-
Distributional Schwarzschild Geometry from. nonsmooth
regularization via Horizon
Jaykov FoukzonIsrael Institute of Technology, Haifa, Israel
[email protected]
Abstract: In this paper we leave the neighborhood of the
singularity at the originand turn to the singularity at the
horizon.Using nonlinear superdistributionalgeometry and super
generalized functions it seems possible to show that thehorizon
singularity is not only a coordinate singularity without
leavingSchwarzschild coordinates.However the Tolman formula for the
total energy ET of astatic and asymptotically flat spacetime,gives
ET m, as it should be.
I.Introduction
I.1.The breakdown of canonical formalism of Riemanngeometry for
the singular solutions of the Einstein fieldequations
Einstein field equations was originally derived by Einstein in
1915 in respectwith canonical formalism of Riemann geometry,i.e. by
using the classicalsufficiently smooth metric tensor, smooth
Riemann curvature tensor, smooth Riccitensor,smooth scalar
curvature, etc.. However have soon been found singularsolutions of
the Einstein field equations with singular metric tensor and
singularRiemann curvature tensor.
These singular solutions was formally acepted beyond rigorous
canonicalformalism of Riemann geometry.
Remark 1.1.Note that if some components of the Riemann curvature
tensorRklmi x become infinite at point x0 one obtain the breakdown
of canonical formalismof Riemann geometry in a sufficiently small
neighborhood of the point x0 , i.e.
-
in such neighborhood Riemann curvature tensor Rklmi x will be
changed byformula (1.7) see remark 1.2.
Remark 1.2.Let be infinitesimal closed contour and let be
thecorresponding surface spanning by , see Pic.1. We assume now
that: (i)christoffel symbol kli x become infinite at singular point
x0 by formulae
kli x klxxi xi0, 1klx C
1.1
and (ii) x0 .Let us derive now to similarly canonical
calculation [3]-[4] thegeneral formula for the regularized change
Ak in a vector Aix after paralleldisplacement around infinitesimal
closed contour . This regularized change Akcan clearly be written
in the form
Ak x x0Ak, 1.2
where x x0 i0
4xi xi02, 1 and where the integral is taken over the
given contour . Substituting in place of Ak the canonical
expressionAk kli xAkdxl (see [4],Eq.(85.5)) we obtain
Ak x x0Ak
x x0 kli xAkdxl , 1.3
where
Aixl kl
i xAk. 1.4
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Pic.1.Infinitesimal closed contour and corresponding singular
surface x0
spanning by .
Now applying Stokes theorem (see [4],Eq.(6.19)) to the integral
(1.3) andconsidering that the area enclosed by the contour has the
infinitesimal value flm,we get
-
Ak x x0 kli xAkdxl
12
kmi xAix x0xl
kli xAix x0xm df
lm
kmi xAix x0
xl kli xAix x0
xmflm2
x x0 kmi x x0Ai
xl kmi xAi x x
0xl
x x0 kli xAixm kl
i xAi x x0
xmflm2
x x0 kmi xAixl x x
0 kli xAixm
Aixx x0 2kmi x
xl xl0 Aixx x0 2kl
i xxm xm0
flm2 .
1.5
Substituting the values of the derivatives (1.4) into Eq.(1.5),
we get finally:
Ak Rklmi Aixx x0flm
2 , 1.6
where Rklmi , is a tensor of the fourth rank
Rklmi Rklmi 2 km
i xxl xl0
kli x
xm xm0 . 1.7
Here Rklmi is the classical Riemann curvature tensor.That Rklmi
is a tensor is clearfrom the fact that in (1.6) the left side is a
vectorthe difference Ak between thevalues of vectors at one and the
same point.
Remark 1.3. Note that similar result was obtained by many
authors [5]-[17] by
-
using Colombeau nonlinear generalized functions
[1]-[2].Definition1.1. The tensor Rklmi is called the generalized
curvature tensor or the
generalized Riemann tensor.Definition1.2. The generalized Ricci
curvature tensor Rkm is defined as
Rkm Rkimi . 1.8
Definition1.3. The generalized Ricci scalar R is defined asR gkm
Rkm. 1.9
Definition1.3. The generalized Einstein tensor Gkm is defined
asGkm Rkm 12 gkmR. 1.10
Remark 1.4. Note that in physical literature the spacetime
singularity usually isdefined as location where the quantities that
are used to measure the gravitationalfield become infinite in a way
that does not depend on the coordinate system.These quantities are
the classical scalar invariant curvatures of singular
spacetime,which includes a measure of the density of matter.
Remark 1.5. In general relativity, many investigations have been
derived withregard to singular exact vacuum solutions of the
Einstein equation and thesingularity structure of space-time. Such
solutions have been formally derivedunder condition
Tx 0, 1.11where Tx 0 represent the energy-momentum densities of
the gravity source.This for example is the case for the well-known
Schwarzschild solution, which isgiven by, in the Schwarzschild
coordinates x0, r,,,
ds2 hrdx02 h1rdr2 r2d2 sin2d2 , 1.12hr 1 rsr ,
where, rs is the Schwarzschild radius rs 2GM/c2 with G,M and c
being theNewton gravitational constant, mass of the source, and the
light velocity in vacuumMinkowski space-time, respectively. The
metric (1.12) describe the gravitationalfield produced by a
point-like particle located at x 0.
Remark 1.6. Note that when we say, on the basis of the canonical
expressionof the curvature square
RrRr 12rs2 1r6 1.13
formally obtained from the metric (1.12), that r 0 is a
singularity of theSchwarzschild space-time, the source is
considered to be point-like and this metric
-
is regarded as meaningful everywhere in space-time.Remark 1.7.
