Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Colombeau Solutions to Einstein Field Equations in Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime Dominance General -Induced Vacuum India . United Kingdom Author(s) Jaykov Foukzon1*, Alexander Alexeevich Potapov2 and Menkova Elena Romanovna3 1Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel. 2Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia. 3All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia. *Corresponding author: E-mail: jaykovfoukzon@list.ru; FIRST EDITION 2019 ISBN 978-93-89562-22-4 (Print) ISBN 978-93-89562-23-1 (eBook) DOI: 10.9734/bpi/mono/978-93-89562-22-4 _________________________________________________________________________________ © Copyright (2019): Authors. The licensee is the publisher (Book Publisher International). Contents Preface i Abstract 1 1 Introduction 2 1.1 Remarks on Linear Distributional Geometry in General Relativity 2 1.2 Basic Notions of Colombeau Generalized Functions and Colom-beau Generalized Numbers. Point Values of Colombeau Gener-alized Functions 6 1.2.1 Basic Notions of Colombeau Generalized Functions 6 1.2.2 The Ring of Colombeau Generalized Numbers . Point Values of Colombeau Generalized Functions 8 1.3 The Point Free Classical Colombeau Geometry 9 1.4 The Point-Free Loop Quantum Gravity 11 1.4.1 Classical Point-Free Phase Space 12 1.4.2 Quantization 13 1.5 Schwarzschild Black Hole 14 1.5.1 Classical Point-like Phase Space Variables 15 1.6 Classical Point-like Loop Quantum Gravity Contradict with a Linear Colombeau Geometry 17 1.6.1 The Point-like Quantum Schwarzschild Geometry 17 1.6.2 The Point Free Quantum Schwarzschild Geometry 19 1.7 Generalized Stokes' Theorem 23 1.7.1 The Colombeau Generalized Curvilinear Coordinates 23 1.7.2 Generalized Stokes' Theorem 26 1.8 The Colombeau Generalized Curvature Tensor 28 1.9 Generalized Einstein's Field Equations 30 1.10 The breakdown of canonical formalism of Riemann geometry for the singular solutions of the Einstein eld equations 32 1.10.1 The Densitized Einstein Field Equations Revisited 32 1.10.2 The Distributional Schwarzschild Geometry 37 1.10.3 The Distributional Schwarzschild Geometry at the Origin. The Smooth Regularization of the Singularity at the Origin 37 1.10.4 The Nonsmooth Regularization of the Singularity at the Origin 38 1.10.5 The Smooth Regularization Via Horizon 39 1.10.6 The Nonsmooth Regularization Via Gorizon 39 1.10.7 Distributional Eddington-Finkelstein Space-time 41 2 Generalized Colombeau Calculus 44 2.1 Notation and Basic Notions from Standard Colombeau Theory 44 2.2 Point Values of Generalized Functions on M. Generalized Numbers 45 2.3 Generalized Colombeau Calculus 47 2.4 Point Values of Supergeneralized Functions on M. Supergeneralized Numbers 48 2.5 Distributional General Relativity 50 3 Distributional Schwarzschild Geometry from Nonsmooth Regularization via Horizon 53 3.1 Calculation of the Stress-tensor by Using Nonsmooth Regularization Via Horizon 53 3.2 Examples of Distributional Geometries. Calculation of the Distributional Quadratic Scalars by Using Nonsmooth Regula-rization via Horizon 62 4 Quantum Scalar Field in Curved Distributional Space-time 66 4.1 Canonical Quantization in Curved Distributional Space-time 66 4.2 Dening Distributional Outgoing Modes 68 4.2.1 Boundary Conditions at the Horizon of the Distributional SAdS BH Geometry 69 5 Energy-Momentum Tensor Calculation by using Colombeau Distributional Modes 70 6 Distributional SAdS BH Space-time-induced Vacuum Dominance 73 6.1 Adiabatic Expansion of Green Functions 73 6.2 Effective Action for the Quantum Matter Fields in Curved Distributional Space-time 76 6.3 Stress-tensor Renormalization 79 7 Novel Explanation of the Active Galactic Nuclei. The Power Source of Quasars as a Result of Vacuum Polarization by the Gravitational Singularities on BHs Horizon 81 7.1 The Current Paradigm for Active Galactic Nuclei. High Energy Emission from Galactic Jets 81 7.2 The Colombeau Distributional Kerr Space-time in BoyerLindquist form 83 8 Conclusions and Remarks 90 Acknowledgement 90 Competing Interests 91 References 91 Appendix A 94 i Preface This book deals with Colombeau solutions to Einstein field equations in general relativity: Gravitational singularities, distributional SAdS BH spacetime-induced vacuum dominance. This book covers key areas of Colombeau nonlinear generalized functions, distributional Riemannian, geometry, distributional schwarzschild geometry, Schwarzschild singularity, Schwarzschild horizon, smooth regularization, nonsmooth regularization, quantum fields, curved spacetime, vacuum fluctuations, vacuum dominance etc. This book contains various materials suitable for students, researchers and academicians of this area. ii Print ISBN: 978-93-89562-22-4, eBook ISBN: 978-93-89562-23-1 Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Jaykov Foukzon1 ∗ , Alexander Alexeevich Potapov2 and Menkova Elena Romanovna3 DOI: 10.9734/bpi/mono/978-93-89562-22-4 Abstract This paper dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [5]-[30] and [15]- [31]. The regularizations of singularities present in some Colombeau solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this paper essentially new class Colombeau solutions to Einstein fild equations is obtained. 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 vacuum energy density of free scalar quantum field Φ with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such distributional BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes. In particular we obtain that the vacuum fluctuations ⟨ Φ2 ⟩ has a singular behavior on BHs horizon r+: ⟨ Φ2 (r) ⟩   |r − r+|−2 . A CHALLENGE TO THE BRIGHTNESS TEMPERATURE LIMIT OF THE QUASAR 3C273 explained successfully. Keywords: Colombeau nonlinear generalized functions; Distributional Riemannian; Geometry; Distributional Schwarzschild Geometry; Schwarzschild singularity; Schwarzschild Horizon; smooth regularization; nonsmooth regularization; quantum fields; curved spacetime; vacuum fluctuations; vacuum dominance. Mathematics Subject Classification : Primary: 83C15; Secondary: 83C15, 83C75, 46F30, 46T30. 1Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel. 2Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia. 3All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia. *Corresponding author: E-mail: jaykovfoukzon@list.ru; Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Physics and Astronomy Classification Scheme: Primary: 04.20.Cv; Secondary: 04.20.Jb, 04.20.Dw, 02.30.Sa, 02.40.Vh. 1 Introduction 1.1 Remarks on Linear Distributional Geometry in General Relativity A degenerate (singular) semi-Riemannian manifold (M ; g) is a differentiable manifold M endowed with a symmetric bilinear form g ∈ T 02M named metric. Note that the metric g is not required to be non-degenerate. In particular, if the metric is non-degenerate, (M ; g) is a semi-Riemannian manifold. If in addition g is positive definite, (M ; g) is a Riemannian manifold. This paper dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions such that Schwarzschild solution etc. of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. Note that in physical literature these singular solutions many years were mistakenly considered as vacuum solutions of the Einstein field equations, see for example [3],[4]. During last 30 years the applications classical linear distributional geometry in general relativity was many developed [5]-[31]. Remark 1.1.1. Let ( Rabcd,ε ) ε ∈ Gδ(R4) be Colombeau generalized function obtained using the standard definition of the Riemann curvature in a coordinate basis, i.e.( Rabcd,ε ) ε = ( Γadb,c,ε ) ε − ( Γacb,d,ε ) ε + ( Γacf,εΓ f ab,ε ) ε − ( Γadf,εΓ f cb,ε ) ε , (1.1.1) where ( Γabc,ε ) ε ∈ Gδ(Rn) and Γabc,ε, ε ∈ (0, δ] , δ ≤ 1 is the regularized Levi-Civita connection coefficients in terms of the regularized metric gab,ε, ε ∈ (0, 1] such that (gab,ε)ε , ( gabε ) ε ∈ G(R4), (det (gab,ε))ε = 0R. It has been shown by many authors (see for example [22]) that under apropriate regularization using the Eq.(1.1.1) one can defines the curvature scalar as a classical Schwartz distribution in D′ (Rn) , [18], [19]. Remark 1.1.2. This is the case even for the well-known Schwarzschild spacetime, which is given in the Schwarzschild coordinates (x0, r, θ, φ), by the metric. ds2 = − ( 1− a r ) (dx0)2 + ( 1− a r )−1 (dr)2 + r2 [ (dθ)2 + sin2 θ(dφ)2 ] . (1.1.2) Here, a is the Schwarzschild radius a = 2GM/c2 with G,M and c being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. Obviously the fundamental tensor corresponding to ds2 has the components which is degenerate or singular: (i) at r = 0 and (ii) at r = a. Remark 1.1.3. Note that in classical papers [5]-[31], etc. (i) the Colombeau distributional metric tensor (gab,ε)ε ∈ Gδ(R 4) related to ( Rabcd,ε ) ε ∈ Gδ(R4) by Eq.(1.1.1) never is not considered as the Colombeau solution of the Einstein field equations, (ii) Colombeau nonlinear distributional geometry never is not considered as the rigorous mathematical model related to really physical spacetime but only as useful purely mathematical tools in order to obtain related to ( Rabcd,ε ) ε ∈ Gδ(R4) classical Schwartz distributions inD′ (Rn) , (iii) there is no any important physical applications of the classical 2 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance linear distributional geometry were obtained. Remark 1.1.4. Originally fundamental physical applications of the Colombeau nonlinear distributional geometry has been obtained in authors papers [33]-[37]. By using now the Cartesian coordinates (x0, x1, x2, x3), which are related to (x0, r, θ, φ) through the canonical relations: x1 = r cosφ sin θ , x2 = r sinφ sin θ , x3 = r cos θ ,the metric (1.1.2) reads ds2 = ĝμνdx μdxν ,where at points r = 0, r = a the metric ĝμν is given by [29]: ĝ00 = −(1− h) , ĝ0α = 0 , ĝαβ = δ αβ + h(1− h)−1 x αxβ r2 , α, β = 1, 2, 3 (1.1.3) with h , a/r. Well known that at points r = 0, r = a : κ  T 0 0 = − h′ r − h r2 , κ  T α 0 = 0 , κ  T 0 α = 0 , κ  T β α = δ β α ( −h ′′ 2 − h ′ r ) + xαxβ r2 ( h′′ 2 − h r2 ) , (1.1.4) where the hatted symbols  T ν μ represent the quantity T ν μ in the coordinate system {xμ;μ = 0, 1, 2, 3}. Also, we have defined h′ , dh/dr and h′′ , d2h/dr2. Remark 1.1.5. We extend now the quantity (1.1.3)-(1.1.4) in point r = 0 as Colombeau generalized functions from Colombeau algebra Gδ ( R3 ) . Regularizing now the function h = a/r as (hε)ε = a/ (√ r2 + ε2 ) ε and the function xαxβ r2 as xαxβ (r2 + ε2)ε with ε ∈ (0, 1], we replace now the the singular metric (1.1.3) by the Colombeau generalized metric ds2 = (ĝμν,εdx μdxν)ε , (1.1.5) where (ĝ00,ε)ε = −(1− hε) , (ĝ0α,ε)ε = 0R , (ĝαβ,ε)ε = δ αβ + (( hε(1− hε)−1 ) ε )( xαxβ r2 + ε2 ) ε , α, β = 1, 2, 3 (1.1.6) and therefore κ (  T 0 0,ε(x) ) ε = − ( aε2 (r2 + ε2)5/2 ) ε , κ (  T α 0 (x; ε) ) ε = 0R, κ (  T 0 α(x; ε) ) ε = 0R, κ (  T β α(x; ε) ) ε = δ βα ( 3aε2 2(r2 + ε2)5/2 ) ε −(( xαxβ r2 + ε2 ) ε )( aε2 (r2 + ε2)5/2 ) ε ( 5 2 + ( ε2 r2 + ε2 ) ε ) . (1.1.7) Note that from Eq. (1.1.7) one obtains  T ν μ(x) , wlimε→0 T ν μ,ε(x) ∼ −Mc2δ 0μ δ ν0 δ(3)(x). (1.1.8) Remark 1.1.6. Note that (  T 0 0(x; ε) ) ε , (  T β α(x; ε) ) ε ∈ G ( R4 ) . Thus the generalized Einstein 3 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance equation [33,37-39] related to Eq.(1.1.6)-Eq.(1.1.7) in Colombeau notations reads:( Ĝνμ,ε ) ε = ( Rνμ,ε ) ε − 1 2 δμ ν ( Rε ) ε = κ ( Tμ,ε ν ) ε , (1.1.9) where ( Rε (r) ) ε = ( Rμμ,ε (r) ) ε = − ( 3aε2 (r2 + ε2)5/2 ) ε + ( 2aε2 (r2 + ε2)5/2 ) ε = − ( aε2 (r2 + ε2)5/2 ) ε (1.1.10) Remark 1.1.7. Note that the regularized scalar curvature Rε has the well-defined weak limit Rw in D′ (Rn) Rw , wlimε→0 Rε = − 4 3 πaδ(3)(x) . (1.1.11) Remark 1.1.8. Note that: (i) for any (rε)ε ∈ R such that st ( (rε)ε ) = rfin = 0,(see Definition 1.2.5) where rfin ∈ R from Eq.(1.1.10) it follows that st (( Rε (rε) ) ε ) = −st (( aε2 (r2ε + ε2)5/2 ) ε ) = −st (( aε2 (r2fin + ε 2)5/2 ) ε ) = 0, (1.1.12) (ii) for any (rε)ε ∈ R such that (rε)ε ≈R 0 (see Definition 1.2.4) from Eq.(1.1.10) it follows that(( Rε (r) ) ε ) ≈R −∞, (1.1.13) (iii) at origin ( rOε ) ε =R 0R (see Definition 1.2.4) one obtains(( Rε ( rOε )) ε ) =R − ( aε2 (ε2)5/2 ) ε =R − a (ε3)ε , (1.1.14) where ε ∈ (0, δ] . Remark 1.1.9. Note that the Eq.(1.1.12) in accordance with Eq.(1.1.11) and by Eqs.