA fast rotating planet has a gravitational eld that is no longer spherically symmetricbut only cylindrically. We here only give the solution:ds2= dt2+ (r2+a2) sin2d2+ 2Mr_dt a sin2d_2r2+a2cos2+ (r2+a2cos2)_d2+dr2r22Mr +a2_.(13.21)This solution was found by Kerr in 1963.To prove that this is indeed a solution of Einsteinsequationsrequirespatiencebutisnotdicult. Foraderivationusingmoreelementaryprinciplesmorepowerfultechniquesandmachineryofmathematicalphysicsareneeded.The free parametera in this solution can be identied with angular momentum.57c) The Newmann et al solutionFor sake of completeness we also mention that rotating planets can also be electricallycharged. The solution for that case was found by Newman et al in 1965. The metric is:ds2= Y_dt a sin2d_2+ sin2Y_adt (r2+a2)d_2+Ydr2+Y d2, (13.22)whereY =r2+a2cos , (13.23)=r22Mr +Q2/4 +a2. (13.24)The vector potential isA0= Qr4Y; A3=Qra sin24Y. (13.25)Exercise: show that whenQ = 0 Eqs. (13.21) and (13.22) coincide.Exercise: ndthenon-rotatingmagneticmonopolesolutionbypostulatingaradialmagnetic eld.Exercise for the advanced student: describe geodesics in the Kerr solution.14. THE ROBERTSON-WALKER METRIC.General relativity plays an important role in cosmology. The simplest theory is thatatacertainmomentt=0theuniversestartedofromasingularity, afterwhichitbegan to expand. We assume maximal symmetry by taking as our metricds2=dt2+F2(t)d2. (14.1)Here d2stands short for some fully isotropic 3-dimensional space, and F(t) describes the(increasing) distance between two neighboring galaxies in space. Although we do embraceheretheCopernicanprinciplethatall pointsinspacelookthesame, weabandontheideathatthereshouldbeinvariancewithrespecttotimetranslationsandalsoLorentzinvariance for this metric the galaxies contain clocks that were set to zero att = 0 andeach provides for a local inertial frame.If we writed2=B()d2+2_d2+ sin2d2_, (14.2)58then in this three dimensional space the Ricci tensor is (by using the same techniques asin chapter 11)R11=B

()/B() , (14.3)R22=1 1B+B

2B2. (14.4)In an isotropic (3-dimensional) space, one must haveRij=gij , (14.5)for some constant, and thereforeB

/B=B , (14.6)1 1B+B

2B2=2. (14.7)Together they give1 1B=122;B=11 122,(14.8)which indeed also obeys (14.6) separately.Exercise:show that with =_2 sin , this gives the metric of the 3-sphere, in termsof its three angular coordinates, , .Often one chooses a new coordinateu:def=_2k/u1 + (k/4)u2. (14.9)One observes thatd=_2k1 14ku2_1 +14ku2_2 du and B=_1 +14ku21 14ku2_2, (14.10)so thatd2=2k du2+u2(d2+ sin2d2)_1 + (k/4)u2_2. (14.11)The parameterk is arbitrary except for its sign, which must be the same as the sign of.The factor in front of Eq. (14.11) may be absorbed in F(t). Therefore we write for (14.1):ds2= dt2+F2(t)dx2_1 +14k x2_2. (14.12)59If k = 1 the spacelike piece is a sphere, if k = 0 it is at, if k = 1 the curvature is negativeand space is unbounded (in spite of the fact that then |x| is bounded, which is an artefactof our coordinate choice). Let us writeF2(t)=eg(t), (14.13)then after some elementary calculationsR00=32 g +34 g2, (14.14)R11=R22=R33=12 g +34 g2+ 2keg, (14.15)R=R=3( g + g2) + 6keg. (14.16)The tensorGbecomes:G00=34 g2+ 3keg, (14.17)G11=G22=G33= eg( g +34 g2) k . (14.18)F(t)tOk = 0k = 1k = 1Fig. 5. The Robertson-Walker universe fork = 1, k = 0, andk = 1.Now what we have to do is to make certain assumptions about matter in the universe,anditsequationsofstate, sothatweknowwhatTtosubstituteinEqs. (14.17)and(14.18). Eq. (14.17)containsthemassdensityandEq. (14.18)thepressure. Letusassume that the pressure vanishes. Then the equation g +34 g2+keg=0 (14.19)60can be solved exactly. In terms ofF(t) we have2FF