From the metric (1.12), the calculation of the canonical
Einstein
tensor proceeds in a straighforward manner gives for r 0Gttr
Grrr h
rr
1 hrr2
0 ,Gr Gr hr2
hrr2
0, 1.14where hr 1 rs/r.Using Eq.(1.14) one formally obtain
boundary conditions
Gtt0 r0lim Gttr 0,Grr0
r0lim Grrr 0,
G0 r0lim Gr 0,G0
r0lim Gr 0. 1.15
However as pointed out above the canonical expression of the
Einstein tensor in asufficiently small neighborhood of the point r
0 and must be replaced by thegeneralized Einstein tensor Gkm
(1.10). By simple calculation easy to see that
Gtt0
r0lim Gttr ,Grr0
r0lim Grrr ,
G0
r0lim Gr ,G0
r0lim Gr .
1.16
and therefore the boundary conditions (1.15) is completely
wrong. But other handas pointed out by many authors [5]-[17] that
the canonical representation of theEinstein tensor, valid only in a
weak (distributional) sense,i.e. [12]:
Gbax 8m0ab03x 1.17and therefore again we obtain Gba0 0ab0.Thus
canonical definition of theEinstein tensor is breakdown in rigorous
mathematical sense for the Schwarzschildsolution at origin r 0.
I.2.The distributional Schwarzschild geometryGeneral relativity
as a physical theory is governed by particular physical
equations; the focus of interest is the breakdown of physics
which need notcoincide with the breakdown of geometry. It has been
suggested to describesingularity at the origin as internal point of
the Schwarzschild spacetime, where theEinstein field equations are
satisfied in a weak (distributional) sense [5]-[22].
1.2.1.The smooth regularization of the singularity at
theorigin.
-
The two singular functions we will work with throughout this
paper (namely thesingular components of the Schwarzschild metric)
are 1r and 1r rs , rs 0.Since1r Lloc1 3, it obviously gives the
regular distribution 1r D3. By convolutionwith a mollifier x
(adapted to the symmetry of the spacetime, i.e. chosen
radiallysymmetric) we embed it into the Colombeau algebra GR3
[22]:
1r
1r 1r 1r , 13 r ,
0,1.1.18
Inserting (1.18) into (1.12) we obtain a generalized Colombeau
object modeling thesingular Schwarzschild spacetime [22]:
ds2 hrdt2 h1rdr2 r2d2 sin2d2 , 1.19hr 1 rs 1r , 0,1.
Remark 1.8.Note that under regularization (1.18) for any 0,1 the
metricds2 hrdt2 h1rdr2 r2d2 sin2d2 obviously is a
classicalRiemannian object and there no exist an the breakdown of
canonical formalism ofRiemannian geometry for these metrics, even
at origin r 0. It has beensuggested by many authors to describe
singularity at the origin as an internal point,where the Einstein
field equations are satisfied in a distributional sense
[5]-[22].From the Colombeau metric (1.19) one obtain in a
distributional sense [22]:
R22 R33 h rr
1 hrr2
8m3,
R00 R11 12hr
2 h rr
4m3,1.20
Hence, the distributional Ricci tensor and the distributional
curvature scalar R are of -type, i.e. R m3.
Remark 1.9. Note that the formulae (1.20) should be contrasted
with what is theexpected result Gbax 8m0ab03x given by Eq.(1.17).
However the equations(1.20) are obviously given in spherical
coordinates and therefore strictly speakingthis is not correct,
because the basis fields r , , are not globally
defined.Representing distributions concentrated at the origin
requires a basis regular at theorigin. Transforming the formulae
for Rij into Cartesian coordinatesassociated with the spherical
ones, i.e., r,, xi, we obtain, e.g., for theEinstein tensor the
expected result Gbax 8m0ab03x given by Eq.(1.17), see[22].
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1.2.2.The nonsmooth regularization of the singularity atthe
origin.
The nonsmooth regularization of the Schwarzschild singularity at
the origin r 0is considered by N. R. Pantoja and H. Rago in paper
[12]. Pantoja non smoothregularization regularization of the
Schwarzschild singularity are
hr 1 rsr r , 0,1, r rs. 1.21Here u is the Heaviside function and
the limit 0 is understood in adistributional sense.Equation (1.19)
with h as given in (1.21) can be considered asan regularized
version of the Schwarzschild line element in curvature
coordinates.From equation (1.21), the calculation of the
distributional Einstein tensor proceedsin a straighforward manner.
By simple calculation it gives [12]:
Gttr, Grrr, h rr
1 hrr2
rs r r2 rsrr2
1.22
and
Gr, Gr, hr2
hrr2
rsr r2
rs r2ddr r rs
rr2
.1.23
which is exactly the result obtained in Ref. [9] using smoothed
versions of theHeaviside function r . Transforming now the formulae
for Gba intoCartesian coordinates associated with the spherical
ones, i.e., r,, xi, weobtain for the generalized Einstein tensor
the expected result given by Eq.(1.17)
Gbax 8m0ab03x , 1.24
see Remark 1.9.
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1.2.3.The smooth regularization via Horizon.The smooth
regularization via Horizon is considered by J.M.Heinzle and
R.Steinbauer in paper [22]. Note that 1r rs Lloc1 3. An
canonical regularizationis the principal value vp 1r rs D3 which
can be embedded into GR3 [22]:
1r rs
vp vp 1r rs vp 1r rs 1r rs GR3. 1.25
Inserting now (1.25) into (1.12) we obtain a generalized
Colombeau objectmodeling the singular Schwarzschild spacetime
[22]:
ds2 hrdt2 h1rdr2 r2d2 sin2d2 , 1.26hr 1 rsr ,h1r 1 rs 1r rs ,
0,1.
Remark 1.10.Note that obviously Colombeau object, (1.26) is
degenerate atr rs, because hr is zero at the horizon. However, this
does not come as asurprise. Both hr and h1r are positive outside of
the black hole and negative inthe interior. As a consequence any
smooth regularization of hr (or h1) must passthrough zero somewhere
and, additionally, this zero must converge to r rs as
theregularization parameter goes to zero.