(1.1.12)- (1.1.14) we have recovered the intuitive meaning about δ-function. For the regularized quadratic scalars one obtains [29]: Rμνε (r) Rμν,ε (r) = 1 2 [ 3aε2 (r2 + ε2)5/2 ]2 + 2 [ aε2 (r2 + ε2)5/2 ]2 = 13 2 [ aε2 (r2 + ε2)5/2 ]2 Rρσμνε (r) Rρσμν,ε (r) = 4a2 r2 + ε2 [ 3 (r2 + ε2)2 ] − 12a 2ε2 (r2 + ε2)4 + + 9a2ε4 (r2 + ε2)5 . (1.1.15) Remark 1.1.10. Note that in contrast with the regularized scalar curvature Rε the regularized quadratic scalars do not have the weak limits, which can be symbolically written as 4 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Rμν(x)Rμν(x) , limε→0 Rμν(x; ε)Rμν(x; ε) ∼ 40π2a2 [ δ(3)(x) ]2 , Rρσμν(x)Rρσμν(x) , limε→0 Rρσμν(x; ε)Rρσμν(x; ε) ∼ 12a 2 r6 + 16πa2 3 1 r3 δ(3)(x) + 16π2a2 [ δ(3)(x) ]2 . (1.1.16) Remark 1.1.11. However Colombeau quadratic scalars ( Rμνε (rε) Rμν,ε (rε) ) ε and( Rρσμνε (rε) Rρσμν,ε (rε) ) ε well defined as Colombeau generalized functions in Gδ ( R3 ) . ( Rμνε (rε) Rμν,ε (rε) ) ε = 1 2 [( 3aε2 (r2ε + ε2)5/2 ) ε ]2 + 2 [( aε2 (r2ε + ε2)5/2 ) ε ]2 ,( Rρσμνε (r) Rρσμν,ε (r) ) ε = 12a2 ((r2ε + ε2)3)ε − ( 12a2ε2 (r2ε + ε2)4 ) ε + + ( 9a2ε4 (r2ε + ε2)5 ) ε . (1.1.17) Remark 1.1.12. Note that Colombeau quadratic scalars ( Rμνε (r) Rμν,ε (r) ) ε and( Rρσμνε (r) Rρσμν,ε (r) ) ε can be triating only nonclassically as Colombeau generalized functions extended on R3 = R× R× R, since at origin ( rOε ) ε = 0R we get( Rμνε ( rOε ) Rμν,ε ( rOε )) ε = 1 2 [( 3aε2 (ε2)5/2 ) ε ]2 + 2 [( aε2 (ε2)5/2 ) ε ]2 ,( Rρσμνε ( rOε ) Rρσμν,ε ( rOε )) ε = 12a2 (ε6))ε − ( 12a2ε2 (ε8) ) ε + + ( 9a2ε4 (ε10) ) ε . (1.1.18) Remark 1.1.13. In the usual Schwarzschild coordinates (t, r, θ, φ), r = a the Schwarzschild metric (1.1.2) takes the form above horizon r > a and below horizon r < a correspondingly above horizon r > 2m : ds+2 = h+(r)dt2 − [ h+(r) ]−1 dr2 + r2dΩ2, h+(r) = −1 + a r = −r − a r below horizon r < 2m : ds−2 = h−(r)dt2 − h−(r)−1dr2 + r2dΩ2, h−(r) = −1 + a r = a− r r (1.1.19) Following the above discussion we consider the metric coefficients h+ (r) , [ h+(r) ]−1 h− (r) , and[ h−(r) ]−1 as an element of D′(R3) and embed it into Gδ(R3) by replacements above horizon r > 2m and below horizon r 6 2m correspondingly r > 2m : r − 2m 7−→ √ (r − 2m)2 + ε2; r < 2m : 2m− r 7−→ √ (2m− r)2 + ε2. (1.1.20) Inserting (1.1.16) into (1.1.2) we obtain Colombeau generalized object modeling the singular Schwarzschild metric above (below) gorizon, i.e.,( ds+2ε ) ε = ( h+ε (r)dt 2 ) ε − ([ h+ε (r) ]−1 dr2 ) ε + r2dΩ2 ,( ds−2ε ) ε = ( h−ε (r)dt 2 ) ε − ([ h−ε (r) ]−1 dr2 ) ε + r2dΩ2 (1.1.21) The generalized Ricci tensor above horizon ([ R+ε ]β α ) ε may now be calculated componentwise using 5 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance the classical formulae([ R+ε (r) ]0 0 ) ε = ([ R+ε (r) ]1 1 ) ε = 1 2 (( h+′′ε ) ε + 2 r ( h+′ε ) ε ) ([ R+ε (r) ]2 2 ) ε = ([ R+ε (r) ]3 3 ) ε = ( h+′ε ) ε r + 1 + ( h+ε ) ε r2 , (1.1.22) where ( h+′ε (r) ) ε = − r − 2m r ([ (r − 2m)2 + ε2 ]1/2) ε + ([ (r − 2m)2 + ε2 ]1/2) ε r2 , (h′′ε (r))ε = − 1( r [ (r − 2m)2 + ε2 ]1/2) ε + (r − 2m)2 r ([ (r − 2m)2 + ε2 ]3/2) ε + + r − 2m r2 ([ (r − 2m)2 + ε2 ]1/2) ε + r − 2m r2 ([ (r − 2m)2 + ε2 ]1/2) ε − − 2 ([ (r − 2m)2 + ε2 ]1/2) ε r3 . (1.1.23) From Eq.(1.1.18)Eq.(1.1.23) we obtain (see sect.3) w -lim ε→0 [ R+ε (r) ]1 1 = w -lim ε→0 [ R+ε (r) ]0 0 = −2mδ(2m). (1.1.24) Remark 1.1.14. Note that the ε-regularization of degenerate and singular metric fields originally has been proposed in A. Einstein and N. Rosen paper [32]. Remark 1.1.15. The full non-linear theory of Colombeau distributional geometry based on Colombeau algebras in general relativity and its various applications to fundamental problems of the quantum gravity in curved Colombeau distributional spacetime originally has been obtained in authors papers [33]-[37]. 1.2 Basic Notions of Colombeau Generalized Functions and Colombeau Generalized Numbers. Point Values of Colombeau Generalized Functions 1.2.1 Basic Notions of Colombeau Generalized Functions In contemporary mathematics, a Colombeau algebra of Colombeau generalized functions is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Remark 1.2.1. Such a multiplication of distributions has been a long time mistakenly believed to be impossible because of Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by J. F. Colombeau [1], [2]. As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, 6 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance geophysics, microlocal analysis and general relativity so far. Basic idea. Definition 1.2.1. The algebra moderate functions C∞M (Rn) on Rn is the algebra of families of smooth functions (fε (x))ε , (fε (x))ε , x ∈ Rn, ε ∈ (0, δ] , δ ≤ 1 (smooth ε-regularisations, where ε is the regularization parameter), such that: (i) for all compact subsets K of Rn and all multiindices 3b1, there is an N > 0 such that sup x∈K ∣∣∣∣ ∂|α|fε (x)(∂x1)α1 * * * (∂xn)αn ∣∣∣∣ = O (ε−N) , ε→ 0, (1.2.1) with addition and multiplication defined by natural way: (fε (x))ε + (gε (x))ε = (fε (x) + gε (x))ε (1.2.2) and (fε (x))ε × (gε (x))ε = (fε (x)× gε (x))ε . (1.2.3) Definition 1.2.2. The ideal Nδ (Rn) of negligible functions is defined in the same way but with the partial derivatives instead bounded by O(εN ) for all N > 0,i.e. sup x∈K ∣∣∣∣ ∂|α|fε (x)(∂x1)α1 * * * (∂xn)αn ∣∣∣∣ = O (εN) , ε→ 0. (1.2.4) Definition 1.2.3. The Colombeau Algebra Gδ (Rn) [1],[2] is defined as the quotient algebra G(Rn) = C∞M (Rn) /Nδ (Rn) . (1.2.5) Elements of calGδ(Rn) are denoted by: u = cl[(uε)ε] , (uε)ε +Nδ(Rn). (1.2.6) Embedding of distributions The space of Schwartz distributions D′ (Rn) can be embedded into the Colombeau algebra G (Rn) by (component-wise) convolution with any element (φ∈)ε of the algebra Gδ (R n) having as representative a δ-net, i.e. a family of smooth functions φ∈ such that φ∈ → δ in D′ (Rn) as ∈→ 0. Remark 1.2.2. Note that the embedding ι : D′ (Rn) ↩→ Gδ (Rn) is non-canonical, because it depends on the choice of the δ-net. Example 1.2.1. Delta function δ (x) ∈ D′ (R) for example has the following different representatives in Colombeau algebra Gδ (R) : 1 2 ( 1√ πε exp ( −x 2 4ε )) ε ∈ Gδ (R) , 1 π ( 1 x sin (x ε )) ε ∈ Gδ (R) , 1 π ( ε x2 + ε2 ) ε ∈ Gδ (R) , 1 π ( 1 x2 sin2 (x ε )) ε ∈ Gδ (R) , (1.2.7) since 1 2 1√ πε exp ( −x 2 4ε ) → δ (x) , 1 π 1 x sin (x ε ) → δ (x) , 1 π ε x2 + ε2 → δ (x) , 1 π 1 x2 sin2 (x ε ) → δ (x) (1.2.8) in D′ as ∈→ 0. 7 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 1.2.3. However note that embeddingD′ (Rn) ↩→ G (Rn) does not meant the full equivalence of the Schwartz distributions and corresponding by embedding Colombeau generalized functions. In contrast with the Schwartz distributions Colombeau generalized functions has well defined walue at any point x ∈ Rn these point values of the Colombeau generalized functions is the Colombeau generalized numbers. Example 1.2.2. Delta function δ (x) ill defined at point x = 0 since δ (0) = ∞.However any Colombeau generalized function defined by Eq.(1.2.7) has well defined point value at point x = 0. For example 1 2 ( 1√ πε exp ( −x 2 4ε )) ε ∣∣∣∣ x=0 = 1 2 √ πε ( 1√ ε ) ε ∈ Rδ, 1 π ( ε x2 + ε2 ) ε ∣∣∣∣ x=0 = 1 π ( 1 ε ) ε ∈ Rδ. (1.2.9) Here R is the ring of real Colombeau generalized numbers [34]. 1.2.2 The Ring of Colombeau Generalized Numbers Rδ. Point Values of Colombeau Generalized Functions Designation 1.2.1. (I) We denote by Rδ, δ ≤ 1 the ring of real Colombeau generalized numbers. Recall that by definition Rδ = ER,δ (R) /Nδ (R) where [34],[36],[37]: ER,δ (R) = { (xε)ε ∈ R (0,δ) ∣∣∣ (∃a ∈ R+) (∃ε0 ∈ (0, 1)) (∀ε ≤ ε0) [|xε| ≤ ε−a]} , Nδ (R) = { (xε)ε ∈ R (0,δ) ∣∣∣ (∀a ∈ R+) (∃ε0 ∈ (0, 1)) (∀ε ≤ ε0) [|xε| ≤ εa]} . (1.2.10) (II) In this subsection we will be write for short R instead Rδ. Notice that the ring R arises naturally as the ring of constants of the Colombeau algebras Gδ (Ω) . Recall that there exists natural embedding r : R ↩→ R such that for all r ∈ R, r = (rε)ε where rε ≡ r for all ε ∈ (0, 1] . We say that r is standard number and abbreviate r ∈ R for short. The ring R can be endowed with the structure of a partially ordered ring: for r, s ∈ R η ∈ R+, η ≤ δ we abbreviate r ≤R,η s or simply r ≤R s if and only if there are representatives (rε)ε and (sε)ε with rε ≤ sε for all ε ∈ (0, η] . Colombeau generalized number r ∈ R with representative (rε)ε we abbreviate cl [ (rε)ε ] . Definition 1.2.4. (i) Let δ = cl [ (δε)ε ] ∈ R. We say that δ is infinite small Colombeau generalized number and abbreviate δ ≈R 0 if there exists representative (δε)ε and some q ∈ N such that |δε| = O (εq) as ε→ 0. (ii) Let ∆ ∈ R.We say that ∆ is infinite large Colombeau generalized number and abbreviate ∆ =R ∞ if ∆ −1 R ≈R 0.(iii) Let R±∞ be R ∪ {±∞} We say that Θ ∈ R±∞ is infinite Colombeau generalized number and abbreviate Θ =R ±∞R if there exists representative (Θε)ε where |Θε| = ∞ for all ε ∈ (0, 1] .Here we abbreviate EM (R±∞) = EM (R ∪ {±∞}) , N (R±∞) = N (R ∪ {±∞}) and R±∞ = EM (R±∞) /N (R±∞) Definition 1.2.5. (Standard Part Mapping). (i) The standard part mapping st : R → R is defined by the formula: st (x) = sup { r ∈ R|r ≤R x } . (1.2.11) If x ∈ R, then st (x) is called the standard part of x. (ii) The mapping st : R → R∪{±∞} is defined by (i) and by st (x) = ±∞ for x ∈ R and for x ∈ 8 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance R±∞,respectively. Definition 1.2.6.[37]. Let (fε (x))ε ∈ G(R) and x ∈ R,then cl [ (fε (x))ε ] ∈ R.We will say that Colombeau generalized number cl [ (fε (x))ε ] is a point values of Colombeau generalized function (fε (x))ε at point x ∈ R. Definition 1.2.7. (Principal value mapping) The principal value mapping p.v. :R → R of Colombeau generalized function (fε (x))ε at point x ∈ R is defined by the formula: p.v. { cl [ (fε (x))ε ]} = sup ε∈(0,1] |fε (x)| . (1.2.12) We will be write for short p.v. [ (fε (x))ε ] . Example 1.2.3. The principal value of the curvature scalar ( Rε (r, a) ) ε (1.1.10) at point r ∈ R reads p.v. [( Rε (r, a) ) ε ] = sup ε∈(0,1] aε2 (r2 + ε2)5/2 . (1.2.13) Fig. 1. Plot of the function R(a, r, ε) = aε2 (r2 + ε2)5/2 , a=1, r=10−3, z = ε ∈ (0, 0.01] . R ( 10−3, 7× 10−4 ) = 1.808× 108. p.v. [( Rε (r, a) ) ε ] ≃ 1.808× 108. 1.3 The Point Free Classical Colombeau Geometry The first definition (prior to the well-known five postulates) of Euclides describes the point as "that of which there is no part" [40]. A huge portion of our mathematics of the physical world is based on the amazingly simple Euclidean geometry. Indeed, starting from very straightforward assumptions and theorems such as those found 9 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance in Euclid's geometry, it is feasible to build also non-Euclidean geometries and complex manifolds able to explain issues such as those in quantum mechanics. One of the main components of Euclidean geometry is the point, that stands for the most fundamental object. The first definition of a point (prior to Euclid) is given by the Pythagoreans: a point is a monad having position. Euclid begins his geometry with the definition of a point [that of which there is no part] (Def. 1, Euclid, 300 BCE) and the extremities of a line are points (Def. 2). Euclid's Def.1 is interpreted by T.L. Heath to mean that a point is that which is indivisible in parts. Therefore, we are confronted with a primitive notion defined only by axioms that it must satisfy, i.e., the point upon which the whole apparatus is built, meaning that geometry cannot be described in terms of previously defined real objects or structures. Here we ask whether the zero-substance point holds true in our physical world and extend our analysis also to other Euclidean objects, such as lines, surfaces, volumes and so on [41]. Definition 1.3.1. Let (fε (x))ε ∈ Gδ(R) and cl [ (xε)ε ] ∈ Rδ.Assume that cl [ (fε (xε))ε ] ∈ Rδ. We will say that Colombeau generalized number cl [ (fε (xε))ε ] is a point values of Colombeau generalized function (fε (x))ε at point (xε)ε ∈ Rδ. Example 1.3.1. For any (rε)ε ∈ Rδ, ε ∈ (0, δ] the point values ( Rε (rε) ) ε of Colombeau generalized function ( Rε (r) ) ε (see Eq.(1.1.10) reads ( Rε (rε) ) ε = ( Rμμ,ε (rε) ) ε = − ( aε2 (r2ε + ε2)5/2 ) ε = −a ( ε2 ) ε[ (r2ε)ε + (ε 2)ε ]5/2 = −a δ2 ( ε21 ) ε1[ (r2ε)ε + δ 2 (ε21)ε1 ]5/2 , (1.3.1) where ε1 ∈ (0, 1] . Remark 1.3.1. We choose now (rε)ε = (r ∗ ε)ε = η (ε)ε = ηδ (ε1)ε1and from Eq.(1.3.1) we get ( Rε (r ∗ ε) ) ε = −a δ2 ( ε21 ) ε1[ (r∗2ε )ε + δ 2 (ε21)ε1 ]5/2 = −a δ2 ( ε21 ) ε1[ ηδ (ε21)ε1 + δ 2 (ε21)ε1 ]5/2 = −a δ2 ( ε21 ) ε1[ η2δ2 (ε21)ε1 + δ 2 (ε21)ε1 ]5/2 = −a δ2 ( ε21 ) ε1 δ5 (ε51)ε1 [η 2 + 1]5/2 = −a δ3 (ε31)ε1 [η 2 + 1]5/2 . (1.3.2) Thus in "point limit" δ ≍ 0 the curvature scalar ( Rε (r ∗ ε) ) ε diverges as δ−3 ( ε−31 ) ε1 . Remark 1.3.2. In order prevent the divergence mentioned above, we assume now that there exists fundamental generalized length l∗ = cl [ (l∗ε1)ε1 ] = ηcl [ (ε1)ε1 ] , ε1 ∈ (0, 1] , η ∈ R, δ ≪ η ≪ 1 , such that: (rε1)ε1 ≥ l ∗,see [36] sec.2.Thus from Eq.(1.3.1) we get now instead Eq.