Distributional Schwarzschild Geometry from Non Smooth Regularization Via Horizon

In this paper we leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon.Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.However the Tolman formula for the total energy ET of a static and asymptotically flat spacetime,gives ET = m, as it should be.


The Breakdown of Canonical Formalism of Riemann Geometry for the Singular Solutions of the Einstein Field Equations
Einstein field equations were originally derived by Einstein in 1915 in respect with canonical formalism of Riemann geometry,i.e. by using the classical sufficiently smooth metric tensor, smooth Riemann curvature tensor, smooth Ricci tensor,smooth scalar curvature, etc.. However have soon been found singular solutions of the Einstein field equations with singular metric tensor and singular Riemann curvature tensor.
These singular solutions were formally accepted beyond rigorous canonical formalism of Riemannian geometry.
Remark 1.1.Note that if some components of the Riemann curvature tensor R i klm (x) become infinite at pointx 0 one obtains the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood Ω of the pointx 0 ∈ Ω,i.e. in such neighborhood Ω Riemann curvature tensor R i klm (x) will be changed by formula (1.7) see remark 1.2.
Remark 1.2.Let Γ be infinitesimal closed contour and let ΣΓ be the corresponding surface spanning by Γ, see Pic.1. We assume now that: (i) christoffel symbol Γ i kl (x) become infinite at singular point x 0 by formulae and(ii) 0 ∈ ΣΓ.Let us derive now to similarly canonical calculation [3]- [4] the general formula for the regularized change ∆A k in a vector Ai (x) after parallel displacement around infinitesimal closed contour Γ. This regularized change ∆A k can clearly be written in the form , δ ≥ 1 and where the integral is taken over the given contour Γ. Substituting in place of δA k the canonical expression δA k = Γ i kl (x) A k dx l (see [4],Eq.(85.5)) we obtain where Pic.1.Infinitesimal closed contour Γ and corresponding singular surface Σ Γ ∋x 0 spanning by Γ Pic.2.Infinitesimal closed contour Γ with a singularity at pointx 0 on Horizon and correspondingsingular surface Σ Γ ∋ x 0 Now applying Stokes' theorem (see [4],Eq. (6.19)) to the integral (1.3) and considering that the area enclosed by the contour has the infinitesimal value ∆f lm , we get ∆f lm 2 . (1.5) Substituting the values of the derivatives (1.4) into Eq. (1.5), we get finally: where R i klm , is a tensor of the fourth rank . (1.7) Here R k im i is the classical Riemann curvature tensor.That R i klm is a tensor is clear from the fact that in (1.6) the left side is a vector-the difference ∆A k between the values of vectors at one and the same point.

Remark 1.4.
Note that in physical literature the spacetime singularity usually is defined as location where the quantities that are used to measure the gravitational field 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. whereT ν µ (x) = 0 represent the energy-momentum densities of the gravity source. This for example is the case for the well-known Schwarzschild solution, which is given by, in the Schwarzschild coordinates (x 0 , r, θ, ϕ), , h (r) = 1 − rs r , (1.12) where, rs is the Schwarzschild radius rs = 2GM/c 2 with G, M and c being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. The metric (1.12) describe the gravitational field produced by a point-like particle located at r = 0.
Remark 1.6. Note that when we say, on the basis of the canonical expression of the curvature squareR formally obtained from the metric (1.12), that r = 0 is a singularity of the Schwarzschild spacetime, 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 ̸ = 0 where h (r) = −1 + rs/r.Using Eq.(1.14) one formally obtain boundary conditions (1.15) However as pointed out above the canonical expression of the Einstein tensor in a sufficiently small neighborhood Ω of the point r = 0 and must be replaced by the generalized Einstein tensor G km (1.10). By simple calculation easy to see that (1. 16) and therefore the boundary conditions (1.15) are completely wrong. But other hand as pointed out by many authors [5]- [17] that the canonical representation of the Einstein tensor, valid only in a weak (distributional) sense,i.e. [12]: and therefore again we obtain .Thus canonical definition of the Einstein tensor is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin r = 0.