Remark 1.11.Note that due to the degeneracy of Colombeau object
(1.26),even the distributional Levi-Civit connection obviously is
not available.
1.2.4.The nonsmooth regularization via GorizonIn this paper we
leave the neighborhood of the singularity at the origin and
turn
to the singularity at the horizon. The question we are aiming at
is the following:using distributional geometry (thus without
leaving Schwarzschild coordinates), is itpossible to show that the
horizon singularity of the Schwarzschild metric is notmerely a
coordinate singularity. In order to investigate this issue we
calculate thedistributional curvature at the horizon in
Schwarzschild coordinates.
The main focus of this work is a (nonlinear) superdistributional
description of theSchwarzschild spacetime. Although the nature of
the Schwarzschild singularity ismuch worse than the quasi-regular
conical singularity, there are severaldistributional treatments in
the literature [8]-[29], mainly motivated by the
followingconsiderations: the physical interpretation of the
Schwarzschild metric is clear aslong as we consider it merely as an
exterior (vacuum) solution of an extended(sufficiently large)
massive spherically symmetric body. Together with the
interiorsolution it describes the entire spacetime. The concept of
point particleswellunderstood in the context of linear field
theoriessuggests a mathematicalidealization of the underlying
physics: one would like to view the Schwarzschildsolution as
defined on the entire spacetime and regard it as generated by a
point
-
mass located at the origin and acting as the gravitational
source.This of course amounts to the question of whether one can
reasonably ascribe
distributional curvature quantities to the Schwarzschild
singularity at the horizon.The emphasis of the present work lies on
mathematical rigor. We derive the
physically expected result for the distributional energy
momentum tensor of theSchwarzschild geometry, i.e., T00 8m3x, in a
conceptually satisfactory way.Additionally, we set up a unified
language to comment on the respective merits ofsome of the
approaches taken so far. In particular, we discuss questions
ofdifferentiable structure as well as smoothness and degeneracy
problems of theregularized metrics, and present possible
refinements and workarounds.Theseaims are accomplished using the
framework of nonlinear supergeneralizedfunctions (supergeneralized
Colombeau algebras GR3,).
Examining the Schwarzschild metric (1.12) in a neighborhood of
the horizon, wesee that, whereas hr is smooth, h1r is not even
Lloc1 (note that the origin is nowalways excluded from our
considerations; the space we are working on is R3\0).Thus,
regularizing the Schwarzschild metric amounts to embedding h1
intoGR3, (as done in (3.2)).Obviously, (3.1) is degenerate at r 2m,
because hr iszero at the horizon. However, this does not come as a
surprise. Both hr andh1r are positive outside of the black hole and
negative in the interior. As aconsequence any (smooth)
regularization hr (hr) [above (below) horizon] ofhr must pass
through small enough vicinityO2m x R3|x 2m,x 2m (O2m x R3|x 2m,x 2m
) of zeros setO02m y R3|y 2m somewhere and, additionally, this
vicinity O2m(O2m) must converge to O02m as the regularization
parameter goes to zero.
Due to the degeneracy of (1.12), the Levi-Civit connection is
not available.Consider, therefore, the following connections kjl
kjlh GR3, andkjl kjlh GR3, :
kjl 12 g1 lmgmk,j gmj,k gkj,m,
kjl 12 g1 lmgmk,j gmj,k gkj,m.
1.27
kjl0,kjl0 coincides with the Levi-Civit connection on 3\r 0, as
g0 g,g0 g and g01 g1, g01 g1 there.Clearly,connections
kjl,kjlrespect the regularized metric g, i.e., gij;k 0. Proceeding
in this manner, weobtain the nonstandard result
R 11 R 00 m2m,R 11 R 00 m2m.
1.28 Investigating
the weak limit of the angular components of the generalized
Ricci tensor using the
-
abbreviation r 0
sind
0
2dx and let (1) x S2m3,k, i.e.
r~r 2mk, r~2m, k 2 (2) x arbitrary function of class C3 classC3
with compact support we get:
w -0lim R 11 w - 0lim R
00 m | m2m,w -
0lim R 11 w - 0lim R
00 m | m2m.1.29
i.e., the spacetime is weakly Ricci-nonflat (the origin was
excluded from ourconsiderations).Furthermore,the Tolman formula
[3],[4] for the total energy of a static andasymptotically flat
spacetime with g the determinant of the four dimensional metricand
d3x the coordinate volume element, gives
ET Trr T T Ttt g d3x m, 1.30as it should be.
The paper is organized in the following way: in section II we
discuss theconceptual as well as the mathematical prerequisites. In
particular we comment ongeometrical matters (differentiable
structure, coordinate invariance) and recall thebasic facts of
nonlinear superdistributional geometry in the context of
algebrasGM, of supergeneralized functions. Moreover, we derive
sensible nonsmoothregularizations of the singular functions to be
used throughout the paper. Section IIIis devoted to these approach
to the problem. We present a new conceptuallysatisfactory method to
derive the main result. In these final section III weinvestigate
the horizon and describe its distributional curvature. Using
nonlinearsuperdistributional geometry and supergeneralized
functions it seems possible toshow that the horizon singularity is
not only a coordinate singularity without leavingSchwarzschild
coordinates.II. Generalized Colombeau CalculusII.1.Notation and
basic notions from standardColombeau theory
We use [1],[2],[7] as standard references for the foundations
and variousapplications of standard Colombeau theory. We briefly
recall the basic Colombeauconstruction. Throughout the paper will
denote an open subset of n. StanfardColombeau generalized functions
on are defined as equivalence classesu u of nets of smooth
functions u C (regularizations) subjected toasymptotic norm
conditions with respect to 0,1 for their derivatives oncompact
sets.