(1.3.2) 10 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ∣∣∣(Rε (rε)) ε ∣∣∣ ≤ ( aε2 (l∗2ε1 + ε 2)5/2 ) ε = a a ( ε2 ) ε[ (l∗2ε1 )ε + (ε 2)ε ]5/2 = a δ2 ( ε21 ) ε1[ η2 (ε21)ε1 + δ 2 (ε21)ε1 ]1/2 1[ η2 (ε21)ε1 + δ 2 (ε21)ε1 ]1/2 1[ (rε1)ε1 + (ε 2)ε ]3/2 = aδ2 [η2 + δ2] 1[ (r2ε1)ε1 + (ε 2)ε ]3/2 = aδ2[η2 + δ2] 1[ (r2r)r∈(0,1] + (ε 2)ε ]3/2 . (1.3.3) 1.4 The Point-Free Loop Quantum Gravity We remind that canonical quantization of GRT can be expressed as an SU(2) gauge theory on the 3 dimensional manifold Σ furnished by canonical point-like geometry, where a topology of space-time M of the form M ∼= R× Σ is assumed, in a background independent manner. In such formulation of GR, the gravitational field is described by a pair of conjugate variables (A,E), where Aia(x) is an SU(2) connection and Eai (x) is the densitised triad vector field, conjugate to A :{ Aia(x), E b j (x ′) } = 8πγδijδ b aδ(x− x′), (1.4.1) with G the gravitational constant and γ the Immirzi parameter.The conjugate pair are constraint to satisfy the system Gi = DaE a i = 0,Hb = E a i F i ab = 0, H = εijkF i abE a jE b k − 2(1 + γ2)Ki[aK j b]E a i E b j = 0, (1.4.2) which are called Gauss, spatial diffeomorphism and Hamiltonian constraints respectively. In fact, the task of finding a metric satisfying the Einstein's equations, describing configuration of a gravitating system, is now replaced by finding a conjugate pair (A,E) satisfying the constraint system (1.4.1). On quantization, one smears the basic fields (A,E) to holonomies of Aai along a curve γ, defined by hγ [A] = P [ exp (∫ γ A )] , and fluxes of Eai through the surface S, defined by Ei(S) = ∫ S d2σnaE a i . They form the holonomy-flux algebra in which holonomies act by multiplication, and fluxes act by derivation. Using a functional representation of quantum field theory and representing states as functionals of the cylindrical functions of holonomies, the kinematical Hilbert space of the theory is constructed. After imposing Gauss and diffeomorphism constraints as operators on such states, the true gauge and diffeomorphism invariant states of the theory turns out to be spin networks acted upon by holonomies and fluxes operators which form a unique representation. More precisely, a spin network is a triplet (Γ, jl, in) consisting of a graph Γ with nodes in Σ, labeled by intertwiners in, and links connecting different nodes, labeled by SU(2) representations jl. Remark 1.4.1. (I)The quantum geometrical picture suggested by canonical LQG [44] is manifest in quantization of geometrical observables, such as area and volume, as quantum operators acting on spin network states which result in discrete spectra and reflect the discrete nature of space-time. (II) In fact singularity resolution occurs as a result of fundamental discreteness of space; while in a classical continuum, divergences emerge as distance goes to zero, there is no room for divergences in quantum level since there is no zero distance below the Planck length. Remark 1.4.2. Canonical quantization of nonlinear distributional GRT can be expressed as an SU(2) gauge theory on the 3 dimensional Colombeau distributional manifold Σ furnished by Colombeau point-free geometry, where a topology of space-time M of the form M ∼= Rδ × Σ is assumed, in a background independent manner. In such formulation of GRT, the gravitational 11 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance field is described by a pair of conjugate variables ((Aε)ε , (Eε)ε), where ( Aia,ε(xε) ) ε is an SU(2) Colombeau distributional connection and ( Eai,ε(xε) ) ε is the distributional densitised triad vector field, conjugate to (Aε)ε :{( Aia,ε(xε) ) ε , ( Ebj (x ′ ε) ) ε } = 8πγδijδ b a (δ(xε − x′ε))ε , (1.4.3) with G the gravitational constant and γ the Immirzi parameter.The conjugate pair are constraint to satisfy the system (Gi,ε)ε =R ( Da,εE a i,ε ) ε = 0, (Hb,ε)ε = ( Eai,εF i ab,ε ) ε =R 0R, (Hε)ε =R εijk ( F iab,εE a j,εE b k,ε ) ε − 2(1 + γ2) ( Ki[a,εK j b,ε]E a i,εE b j,ε ) ε =R 0R, (1.4.4) In fact, the task of finding Colombeau metric satisfying the generalized Einstein's field equations (see subsect.1.8), describing configuration of a gravitating system, is now replaced by finding a conjugate pair ((Aε)ε , (Eε)ε) satisfying the constraint system (1.4.3). On quantization, one smears the basic Colombeau generalized fields ((Aε)ε , (Eε)ε) to holonomies of ( Aai,ε ) ε along a curve γ = (γε)ε, defined by (hγ [Aε])ε = P [ exp (∫ γ Aε ) ε ] , and fluxes of ( Eai,ε ) ε through the surface S, defined by( Ei,ε(S) ) ε = (∫ S d2σnaE a i,ε ) ε . A spin network is a triplet (Γ, jl, in) consisting of a graph Γ with nodes in Σ, labeled by intertwiners in, and links connecting different nodes, labeled by SU(2) representations jl. 1.4.1 Classical Point-Free Phase Space Definition 1.4.1.(1) The general linear group over Colombeau algebras R, C (the set of real,complex Colombeau numbers) is the group of n×n invertible matrices of real (complex) Colombeau numbers, and is denoted by GLn ( R ) , GLn ( C ) or GL ( n, R ) , GL ( n, C ) . (2) The unitary group of degree n over Colombeau algebra C, denoted Ũ(n), or U(n, C) is the group of n× n unitary matrices over C. (3) The unitary group is a subgroup of the general linear group GL ( n, C ) . (4) In the simple case n = 1, the group U(n, C) corresponds to the circle group T, consisting of all Colombeau complex numbers with absolute value 1 under multiplication,i.e. T = { z ∈ C ∣∣∣ |z| = 1} . (5) The special unitary group of degree n, denoted SU(n), is the Lie group of n×n unitary matrices over Colombeau algebra R with determinant 1. The Colombeau distributional manifold Σ over Colombeau algebra R having the symmetry group S with an isotropy subgroup F , can be decomposed as Σ ∼= Σ/S × S/F . The connection can generally be written as (Aε)ε = ( AΣ/S,ε ) ε + ( AS/F ,ε ) ε . Then ( AΣ/S,ε ) ε can be considered as the connection of the reduced theory and its holonomies along curves in Σ/S can be quantized.For the spherically symmetric case, Σ ∼= Rδ × S2, and the symmetry group is S = SU(2). This implies identifying Σ/S with Rδ and the gauge group of the reduced theory F with U(1). Therefore, reduced connections are Ũ(1) gauge fields on R. Roughly speaking, spherical symmetry implies that our basic fields, in the spherical coordinate ((xε)ε , θ, φ), are independent of angular variables. 12 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Thus, the Colombeau generalized connection (Aε(x⃗ε))ε is just a function of the radial coordinate; (Aε)ε = (Aε(xε))ε.These connections and triads of the reduced spherically symmetric phase space have the general form: (Aε)ε = [ (Axε(xε))ε ] τ3 [ (dxε)ε ] + ( [ (A1(xε))ε ] τ1 + [ (A2(xε))ε ] τ2) [ (dθε)ε ] + (((A1,ε(xε)τ2)ε − (A2,ε(xε)τ1 sin θε)ε) + τ3 (cos θε)ε) [ (dφε)ε ] (1.4.5) and (Eε)ε = (E xε ε (xε)τ3 sin θε∂xε)ε + ( ( E1ε (xε)τ1 ) ε +( E2ε (xε)τ2) sin θε∂θε ) ε + ( (E1ε (xε)τ2 ) ε − ( E2ε (xε)τ1)∂φε ) ε (1.4.6) correspondingly, where τi = − i2σi are the generators of su(2) algebra. They define the Colombeau generalized symplectic structure: (Ωε)ε = 1 2γG (∫ dxε(dAxε,ε ∧ dExεε + 2dA1,ε ∧ dE1ε + 2dA2,ε ∧ dE2ε ) ) ε . (1.4.7) However, a suitable canonical transformation can be made resulting in Colombeau generalized canonical variables ((Axε(xε))ε , (E xε ε (xε))ε), (γ (Kφε,ε(xε))ε , ( Eφεε (xε) ) ε ) and ((ηε(xε))ε , (P ηε ε (xε))ε): (Ωε)ε = 1 2γG ∫ dxε(dAx ∧ dExε + d(γKφε) ∧ dEφε + 2dηε ∧ dP ηε )ε, (1.4.8) with (Kφε,ε)ε being the (φε)ε component of the extrinsic Colombeau generalized curvature.The Gauss constraint, generating Ũ(1) gauge transformations, takes the form: (Gε[λε])ε = (∫ dxελε(E x′ε ε + P ηε ε ) ) ε =R 0R, (1.4.9) where prime denotes differentiation with respect to xε. Note that in terms of these variables, conjugate pair is not simply the connection-flux pair which suggests a different situation than the full theory.The Colombeau generalized Hamiltonian constraint can be written as (Hε[Nε])ε = − 1 2G ×∫ dxεNε(xε) 1√ |Exεε | ( (1 − Γ2φε,ε + K 2 φε,ε )Eφεε + 2 γ Kφε,εE xε ε (Axε,ε + η ′ ε) + 2E xε ε Γ ′ φε,ε ) ε . (1.4.10) 1.4.2 Quantization Along the standard lines of constructing basic operators and states in the kinematical Hilbert space of classical LQG, we start with holonomies of the connections. Holonomies of (Ax,ε)ε along curves (γε)ε in R are defined as ( h (γε) ε ) ε ≡ exp [ i 2 (∫ γε Axε(xε) ) ε ] which are elements in Ũδ(1) = Rδ/Z. For (Aφε,ε)ε point holonomies exp [ iμ (Aφε,ε(xε))ε ] are used which belongs to the space of continuous almost periodic functions on the Bohr compactification of real line Rδ , and point holonomies of (ηε)ε ∈ S 1, have the form exp [ (iηε(xε))ε ] which are elements of Ũ(1). The kinematical Hilbert space of the present reduced theory is the space spanned by spin network state (Tg,k,μ,ε)ε : 13 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (Tg,k,μ,ε)ε =∏ e∈g exp ( i 2 ke (∫ e dxεAxε(xε) ) ε )(∏ ν∈V (g) exp (iμν,εγKφε,ε(ν)) exp (ikνηε(ν)) ) ε . (1.4.11) For a given graph g, these are cylindrical functions of holonomies along edges e of g. Vertices V (g) of such spin networks are labeled by irreducible  RBohr representations (μν,ε)ε ∈ Rδ and irreducible S1 representation kν ∈ Z, while edges are labeled by irreducible representations of Ũδ(1). Holonomies act on spin network states by multiplication. Their corresponding momenta, on the other hand, act by differentiation( Êxεε (xε)Tg,k,μ,ε ) ε = γ l2p 2 ( (ke+(xε) + ke−(xε))Tg,k,μ,ε ) ε , (1.4.12) ∫ dxεÊ φε ε (xε)Tg,k,μ = γl 2 p ∑ v μν,εTg,k,μ,ε, (1.4.13)∫ dxεP ηε ε (xε)Tg,k,μ,ε = 2γl 2 p ∑ v kν,εTg,k,μ,ε. (1.4.14) The generalized volume operator can be expressed as( Vε ) ε = 4π ∫ dxε|Êφεε (xε)| √ |Êxεε (xε)| (1.4.15) which is diagonal in spin network representation( VεTg,k,μ,ε ) ε = (Vk,m,εTg,k,μ,ε)ε , (1.4.16) where (Vk,m,ε)ε = 4πγ 3/2l3p (∑ ν |μν | √ 1 2 |ke+(xε) + ke−(xε)| ) ε . (1.4.17) Implementing the Gauss constraint as an operator on spin networks to select the gauge invariant states, leads to a restriction on labels( Ĝε[λε]Tg,k,μ,ε ) ε = γl2p (∑ ν λε(ν)(ke+(xε) − ke−(xε) + 2kν)Tg,k,μ,ε ) ε (1.4.18) ( Ĝε[λε]Tg,k,μ ) ε = 0R =⇒ kν = − 1 2 (ke+(xε) − ke−(xε))ε. (1.4.19) Imposing now this on (1.4.11) results in the gauge invariant states (Tg,k,μ,ε)ε =∏ e∈g exp [ i 2 ke (∫ e dxε(Axε(xε) + η ′ ε) ) ε ](∏ ν∈V (g) exp (iμνγKφε,ε(ν)) ) ε . (1.4.20) 1.5 Schwarzschild Black Hole Remind that the Schwarzschild metric is a spherically symmetric solution to Einstein equations describing the space-time of a source with massm in coordinate system (x, θ, φ) reads 14 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ds2 = − ( 1− 2m x ) dt2 + ( 1− 2m x )−1 dx2 + x2dΩ2. (1.5.1) Horizon x+ appear where g00 = 0: x+ − 2m = 0. (1.5.2) The event horizons partition space-time into 2 regions: I (x > x+), and II (0 < x < x+). By inspecting the sign of g00, one observes that in region II, x and t interchange their roles and becomes time-like and space-like respectively. 1.5.1 Classical Point-like Phase Space Variables In region II, the metric of space-time takes the form ds2 = − ( 2m t − 1 )−1 dt2 + ( 2m t − 1 ) dx2 + t2dΩ2. (1.5.3) According to definition of tetrad (frame) fields gμν = ηIJe I μe J ν , they can be determined only up to a Lorentz transformation. This leaves us with an SO(3, 1) freedom in choosing tetrad. In fact, given the metric gμν = ηIJe I μe J ν we are free to choose their sign and Minkowski indices, which can be viewed as sort of a labeling 4 tetrad fields. However, in order to serve as the fundamental fields for constructing the conjugate pair (A,E), a particular labeling must be chosen which will be clear below.The suitable choice for labeling 4 orthogonal frame fields reads e0 = ± ( 2m t − 1 )−1/2 dt ; e1 = ±t sin θdφ; e2 = ±tdθ ; e3 = ± ( 2m t − 1 )1/2 dx, (1.5.4) which gives the compatible spin connection components ω30 = −ω03 = ( −m t2 ) dx ; ω20 = −ω02 = ( 2m t − 1 )1/2 dθ, ω10 = −ω01 = ( 2m t − 1 )1/2 sin θdφ ; ω12 = −ω21 = cos θdφ. (1.5.5) The A field can be constructed using spin connections: A3 = ±γ ( −m t2 ) dx,A2 = ±γ ( 2m t − 1 )1/2 dθ, A1 = ±γ ( 2m t − 1 )1/2 sin θdφ,A3 = ± cos θdφ. (1.5.6) To construct the E field on Σin we choose a gauge in which e 0 μ = nμ, the normal vector field to the spatial slice. This way we are in fact breaking the SO(3, 1) symmetry into SO(3) on a hypersurface with topology Σ = R× S2. The 3 triad fields become: e1 = ±t sin θdφ ; e2 = ±tdθ ; e3 = ± ( 2m t − 1 )1/2 dx, (1.5.7) with determinant 15 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance det (e) = t2 sin θ ( 2m t − 1 )1/2 , (1.5.8) and inverse triad e1 = ± 1 t sin θ ∂φ ; e2 = ± 1 t ∂θ ; e3 = ± ( 2m t − 1 )−1/2 ∂x, (1.5.9) The E fields become E1 = ±t ( 2m t − 1 )1/2 ∂φ, E2 = ±t ( 2m t − 1 )1/2 sin θ∂θ, E3 = ±t2 sin θ∂x. (1.5.10) The 3 triad fields (1.5.7) define their compatible spin connection, Γij ∧ ej + dei = 0 : Γ12 = −Γ21 = cos θdφ, (1.5.11) and Γ3 = 1 2 ( ε312Γ12 + ε321Γ21 ) = cos θdφ. (1.5.12) Extrinsic curvature is related to A via γK = A− Γ reads K3r = 1 γ A3r = ± ( −m t2 ) dx,K2θ = 1 γ A2θ = ± ( 2m t − 1 )1/2 , (1.5.13) K1φ = 1 γ A1φ = ± ( 2m t − 1 )1/2 sin θ. (1.5.14) Note that had we chosen other Minkowski indices for tetrad (1.5.4) we would not have obtained the conjugate pair(A,E) with correct indices satisfying { Aia(x), E b j ( x′j )} = δijδ b aδ(x− x′). The phase space variables are determined up to a sign freedom. By demanding E and A to satisfy the diffeomorphism, Gauss and Hamiltonian constraints, their signs can be fixed relative to each other. All components of diffeomorphism and Gauss constraints are zero except Hθ = γt ( 2m t − 1 ) cos θ { sgn(A1φ) + sgn(A 2 θA 3 φ) } , (1.5.15) G2 = t ( 2m t − 1 )1/2 cos θ { sign(Eθ2 ) + sign(A 3 φE φ1) } , (1.5.16) and Hamiltonian constraint gives: C = t ( 2m t − 1 ) sin2 θ { sign(Eθ2 ) + sign(E φ 1 ) } . (1.5.17) For the above constraints to be zero we must have sign(Eθ2 ) = −sign(Eφ1 ), sign(A3φ) = +1, sign(A1φ) = −sign(A2θ). (1.5.18) This leaves us with two alternatives corresponding to the residual gauge freedom (b, pb) → (−b,−pb). Aia = cτ3dr + bτ2dθ + (cos θτ3 − b sin θτ1)dφ Eai = pcτ3 sin θ∂r + pbτ2 sin θ∂θ − pbτ1∂φ, (1.5.19) 16 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance and Aia = cτ3dr − bτ2dθ + (cos θτ3 + b sin θτ1)dφ Eai = pcτ3 sin θ∂r − pbτ2 sin θ∂θ + pbτ1∂φ, (1.5.20) where, b = ±γ ( 2m t − 1 )1/2 ; c = ±γ ( −m t2 ) , (1.5.21) pc = ±t2 ; pb = t ( 2m t − 1 )1/2 . (1.5.22) The momentum pc = ±t2 is a monotonic function and can be interpreted as an internal time parameter (as is interpreted in [44] for the case of the Kantowski-Sachs minisuperspace of Schwarzshild black hole). Region I. The analoguos calculations for region I with line element (1.5.1) leads to the folowing phase space coordinates Ãia = cτ3dr + bτ2dθ + (cosθτ3 − bsinθτ1)dφ Ẽai = pcτ3sinθ∂r + pbτ2sinθ∂θ − pbτ1∂φ, (1.5.23) Ãia = cτ3dr − bτ2dθ + (cos θτ3 + b sin θτ1)dφ Ẽai = pcτ3 sin θ∂r − pbτ2 sin θ∂θ + pbτ1∂φ, (1.5.24) where, b = ±γ ( 1− 2m x )1/2 ; c = ∓γ (m x2 ) ; pc = ±x2 ; pb = x ( 1− 2m x )1/2 . (1.5.25) This defines variables (1.4.5)-(1.4.6) introduced above in subsection 1.4 as Ax = c , E x = pc; γKφ = b , E φ = pb; η = (2n+ 1)π, P η = 0 (1.5.26) which constitute a 4 dimensional phase space. 