+F

2+k=0 . (14.20)Write this as_F(F

)2_

+kF

=0 , (14.21)then we see thatFF

2+kF=D=constant; (14.22)F

2=D/F k , (14.23)and from (14.20):F

= D/2F2. (14.24)Write Eq. (14.23) asdtdF=_FD kF, (14.25)then we tryF=Dksin2, (14.26)dtd=dFddtdF=2Dkksin cos sin cos , (14.27)t()=Dkk( 12 sin 2), (14.28)F()=D2k(1 cos2). (14.29)These are the equations for a cycloid. Note thatG00=3( F2+k)F2=3DF3. (14.30)Therefore D is something like the mass density of the universe, hence positive. Since t > 0andF> 0 we demandk> 0 real ;k < 0 imaginary ;k = 0 innitesimal .(14.31)See Fig. 5. All solutions start with a big bang att = 0. Only the cycloid in thek = 1casealsoshowsabigcrunchintheend. If k 0notonlyspacebutalsotimeareunbounded. This relationship between the boundedness of space and the boundedness oftime alsoholdsifwedonotassumethatthepressurevanishes(itonlyhastovanishasthe mass density aproaches zero). The solution of the caseG00=eg

iGiiis a good exercise.6115. GRAVITATIONAL RADIATION.Fast movingobjects formatimedependent sourceof thegravitational eld, andcausality arguments (information in the gravitational elds should not travel fasterthanlight)thensuggestthatgravitationaleectsspreadlikewavesinalldirectionsfromthesource. Far fromthe sourcethe metricgwill stayclose to thatof atspace-time. Tocalculate this eect one can adopt a linearized approximation. In contrast to what we didin previous chapters it is now convenient to choose units such that16GN=1 . (15.1)The linearized Einstein equations were already treated in chapter 7, and in chapter 9 wesee that, after gauge xing, wave equations can be derived (in the absence of matter, Eq.(9.17)canbesettozero). ItisinstructivetorecasttheseequationsinEuler-Lagrangeform. TheLagrangianforalinearequationhoweverisitself quadratic. Sowehavetoexpand the Einstein-Hilbert action to second order in the perturbations hin the metric:g= +h , (15.2)and after some calculations we nd that the terms quadratic inhcan be written as:g_R +Lmatter_=18(h)214(h)(h) 12Th+12A2+total derivative +higher orders inh,(15.3)whereA=h12h, (15.4)and Tis the energy momentum tensor of matter when present. Indices are summed overwith the at metric, Eq. (7.2).The Lagrangian is invariant under the linearized gauge transformation (compare (8.16)and (8.17))hh +u +u, (15.5)which transforms the quantityAintoA A +2u . (15.6)One possibility to x the gauge is to chooseA=0 (15.7)62(the linearized De Donder gauge). For calculations this is a convenientgauge. But for abetterunderstandingoftherealphysicaldegreesoffreedominaradiatinggravitationalelditisinstructiverst tolookattheradiationgauge(whichisanalogoustotheelectromagnetic caseiAi = 0):ihij=0 ; ihi4=0 , (15.8)wherewesticktotheearlieragreement that indicesfromthemiddleof thealphabet,i, j, . . ., in a summation run from 1 to 3. So we do not impose (15.7).First go to momentum representation:h(x, t)=(2)3/2_d3

kh(

k, t) ei

kx; (15.9)i iki. (15.10)Wewillhenceforthomitthehat()sinceconfusionishardlypossible. Theadvantageofthemomentumrepresentationisthatthedierentvaluesof