The basic idea of classical Colombeaus theory of nonlinear
generalizedfunctions [1],[2] is regularization by sequences (nets)
of smooth functions and theuse of asymptotic estimates in terms of
a regularization parameter . Let u0,1
-
with u CM for all ,where M a separable, smooth
orientableHausdorff manifold of dimension n.Definition 2.1.The
classical Colombeaus algebra of generalized functions on M
isdefined as the quotient:
GM EMM/NM 2.1of the space EMM of sequences of moderate growth
modulo the space NM ofnegligible sequences. More precisely the
notions of moderateness resp.negligibility are defined by the
following asymptotic estimates (where XMdenoting the space of
smooth vector fields on M):
EMM u| KK Mkk NN 1,,k1,,k XM
pKsup |L1Lk up| ON as 0 ,
2.2
NM u| KK M, kk 0qq N1,,k1,,k XM
pKsup |L1Lk up| Oq as 0 .
2.3
Remark 2.1. In the definition the Landau symbol a O appears,
having thefollowing meaning: CC 000 0,1 0a C.
Definition 2.3. Elements of GM are denoted by:u clu u NM.
2.4
Remark 2.2.With componentwise operations (, ) GM is a fine sheaf
ofdifferential algebras with respect to the Lie derivative defined
by Lu clLu.
The spaces of moderate resp. negligible sequences and hence the
algebraitself may be characterized locally, i.e., u GM iff u GV for
all chartsV,, where on the open set V Rn in the respective
estimates Liederivatives are replaced by partial derivatives.
The spaces of moderate resp. negligible sequences and hence the
algebraitself may be characterized locally, i.e., u GM iff u GV for
all chartsV,, where on the open set V Rn in the respective
estimates Liederivatives are replaced by partial derivatives.
Remark 2.4.Smooth functions f CM are embedded into GM simply by
theconstant embedding , i.e., f clf, hence CM is a faithful
subalgebraof GM.
Point Values of a Generalized Functions on M.Generalized
Numbers.
Within the classical distribution theory, distributions cannot
be characterized bytheir point values in any way similar to
classical functions. On the other hand, there
-
is a very natural and direct way of obtaining the point values
of the elements ofColombeaus algebra: points are simply inserted
into representatives. The objectsso obtained are sequences of
numbers, and as such are not the elements in thefield or . Instead,
they are the representatives of Colombeaus generalizednumbers. We
give the exact definition of these numbers.
Definition 2.5.Inserting p M into u GM yields a well defined
element of thering of constants (also called generalized numbers) K
(corresponding to K Rresp. C), defined as the set of moderate nets
of numbers (r K0,1 with|r| ON for some N) modulo negligible nets
(|r| Om for each m);componentwise insertion of points of M into
elements of GM yields well-definedgeneralized numbers,
i.e.,elements of the ring of constants:
K EcM/NcM 2.5(with K or K for K or K ), where
EcM r KI|nn |r | On as 0NcM r KI|mm |r | Om as 0
I 0,1.2.6
Generalized functions on M are characterized by their
generalized point values,i.e., by their values on points in Mc, the
space of equivalence classes of compactlysupported nets p M0,1 with
respect to the relationp p : dhp,p Om for all m, where dh denotes
the distance on Minduced by any Riemannian metric.
Definition 2.6. For u GM and x0 M, the point value of u at the
pointx0,ux0, is defined as the class of ux0 in K.
Definition 2.7.We say that an element r K is strictly nonzero if
there exists arepresentative r and a q such that |r| q for
sufficiently small. If r isstrictly nonzero, then it is also
invertible with the inverse 1/r. The converse istrue as well.
Treating the elements of Colombeau algebras as a generalization
of classicalfunctions, the question arises whether the definition
of point values can beextended in such a way that each element is
characterized by its values. Such anextension is indeed
possible.
Definition 2.8. Let be an open subset of n. On a set :
x I|pp 0|x | Op x I|pp 000 0 |x | p, for 0 0
2.7
we introduce an equivalence relation:
-
x y qq 0 0 |x y | q, for 0 0 2.8and denote by / the set of
generalized points. The set of points withcompact support is
c x clx |KK 00 0 x K for 0 0 2.9
Definition 2.9. A generalized function u GM is called associated
to zero,u 0 on M in L. Schwartzs sense if one (hence any)
representative uconverges to zero weakly,i.e.
w - lim0 u 0 2.10We shall often write:
uSch 0. 2.11
The GM-module of generalized sections in vector
bundlesespecially the spaceof generalized tensor fields Ts rMis
defined along the same lines usinganalogous asymptotic estimates
with respect to the norm induced by anyRiemannian metric on the
respective fibers. However, it is more convenient to usethe
following algebraic description of generalized tensor fields
GsrM GM Ts rM , 2.12where Ts rM denotes the space of smooth
tensor fields and the tensor product istaken over the module CM.
Hence generalized tensor fields are just given byclassical ones
with generalized coefficient functions. Many concepts of
classicaltensor analysis carry over to the generalized setting
[1]-[2], in particular Liederivatives with respect to both
classical and generalized vector fields, Liebrackets, exterior
algebra, etc. Moreover, generalized tensor fields may also beviewed
as GM-multilinear maps taking generalized vector and covector
fields togeneralized functions, i.e., as GM-modules we have
GsrM LMG10Mr,G01Ms;GM. 2.13In particular a generalized metric is
defined to be a symmetric, generalized0,2-tensor field gab gab
(with its index independent of and) whosedeterminant detgab is
invertible in GM. The latter condition is equivalent to
thefollowing notion called strictly nonzero on compact sets: for
any representativedetgab of detgab we have K M m infpK|detgab | m
for all small enough. This notion captures the intuitive idea of a
generalized metric to be asequence of classical metrics approaching
a singular limit in the following sense:gab is a generalized metric
iff (on every relatively compact open subset V of M)there exists a
representative gab of gab such that for fixed (smallenough)gab gab
(resp. gab |V) is a classical pseudo-Riemannian metricand detgab is
invertible in the algebra of generalized functions. A
generalizedmetric induces a GM-linear isomorphism from G01M to G10M
and the inverse
-
metric gab gab1 is a well defined element of G02M (i.e.,
independent of therepresentative gab ). Also the generalized
Levi-Civita connection as well as thegeneralized Riemann-, Ricci-
and Einstein tensor of a generalized metric aredefined simply by
the usual coordinate formulae on the level of representatives.
