Distributional Schwarzschild Geometry fromDistribut

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

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].

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.

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


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 -


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


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


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 -


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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