1.6 Classical Point-like Loop Quantum Gravity Contradict with a Linear Colombeau Geometry 1.6.1 The Point-like Quantum Schwarzschild Geometry We remind that in accordance with a linear Colombeau geometry approach [30], the Schwarzschild black hole,etc. has a distributional source ∼ δ (x) ∈ D′ (R) ,see Eq. (1.1.8) and Eq. (1.1.11). This result as well established and acceptet by scientific community as physical reality [29]-[31]. Remark 1.6.1.However under local singularity resolution based on canonical LQG approach [44], these distributional sources vanishes and we go bak to ubnormal and mistaken results from classical handbooks, see for example [3],[4].Obviously this is a contradiction. Thus by using canonical LQG approach we can not quantized the well established classically distributional Schwarzschild black hole,etc. 17 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Viewing LQG as a method to quantize connections, one would be able to impose a symmetry through two avenues: (i) to pick, in the classical level, only those connections which are invariant under symmetry group action and consequently reduce the phase space, and (ii) to restrict the distributional states of the quantum theory, at the kinematical level, only to invariant connection [42]-[45]. We will consider the simplest case of a spin network that is equispaced in normal coordinates with lattice spacing ∆ ∼ lPl. Remind that under naive formal calculation the Kretschmann scalar curvature of the Schwarzschild black hole reads [43]: Rρσμν (r) Rρσμν (r) = 48M2 t6 . (1.6.1) Obviously (1.6.1) indicates that in this case the singularity of space-time lies at r = 0 as well. The classical phase space variables calculated in subsect.1.5 {c, pc} used in this section are given by Eq.(1.5.21)-Eq.(1.5.22) and therefore b = ±γ ( 2m t − 1 )1/2 ; c = ∓γ (m t2 ) ; pc = ±t2 ; pb = t ( 2m t − 1 )1/2 . (1.6.2) Let us consider the following quantity on the classical point-like phase space [43]: R ≡ 1 2πγG { c, √ |pc| } = sgn(pc)√ |pc| = 1 t . (1.6.3) Following the methods presented in [44], we expand now the holonomy along x direction of Σ = R× S2 with oriented length τ as h (τ) x = 1 + ε ∫ τ 0 dxcτ3 +O(ε2), (1.6.4) and rewrite R as R = 1 2πγG tr ( τ3h (τ) x { h (τ)−1 x , √ |pc| }) . (1.6.5) Now, quantization would be straightforward: R = 1 2πγl2Pl tr ( τ3ĥ (τ) x [ ĥ (τ)−1 x , √ |pc| ]) = 1 2πγl2Pl ( cos (τc 2 )√ |pc| sin (τc 2 ) − sin (τc 2 )√ |pc| cos (τc 2 )) . (1.6.6) Its action on |τ, μ⟩ which are the simplified version of the spin network states in this reduced model (with μ being the oriented length along the equator of S2), then becomes: R |τ, μ⟩ = 1 2π √ γlPl (√ |τ + 1| − √ |τ − 1| ) |τ, μ⟩ . (1.6.7) Such operator R |τ, μ⟩ has a bounded spectrum with maximum value of (√ 2π √ γlPl )−1 . Thus the Kretschmann scalar curvature, which is classically divergent, at quantum level has a maximum value of [43]: Rρσμν (r) Rρσμν (r) ∣∣∣ max = 48M2 r6 ∣∣∣∣ max = 48M2 γ3π6l6Pl . (1.6.8) 18 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 1.6.2. Note that a quantity ( Rε (r) ) ε which is classically has a weak distributional limit, at quantum level obtained by canonical LQG has a maximum value of ∣∣∣Rε (r)∣∣∣∣∣∣ max ∼ Mε 2 l5Pl ∣∣∣∣ max . (1.6.9) Thus limε→0 Rε (r) = 0 since RHS of the Eq.(1.6.9) wanishes in the limit ε→ 0. 1.6.2 The Point Free Quantum Schwarzschild Geometry In region II, the Colombeau metric of point-free Schwarzschild space-time takes the form ( ds2ε ) ε = − [ 2m (tε)ε − 1 ]−1 ( dt2ε ) ε + [ 2m (tε)ε − 1 ] ( dx2ε ) ε + ( t2εdΩ 2 ε ) ε , (1.6.10) where cl [ (tε)ε ] ∈ R.According to definition of Colombeau tetrad (frame) fields (gμν,ε)ε = ηIJ ( eIμ,εe J ν,ε ) ε , they can be determined only up to a Lorentz transformation. This leaves us with an SO(3, 1) freedom in choosing tetrad. In fact, given the Colombeau metric (gμν,ε)ε = ηIJ ( eIμ,εe J ν,ε ) ε we are free to choose their sign and Minkowski indices, which can be viewed as sort of a labeling 4 tetrad fields. However, in order to serve as the fundamental fields for constructing the conjugate pair ((Aε)ε , (Eε)ε), a particular labeling must be chosen which will be clear below. The suitable choice for labeling 4 orthogonal Colombeau generalized frame fields reads ( e0ε ) ε = ± [ 2m (tε)ε − 1 ]−1/2 (dtε)ε ; ( e1ε ) ε = ± [ (tε)ε ] [ (sin θε)ε ] [ (dφε)ε ] ;( e2ε ) ε = ± [ (tε)ε ] [ (dθε)ε ] ; ( e3ε ) ε = ± [ 2m (tε)ε − 1 ]1/2 [ (dxε)ε ] , (1.6.11) which gives the compatible Colombeau generalized spin connection components( ω30ε ) ε = − ( ω03ε ) ε = − m[ (t2ε)ε ] [(dxε)ε] ;( ω20ε ) ε = − ( ω02ε ) ε = [ 2m (tε)ε − 1 ]1/2 [ (dθε)ε ] ,( ω10ε ) ε = − ( ω01ε ) ε = [ 2m (tε)ε − 1 ]1/2 [ (sin θε)ε ] [ (dφε)ε ] ;( ω12ε ) ε = − ( ω21ε ) ε = [ (cos θε)ε ] [ (dφε)ε ] . (1.6.12) The (Aε)ε field can be constructed using spin connections: ( A3ε ) ε = ±γ ( − m (t2ε)ε )[ (dxε)ε ] , ( A2ε ) ε = ±γ [ 2m (tε)ε − 1 ]1/2 [ (dθε)ε ] ,( A1ε ) ε = ±γ [ 2m (tε)ε − 1 ]1/2 [ (sin θε)ε ] [ (dφε)ε ] , ( A3ε ) ε = ± [ (cos θε)ε ] [ (dφε)ε ] . (1.6.13) To construct Colombeau generalized field (Eε)ε on Σ we choose a gauge in which e 0 μ = nμ, the normal vector field to the spatial slice. This way we are in fact breaking the SO(3, 1) symmetry into SO(3) on a hypersurface with topology Σ = R× S2. The Colombeau generalized 3 triad fields become: 19 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ( e1ε ) ε = ± [( t 1/2 ε ) ε ] [ (sin θε)ε ] [ (dφε)ε ] ; ( e2ε ) ε = ± [ (tε)ε ] [ (dθε)ε ] ;( e3ε ) ε = ± [ 2m (tε)ε − 1 ]1/2 [ (dxε)ε ] , (1.6.14) with determinant (det (eε))ε = [( t2ε ) ε ] [ (sin θε)ε ] [ 2m (tε)ε − 1 ]1/2 , (1.6.15) and inverse triad (e1,ε)ε = ± 1[ (tε)ε ] [ (sin θε)ε ] [(∂φε)ε] ; (e2,ε)ε = ± 1[(tε)ε] [(∂θε)ε] ; (e3,ε)ε = ± [ 2m (tε)ε − 1 ]−1/2 [ (∂x,ε)ε ] . (1.6.16) The (Eε)ε fields become (E1,ε)ε = ± [ (tε)ε ] [ 2m (tε)ε − 1 ] [ 1/2 (∂φε)ε ] , (E2,ε)ε = ± [ (tε)ε ] [ 2m (tε)ε − 1 ]1/2 [ (sin θε)ε ] [ (∂θε)ε ] , (E3,ε)ε = ± [( t2ε ) ε ] [ (sin θε)ε ] [ (∂x,ε)ε ] . (1.6.17) The Colombeau generalized 3 triad fields (1.6.14) define their generalized compatible spin connection,( Γijε ∧ ejε ) ε + ( deiε ) ε = 0R : ( Γ12ε ) ε = − ( Γ21ε ) ε = [ (cos θε)ε ] [ (dφε)ε ] , (1.6.18) and( Γ3ε ) ε = 1 2 ( ε312Γ12ε + ε 321Γ21ε ) ε = [ (cos θε)ε ] [ (dφε)ε ] . (1.6.19) Extrinsic distributional curvature is related to (Aε)ε via γ (Kε)ε = (Aε)ε−(Γε)ε reads( K3rε,ε ) ε = 1 γ ( A3rε,ε ) ε = ± ( − m (t2ε)ε )[ (dxε)ε ] ,( K2θε,ε ) ε = 1 γ ( A2θε,ε ) ε = ± [ 2m (tε)ε − 1 ]1/2 , (1.6.20) ( K1φε,ε ) ε = 1 γ ( A1φε,ε ) ε = ± [ 2m (tε)ε − 1 ]1/2 [ (sin θε)ε ] . (1.6.21) Note that had we chosen other Minkowski indices for tetrad (1.6.11) we would not have obtained the conjugate pair(A,E) with correct indices satisfying ({ Aia,ε(xε), E b j (x ′ ε) }) ε = δij ( δbaδ(xε − x′ε) ) ε . The phase space variables are determined up to a sign freedom. By demanding E and A to satisfy the diffeomorphism, Gauss and Hamiltonian constraints, their signs can be fixed relative to each other. All components of diffeomorphism and Gauss constraints are zero except (Hθε,ε)ε = γ [ (tε)ε ] [ 2m (tε)ε − 1 ] [ (cos θε)ε ] { sign [ (A1φε,ε)ε ] + sign [ (A2θε,εA 3 φε,ε)ε ]} , (1.6.22) 20 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (G2,ε)ε = [ (tε)ε ] [ 2m (tε)ε − 1 ]1/2 [ (cos θε)ε ] { sign [ (Eθε2,ε)ε ] + sign [ (A3φεE φε1 ε )ε ]} , (1.6.23) and Hamiltonian constraint gives: (Cε)ε = [ (tε)ε ]( 2m[ (tε)ε ] − 1)[(sin2 θε)ε] {sign [(Eθε2,ε)ε]+ sign [(Eφε1,ε)ε]} . (1.6.24) For the above constraints to be zero we must have sign [ (Eθε2,ε)ε ] = −sign [ (Eφε1,ε)ε ] , sign [ (A3φε,ε)ε ] = +1, sign [ (A1φε,ε)ε ] = −sign [ (A2θε,ε)ε ] . (1.6.25) This leaves us with two alternatives corresponding to the residual gauge freedom ((bε)ε , (pbε,ε)ε) → (− (bε)ε ,− (pbε,ε)ε).( Aia,ε ) ε = [ (cε)ε ] τ3 [ (drε)ε ] + [ (bε)ε ] τ2 [ (dθε)ε ] +([ (cos θε)ε ] τ3 − [ (bε)ε ] [ (sin θε)ε ] τ1 ) [ (dφε)ε ]( Eai,ε ) ε = [ (pcε,ε)ε ] τ3 [ (sin θε)ε ] [ (∂rε)ε ] +[ (pbε,ε)ε ] τ2 [ (sin θε)ε ] [ (∂θε)ε ] − [ (pbε,ε)ε ] τ1 [ (∂φε)ε ] , (1.6.26) and ( Aia,ε ) ε = [ (cε)ε ] τ3 [ (drε)ε ] − [ (bε)ε ] τ2 [ (dθε)ε ] + ( [ (cos θε)ε ] τ3 + [ (bε)ε ] [ (sin θε)ε ] τ1)dφ( Eai,ε ) ε = [ (pcε,ε)ε ] τ3 [ (sin θε)ε ] [ (∂rε)ε ] − [ (pcε,ε)ε ] τ2 [ (sin θε)ε ] [ (∂θε)ε ] + + [ (pbε,ε)ε ] τ1 [ (∂φε)ε ] , (1.6.27) where, (bε)ε = ±γ [ 2m (tε)ε − 1 ]1/2 ; (cε)ε = ∓γ m[ (t2ε)ε ] , (1.6.28) (pcε,ε)ε = ± ( t2ε ) ε ; (pbε,ε)ε = [ (tε)ε ]([ 2m (tε)ε − 1 ])1/2 . (1.6.29) The momentum (pcε,ε)ε = ± ( t2ε ) ε is a monotonic generalized function on R and can be interpreted as an internal generalized time parameter. Region I. The analoguos calculations for region I with Colombeau generalized line element (1.6.10) leads to the folowing phase space coordinates( Ãia,ε ) ε = [ (cε)ε ] τ3 [ (drε)ε ] + [( bε ) ε ] τ2 [ (dθε)ε ] + ( [ (cos θε)ε ] τ3 − [( bε ) ε ] [ (sin θε)ε ] τ1) [ (dφε)ε ] ,( Ẽai,ε ) ε = [ (pcε,ε)ε ] τ3 [ (sin θε)ε ] [ (∂rε)ε ] +[ (pbε,ε)ε ] τ2 [ (sin θε)ε ] [ (∂θε)ε ] − [ (pbε,ε)ε ] τ1 [ (∂φε)ε ] , (1.6.30) 21 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ( Ãia,ε ) ε = [ (cε)ε ] τ3 [ (drε)ε ] − [( bε ) ε ] τ2 [ (dθε)ε ] + ( [ (cos θε)ε ] τ3 + [( bε ) ε ] [ (sin θε)ε ] τ1) [ (dφε)ε ] ,( Ẽai,ε ) ε = [ (pcε,ε)ε ] τ3 [ (sin θε)ε ] [ (∂rε)ε ] −[ (pbε,ε)ε ] τ2 [ (sin θε)ε ] [ (∂θε)ε ] + [ (pbε,ε)ε ] τ1 [ (∂φε)ε ] , (1.6.31) where,( bε ) ε = ±γ [ 1− 2m (xε)ε ]1/2 ; (cε)ε = ∓γ m[ (x2ε)ε ] ; (pcε,ε)ε = ± (x2ε)ε ; (pbε)ε = [ (xε)ε ] [ 1− 2m (xε)ε ]1/2 . (1.6.32) This defines variables (1.4.5)-(1.4.6) introduced above as (Axε,ε)ε = (cε)ε , (E xε ε )ε = (pcε,ε)ε ; γ (Kφε,ε)ε = ( bε ) ε ,( Eφεε ) ε = (pbε,ε)ε ; η = (2n+ 1)π, (P ηε ε )ε = 0R (1.6.33) which constitute a 4 dimensional phase space. Let us consider the following quantity on the point-free phase space mentioned above (Rε)ε ≡ 1 2πγG {[ (cε)ε ] , [√[ (|pcε,ε|)ε ]]} = sign [(pcε,ε)ε]√[ (|pcε,ε|)ε ] = 1[(tε)ε] . (1.6.33a) Following the canonical methods presented in [44], we expand now the holonomy along x direction of Σ = R× S2 with oriented generalized length (τε)ε as( h (τε) xε,ε ) ε = 1 + ε (∫ τε 0 dxεcετ3 ) ε + ( Oε(ε2) ) ε , (1.6.34) and rewrite (Rε)ε as (Rε)ε = 1 2πγG tr ( τ3 [( h (τε) xε,ε ) ε ]{[( h (τε)−1 xε,ε ) ε ] , √[ (|pcε,ε|)ε ]}) . (1.6.35) Now, quantization would be straightforward: ( Rε ) ε = 1 2πγl2Pl tr ( τ3 [( ĥ (τε) xε,ε ) ε ]{[( ĥ (τε)−1 xε,ε ) ε ] , √[ (|pcε,ε|)ε ]}) = 1 2πγl2Pl ([ cos (τεcε 2 ) ε ]√[ (|pcε,ε|)ε ] [ sin (τεcε 2 ) ε ] −[ sin (τεcε 2 ) ε ]√[ (|pcε,ε|)ε ] [ cos (τεcε 2 ) ε ]) . (1.6.36) Its action on ∣∣(τε)ε , (με)ε⟩ which are the simplified version of the spin network states in this reduced model (with (με)ε being the oriented length along the equator of S 2), then becomes: ( Rε |τε, με⟩ ) ε = 1 2π √ γlPl (√ | (τε)ε + 1| − √ | (τε)ε − 1| ) ∣∣(τε)ε , (με)ε⟩ . (1.6.37) Such operator ( Rε |τε, με⟩ ) ε has a bounded spectrum with maximum value of (√ 2π √ γlPl )−1 . 22 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 1.6.3. Thus the Colombeau generalized Kretschmann scalar curvature( Rρσμνε (tε) Rρσμν,ε (tε) ) ε , which is classically has infinite large point value ∼ cl [( ε−6 ) ε ] ∈ R (see Eq.