II.2. Generalized Colombeau Calculus.We briefly recall the basic
generalized Colombeau construction. Colombeau
supergeneralized functions on n, where dim n are defined
asequivalence classes u u of nets of smooth functions u C\,wheredim
n (regularizations) subjected to asymptotic norm conditions with
respectto 0,1 for their derivatives on compact sets.
The basic idea of generalized Colombeaus theory of
nonlinearsupergeneralized functions [1],[2] is regularization by
sequences (nets) of smoothfunctions and the use of asymptotic
estimates in terms of a regularizationparameter . Let u0,1 with u
CM for all ,where M a separable,smooth orientable Hausdorff
manifold of dimension n.
Definition 2.10.The generalized Colombeaus algebra G GM,
ofsupergeneralized functions on M, where M, dimM n, dim n , is
definedas the quotient:
GM, EMM,/NM, 2.14of the space EMM, of sequences of moderate
growth modulo the space NM,of negligible sequences. More precisely
the notions of moderateness resp.negligibility are defined by the
following asymptotic estimates (where XMdenoting the space of
smooth vector fields on M):
EMM, u| KK M\kk NN 1,,k1,,k XM
pKsup |L1Lk up| ON as 0 ,
2.15
NM, u| KK M\, kk 0qq N1,,k1,,k XM
pKsup |L1Lk up| Oq as 0 .
2.16
In the definition the Landau symbol a O appears, having the
followingmeaning: CC 000 0,1 0a C.
Definition 2.11. Elements of GM, are denoted by:u clu u NM,.
2.17
Remark 2.5.With componentwise operations (, ) GM, is a fine
sheaf ofdifferential algebras with respect to the Lie derivative
defined by Lu clLu.
-
The spaces of moderate resp. negligible sequences and hence the
algebraitself may be characterized locally, i.e., u GM, iff u GV
for allcharts V,, where on the open set V Rn in the respective
estimates Liederivatives are replaced by partial derivatives.
The spaces of moderate resp. negligible sequences and hence the
algebraitself may be characterized locally, i.e., u GM, iff u GV
for allcharts V,, where on the open set V Rn in the respective
estimates Liederivatives are replaced by partial derivatives.
Remark 2.6.Smooth functions f CM\ are embedded into GM, simply
bythe constant embedding , i.e., f clf, hence CM\ is a
faithfulsubalgebra of GM,.
Point Values of a Supergeneralized Functions on
M.Supergeneralized Numbers
Within the classical distribution theory, distributions cannot
be characterized bytheir point values in any way similar to
classical functions. On the other hand, thereis a very natural and
direct way of obtaining the point values of the elements
ofColombeaus algebra: points are simply inserted into
representatives. The objectsso obtained are sequences of numbers,
and as such are not the elements in thefield or . Instead, they are
the representatives of Colombeaus generalizednumbers. We give the
exact definition of these numbers.
Definition 2.12.Inserting p M into u GM, yields a well defined
element ofthe ring of constants (also called generalized numbers) K
(corresponding to K Rresp. C), defined as the set of moderate nets
of numbers (r K0,1 with|r| ON for some N) modulo negligible nets
(|r| Om for each m);componentwise insertion of points of M into
elements of GM, yields well-definedgeneralized numbers,
i.e.,elements of the ring of constants:
K EcM,/NcM, 2.18(with K or K for K or K ), where
EcM, r KI|nn |r | On as 0NcM, r KI|mm |r | Om as 0
I 0,1.2.19
Supergeneralized functions on M are characterized by their
generalized pointvalues, i.e., by their values on points in Mc, the
space of equivalence classes ofcompactly supported nets p M\0,1
with respect to the relationp p : dhp,p Om for all m, where dh
denotes the distance on M\induced by any Riemannian metric.
-
Definition 2.13. For u GM, and x0 M\, the point value of u at
the pointx0,ux0, is defined as the class of ux0 in K.
Definition 2.14.We say that an element r K is strictly nonzero
if there exists arepresentative r and a q such that |r| q for
sufficiently small. If r isstrictly nonzero, then it is also
invertible with the inverse 1/r. The converse istrue as well.
Treating the elements of Colombeau algebras as a generalization
of classicalfunctions, the question arises whether the definition
of point values can beextended in such a way that each element is
characterized by its values. Such anextension is indeed
possible.
Definition 2.15. Let be an open subset of n\. On a set :
x \I|pp 0|x | Op x \I|pp 000 0 |x | p, for 0 0
2.20
we introduce an equivalence relation:x y qq 0 0 |x y | q, for 0
0 2.21
and denote by / the set of supergeneralized points. The set of
pointswith compact support is
,c x clx |KK \00 0 x K for 0 0 2.22
Definition 2.16. A supergeneralized function u GM, is called
associated tozero, u 0 on M in L. Schwartzs sense if one (hence
any) representativeu converges to zero weakly,i.e.
w - lim0 u 0 2.23We shall often write:
uSch 0. 2.24
Definition 2.17.The GM,-module of supergeneralized sections in
vectorbundles especially the space of generalized tensor fields Ts
rM\is definedalong the same lines using analogous asymptotic
estimates with respect to thenorm induced by any Riemannian metric
on the respective fibers. However, it ismore convenient to use the
following algebraic description of generalized tensorfields
GsrM, GM, Ts rM\ , 2.25where Ts rM\ denotes the space of smooth
tensor fields and the tensor product istaken over the module CM\.