The Distributional Schwarzschild Geometry
General relativity as a physical theory is governed by particular physical equations; the focus of interest is the breakdown of physics which need not coincide with the breakdown of geometry. It has been suggested to describe singularity at the origin as internal point of the Schwarzschild spacetime, where the Einstein field equations are satisfied in a weak (distributional) sense [5]- [22].

The smooth regularization of the singularity at the origin
The two singular functions we will work with throughout this paper (namely the singular components of the Schwarzschild metric) are 1 r and 1 r − rs , rs ≥ 0.Since . By convolution with a mollifier (ρ) (x) (adapted to the symmetry of the spacetime, i.e. chosen radially symmetric) we embed it into the Colombeau algebra G ( Inserting (1.18) into (1.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]: obviously is a classical Riemannian object and there no exists an the breakdown of canonical formalism of Riemannian geometry for these metrics, even at origin r = 0. It has been suggested 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]: (1.20) Hence, the distributional Ricci tensor and the distributional curvature scalar (Rε (r)) ε are of δ-type, i.e. (Rε (r)) ε = πm δ (r) r 2 .

The nonsmooth regularization of the singularity at the origin
The nonsmooth regularization of the Schwarzschild singularity at the origin r = 0 is considered by N. R. Pantoja and H. Rago in paper [12]. Pantoja non smooth regularization regularization of the Schwarzschild singularity are Here Θ (u) is the Heaviside function and the limit ε → 0 is understood in a distributional sense.Equation (1.19) with hε as given in (1.21) can be considered as an regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.21), the calculation of the distributional Einstein tensor proceeds in a straighforward manner. By simple calculation it gives [12]: (1. 23) which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function Θ(r − ε). Transforming now the formulae for (G a b (r, ε))ε into Cartesian coordinates associated with the spherical ones, i.e., r, φ, θ} ↔ {x i }, we obtain for the generalized Einstein tensor the expected result given by Eq.(1.17) see Remark 1.9.

The smooth regularization via Horizon
The smooth regularization via Horizon is considered by J.M.Heinzle and R.Steinbauer in paper [22]. Note ). An canonical regularization is the principal value vp Inserting now (1.25) into (1.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]: (1.

The nonsmooth regularization via Gorizon
In 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 it possible to show that the horizon singularity of the Schwarzschild metric is not merely a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at the horizon in Schwarzschild coordinates.
The main focus of this work is a (nonlinear) superdistributional description of the Schwarzschild spacetime. Although the nature of the Schwarzschild singularity is much "worse" than the quasiregular conical singularity, there are several distributional treatments in the literature [8]- [29], mainly motivated by the following considerations: the physical interpretation of the Schwarzschild metric is clear as long as we consider it merely as an exterior (vacuum) solution of an extended (sufficiently large) massive spherically symmetric body. Together with the interior solution it describes the entire spacetime. The concept of point particles-well understood in the context of linear field theories-suggests a mathematical idealization of the underlying physics: one would like to view the Schwarzschild solution 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 the Schwarzschild geometry, i.e., T 0 0 = 8πmδ (3) (⃗ x), in a conceptually satisfactory way. Additionally, we set up a unified language to comment on the respective merits of some of the approaches taken so far. In particular, we discuss questions of differentiable structure as well as smoothness and degeneracy problems of the regularized metrics, and present possible refinements and workarounds.These aims are accomplished using the framework of nonlinear supergeneralized functions (supergeneralized Colombeau algebras G(R 3 , Σ)).Examining the Schwarzschild metric (1.12) in a neighborhood of the horizon, we see that, whereas h(r) is smooth, h −1 (r) is not even L 1 loc (note that the origin is now always excluded from our considerations; the space we are working on is R 3 \{0}). Thus, regularizing the Schwarzschild metric amounts to embedding h −1 into G(R 3 , Σ) (as done in (3.2)).Obviously, (3.1) is degenerate at r = 2m, because h(r) is zero at the horizon. However, this does not come as a surprise. Both h(r) and h −1 (r) are positive outside of the black hole and negative in the interior. As a consequence any (smooth) ) must converge to O0 (2m) as the regularization parameter ϵ goes to zero.Due to the degeneracy of (1.12), the Levi-Cività connection is not available. Consider, therefore, the following connections Γ +l (1.28) Investigating the weak limit of the angular components of the generalized Ricci tensor using the abbreviatioñ i.e., the Schwarzschild spacetime is weakly Ricci-nonflat (the origin was excluded from our considerations). Furthermore,the Tolman formula [3], [4] for the total energy of a static and asymptotically flat spacetime with g the determinant of the four dimensional metric and d 3 x the coordinate volume element, gives as it should be.
The paper is organized in the following way: in section2-6 we discuss the conceptual as well as the mathematical prerequisites. In particular we comment on geometrical matters (differentiable structure, coordinate invariance) and recall the basic facts of nonlinear superdistributional geometry in the context of algebras G(M, Σ) of supergeneralized functions. Moreover, we derive sensible nonsmooth regularizations of the singular functions to be used throughout the paper. Section 7 is devoted to this approach to the problem. We present a new conceptually satisfactory method to derive the main result. In this final section 7 we investigate the horizon and describe its distributional curvature. Using nonlinear superdistributional geometry and supergeneralized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.