(1.1.18)), at quantum level has a maximum value of : cl [( Rρσμνε (tε) Rρσμν,ε (tε) ) ε ]∣∣∣ max ∼ M2 cl [ (t6ε)ε ] ∣∣∣∣∣ max ≤R st ( M2 cl [ (t6ε)ε ] ∣∣∣∣∣ max ) ≤ M 2 γ3π6l6Pl . (1.6.38) Remark 1.6.4. Note that the Colombeau generalized curvature scalar ( Rε (tε) ) ε obtained at quantum level by point-free LQG by using similarly calculation as it has been applied above, has nonzero maximum value(∣∣∣Rε (tε)∣∣∣) ε ∣∣∣ max ∼ M l3Pl ∣∣∣∣ max . (1.6.39) Remark 1.6.5. We emphasize that in contrast with trivial (zero valued) result obtained at quantum level for Colombeau generalized curvature scalar ( Rε (tε) ) ε by using canonical LQG, see Remark 1.6.2, Colombeau generalized curvature scalar ( Rε (tε) ) ε obtained at quantum level by point-free LQG has nonzero maximum value given by Eq.(1.6.39). . 1.7 Generalized Stokes' Theorem 1.7.1 The Colombeau Generalized Curvilinear Coordinates Let us consider now the Colombeau generalized transformation from one generalized coordinate system,( x0ε ) ε , ( x1ε ) ε , ( x2ε ) ε , ( x3ε ) ε ,to another generalized coordinate system ( x′0ε ) ε , ( x′1ε ) ε , ( x′2ε ) ε , ( x′3ε ) ε : transform according to the relation( xiε ) ε = ( f iε ( x′0ε , x ′1 ε , x ′2 ε , x ′3 ε )) ε , (1.7.1) where the ( f iε ) ε are certain Colombeau generalized functions and where ( Jε ( x′0, x′1, x′2, x′3 )) ε ( Jε ( x′0ε , x ′1 ε , x ′2 ε , x ′3 ε )) ε = ( ∂ ( x0ε, x 1 ε, x 2 ε, x 3 ε ) ∂ (x′0ε , x′1ε , x′2ε , x′3ε ) ) ε = 0R (1.7.2) is the Jacobian of the Colombeau generalized transformation (1.7.1). Remark 1.7.1. When we transform the coordinates, their Colombeau differentials ( dxiε ) ε transform according to the relation ( dxiε ) ε = ( ∂xiε ∂x′kε dx′kε ) ε = [( ∂xiε ∂x′kε ) ε ] ( dx′kε ) ε . (1.7.3) Definition 1.7.1. Every tuple of four Colombeau quantities ( Aiε ) ε , i = 0, 1, 2, 3, which under a transformation (1.7.1) of coordinates, transform like the Colombeau coordinate differentials (1.7.2), is called Colombeau contravariant four-vector:( Aiε ) ε = ( ∂xiε ∂x′kε A′kε ) ε = [( ∂xiε ∂x′kε ) ε ] ( A′kε ) ε . (1.7.4) 23 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Let (φε)ε be the Colombeau scalar. Under a coordinate transformation (1.7.1), the four Colombeau quantities ( ∂φε ∂xiε ) ε , i = 0, 1, 2, 3 transform according to the formula ( ∂φε ∂xiε ) ε = ( ∂φε ∂x′kε ∂x′kε ∂xiε ) ε = ( ∂φε ∂x′kε ) ε ( ∂x′kε ∂xiε ε ) . (1.7.5) Definition 1.7.2. Every tuple of four Colombeau generalized functions (Ai,ε)ε which, under a coordinate transformation (1.7.1), transform like the Colombeau derivatives of a scalar, is called Colombeau generalized covariant four-vector (Ai,ε)ε = ( ∂x′kε ∂xiε A′k,ε ) ε = ( ∂x′kε ∂xiε ) ε ( A′k,ε ) ε . (1.7.6) Definition 1.7.3. We call the Colombeau generalized contravariant tensor of the second rank,( Aikε ) ε ,any tuple of sixteen Colombeau generalized functions which transform like the products of the components of two Colombeau generalized contravariant vectors, i.e. according to the law ( Aikε ) ε = ( ∂x′kε ∂xiε ∂x′mε ∂xiε A′im,ε ) ε = ( ∂x′kε ∂xiε ∂x′mε ∂xiε ) ε ( A′im,ε ) ε (1.7.7) and a mixed Colombeau generalized tensor transforms as follows ( Aik,ε ) ε = ( ∂xiε ∂x′lε ∂x′mε ∂xkε A′lm,ε ) ε = ( ∂xiε ∂x′lε ∂x′mε ∂xkε ) ε ( A′lm,ε ) ε . (1.7.8) Remark 1.7.2. Note that the scalar product of two four-vectors ( AiεBi,ε ) is invariant since ( AiεBi,ε ) ε = ( ∂xiε ∂x′lε ∂x′mε ∂xiε A′lεB ′ m,ε ) ε = ( ∂x′mε ∂x′lε A′lεB ′ m,ε ) ε = ( A′lεB ′ l,ε ) ε . (1.7.9) The unit four-tensor δik is defined the same as in classical case: δ i k = 0 for i = k and δik = 1 for i = k.If ( Akε ) ε is a Colombeau generalized four-vector,then multiplying by δik we obtain ( Akεδ i k ) ε = ( Aiε ) ε , (1.7.10) i.e. again Colombeau generalized four-vector; this proves that δik is a tensor. Remark 1.7.3. The square of the Colombeau generalized line element ( ds2ε ) ε in curvilinear coordinates is a quadratic form in the differentials dxi, i = 0, 1, 2, 3 :( ds2ε ) ε = ( gik,εdx idxk ) ε = [ (gik,ε)ε ] dxidxk. (1.7.11) where the (gik,ε)ε are Colombeau generalized functions of the coordinates; (gik,ε)ε is symmetric in the indices i and k : (gik,ε)ε = (gki,ε)ε . (1.7.12) Definition 1.7.4.Since the (contracted) product of (gik,ε)ε and the contravariant tensor dxidxkis a scalar, the (gik,ε)ε form a covariant tensor; it is called the Colombeau generalized metric tensor. Definition 1.7.5. Two tensors (Aik,ε)ε and ( Bikε ) ε are said to be reciprocal to each other if 24 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ( Aik,εB ik ε ) ε = [ (Aik,ε)ε ] [( Bikε ) ε ] = δik. (1.7.13) In particular the contravariant metric tensor is the tensor (gik,ε)εreciprocal to the tensor( gikε ) ε ,that is,{ (gik,ε)ε }{( gikε ) ε } = δik. (1.7.14) The same physical quantity can be represented in contravariant or covariant components. It is obvious that the only quantities that can determine the connection between the different forms are the components of the metric tensor. This connection is given by the formulas:( Aiε ) ε = ( gikε Ak,ε ) ε , (Ai,ε)ε = ( gik,εA k ε ) ε . (1.7.15) These remarks also apply to Colombeau generalized tensors. The transition between the different forms of a given physical generalized tensor is accomplished by using the metric tensor according to the formulas:( Aik,ε ) ε = ( gilε Alk,ε ) ε , ( Aikε ) ε = ( gilε g km ε Alm,ε ) ε , etc. (1.7.16) The completely antisymmetric unit pseudotensor in galilean coordinates we denote by eiklm.Let us transform it to an arbitrary system of Colombeau generalized coordinates, and now denote it by ( Eiklmε ) ε . We keep the notation eiklm for the quantities defined as before by e0123 = 1 (or e0123 = −1).Let the x′i, i = 0, 1, 2, 3 be galilean, and the ( xiε ) ε , i = 0, 1, 2, 3 be arbitrary Colombeau generalized curvilinear coordinates. According to the general rules for transformation of Colombeau generalized tensors, we have ( Eiklmε ) ε = [( ∂xiε ∂x′p ∂xkε ∂x′r ∂xlε ∂x′s ∂xmε ∂x′t ) ε ] eprst, (1.7.17) or( Eiklmε ) ε = {( Jε ( x′0, x′1, x′2, x′3 )) ε } eprst, (1.7.18) where ( Jε ( x′0, x′1, x′2, x′3 )) ε = 0R is the determinant formed from the derivatives ∂x i/∂x′p, i.e. it is just the Colombeau generalized Jacobian of the Colombeau generalized transformation from the galilean to the Colombeau generalized curvilinear coordinates: ( Jε ( x′0, x′1, x′2, x′3 )) ε = ( ∂ ( x0ε, x 1 ε, x 2 ε, x 3 ε ) ∂ (x′0, x′1, x′2, x′3) ) ε . (1.7.19) This Colombeau generalized Jacobian can be expressed in terms of the determinant of the Colombeau generalized metric tensor (gik,ε)ε (in the system ( xiε ) ε ). To do this we write the formula for the transformation of the metric tensor:( gikε ) ε = [( ∂xiε ∂x′i ∂xkε ∂x′m ) ε ] g(0)im, (1.7.20) where g(0)im = g (0) im =  1 1 1 1 1 −1 1 1 1 1 −1 1 1 1 1 −1  , (1.7.21) 25 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance and equate the determinants of the two sides of this equation. The determinant of the reciprocal tensor det ∣∣(gikε )ε∣∣ = 1/ (gε)ε. The determinant det |g(0)im| = −1. Thus we have 1/ (gε)ε = − ( J2ε ( x′0, x′1, x′2, x′3 )) ε , and so( J2ε ( x′0, x′1, x′2, x′3 )) ε = 1/ √ (gε)ε. (1.7.22) Thus, in curvilinear coordinates the antisymmetric unit tensor of rank four must bedefined as( Eiklmε ) ε = 1√ − (gε)ε eiklm (1.7.23) and its covariant components are (Eiklm,ε)ε = √ − (gε)εeiklm. (1.7.24) In a galilean coordinate system x′i, i = 0, 1, 2, 3 the integral of a scalar with respect to dΩ′ = dx′0dx′1dx′2dx′3 is also a scalar, i.e. the element dΩ′ behaves like a scalar in the integration. On transforming to Colombeau generalized curvilinear coordinates ( xiε ) ε , i = 0, 1, 2, 3, the element of integration dΩ′ goes over into dΩ′ := {( J−1ε ) ε } dΩ = √ − (gε)ε (dΩε)ε , (1.7.25) where (dΩε)ε = {( dx0ε ) ε }{( dx1ε ) ε }{( dx2ε ) ε }{( dx3ε ) ε } . Thus, in Colombeau generalized curvilinear coordinates, when integrating over a four-volume the quantity √ − (gε)ε (dΩε)ε behaves like an invariant. Remark 1.7.4. The element of "area" of the Colombeau generalized hypersurface spanned by three infinitesimal Colombeau generalized displacements is the contravariant antisymmetric Colombeau generalized tensor ( dSiklε ) ε ; the vector dual to it is gotten by multiplying by the tensor √ − (gε)εeiklm, so it is equal to√ − (gε)ε (dSε,i)ε = − 1 6 √ − (gε)εeiklm ( dSkimε ) ε . (1.7.26) Remark 1.7.5. Let ( df ikε ) ε be the element of two-dimensional Colombeau generalized surface spanned by two infinitesimal Colombeau generalized displacements, the dual Colombeau generalized tensor is defined as√ − (gε)ε ( df ∗ik,ε ) ε = 1 2 √ − (gε)εeiklm ( df lmε ) ε . (1.7.27) We will be use the designations (dSε,i) and ( df ∗ki,ε ) ε for eiklm ( dSkimε ) ε and eiklm ( df lmε ) ε (and not for their products by √ − (gε)ε). 1.7.2 Generalized Stokes' Theorem Remark 1.7.6. Note that the canonical rules for transforming the various integrals into one another remain the same, since their derivation was formal in character and not related to the tensor properties of the different quantities. Of particular importance is the rule for transforming the integral over a hypersurface into an integral over a four-volume (Gauss' theorem), which is accomplished by the substitution (dSi,ε)ε := [ (dΩε)ε ]( ∂ ∂xiε ) ε . (1.7.28) 26 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 1.7.7. (Stokes' theorem) Note that for the integral of Colombeau generalized vector ( Aiε ) ε we have(∮ AiεdSi,ε ) ε = (∫ ∂Aiε ∂xiε dΩε ) ε = ∫ [(∂Aiε ∂xiε ) ε ] [ (dΩε)ε ] . (1.7.29) which is the generalization of Stokes' theorem. Note that in galilean coordinates the Colombeau generalized differentials (dAi,ε)ε of a vector (Ai,ε)ε form the Colombeau generalized vector, and the derivatives ( ∂Ai,ε/∂x k ε ) ε of the components of a vector with respect to the coordinates form the Colombeau generalized tensor. In Colombeau generalized curvilinear coordinates this is not so; (dAi,ε)ε is not a vector, and ( ∂Ai,ε/∂x k ε ) ε is not the Colombeau generalized tensor.This is due to the fact that (dAi,ε)ε is the difference of vectors located at different (infinitesimally separated) points of space; at different points in space vectors transform differently, since the coefficients in the transformation formulas (1.7.3), (1.7.4) are Colombeau generalized functions of the generalized coordinates.Thus in order to compare two infmitesimally separated generalized vectors we must subject one of them to a parallel translation to the point where the second is located. Let us consider an arbitrary generalized contravariant vector ; if its value at the point xi is ( Aiε ) ε , then at the neighboring point xi + dxi it is equal to ( Aiε ) ε + ( dAiε ) ε = ( Aiε + dA i ε ) ε . We subject the vector ( Aiε ) ε to an infinitesimal parallel displacement to the point xi + dxi; the change in the vector which results from this we denote by ( δAiε ) ε Then the difference ( DAiε ) ε between the two Colombeau generalized vectors which are now located at the same point is( DAiε ) ε = ( dAiε ) ε − ( δAiε ) ε . (1.7.30) The change ( δAiε ) ε in the components of Colombeau generalized vector under an infinitesimal parallel displacement depends on the values of the components themselves, where the dependence must clearly be linear. This follows directly from the fact that the sum of two Colombeau generalized vectors must transform according to the same law as each of the constituents. Thus ( δAiε ) ε has the form( δAiε ) ε = − ( Γikl,εA k εdx l ) ε , (1.7.31) where ( Γikl,ε ) ε the are certain Colombeau generalized functions of the coordinates. Their form depends, of course, on the coordinate system; for a galilean coordinate system ( Γikl,ε ) ε = 0R. From this it is already clear that the quantities ( Γikl,ε ) ε do not form Colombeau generalized tensor, since a tensor which is equal to zero in one coordinate system is equal to zero in every other one. In a curvilinear space it is, of course, impossible to make all the ( Γikl,ε ) ε vanish over all of space. But we can choose a coordinate system for which the ( Γikl,ε ) ε become 0R over a given infinitesimal region. The quantities ( Γikl,ε ) ε , are called generalized Christoffel symbols. In addition to the quantities ( Γikl,ε ) ε we shall later also use Colombeau generalized quantities (Γi,kl,ε)ε defined as follows (Γi,kl,ε)ε = ( gim,εΓ m km,ε ) ε . (1.7.32) Conversely,( Γikl,ε ) ε = ( gimε Γm,kl,ε ) ε . (1.7.33) It is also easy to relate the change in the components of a covariant vector under a parallel displacement to the Christoffel symbols. To do this we note that under a parallel displacement, a scalar is unchanged. In particular, the scalar product of two vectors does not change under a 27 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance parallel displacement.Let (Ai,ε)ε and ( Biε ) ε be any covariant and contravariant vectors. Then from δ ( Ai,εB i ε ) ε = 0R, we have( BiεδAi,ε ) = − ( Ai,εδB i ε ) ε = ( Γikl,εB k εAi,εdx l ) ε (1.7.34) or, changing the indices,( BiεδAi,ε ) = ( Γkil,εB i εAk,εdx l ) ε (1.7.35) From this, by the arbitrariness of the ( Biε ) ε one obtains (δAi,ε)ε = (( Γkil,εAk,ε ) ε ) dxl (1.7.36) which determines the change in a covariant vector under a parallel displacement. Substituting (1.7.31) and ( dAiε ) ε = ( (∂Aiε/∂x l)ε ) dxl in (1.7.30), we obtain ( DAiε ) ε = [( ∂Aiε ∂xl ) ε + ( Γikl,εA k ε ) ε ] dxl. (1.7.37) . 