Hence generalized tensor fields are just given byclassical ones
with generalized coefficient functions. Many concepts of
classicaltensor analysis carry over to the generalized setting [],
in particular Lie derivatives
-
with respect to both classical and generalized vector fields,
Lie brackets, exterioralgebra, etc. Moreover, generalized tensor
fields may also be viewed asGM,-multilinear maps taking generalized
vector and covector fields togeneralized functions, i.e., as
GM,-modules we have
GsrM, LMG10M,r,G01M,s;GM,. 2.26In particular a supergeneralized
metric is defined to be a symmetric,supergeneralized 0,2-tensor
field gab gab (with its index independent of and) whose determinant
detgab is invertible in GM\. The latter condition isequivalent to
the following notion called strictly nonzero on compact sets: for
anyrepresentative detgab of detgab we haveK M\ m infpK|detgab | m
for all small enough. This notioncaptures the intuitive idea of a
generalized metric to be a sequence of classicalmetrics approaching
a singular limit in the following sense: gab is a generalizedmetric
iff (on every relatively compact open subset V of M) there exists
arepresentative gab of gab such that for fixed (small enough)gab
gab (resp. gab |V) is a classical pseudo-Riemannian metric and
detgab is invertible inthe algebra of generalized functions. A
generalized metric induces a GM,-linearisomorphism from G01M, to
G10M, and the inverse metric gab gab1 is awell defined element of
G02M, (i.e., independent of the representative gab ).Also the
supergeneralized Levi-Civita connection as well as the
supergeneralizedRiemann-, Ricci- and Einstein tensor of a
supergeneralized metric are definedsimply by the usual coordinate
formulae on the level of representatives.
II.3.Superdistributional general relativityWe briefly summarize
the basics of superdistributional general relativity, as a
preliminary to latter discussionIn the classical theory of
gravitation one is led to consider the Einstein field
equations which are, in general,quasilinear partial differential
equations involvingsecond order derivatives for the metric tensor.
Hence, continuity of the firstfundamental form is expected and at
most, discontinuities in the secondfundamental form, the coordinate
independent statements appropriate to consider3-surfaces of
discontinuity in the spacetime manifolfd of General Relativity.
In standard general relativity, the space-time is assumed to be
afour-dimensional differentiable manifold M endowed with the
Lorentzian metricds2 gdxdx , 0,1,2,3. At each point p of space-time
M, the metric can bediagonalized as dsp2 dXpdXp with 1,1,1,1, by
choosing thecoordinate system X; 0,1,2,3 appropriately.
In superdistributional general relativity the space-time is
assumed to be afour-dimensional differentiable manifold M\, where
dimM 4,dim 3
-
endowed with the Lorentzian supergeneralized metricds2 gdxdx ;,
0,1,2,3. 2.27
At each point p M\, the metric can be diagonalized asdsp2 dXpdXp
with 1,1,1,1, 2.28
by choosing the generalized coordinate system X; 0,1,2,3
appropriately.The classical smooth curvature tensor is given by
R
2.29
with being the smooth Christoffel symbol.The supergeneralized
nonsmoothcurvature tensor is given by
R
2.30
with being the supergeneralized Christoffel symbol.The
fundamentalclassical action integral I is
I 1c LG LMd4x, 2.31where LM is the Lagrangian density of a
gravitational source and LG is thegravitational Lagrangian density
given by
LG 12 G . 2.32Here is the Einstein gravitational constant 8G/c4
and G is defined by
G g g
2.33
with g detg. There exists the relationg R G D , 2.34
withD g g g
. 2.35
Thus the supergeneralized fundamental action integral I isI 1c
LG LMd4x , 2.36
where LM is the supergeneralized Lagrangian density of a
gravitationalsource and LG is the supergeneralized gravitational
Lagrangian density given
-
byLG 12 G . 2.37
Here is the Einstein gravitational constant 8G/c4 and G is
defined byG g g
2.38with g detg . There exists the relation
g R G D , 2.39with
D g g g . 2.40
Also, we have defined the classical scalar curvature byR R
2.41
with the smooth Ricci tensorR R . 2.42
From the action I, the classical Einstein equationG R 12 R T ,
2.43
follows, where T is defined byT T
g 2.44with
T 2g LMg 2.45being the energy-momentum density of the classical
gravity source. Thus we havedefined the supergeneralized scalar
curvature by
R R 2.46with the supergeneralized Ricci tensor
R R . 2.47From the action I, the generalized Einstein
equation
G R 12 R T , 2.48follows, where T is defined by
T T g
2.49
with
-
T 2g
LMg
2.50being the supergeneralized energy-momentum density of the
supergeneralizedgravity source.The classical energy-momentum
pseudo-tensor density t of thegravitational field is defined by
t LG LGg, g, 2.51with g, g/x.The supergeneralized
energy-momentum pseudo-tensordensity t of the gravitational field
is defined by
t LG
LGg,
g, 2.52with g, g/x.
III.Distributional Schwarzschild Geometry fromnonsmooth
regularization via Horizon
In this last section we leave the neighborhood of the
singularity at the origin andturn to the singularity at the
horizon. The question we are aiming at is the following:using
distributional geometry (thus without leaving Schwarzschild
coordinates), is itpossible to show that the horizon singularity of
the Schwarzschild metric is notmerely only a coordinate
singularity. In order to investigate this issue we calculatethe
distributional curvature at horizon (in Schwarzschild coordinates).
In the usualSchwarzschild coordinates t, r 0,, the metric takes the
form
ds2 hrdt2 hr1dr2 r2d2,hr 1 2mr .