Notation and Basic Notions from Standard Colombeau Theory
We use [1], [2], [7] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper Ω will denote an open subset of R n . Stanfard Colombeau generalized functions on Ω are defined as equivalence classes u = [(uε)ε] of nets of smooth functions uε ∈ C ∞ (Ω) (regularizations) subjected to asymptotic norm conditions with respect to ε ∈ (0, 1] for their derivatives on compact sets. The basic idea of classical Colombeau's theory of nonlinear generalized functions [1], [2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter ε.
In the definition the Landau symbol aε = O (ψ (ε)) appears, having the following meaning:

Definition 2.3. Elementsof calG(M)aredenotedby
: The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., u ∈ G(M ) iff u • ψα ∈ G(ψα(Vα)) for all charts (Vα, ψα), where on the open set ψα(Vα) ⊂ R n in the respective estimates Lie derivatives are replaced by partial derivatives.

Point Values of a Generalized Functions on M . generalized Numbers
Within the classical distribution theory, distributions cannot be characterized by their 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 of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field R or C. Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".
where d h denotes the distance on M induced by any Riemannian metric. Definition 2.6. For u ∈ G(M ) and x0 ∈ M, the point value of u at the point x0, u(x0),is defined as the class of (uε(x0)) ε in K. Definition 2.7.We say that an element r ∈ K is strictly nonzero if there exists a representative (rε) ε and a q ∈ N such that |rε| ε q for ε sufficiently small. If r is strictly nonzero, then it is also invertible with the inverse [(1/rε)ε]. The converse is true as well.
Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.

Definition 2.8.
Let Ω be an open subset of R n . On a setΩ : we introduce an equivalence relation: anddenoteby Ω =Ω/ ∼ the set of generalized points. The set of points with compact support is Definition 2.9. A generalized f unction u ∈ G(M ) is called associated to zero, u ≈ 0 on Ω ⊆ M in L.Schwartz sense if one (hence any) representative (uϵ)ϵ converges to zero weakly,i.e.
w -limϵ→0 uϵ = 0 (2.10) We shall often write: The G(M )-module of generalized sections in vector bundles-especially the space of generalized tensor fields T r s (M )-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