kwill decouple, sowecanconcentrate on just one k vector, and choose coordinates such that it is in thez direction:k1 = k2 = 0, k3 = k . We now decide to let indices from the beginning of the alphabet runfrom 1 to 2. Then one has in the radiation gauge (15.8):h3a=h33=h30=0 . (15.11)FurthermoreAa= h0a,A3= 12ik(haah00) ,A0=12(h00 haa).(15.12)Let us split o the trace ofhab:hab=hab +12abh, (15.13)withh=haa;haa=0 . (15.14)Then we nd thatL= L1 +L2 +L3,L1=14_hab_214k2h2ab12Tabhab, (15.15)L2=12k2h20a +h0aT0a, (15.16)L3= 18h2+18k2h212k2hh0012h00T0014hTaa . (15.17)63Here we used the abbreviated notation:h2=_d3

k h(

k, t)h(

k, t) ,k2h2=_d3

k k2h(

k, t)h(

k, t) .(15.18)TheLagrangian L1hastheusual formofaharmonicoscillator. Sincehab=hbaandhaa=0 , thereareonlytwodegreesoffreedom(formingaspin2representationoftherotation group around the k axis: gravitons are particles with spin 2). L2 has no kineticterm. It generates the following Euler-Lagrange equation:h0a= 1k2Toa . (15.19)We can substitute this back into L2:L2= 12k2T20a. (15.20)Sincethereare nofurtherkinetic terms thisLagrangian producesdirectlyatermintheHamiltonian:H2= _L2d3

k=_12k2T20ad3

k=_ _ij kikj/k22k2_T0i(

k)Toj(

k)d3

k==12_T0i(x)_1(x y)ij + 2(x y)ij_T0j(y)d3xd3y ;with 2(x y)= 3(x y) ; =14|x y|.(15.21)In L3we nd thath00acts as a Lagrange multiplier. So the Euler-Lagrange equation itgenerates is simply:h= 1k2T00, (15.22)leading toL3= T200/8k4+T200/8k2+T00Taa/4k2. (15.23)Now for the source we have in a good approximationT=0 , (15.24)ikT3=T0, so ikT30=T00, (15.25)and therefore one can writeL3= T230/8k2+T200/8k2+T00Taa/4k2; (15.26)H3= _L3d3

k . (15.27)64Here the second term is the dominant one:_d3

kT200/8k2= _T00(x)T00(y)d3xd3y8 4|x y|= GN2_d3xd3y|x y|T00(x)T00(y) ,(15.28)where we reinsertedNewtons constant. This is the linearized gravitational potential forstationary mass distributions.We observe that in the radiation gauge, L2 and L3 generate contributions to the forcesbetween the sources. It looks as if these forces are instantaneous, without time delay, butthis is an artefact peculiar to this gauge choice. There is graviational radiation, but it isall described by L1. We see thatTab, the traceless, spacelike, transverse part of the energymomentum tensor acts as a source. Let us now consider a small, localized source; only ina small regionVwith dimensions much smaller than 1/k. Then we can use:_Tijd3x=_Tkj(kxi)d3x= _xikTkjd3x=0_xiT0jd3x=0_xi(kxj)T0kd3x=120_k(xixj)T0kd3x= 12_xixjkT0kd3x=1220_xixjT00d3x.(15.29)This means that, when integrated, the space-space components of the energy momentumtensor can be identied with the second time derivative of the quadrupolemoment of themass distributionT00.We would like to know how much energy is emitted by this radiation. To do this letus momentarily return to electrodynamics,or even simpler,a scalar eld theory. Take aLagrangian of the formL=12 212k22J . (15.30)LetJbe periodic in time:J(x, t)=J(x)eit, (15.31)then the solution of the eld equation (see the lectures about classical electrodynamics) isat larger:(x, t)= eikr4r_J(x

)d3x

; k= , (15.32)wherex

is the retarded position where one measuresJ. Since we took the supportV ofour source to be very small compared to 1/k the integral here is just a spacelike integral.65The energyPemitted per unit of time isdEdt=P=4r2_12 2+k222_=k24_J(x

)d3x

2=14_0J(x)d3x2.(15.33)Now this derivation was simple because we have been dealing with a scalar eld. How doesone handle the more complicated Lagrangian L