3.1
Following the above discussion we consider the singular metric
coefficient hr asan element of D3 and embed it into G3 by
replacementr 2m r 2m2 2 . Note that, accordingly, we have fixed the
differentiablestructure of the manifold: the Cartesian coordinates
associated with the sphericalSchwarzschild coordinates in (3.1) are
extended through the origin. We have above(below) horizon where r
2m (r 2m):
-
hr 1 2mr r 2mr , r 2m
0, r 2m hr
r 2m2 2
r G3,B2m,R ,
where B2m,R x 3|2m x R.h1r 1 2mr
1 rr 2m , r 2m
, r 2m h1r
rr 2m2 2
G3,B2m,R .
r 2mr , r 2m0, r 2m
hr
2m r2 2
r G3,B0,2m ,
where B0,2m x 3|0 x 2m rr 2m , r 2m
, r 2m h1r
rr 2m2 2
G3,B0,2m
3.2
Inserting (3.2) into (3.1) we obtain a generalized object
modeling the singularSchwarzschild metric above (below) gorizon,
i.e.,
ds2 hrdt2 hr1dr2 r2d2 ,ds2 hrdt2 hr1dr2 r2d2
3.3
The generalized Ricci tensor above horizon R may now be
calculatedcomponentwise using the classical formulae
R 00 R 11 12 h 2r hR 22 R 33
hr
1 hr2
.3.4
-
From (3.2) we obtain
-
hr r 2m2 2
r .
hr r 2mr r 2m2 2 1/2
r 2m2 2 1/2r2
,
rh 1 h
r r 2mr r 2m2 2 1/2
r 2m2 2 1/2r2
1 r 2m2 2
r
r 2mr 2m2 2 1/2
r 2m2 2 1/2r 1
r 2m2 2r
r 2mr 2m2 2 1/2 1.
hr r 2mr r 2m2 2 1/2
r 2m2 2 1/2r2
1r r 2m2 2 1/2
r 2m2r r 2m2 2 3/2
r 2mr2 r 2m2 2 1/2
r 2mr2 r 2m2 2 1/2
2 r 2m2 2 1/2r3
.
r2h 2rh
r2 1r r 2m2 2 1/2
r 2m2r r 2m2 2 3/2
r 2mr2 r 2m2 2 1/2
r 2mr2 r 2m2 2 1/2
2 r 2m2 2 1/2r3
2r r 2mr r 2m2 2 1/2
r 2m2 2 1/2r2
rr 2m2 2 1/2
rr 2m2r 2m2 2 3/2
r 2mr 2m2 2 1/2
r 2mr 2m2 2 1/2
2 r 2m2 2 1/2r
2r 2mr 2m2 2 1/2
2 r 2m2 2 1/2r
rr 2m2 2 1/2
rr 2m2r 2m2 2 3/2 .
3.5
-
Investigating the weak limit of the angular components of the
Ricci tensor (usingthe abbreviation r
0
sind
0
2dx and let (1)
x S2m3,k, i.e.r~r 2mk, k 2 (2) x arbitrary function of classC3
with compact support K,such that K B2m,R x 3|2m x Rwe get:
K
R 22 xd3x K R 33 xd3x 2m
R
rh 1 hrdr
2m
R
r 2mr 2m2 2 1/2 rdr 2m
R
rdr.
3.6
By replacement r 2m u, from (3.6) we obtainK
R 22 xd3x K R 33 xd3x
0
R2muu 2mduu2 21/2 0
R2mu 2mdu.
3.7
By replacement u , from (3.7) we obtainI3
KR 33 xd3x I2 K R 22 xd3x
0
R2m 2md2 11/2 0
R2m
2md .3.8
which is calculated to give
I3 I2 2m0! 0R2m
2 11/2 1 d
21! 0
R2m 2 11/2 1
1d
2m R 2m2 1 1 R 2m
21 0
R2m 2 11/2 1
1d,
3.9
where we have expressed the function 2m as
-
2m l0n1 l2ml! l 1n! nn ,
2m , 1 0 , n 13.10
with l dl/d l. Equation (3.9) gives
0lim I3
0lim I2
0lim 2m R 2m
2 1 1 R 2m
0lim 21
0
R2m 2 11/2 1
1d 0
3.11
Thus in S2m BR2m,k S2m 3,k D3, whereB2m,R x 3|2m x R from
Eq.(3.11) we obtein
w 0lim R 33 0lim I3
0,w
0lim R 22 0lim I2
0. 3.12
For R 11 , R 00 we get:2
KR 11 xd3x 2 K R 00 xd3x 2m
R
r2h 2rhrdr
2m
R
rr 2m2 2 1/2
rr 2m2r 2m2 2 3/2 rdr.
3.13
By replacement r 2m u, from (3.13) we obtainI1 2
KR 11 xd3x I2 2 K R 00 xd3x
2m
R
r2h 2rhrdr
0
R2m u 2mu2 21/2
u2u 2mu2 23/2 u 2mdu.
3.14
By replacement u , from (3.14) we obtain
-
2 K
R 11 xd3x 2 K R 00 xd3x
2m
R
r2h 2rhrdr
0
R2m
2m22 21/2 22 2m22 23/2 2md
0
R2m 2 2md22 21/2 2m 0
R2m 2md22 21/2
0
R2m 43 2md22 23/2 2m 0
R2m 32 2md22 23/2
0
R2m 2md2 11/2 0
R2m 3 2md2 13/2
2m 0
R2m 2md2 11/2 0
R2m 2 2md2 13/2 .
3.15
which is calculated to give
I0 I1 2m 2m0! 0R2m
12 11/2 2
2 13/2 d
1! 0
R2m
1 12 11/2 2
2 13/2 d
2m0! 2m
R2m
12 11/2 2
2 13/2 d
21! 0
R2m
1 12 11/2 2
2 13/2 d.
3.16
where we have expressed the function 2m as 2m l0n1
l2ml! l 1n! nn ,
2m , 1 0 , n 13.17
with l dl/d l.Equation (3.17) gives
-
w -0lim I0 w -
0lim I1
2m2m0lim
0
R2m
12 11/2 2
2 13/2 d
2m2mslim 0
s 2d2 13/2 0
s d2 11/2
2m2m.