12)
whereT r s (M ) denotes the space of smooth tensor fields and the tensor product is taken over the module C ∞ (M ). Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [1]- [2], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as G(M )multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as G(M )-modules we have In particular a generalized metric is def ined to be a symmetric, generalized (0,2)−tensor f ield g ab = [((gϵ) ab )ϵ] (with its index independent of ε and) whose determinant det(g ab ) is invertible in G(M ). The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative det((gϵ) ab )ϵ of det(g ab ) we have ∀K ⊂ M ∃m ∈ N [infp∈K | det(g ab (ϵ))| ≥ ϵ m ] for all ϵ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: g ab is a generalized metric iff (on every relatively compact open subset V of M ) there exists a representative ((gϵ) ab )ϵ of g ab such that for fixed ϵ (small enough)(gϵ) ab = g ab (ϵ) (resp. (gϵ) ab |V ) is a classical pseudo-Riemannian metric and det(g ab ) is invertible in the algebra of generalized functions. A generalized metric induces a G(M )-linear isomorphism from G 1 0 (M ) to G 0 1 (M ) and the inverse metric g ab [(g −1 ab (ϵ))ϵ] is a well defined element of G 2 0 (M ) (i.e., independent of the representative ((gϵ) ab )ϵ). Also the generalized Levi-Civita connection as well as the generalized Riemann-, Ricci-and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives.

16)
whereL w ξ k denoting the weak Lie derivative in L.Schwartz sense.In the definition the Landau symbol aε = O (ψ (ε)) appears, having the following meaning:

Point Values of a Supergeneralized Functions on M . Supergeneralized Numbers
Within the classical distribution theory, distributions cannot be characterized by their 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 of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field R or C. Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".  Definition 2.14.We say that an element r ∈ K is strictly nonzero if there exists a representative (rε) ε and a q ∈ N such that |rε| ε q for ε sufficiently small. If r is strictly nonzero, then it is also invertible with the inverse [(1/rε)ε]. The converse is true as well.
Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.

Definition 2.15.
Let Ω be an open subset of R n \Σ. On a setΩΣ : we introduce an equivalence relation: anddenoteby ΩΣ =ΩΣ/ ∼ the set of supergeneralized points. The set of points with compact support is where T r s (M \Σ) denotes the space of smooth tensor fields and the tensor product is taken over the module C ∞ (M \Σ). Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [21][22][23], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as G(M, Σ)multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as G(M, Σ)-modules we have In particular a supergeneralized metric is def ined to be a symmetric, supergeneralized (0,2)− tensorf ieldg ab = [((gε) ab )ε] (with its index independent of ε and) whose determinant det(g ab ) is invertible in G(M \Σ). The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative det((gε) ab )ε of det(g ab ) we have ∀K ⊂ M \Σ ∃m ∈ N [infp∈K | det(g ab (ε))| ≥ ε q ] for all ε small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: g ab is a generalized metric iff (on every relatively compact open subset V of M ) there exists a representative ((gε) ab )ε of g ab such that for fixed ε (small enough)(gε) ab = g ab (ε) (resp. (gε) ab |V ) is a classical pseudo-Riemannian metric and det(g ab ) is invertible in the algebra of generalized functions. A generalized metric induces a G(M, Σ)-linear isomorphism from G 1 0 (M, Σ) to G 0 1 (M, Σ) and the inverse metric g ab [(g −1 ab (ε))ε] is a well defined element of G 2 0 (M, Σ) (i.e., independent of the representative ((gε) ab )ε). Also the supergeneralized Levi-Civita connection as well as the supergeneralized Riemann, Ricci and Einstein tensor of a supergeneralized metric are defined simply by the usual coordinate formulae on the level of representatives.