1.8 The Colombeau Generalized Curvature Tensor In this subsection we derive the general formula for the change in a vector after parallel displacement around any infinitesimal closed contour γ. This generalized change (∆Ak,ε)ε ∈ R can clearly be written in the form (∮ δAk,ε ) ε , where the Colombeau integral is taken over the given regular contour γ. Substituting in place of (δAk,ε)εthe expression (1.7.36), we get (∆Ak,ε)ε = (∮ γ Γikl,ε (x)Ai (x) dx l ) ε ∈ R, (1.8.1) where for any i, k, l = 0, 1, 2, 3 : ( Γikl,ε (x) ) ε ∈ G(R4), x = ( x0, x1, x2, x3 ) , Ai (x) ∈ D (G) , G ⊆ R4. Note that the vector Ai which appears in the integrand obviously changes as we move along the contour γ. Definition 1.8.1.We will say that generalized change (∆Ak,ε)ε exists in the sense of the Schwartz distributions if for any Ai (x) ∈ D (G) the limit: limε→0 ∆Ak,ε exists. Of course in this case obviously cl [( Γikl,ε (x) ) ε ] ∈ D′ (G) and cl [ (∆Ak,ε)ε ] ∈ R. For the further transformation of this Colombeau integral, we must note the following. The values of the vector Ai at points inside the contour are not unique; they depend on the path along which we approach the particular point. However, as we shall see from the result obtained below, this non-uniqueness is related to terms of second order. We may therefore, with the first-order accuracy which is sufficient for the transformation, regard the components of the vector (Ai,ε)ε at points inside the infinitesimal contour γ as being uniquely determined by their values on the contour itself by the formulas (δAi (x))ε = ( Γnil,ε (x)An,ε (x) dx l ) ε , (1.8.2) i.e., by the derivatives ∂Ai,ε (x) ∂xl = ( Γnil,ε (x)An,ε (x) ) ε . (1.8.3) Now applying generalized Stokes' theorem (see Theorem 1.7.1) to the integral (1.8.1) and considering 28 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance that the area enclosed by the contour has the infinitesimal value ( ∆f imε ) ε , we get: (∆Ak,ε)ε = 1 2 [( ∂ ( Γikm,ε (x)Ai (x) ) ∂xl ) ε − ( ∂ ( Γikl,ε (x)Ai (x) ) ∂xm ) ε ] ( ∆f imε ) ε = = 1 2 [ Ai (x) ( ∂ ( Γikm,ε (x) ) ∂xl ) ε −Ai (x) ( ∂ ( Γikl,ε (x) ) ∂xm ) ε +( ∂Ai (x) ∂xl )( Γikm,ε (x) ) ε − ( ∂Ai (x) ∂xm )( Γikl,ε (x) ) ε ] ( ∆f imε ) ε . (1.8.4) Definition 1.8.2. Colombeau generalized k-form (ωε)ε on a differentiable manifold M is a smooth section of the bundle of alternating Colombeau generalized k-tensors on M . Equivalently, (ωε)ε associates to each x ∈M an alternating Colombeau generalized k-tensor (ωx,ε)ε, in such a way that in any chart for M , the coefficients(ωi1...ik,ε)ε are Colombeau generalized functions. Theorem 1.8.1. (Generalized Stokes' Theorem) Let (ωε)ε be Colombeau generalized differential form.Then the Colombeau integral of a differential form (ωε)ε over the boundary of some orientable manifold Σ ⊂M is equal to the integral of its exterior Colombeau derivative (dωε)ε over the whole of Σ, i.e., ∫ ∂Σ (ωε)ε = ( ∫ ∂Σ ωε ) ε = (∫ Σ dωε ) ε = ∫ Σ (dωε)ε . (1.8.5) Proof. Immediately from the classical Stokes' Theorem and definitions. Example 1.8.1. For example, for the integral of Colombeau generalized vector (Ai,ε (x))ε we have(∮ Γ Ai,εdx i ) ε = (∫ Σ df ki ∂Ai,ε ∂xk ) ε = 1 2 (∫ [( df kiε ) ε ](∂Ak,ε ∂xi − ∂Ai,ε ∂xk )) ε = 1 2 ∫ [( df kiε ) ε ](∂Ak,ε ∂xi − ∂Ai,ε ∂xk ) ε = 1 2 ∫ [( df kiε ) ε ] [(∂Ak,ε ∂xi ) ε − ( ∂Ai,ε ∂xk ) ε ] , (1.8.6) where Γ = ∂Σ and ( df kiε ) ε = ( dxiεdx ′k ε ) ε − ( dxkεdx ′i ε ) ε is the infinitesimal element of surface which is given by the antisymmetric tensor of second rank ( df kiε ) ε . Substituting the values of the derivatives (1.4.3) into Eq.(1.4.4), we get (∆Ak,ε)ε = 1 2 ( Riklm,ε (x)Ai,ε (x)∆f im ε ) ε , (1.8.7) where ( Riklm,ε (x) ) ε is a Colombeau generalized tensor field of the fourth rank: ( Riklm,ε (x) ) ε = ( ∂ ( Γikm,ε (x) ) ∂xl ) ε − ( ∂ ( Γikl,ε (x) ) ∂xm ) ε +( Γini,ε (x) Γ n km,ε (x) ) ε − ( Γinm,ε (x) Γ n kl,ε (x) ) ε . (1.8.8) Definition 1.8.3. The tensor field ( Rlkim,ε (x) ) ε is called the distributional curvature tensor or the distributional Riemann tensor. Remark 1.8.1. Note that in general case for any i, k, l = 0, 1, 2, 3 : cl [( Riklm,ε (x) ) ε ] ∈ G(R4). 29 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Definition 1.8.4. We will say that the distributional Riemann tensor ( Riklm,ε (x) ) ε exists in the sense of the Schwartz distributions if for any i, k, l = 0, 1, 2, 3 and for any Ai (x) ∈ D (G) the limit: limε→0 ∫ G Riklm,ε (x)Ai (x) d 4x (1.8.9) exists. Definition 1.8.5. We will say that the distributional Riemann tensor ( Riklm,ε (x) ) ε exists in the classical sense at point x ∈ R if there exists standard part of point value of Colombeau generalized function ( Riklm,ε (x) ) ε at point x ∈ R,i.e. st ( cl [( Riklm,ε (x) ) ε ]) ∈ R. From the expression (1.8.8) it follows directly that ∀x ∈ R the distributional curvature tensor is antisymmetric in the indices l and m :( Riklm,ε (x) ) ε = − ( Rikml,ε (x) ) ε (1.8.10) and therefore ∀z = (xε)ε ∈ R the following identity holds( Riklm,ε (xε) ) ε = − ( Rikml,ε (xε) ) ε . (1.8.11) 1.9 Generalized Einstein's Field Equations The action functional for the gravitational field reads [33,36,37](∫ Rε √ −gεdΩ ) ε . (1.9.1) The invariant Colombeau integral (1.9.1) can be transformed by means of Gauss'theorem to the integral of an expression not containing the second derivatives. Thus Colombeau integral (1.9.1) can be presented in the following form (∫ Rε √ −gεdΩ ) ε = (∫ Gε √ −gεdΩ ) ε + (∫ ∂ (√−gεwiε) ∂xi dΩ ) ε , (1.9.2) where (Gε)ε contains only the tensor (gik,ε)ε and its first derivatives, and the integrand of the second integral has the form of a divergence of a certain quantity ( wiε ) ε .According to Gauss' theorem, this second integral can be transformed into an integral over a hypersurface surrounding the four-volume over which the integration is carried out in the other two integrals. When we vary the action, the variation of the second term on the right vanishes, since in the principle of least action, the variations of the field at the limits of the region of integration are zero. Consequently, we may write δ (∫ Rε √ −gεdΩ ) ε = ( δ ∫ Rε √ −gεdΩ ) ε = ( δ ∫ Gε √ −gεdΩ ) ε . (1.9.3) The left side is Colombeau scalar; therefore the expression on the right is also Colombeau scalar (the quantity (Gε)ε itself is, of course, not Colombeau scalar). The quantity (Gε)ε satisfies the condition imposed above, since it contains only the (gik,ε)ε and its Colombeau derivatives. Thus finally we obtain δS [ (gε)ε ] = − c 3 16πκ ( δ ∫ Gε √ −gεdΩ ) ε = − c 3 16πk ( δ ∫ Rε √ −gεdΩ ) ε . (1.9.4) The constant κ is called the gravitational constant. The dimensions of κ follow from (1.9.4). Its numerical value is κ = 6.67× 10−8sm3×gr−1 × sec−2 . We now proceed to the derivation of the equations of the gravitational field. These equations are obtained from the principle of least action δ((Sm,ε)ε + (Sgε)ε) = 0R, where (Sm,ε)ε and (Sgε)εare the distributional actions of the gravitational field and matter respectively. We now subject the 30 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance gravitational Colombeau metric field, that is,the quantities gik, to variation. Calculating the variation δ (Sgε)ε, we get δ (∫ Rε √ −gεdΩ ) ε = ( δ ∫ Rε √ −gεdΩ ) ε = ( δ ∫ gikε Rik,ε √ −gεdΩ ) ε ={(∫ Rik,ε √ −gεδgikε dΩ ) ε + (∫ Rik,εg ik ε δ √ −gεdΩ ) ε + (∫ gikε √ −gεδRik,εdΩ ) ε }∫ {( Rik,ε √ −gεδgikε ) ε + ( Rik,εg ik ε δ √ −gε ) ε + ( gikε √ −gεδRik,ε ) ε } dΩ. (1.9.5) Thus, the variation S [ (gε)ε ] is equal to S [ (gε)ε ] = − c 3 16πκ (∫ { Rik,ε − 1 2 gik,εRε } √ −gεδgikε dΩ ) ε . (1.9.6) Remark 1.9.1. We note that if we had started from the expression δSg [ (gε)ε ] = − c 3 16πκ ( δ ∫ Gε √ −gεdΩ ) ε (1.9.7) for the action of the field, then we get δS [ (gε)ε ] = − c 3 16πκ ∫ δ ( gikε ) ε dΩ  ( ∂ {Gε √ −gε} ∂gikε ) ε −  ∂∂xl ∂ {Gε √ −gε} ∂ ∂gikε ∂xl  ε  . (1.9.8) Comparing Eq.(1.9.8) with Eq.(1.9.6), we get (Rik,ε)ε − 1 2 (gik,εRε)ε ={( 1√ −gε ) ε } ( ∂ {Gε √ −gε} ∂gikε ) ε −  ∂∂xl ∂ {Gε √ −gε} ∂ ∂gikε ∂xl  ε  . . (1.9.9) For the variation of the action of the matter we can write (δSm,ε)ε = 1 2c (∫ Tik,ε √ −gεδgikε dΩ ) ε , (1.9.10) where (Tik,ε)ε ∈ G(R 4) is the generalized energy-momentum tensor of the matter fields. Thus, from the principle of least action δ { S [ (gε)ε ] + (Sm,ε)ε } = 0R (1.9.11) one obtains − c 3 16πκ (∫ { Rik,ε − 1 2 gik,εRε − 8πκ c4 Tik,ε } √ −gεδgikε dΩ ) ε . = 0R (1.9.12) From Eq.(1.9.12), since of the arbitrariness of the ( δgikε ) ε ∈ G(R4) finally we get (Rik,ε)ε − 1 2 (gik,εRε)ε = 8πκ c4 (Tik,ε)ε (1.9.13) 31 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance or, in mixed components, ( Rki,ε ) ε − 1 2 δki (Rε)ε = 8πκ c4 ( T ki,ε ) ε . (1.9.14) They are called the generalized Einstein equations. Contracting (1.9.14) on the indices i and k,we get (Rε)ε = − 8πκ c4 ( T ii,ε ) ε = −8πκ c4 (Tε)ε . (1.9.15) Therefore the generalized Einstein equations of the field can also be written in the form (Rik,ε)ε = 8πκ c4 { (Tik,ε)ε − 1 2 (gik,εTε)ε } . (1.9.16) Note that the generalized Einstein equations of the gravitational field are nonlinear Colombeau equations.[37]. 1.10 The breakdown of canonical formalism of Riemann geometry for the singular solutions of the Einstein field equations 1.10.1 The Densitized Einstein Field Equations Revisited The densitized Einstein field equations originally considered in A. Einstein and N. Rosen paper [32], see also [46]. Remark 1.10.1. Note that if some components of the Riemann curvature tensor Riklm (x) become infinite at point x0 one obtain the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood Ω of the point x0 ∈ Ω,i.e. in such neighborhood Ω Riemann curvature tensor Riklm (x) will be changed by formula (1.10.7) see remark 1.10.2. Remark 1.10.2. Let Γ be infinitesimal closed contour and let ΣΓ be the corresponding surface spanning by Γ, see Fig.1. We assume now that: (i) christoffel symbol Γikl (x) become infinite at singular point x0 by formulae{ Γikl (x) ≍ Ξkl (x) ( xi − x0i )−δ , δ ≥ 1 Ξkl (x) ∈ C∞ (ΣΓ) (1.10.1) and (ii) x0 ∈ ΣΓ.Let us derive now to similarly canonical calculation [3]-[4] the general formula for the regularized change ∆Ak in a vector Ai (x) after parallel displacement around infinitesimal closed (1.9.16) contour Γ. This regularized change ∆Ak can clearly be written in the form ∆Ak = ∮ Γ Φ ( x− x0 ) δAk, (1.10.2) where Φ ( x− x0 ) = Π4i=0 ( xi − x0i )2δ , δ ≥ 1 and where the integral is taken over the given contour Γ. Substituting in place of δAk the canonical expression δAk = Γ i kl (x)Akdx l (see [9], Eq. (85.5)) we obtain ∆Ak = ∫ Γ Φ ( x− x0 ) δAk = ∫ Γ Φ ( x− x0 ) Γikl (x)Akdx l , (1.10.3) where 32 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Fig. 2. Infinitesimal closed contour Γand corresponding singular surface ΣΓ ∋ x0 spanning by Γ. Due to the degeneracy of the metric (1.10.12) at point r=0, the Levi-Civita' connection Γ+lkj ({}) = = 1 2 [ glm ({}) ] [(gmk,j ({}) + gmj,k ({})− gkj,m ({})] is not available on R 3+ ∪ {0} in canonical sense but only in an distributional sense. ∂Ai ∂xl = Γikl (x)Ak. (1.10.4) Now applying Stokes' theorem (see [4],Eq.(6.19)) to the integral (1.10.3) and considering that the area enclosed by the contour has the infinitesimal value ∆f lm, we get ∆Ak = ∮ Γ Φ ( x− x0 ) Γikl (x)Akdx l = = 1 2 ∫ ΣΓ [ ∂ ( Γikm (x)AiΦ ( x− x0 )) ∂xl − ∂ ( Γikl (x)AiΦ ( x− x0 )) ∂xm ] df lm ≈ ≈ [ ∂ ( Γikm (x)AiΦ ( x− x0 )) ∂xl − ∂ ( Γikl (x)AiΦ ( x− x0 )) ∂xm ] ∆f lm 2 =[ Φ ( x− x0 ) ∂ (Γikm (x− x0)Ai) ∂xl + ( Γikm (x)Ai ) ∂Φ (x− x0) ∂xl − −Φ ( x− x0 ) ∂ (Γikl (x)Ai) ∂xm − ( Γikl (x)Ai ) ∂Φ (x− x0) ∂xm ] ∆f lm 2 =[ Φ ( x− x0 ) ∂ (Γikm (x)Ai) ∂xl − Φ ( x− x0 ) ∂ (Γikl (x)Ai) ∂xm − Ai (x)Φ ( x− x0 ) 2δΓikm (x) xl − x0l −Ai (x)Φ ( x− x0 ) 2δΓikl (x) xm − x0m ] ∆f lm 2 . (1.10.5) Substituting the values of the derivatives (1.10.4) into Eq.