3.18
where use is made of the relation
slim
0
s 2d2 13/2 0
sd
u2 11/2 1 3.19
Thus in S2m B2m,R,k S2m 3,k we obtainw -
0lim R 11 w - 0lim R
00 m2m. 3.20The supergeneralized Ricci tensor below horizon R R
may now becalculated componentwise using the classical formulae
R 00 R 11 12 h 2r hR 22 R 33
hr
1 hr2
.3.21
From (3.2) we obtain
hr r 2mr hr 2m r2 2
r hr, r 2m.
hr hr r 2mr r 2m2 2 1/2
r 2m2 2 1/2r2
,
rh 1 h rh 1 h r 2m
r 2m2 2 1/2 1.
hr hr
r 2mr2 r 2m2 2 1/2
2 r 2m2 2 1/2r3
.
r2h 2rh r2h 2rh r
r 2m2 2 1/2 rr 2m2
r 2m2 2 3/2 .
3.22
Investigating the weak limit of the angular components of the
Ricci tensor (using
-
the abbreviation r 0
sind
0
2dx and let (1)
x S2m3,k, i.e.r~r 2mk, k 2 (2) x arbitrary function of classC3
with compact support K,such that K B0,2m x 3|0 x 2mwe get:
K
R 22 xd3x K R 33 xd3x 0
2mrh 1 hrdr
0
2mr 2m
r 2m2 2 1/2 rdr 02mrdr.
3.23
By replacement r 2m u, from (3.23) we obtainK
R 22 xd3x K R 33 xd3x 2m
0uu 2mduu2 21/2 2m
0u 2mdu.
3.24
By replacement u , from (3.23) we obtainI3
KR 33 xd3x I2 K R 22 xd3x
2m
0 2md2 11/2 2m
0 2md .
3.25
which is calculated to give
I3 I2 2m0! 2m
0 2 11/2 1 d
21! 2m
0 2 11/2 1
1d
2m 1 2m2 1 2m
21 2m
0 2 11/2 1
1d,
3.26
where we have expressed the function 2m as
-
2m l0n1 l2ml! l 1n! nn ,
2m , 1 0 , n 13.27
with l dl/drl. Equation (3.27) gives
0lim I3
0lim I2
0lim 2m 1 2m
2 1 2m
0lim 22
2m
0 2 11/2 1
1d 0.
3.28
Thus in S2m BR2m,k S2m 3,k, where B0,2m x 3|0 x 2m fromEq.(3.28)
we obtein
w 0lim R 33 0lim I3
0.w
0lim R 22 0lim I2
0. 3.29
For R 11 , R 00 we get:2
KR 11 xd3x 2 K R 00 xd3x 0
2mr2h 2rhrdr
0
2mr
r 2m2 2 1/2 rr 2m2
r 2m2 2 3/2 rdr.
3.30
By replacement r 2m u, from (3.30) we obtainI1 2 R 11 xd3x I2 2
R 00 xd3x
0
2mr2h 2rhrdr
2m
0u 2m
u2 21/2 u2u 2mu2 23/2 u 2mdu.
3.31
By replacement u , from (3.31) we obtain
-
2 K
R 11 xd3x 2 K R 00 xd3x
2m
0r2h 2rhrdr
2m
0 2m22 21/2
22 2m22 23/2 2md
2m
0 2 2md22 21/2 2m 2m
0 2md22 21/2
2m
0 43 2md22 23/2 2m 2m
0 32 2md22 23/2
2m
0 2md2 11/2 2m
0 3 2md2 13/2
2m 2m
0 2md2 11/2 2m
0 2 2md2 13/2 .
3.32
which is calculated to give
I0 I1 2m 2m0! l 2m
01
2 11/2 2
2 13/2 d
1! 0
2m
1 12 11/2 2
2 13/2 d O2.
3.33
where we have expressed the function 2m as 2m l0n1
l2ml! l 1n! nn ,
2m , 1 0 , n 13.34
with l dl/d l.Equation (3.34) gives
-
0lim I0
0lim I1
2m0lim 2m0!
2m
01
2 11/2 2
2 13/2 d
2m2ms0lim
s
0 d2 11/2 s
0 2d2 13/2
2m2m.
3.35
where use is made of the relation
slim
s
0d
u2 11/2 s
0 2d2 13/2 1. 3.36
Thus in S2m B0,2m,k S2m 3,k we obtainw -
0lim R 11 w - 0lim R
00 m2m. 3.37Using Egs. (3.12),(3.20),(3.29),(3.37) we obtain
Trr T T Ttt Trr T T Ttt g d3x 0 3.38Thus the Tolman formula
[3],[4] for the total energy of a static and asymptoticallyflat
spacetime with g the determinant of the four dimensional metric and
d3x thecoordinate volume element, gives
ET Trr T T Ttt g d3x m, 3.39as it should be.
IV.Conclusions and remarks.
We have shown that a succesfull approach for dealing with
curvature tensorvalued distribution is to first impose admisible
the nondegeneracy conditions on themetric tensor, and then take its
derivatives in the sense of classical distributions inspace S2m
3,k,k 2.
The distributional meaning is then equivalent to the junction
conditionformalism. Afterwards, through appropiate limiting
procedures, it is then possible toobtain well behaved
distributional tensors with support on submanifolds of d 3, aswe
have shown for the energy-momentum tensors associated with the
-
Schwarzschild spacetimes. The above procedure provides us with
what isexpected on physical grounds. However, it should be
mentioned that the use ofnew supergeneralized functions
(supergeneralized Colombeau algebras GR3,).in order to obtain
superdistributional curvatures, may renders a more rigoroussetting
for discussing situations like the ones considered in this
paper.
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