Superdistributional General Relativity
We briefly summarize the basics of superdistributional general relativity, as a preliminary to latter discussion.In the classical theory of gravitation one is led to consider the Einstein field equations which are, in general, quasilinear partial differential equations involving second order derivatives for the metric tensor. Hence, continuity of the first fundamental form is expected and at most, discontinuities in the second fundamental form, the coordinate independent statements appropriate to consider 3-surfaces of discontinuity in the spacetime manifolfd of General Relativity.
The classical smooth curvature tensor is given by being the smooth Christoffel symbol.The supergeneralized nonsmooth curvature tensor is given by ϵ being the supergeneralized Christoffel symbol.The fundamental classical action integral I is where LM is the Lagrangian density of a gravitational source andLG is the gravitational Lagrangian density given bȳ Here κ is the Einstein gravitational constant κ = 8πG/c 4 and G is defined by with g = det(gµν ). There exists the relation Thus the supergeneralized fundamental action integral (Iϵ) ϵ is where (LM (ϵ)) ϵ is the supergeneralized Lagrangian density of a gravitational source and (L G (ϵ) ) ϵ is the supergeneralized gravitational Lagrangian density given by Hereκ is the Einstein gravitational constant κ = 8πG/c 4 and (Gϵ) ϵ is defined by .
(2.40) Also, we have defined the classical scalar curvature by (2.42) From the action I, the classical Einstein equation follows, where Tµ ν is defined by being the energy-momentum density of the classical gravity source. Thus we have defined the supergeneralized scalar curvature by (2.47) From the action (Iϵ) ϵ , the generalized Einstein equation follows, where (Tµ ν (ϵ)) ϵ is defined by being the supergeneralized energy-momentum density of the supergeneralized gravity source.The classical energy-momentum pseudo-tensor density t ν µ of the gravitational field is defined by with gστ,ν = ∂gστ /∂x ν .The supergeneralized energy-momentum pseudo-tensor density t ν µ of the gravitational field is defined by with (gστ,ν (ϵ)) ϵ = (∂gστ (ϵ) /∂x ν ) ϵ .

Distributional Schwarzschild Geometry from Nonsmooth Regularization via Horizon
In this last section 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 it possible to show that the horizon singularity of the Schwarzschild metric is not merely only a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at horizon (in Schwarzschild coordinates). In the usual Schwarzschild coordinates (t, r > 0, θ, ϕ) the metric takes the form the Following the above discussion we consider the singular metric coefficient h(r) as an element of D ′ (R 3 ) and embed it into (G(R 3 )) by replacement Note that, accordingly, we have fixed the differentiable structure of the manifold: the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1) are extended through the origin. We have above r 2m (below (r 2m)) horizon Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) gorizon, i.e., The generalized Ricci tensor above horizon [ R + ] β α may now be calculated componentwise using the classical formulae with compact support such that: Then for any function Φ(⃗ x) ∈ S2m(R 3 ) we get: (r)dr. (3.6) By replacement r − 2m = u, from (3.6) we obtain By replacement u = ϵη, from (3.7) we obtain the expression          From Eq.(3.8) we obtain where we have expressed the function Φ(ϵη + 2m) as (3.14) By replacement u = ϵη, from (3.14) we obtain (3.15) From Eq.(3.15) we obtain ηdη, (3.16) where we have expressed the function Φ(ϵη + 2m) as where use is made of the relation The supergeneralized Ricci tensor below horizon α may now be calculated componentwise using the classical formulae (3.21) (3.22) Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviatioñ (3.23) By replacement r − 2m = u, from Eq.(3.23) we obtain (u + 2m)du. (3.24) By replacement u = ϵη, from (3.23) we obtain which is calculated to give where we have expressed the function Φ(ϵη + 2m) as (3.30) By replacement r − 2m = u, from (3.30) we obtain (u 2 + ϵ 2 ) 1/2 − u 2 (u + 2m) (u 2 + ϵ 2 ) 3/2 }Φ (u + 2m)du. (3.31) By replacement u = ϵη, from (3.31) we obtain (3.32) which is calculated to give Thus the Tolman formula [3], [4] for the total energy of a static and asymptotically flat spacetime with g the determinant of the four dimensional metric and d 3 x the coordinate volume element, gives

Conclusions and Remarks
We have shown that a succesfull approach for dealing with curvature tensor valued distribution is to first imposes admissiible the nondegeneracy conditions on the metric tensor, and then take its derivatives in the sense of classical distributions in space S ′ 2m (R 3 ) The distributional meaning is then equivalent to the junction condition formalism. Afterwards, through appropiate limiting procedures, it is then possible to obtain well behaved distributional tensors with support on submanifolds of d ≤ 3, as we have shown for the energy-momentum tensors associated with the Schwarzschild spacetimes. The above procedure provides us with what is expected on physical grounds. However, it should be mentioned that the use of new supergeneralized functions (supergeneralized Colombeau algebras G(R 3 , Σ)). in order to obtain superdistributional curvatures, may renders a more rigorous setting for discussing situations like the ones considered in this paper.