(1.10.5), we get finally: ∆Ak = R i klm Ai (x)Φ ( x− x0 ) ∆f lm 2 , (1.10.6) 33 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Fig. 3. Infinitesimal closed contour Γ with a singularity at point x0 on Horizon and corresponding singular surface ΣΓ ∋ x0 spanning by Γ. Due to the deheneracy of the metric (1.12) at r=2m, the Levi-Civita' connection Γ+lkj ({}) = = 1 2 [ glm ({}) ] [(gmk,j ({}) + gmj,k ({})− gkj,m ({})] is not available on horizon in canonical sense but only in a distributional sense. where Riklm, is a tensor of the fourth rank Riklm = R i klm + 2δ [ Γikm (x) xl − x0l − Γ i kl (x) xm − x0m ] . (1.10.7) Here Riklm is the classical Riemann curvature tensor.That R i klm is a tensor is clear from the fact that in (1.10.6) the left side is a vector-the difference ∆Ak between the values of vectors at one and the same point. Note that an similar results was obtained by many authors [5]-[17] by using Colombeau nonlinear generalized functions [1]-[2]. Definition1.10.1. The tensor Riklm is called the generalized curvature tensor or the generalized Riemann tensor. Definition1.10.2. The generalized Ricci curvature tensor Rkm is defined as Rkm = R i kim. (1.10.8) Definition1.10.3. The generalized Ricci scalar R is defined as R = gkm Rkm. (1.10.9) Definition1.10.4. The generalized Einstein tensor Gkm is defined as Gkm = Rkm − 1 2 gkmR. (1.10.10) Remark 1.10.3. (I) Note that the Schwarzschild spacetime is well defined only for r > 2m. 34 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance The boundary of the manifold {r > 2m} in R3×R is the submanifold {r = 2m} of R3×R, diffeomorfic to a product S2 × R. This submanifold is colled the event horizon, or simply the horizon [33],[34]. (II) The Schwarzschild metric (1.10.12) in canonical coordinates (x0, r, θ, φ), with m > 0, ceases to be a smooth Lorentzian metric for r = 2m, because for such a value of r the coefficient g00 becomes zero while g11 becomes infinite. For 0 < r < 2m the metric (1.10.12) again a smooth Lorentzian metric but t is a space coordinate while r is a time coordinate. Hence the metric (1.10.12) cannot be said to be either spherically symmetric or static for r < 2m [33]. (III) From consideration above obviously it follows the metric (1.10.12) that on Schwarzschild spacetime Sh = ( S2 × {r > 2m} ∪ {0 < r < 2m} ) × R the Levi-Civita connection { Γ+lkj ({}) == 1 2 [ glm ({}) ] [(gmk,j ({}) + gmj,k ({})− gkj,m ({})] (1.10.11) is not available in classical sense and that is well known many years from mathematical literature, see for example [22] section 6 and Remark 1.10.1 Remark 1.10.2 above. (IV) Note that [4] : (i) The determinat det (glm({})) = −r4 sin2 θ of the metric (1.10.12) is reqular on horizon,i.e., smooth and non-vanishing for r = 2m. In addition: (ii) The curvature scalar R({}) = gμνRμν({}) is zero for r = 2m. (iii) The none of higher-order scalars such as Rμν({})Rμν({}),etc. blows up. For example the quadratic scalarRρσμν({})Rρσμν({}) = 48m2/r6 is reqular on horizon,i.e.,smooth and non-vanishing for r = 2m. (V) Note that: (i) In physical literature (see for example [4],[33],[35],) it was wrongly assumed that a properties (i)-(iii) is enough to convince us that r = 2m represent an non honest physical singularity but only coordinate singularity. (VI) Such assumption based only on formal extensions R({}), Rμν({})Rμν({}), ..., Rρσμν({})Rρσμν({}) of the curvature scalar R({}) and higher-order scalars such as Rμν({})Rμν({}), ...,Rρσμν({})Rρσμν({}) on horizon r = 2m and on origin r = 0 by formulae R(r) ∣∣∣ r=2m = lim r→2m R(r) = 0, R(r) ∣∣∣ r=0 = lim r→0 R(r) = 0 ............................ Rρσμν(r)Rρσμν(r) ∣∣∣ r=2m = lim r→2m (Rρσμν(r)Rρσμν(r)) = lim r→2m 48m2 r6 = 48m2 r6 ∣∣∣∣ r=2m , Rρσμν(r)Rρσμν(r) ∣∣∣ r=0 = lim r→0 (Rρσμν(r)Rρσμν(r)) = lim r→0 48m2 r6 = ∞. 35 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance However in the limit r → 2m the Levi-Civitá connection Γ+lkj ({}) becomes infinite [4]: Γ100 (r) ∣∣∣ r=2m = lim r→2m m (r − 2m) r3 = 0, Γ111 (r) ∣∣∣ r=2m = lim r→2m −m r (r − 2m) = ∞, Γ001 (r) ∣∣∣ r=2m = lim r→2m m r (r − 2m) = ∞, Γ212 (r) ∣∣∣ r=2m = lim r→2m 1 r = 2−1m−1, Γ122 ∣∣∣ r=2m = − lim r→2m (r − 2m) = 0, Γ313 ∣∣∣ r=2m = lim r→2m 1 r = 2−1m−1, Γ133 ∣∣∣ r=2m = − lim r→2m (r − 2m) sin2 θ = 0, Γ100 (r) ∣∣∣ r=0 = lim r→0 m (r − 2m) r3 = ∞ Γ111 (r) ∣∣∣ r=0 = lim r→0 −m r (r − 2m) = ∞, .................................................................... Γ233 = − sin θ cos θ,Γ 3 23 = cos θ sin θ . Thus obviously by consideration above (see Remark 1.10.1-Remark 1.10.2) this extension given by Eq.(1.10.15) has no any sense in respect of the canonical Riemannian geometry. (VII) From consideration above (see Remark 1.10.1-Remark 1.10.2) it obviously follows that the scalars such as R({}), Rμν({})Rμν({}), ..., Rρσμν({})Rρσμν({}) have no any rigorous sense in respect to the canonical Levi-Civitá connection (1.10.11) and therefore cannot be said to be either honest physical singularity or only coordinate singularity in respect of the canonical Riemannian geometry. Remark 1.10.4. 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. Remark 1.10.5. In general relativity, many investigations have been derived with regard to singular exact vacuum solutions of the Einstein equation and the singularity structure of spacetime. Such solutions have been formally derived under condition Tνμ(x) = 0,where T ν μ(x) represent the energy-momentum densities of the gravity source. This for example is the case for the wellknown Schwarzschild solution, which is given by, in the Schwarzschild coordinates (x0, r, θ, φ), ds2 = −h (r) (dx0)2 + h−1 (r) (dr)2 + r2 [ (dθ)2 + sin2 θ(dφ)2 ] , h (r) = 1− rs r , (1.10.12) 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 (1.3.12) describes the gravitational field produced by a point-like particle located at r = 0. Remark 1.10.6. Note that when we say, on the basis of the canonical expression of the curvature square Rρσμν(r)Rρσμν(r) = 12r2s r6 (1.10.13) formally obtained from the metric (1.3.12), 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 1.10.7. From the metric (1.10.12), the calculation of the canonical Einstein tensor proceeds in a straighforward manner gives for r = 0 36 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Gtt (r) = G r r (r) = − h′ (r) r − 1 + h (r) r2 ≡ 0 , Gθθ (r) = Gφφ (r) = − h′′ (r) 2 − h (r) r2 ≡ 0, (1.10.14) Where h (r) = −1 + rs/r.Using Eq.(1.10.14) one formally obtains a boundary conditions{ Gtt (0) , lim r→0 Gtt (r) = 0, G r r (0) , lim r→0 Grr (r) = 0, Gθθ (0) , lim r→0 Gθθ (r) = 0, G φ φ (0) , lim r→0 Gφφ (r) = 0. (1.10.15) 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 Gkm (1.10.10). By simple calculation easy to see that G t t (0) , lim r→0 Gtt (r) = −∞, Grr (0) , lim r→0 Grr (r) = −∞, Gθθ (0) , lim r→0 Gθθ (r) = −∞, Gφφ (0) , lim r→0 Gφφ (r) = −∞. (1.10.16) and therefore the boundary conditions (1.15) is completely wrong. But on the other hand as pointed out by many authors [5]-[17] that the canonical representation of the Einstein tensor, is valid only in a weak (distributional) sense,i.e. [12]: Gab (x) = −8πmδa0δ0bδ3 (x) (1.10.17) and therefore again we obtain Gab (0) = −∞ × ( δa0δ 0 b ) .Thus canonical definition of the Einstein tensor is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin r = 0. 1.10.2 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]. 1.10.3 The Distributional Schwarzschild Geometry at the Origin. 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 1 r ∈ L1loc(R3), it obviously gives the regular distribution 1 r ∈ D′(R3). 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 ( R3 ) [22]: 1 r ι→ ι ( 1 r ) , ( 1 r ) ∗ ρε , ( 1 r ) ε , ρε = 1 ε3 ρ (r ε ) , ε ∈ (0, 1] . (1.10.18) Inserting (1.10.18) into (1.3.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]: ( ds2ε ) ε = ( hε (r) (dt) 2 ) ε − ( h−1ε (r) (dr) 2 ) ε + r2 [ (dθ)2 + sin2 θ(dφ)2 ] , hε (r) = −1 + rs ( 1 r ) ε , ε ∈ (0, 1] . (1.10.19) 37 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 1.10.8. Note that under regularization (1.10.18) for any ε ∈ (0, 1] the metric ds2ε = hε (r) (dt) 2 − h−1ε (r) (dr)2 + r2 [ (dθ)2 + sin2 θ(dφ)2 ] obviously is a classical Riemannian object and there no exist 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 (1.10.19) one obtains in a distributional sense [22]: ( R22 (r, ε) ) ε = ( R33 ) ε = ( h′ε (r) r + 1 + hε (r) r2 ) ε = 8πm δ (r) r2 ,( R00 (r, ε) ) ε = ( R11 ) ε = 1 2 ( h′′ε (r) 2 + h′ε (r) r ) ε = −4πmδ δ (r) r2 . (1.10.20) 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) r2 . Remark 1.10.9. Note that the formulae (1.10.20) should be contrasted with what is the expected result Gab (x) = −8πmδa0δ0bδ3 (x) given by Eq.(1.10.17). However the equations (1.10.20) are obviously given in spherical coordinates and therefore strictly speaking this is not correct, because the basis fields { ∂ ∂r , ∂ ∂φ , ∂ ∂θ } are not globally defined. Representing distributions concentrated at the origin requires a basis regular at the origin. Transforming the formulae for (Rε(r))∈ into Cartesian coordinates associated with the spherical ones, i.e., {r, θ, φ} ↔ {xi}, we obtain, e.g., for the Einstein tensor the expected result Gab (x) = −8πmδa0δ0bδ3 (x) given by Eq.(1.10.17), see [22]. 1.10.4 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 reads (hε (r))ε = −1 + (rs r Θ(r − ε) ) ε , ε ∈ (0, 1] , r < rs. (1.10.21) Here Θ (u) is the Heaviside function and the limit ε → 0 is understood in a distributional sense. Equation (1.10.19) with hε as given in (1.10.21) can be considered as an regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.10.21), the calculation of the distributional Einstein tensor proceeds in a straighforward manner. By simple calculation it gives [12]:  ( Gtt (r, ε) ) ε = (Grr (r, ε))ε = − ( h′ε (r) r ) ε − ( 1 + hε (r) r2 ) ε = = −rs ( δ (r − ε) r2 ) ε = −rs δ (r) r2 (1.10.22) and ( Gθθ (r, ε) ) ε = ( Gφφ (r, ε) ) ε = − ( h′′ε (r) 2 ) ε − ( hε (r) r2 ) ε = rs ( δ (r − ε) r2 ) ε − rs ( ε r2 d dr δ (r − ε) ) ε = −rs δ (r) r2 . (1.10.23) 38 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function Θ(r−ε). Transforming now the formulae for (Gab (r, ε))∈ into Cartesian coordinates associated with the spherical ones, i.e., {r, θ, φ} ↔ {xi}, we obtain for the generalized Einstein tensor the expected result given by Eq.(1.10.17) Gab (x) = −8πmδa0δ0bδ3 (x) , (1.10.24) see Remark 1.10.9. 1.10.5 The Smooth Regularization Via Horizon The smooth regularization via Horizon is considered by J.M.Heinzle and R.Steinbauer in paper [22]. Note that 1 r − rs /∈ L1loc(R3). An canonical regularization is the principal value vp ( 1 r − rs ) ∈ D′(R3) which can be embedded into G ( R3 ) [22]: 1 r − rs vp→ vp ( 1 r − rs ) ι→ ι [ ρε ∗ vp ( 1 r − rs )] , ( 1 r − rs ) ε ∈ G ( R3 ) . (1.10.25) Inserting now (1.10.25) into (1.10.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:( ds2ε ) ε = ( h (r) (dt)2 ) ε − ( h−1ε (r) (dr) 2 ) ε + r2 [ (dθ)2 + sin2 θ(dφ)2 ] , (1.10.26) where h (r) = −1 + rs r , h−1ε (r) = −1− rs ( 1 r − rs ) ε , ε ∈ (0, 1] . (1.10.27) Remark 1.10.10. Note that obviously Colombeau object, (1.10.27) is degenerate at r = rs, 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 regularization of h(r) (or h−1) must pass through zero somewhere and, additionally, this zero must converge to r = rs as the regularization parameter goes to zero. Remark 1.10.11. Note that due to the degeneracy of Colombeau object (1.10.26), even the distributional Levi-Civitá connection obviously is not available by using the smooth regularization via horizon [22]. 1.10.6 The Nonsmooth Regularization Via Gorizon In this Book 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 39 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance 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 00 = 8πmδ(3)(x⃗), 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(R3,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 L1loc (note that the origin is now always excluded from our considerations; the space we are working on is R3\{0}). Thus, regularizing the Schwarzschild metric amounts to embedding h−1 into G(R3,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) regularization h+ε (r) (h − ε (r)) [above (below) horizon] of h(r) must pass through small enough vicinity O+ε (2m) = { x⃗ ∈ R3| ∥x⃗∥ > 2m, ∥x⃗− 2m∥ 6 ε } (O−ε (2m) = { x⃗ ∈ R3| ∥x⃗∥ < 2m, ∥x⃗− 2m∥ 6 ε } ) of zeros set O0 (2m) = { y⃗ ∈ R3| ∥y⃗∥ = 2m } somewhere and, additionally, this vicinity O+ε (2m) (O−ε (2m)) must converge to O0 (2m) as the regularization parameter ε goes to zero.Due to the degeneracy of (1.10.12), the Levi-Cività connection is not available. By apriporiate nonsmooth regularization (see section 3) we obtain an Colombeau generalized object modeling the singular Schwarzschild metric above and below horizon, i.e., ( ds+2ε ) ε = ( h+ε (r)dt 2 ) ε − ([ h+ε (r) ]−1 dr2 ) ε + r2dΩ2 ,( ds−2ε ) ε = ( h−ε (r)dt 2 ) ε − ([ h−ε (r) ]−1 dr2 ) ε + r2dΩ2, ε ∈ (0, 1] . (1.10.28) Consider corresponding distributional connections( Γ+lkj (ε) ) ε = ( Γ+lkj [ h+ε ]) ε ∈ G(R3,Σ) and ( Γ−lkj (ε) ) ε = ( Γ−lkj [ h+ε ]) ε ∈ G(R3,Σ) : ( Γ+lkj (ε) ) ε = 1 2 (( g+lmε ) [(g+ε )mk,j + (g + ε )mj,k − (g+ε )kj,m] ) ε ,( Γ−lkj (ε) ) ε = 1 2 ( (g− lmε )[(g − ε )mk,j + (g − ε )mj,k − (g−ε )kj,m] ) ε . (1.10.29) Obviously ( Γ+lkj [ h+ε ]) ε , ( Γ−lkj [ h+ε ]) ε coincides with the corresponding Levi-Cività connection on R3\{(r = 0) ∪ (r = 2m)}, as (h+ε )ε = h+0 , (h−ε )ε = h − 0 , and ( g+lmε ) ε = g+lm0 , ( g−lmε ) ε = g−lm0 there. Clearly, connections Γ+lkj (ε) ,Γ −l kj (ε) , ε ∈ (0, 1] in respect the regularized metric g ± ε , ε ∈ (0, 1] ,i.e., (g±ε )ij;k = 0. Proceeding in this manner, we obtain the nonstandard result ([ R+ε ]1 1 ) ε = ([ R+ε ]0 0 ) ε ≈ −mΦ(2m),([ R−ε ]1 1 ) ε = ([ R−ε ]0 0 ) ε ≃ mΦ(2m). (1.10.30) Investigating the weak limit of the angular components of the generalized Ricci tensor using the abbreviation 40 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Φ(r) = π∫ 0 sin θdθ 2π∫ 0 dφΦ(x) and let Φ(x) be the function Φ(x) ∈ S+2m(R3) (Φ(x) ∈ S − 2m(R 3)), where by S+2m(R3) (S − 2m(R 3)) we denote the class of all functions Φ(x) with compact support such that (i) supp(Φ(x)) ⊂ {x| ∥x∥ ≥ 2m} (supp(Φ(x)) ⊂ {x| ∥x∥ ≤ 2m}) (ii) Φ(r) ∈ C∞ (R) . Then for any function Φ(x) ∈ S±2m(R3) we get: w -limε→0 [ R+ε ]1 1 = w -lim ε→0 [ R+ε ]0 0 = m ⟨ δ|Φ ⟩ = −mΦ(2m), w -lim ε→0 [ R−ε ]1 1 = w -lim ε→0 [ R−ε ]0 0 = m ⟨ δ|Φ ⟩ = mΦ(2m), (1.10.31) 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 d3x the coordinate volume element, gives ET = ∫ ( Trr +T θ θ +T φ φ +T t t )√ −gd3x = m, (1.10.32) as it should be. The paper is organized in the following way: in chapter II 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. chapter III is devoted to these approach to the problem. We present a new conceptually satisfactory method to derive the main result. In this final chapter III 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. 1.10.7 Distributional Eddington-Finkelstein Space-time In physical literature many years a belief exist that Schwarzschild spacetime ( S2 × {r > 2m} ) ×R is extendible, in the sense that it can be immersed in a larger spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with r > 2m.In physical literature [4],[33], [34],[35] one considers the formal change of coordinates obtained by replacing the canonical Schwarzschild time by "retarded time" above horizon ν+ given when r > 2m by ν+ = t+ r + 2m ln ( r 2m − 1 ) . (1.10.33) From (1.10.31) it follows for r > 2m dt = − dr 1− 2m r + dv+. (1.10.34) The Schwarzschild metric (1.10.12) above horizon ds+2 (see section 3) in this coordinate obviously takes the form 41 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ds+2 = − ( 1− 2m r ) dv+2 + 2drdv− + r2 [ (dθ)2 + sin2 θ(dφ)2 ] . (1.10.35) When r < 2m we replace (1.10.33) below horizon by ν− = t+ r + 2m ln ( 1− r 2m ) . (1.10.36) From (1.10.36) it follows for r < 2m dt = dr 2m r − 1 + dv−. (1.10.37) The Schwarzschild metric (1.10.12) below horizon ds2− (see section 3) in this coordinate obviously takes the form ds−2 = ( 2m r − 1 ) dv−2 − 2drdv− + r2 [ (dθ)2 + sin2 θ(dφ)2 ] . (1.10.38) Remark 1.10.12.(i) Note that the metric (1.10.33) is defined on the manifold S2 × (r > 0)× R and obviously it is regular Lorentzian metric: its coefficients are smooth. (ii) The term 2drdv ensures its non-degeneracy for r = 2m. (iii) Due to the nondegeneracy of the metric (1.10.32) the Levi-Civita connection{ Γ+lkj ({}) == 1 2 [ glm ({}) ] [(gmk,j ({}) + gmj,k ({})− gkj,m ({})] (1.10.39) obviously now available and therefore nonsingular on horizon in contrast with Schwarzschild metric (1.10.12) one obtains [3]: Γννν = rs 2r2 ,Γrνν = rs (r − rs) 2r3 ,Γrνr = − rs 2r2 ,Γθrθ = 1 r , Γφrφ = 1 r ,Γνθθ = −r,Γrθθ = −r (r − rs) ,Γφθφ = cot θ, Γνφφ = −r sin2 θ,Γrφφ = −r (r − rs) sin2 θ,Γθφφ = − sin θ cos θ. (1.10.40) (iv) In physical literature [3],[4] by using properties (i)-(iii) this spacetime wrongly convicted as an rigorous mathematical extension of the Schwarzschild spacetime. Remark 1.10.13. Let us consider now the coordinates: (i) v+, r′ = r, θ′ = θ, φ′ = φ and (ii) v−, r′ = r, θ′ = θ, φ′ = φ. Obviously both transformations given by Eq.(1.10.33) and Eq.(1.10.36 ) are singular because the both Jacobian of these transformations is singular at r = 2m :∂v + ∂t ∂v+ ∂r ∂r′ ∂t ∂r′ ∂r  = (1 rr − 2m 0 1 ) (1.10.41) and ∂v − ∂t ∂v− ∂r ∂r′ ∂t ∂r′ ∂r  = (1 − r2m− r 0 1 ) . (1.10.42) Remark 1.10.14. Note first (i) such singular transformations are not allowed in conventional Lorentzian geometry and second (ii) both Eddington-Finkelstein metrics given by Eq.(1.10.35) and 42 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance by Eq.(1.10.38 ) are well defined in rigorous mathematical sense at r = 2m. Remark 1.10.15. (I) From consideration above it follows that Schwarzschild spacetime( S2 × {r > 2m} ) × R is not extendible, in the sense that it can be immersed in a larger spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with r > 2m. Thus Eddington-Finkelstein spacetime cannot be considered as an extension of the Schwarzschild spacetime in natural way in respect with conventional Lorentzian geometry. Such an "extension" is the extension by abnormal definition and nothing more. (II) However distributional EddingtonFinkelstein spacetime (1.10.53) is equivalent of the distributional Schwarzschild spacetime in natural way. Remark 1.10.16. From consideration above it follows that it is necessary a regularization of the Eq.(1.10.34) and Eq.(1.10.37) on horizon. However obviously only nonsmooth regularization via horizon r = 2m is possible. Under nonsmooth regularization (see section 3) Eq.(1.10.34) and Eq.(1.10.37) take the form dt = − dr 1 r √ (r − 2m)2 + ε2 + dv+ε , ε ∈ (0, 1] (1.10.43) and dt = dr 1 r √ (2m− r)2 + ε2 + dv−ε , ε ∈ (0, 1] (1.10.44) correspondingly. Therefore Eq.(1.10.41)-Eq.(1.10.42) take the form∂v + ε ∂t ∂v+ε ∂r ∂r′ ∂t ∂r′ ∂r  = 1 r√ (r − 2m)2 + ε2 0 1  (1.10.45) and∂v − ε ∂t ∂v−ε ∂r ∂r′ ∂t ∂r′ ∂r  = 1 − r√ (2m− r)2 + ε2 0 1  . (1.10.46) From Eq.(1.10.43)-Eq.(1.10.44) one obtain generalized Eddington-Finkelstein transformatios such that dt = − rdr(√ (r − 2m)2 + ε2 ) ε + ( dv+ε ) ε , ε ∈ (0, 1] (1.10.47) and dt = rdr(√ (2m− r)2 + ε2 ) ε + dv−ε , ε ∈ (0, 1] . (1.10.48) 43 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Therefore Eq.(1.10.45)-Eq.(1.10.46) take the form ( ∂v+ε ∂t ) ε ( ∂v+ε ∂r ) ε ∂r′ ∂t ∂r′ ∂r  = 1 r(√ (r − 2m)2 + ε2 ) ε 0 1  (1.10.49) and ( ∂v−ε ∂t ) ε ( ∂v−ε ∂r ) ε ∂r′ ∂t ∂r′ ∂r  = 1 − r(√ (2m− r)2 + ε2 ) ε 0 1  . (1.10.50) At point r = 2m one obtains ( ∂v+ε ∂t ) ε ( ∂v+ε ∂r ) ε ∂r′ ∂t ∂r′ ∂r  ∣∣∣∣∣∣∣ r=2m = ( 1 r ( ε−1 ) ε 0 1 ) (1.10.51) and ( ∂v−ε ∂t ) ε ( ∂v−ε ∂r ) ε ∂r′ ∂t ∂r′ ∂r  ∣∣∣∣∣∣∣ r=2m = ( 1 −r ( ε−1 ) ε 0 1 ) , (1.10.52) where ( ε−1 ) ε ∈ R. Thus generalized Eddington-Finkelstein transformations (1.10.47)- (1.10.48) well defined in sense of Colombeau generalized functions. Therefore Colombeau generalized object modeling the classical Eddington-Finkelstein metric (1.10.35)-(1.10.36) above and below horizon take the form( ds+2ε ) ε = 1 r √ (r − 2m)2 + ε2dv+2ε + 2drdv+ε + r2 [ (dθ)2 + sin2 θ(dφ)2 ] .( ds−2ε ) ε = 1 r √ (2m− r)2 + ε2dv−2ε + 2drdv−ε + r2 [ (dθ)2 + sin2 θ(dφ)2 ] . (1.10.53) It easily to verify by using formula A.2 (see appendix) that the distributional curvature scalar (R (ε))ε again singular at r = 2m as in the case of the distributional Schwarzschild spacetime given by Eq.(1.10.28). However this is not surprising because the classical EddingtonFinkelstein spacetime and generalized Eddington-Finkelstein specetime given by Eq.(1.10.53) that is essentially different geometrical objects. 2 Generalized Colombeau Calculus 2.1 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 Rn. 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 44 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance 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. Let (uε)ε∈(0,1] with (uε)ε ∈ C ∞(M) for all ε ∈ R+,where M a separable, smooth orientable Hausdorff manifold of dimension n. Definition 2.1.1. The classical Colombeau's algebra of generalized functions on M is defined as the quotient: G(M) , EM (M)/N (M) (2.1) of the space EM (M) of sequences of moderate growth modulo the spaceN (M) of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where X(M) denoting the space of smooth vector fields on M):  EM (M) , {(uε)ε| ∀K (K $M) ∀k (k ∈ N) ∃N (N ∈ N) ∀ξ1, . . . , ξk (ξ1, . . . , ξk ∈ X(M)) [ sup p∈K |Lξ1 . . . Lξk uε(p)| = O(ε −N ) as ε→ 0 ]} , (2.2)  N (M) , {(uε)ε| ∀K (K $M) , ∀k (k ∈ N0) ∀q (q ∈ N) ∀ξ1, . . . , ξk (ξ1, . . . , ξk ∈ X(M)) [ sup p∈K |Lξ1 . . . Lξk uε(p)| = O(ε q) as ε→ 0 ]} . (2.3) Remark 2.1.1. In the definition the Landau symbol aε = O (ψ (ε)) appears, having the following meaning: ∃C (C > 0)∃ε0 (ε0 ∈ (0, 1]) ∀ε (ε < ε0) [aε ≤ Cψ (ε)] . Definition 2.3. Elements of calG(M) are denoted by: u = cl[(uε)ε] , (uε)ε +N (M). (2.4) Remark 2.1.2. With componentwise operations (*,± ) G(M) is a fine sheaf of differential algebras with respect to the Lie derivative defined by Lξu , cl[(Lξuε)ε]. 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) ⊂ Rn in the respective estimates Lie derivatives are replaced by partial derivatives. 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) ⊂ Rn in the respective estimates Lie derivatives are replaced by partial derivatives. Remark 2.1.3. Smooth functions f ∈ C∞(M) are embedded into G(M) simply by the "constant" embedding σ, i.e., σ(f) = cl[(f)ε], hence C∞(M) is a faithful subalgebra of G(M). 2.2 Point Values of 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. 45 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers". Definition 2.2.1. Inserting p ∈ M into u ∈ G(M) yields a well defined element of the ring of constants (also called generalized numbers) K (corresponding to K = R resp. C), defined as the set of moderate nets of numbers ((rε)ε ∈ K(0,1] with |rε| = O(ε−N ) for some N) modulo negligible nets (|rε| = O(εm) for each m); componentwise insertion of points of M into elements of G(M) yields well-defined generalized numbers, i.e.,elements of the ring of constants: K = Ec (M) /Nc (M) (2.5) (with K = R or K = C for K = R or K = C), where Ec (M) = { (rε)ε ∈ K I |∃n (n ∈ N) [ |rε| = O ( ε−n ) as ε→ 0 ]} Nc (M) = { (rε)ε ∈ K I |∀m (m ∈ N) [|rε| = O (εm) as ε→ 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 compactly supported nets (pε)ε ∈ M (0,1] with respect to the relation pε ∼ p′ε :⇔ dh(pε, p′ε) = O(εm) for all m, where dh denotes the distance on M induced by any Riemannian metric. Definition 2.2.2. 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| > εq 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 2.2.3. Let Ω be an open subset of Rn. On a set Ω : { Ω = { (xε)ε ∈ Ω I |∃p (p > 0) [|xε| = O (εp)] } ={ (xε)ε ∈ Ω I |∃p (p > 0) ∃ε0 (ε0 > 0) [|xε| ≤ εp, for 0 < ε < ε0] } . (2.7) We introduce an equivalence relation: (xε)ε ∼ (yε)ε ⇐⇒ ∀q (q > 0) ∀ε (ε > 0) [|xε − yε| ≤ ε q, for 0 < ε < ε0] . (2.8) and denote by Ω = Ω/ ∼ the set of generalized points. The set of points with compact support is Ωc = { x = cl[(xε)ε] ∈ Ω|∃K (K ⊂ Ω) ∃ε0 (ε0 > 0) [xε ∈ K for 0 < ε < ε0] } . (2.9) Definition 2.2.5 A generalized function 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) 46 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance We shall often write: u ≈ Sch 0. (2.11) The G(M)-module of generalized sections in vector bundles-especially the space of generalized tensor fields T rs (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 Grs(M) = G(M)⊗ T rs (M) , (2.12) where T rs (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 Grs(M) ∼= L(M)(G01(M)r,G10(M)s;G(M)). (2.13) In particular a generalized metric is defined to be a symmetric, generalized (0, 2)-tensor field gab = [((gε)ab )ε] (with its index independent of ε and) whose determinant det(gab) 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(gab) we have ∀K ⊂M ∃m ∈ N [infp∈K | det(gab (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: gab is a generalized metric iff (on every relatively compact open subset V of M) there exists a representative ((gε)ab )ε of gab such that for fixed ε (small enough)(gε)ab = gab (ε) (resp. (gε)ab |V ) is a classical pseudo-Riemannian metric and det(gab) is invertible in the algebra of generalized functions. A generalized metric induces a G(M)-linear isomorphism from G10(M) to G01(M) and the inverse metric gab , [(g−1ab (ε))ε] is a well defined element of G20(M) (i.e., independent of the representative ((gε)ab )ε). Also the generalized Levi-Civita connection as well as the generalized Riemann-, Ricciand Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives. 2.3 Generalized Colombeau Calculus We briefly recall the basic generalized Colombeau construction. Colombeau supergeneralized functions on Ω ⊆ Rn, where dim (Ω) = n are defined as equivalence classes u = [(uε)ε] of nets of smooth functions uε ∈ C∞(Ω\Σ),where dim (Σ) < n (regularizations) subjected to asymptotic norm conditions with respect to ε ∈ (0, 1] for their derivatives on compact sets. The basic idea of generalized Colombeau's theory of nonlinear supergeneralized functions [1],[2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter ε. Let (uε)ε∈(0,1] with uε such that: (i) uε ∈ C∞(M\Σ) and (ii) uε ∈ D′(M),for all ε ∈ (0, 1] ,where M a separable, smooth orientable Hausdorff manifold of dimension n. Definition 2.3.1. The supergeneralized Colombeau's algebra G = G(M,Σ) of supergeneralized functions on M, where Σ ⊂M, dim (M) = n,dim (Σ) < n , is defined as the quotient: G(M,Σ) , EM (M,Σ)/N (M,Σ) (2.14) 47 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance of the space EM (M,Σ) of sequences of moderate growth modulo the space N (M,Σ) of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where X(M\Σ) denoting the space of smooth vector fields on M\Σ): EM (M,Σ) , {(uε)ε| ∀K (K $M\Σ) ∀k (k ∈ N) ∃N (N ∈ N) ∀ξ1, . . . , ξk (ξ1, . . . , ξk ∈ X(M\Σ)) [ sup p∈K |Lξ1 . . . Lξk uε(p)| = O(ε −N ), ε→ 0 ] & ∀K (K $M) ∀k (k ∈ N) ∃N (N ∈ N) ∀ (f ∈ C∞(M))∀ξ1, . . . , ξk (ξ1, . . . , ξk ∈ X(M))[∥∥Lwξ1 . . . Lwξk uε∥∥ = ( sup f∈C∞(M) ∣∣Lwξ1 . . . Lwξkuε(f)∣∣ ) = O(ε−N ), ε→ 0 ]} , (2.15) N (M,Σ) , {(uε)ε| ∀K (K $M\Σ) , ∀k (k ∈ N0) ∀q (q ∈ N) ∀ξ1, . . . , ξk (ξ1, . . . , ξk ∈ X(M\Σ)) [ sup p∈K |Lξ1 . . . Lξk uε(p)| = O(ε q), ε→ 0 ]} & ∀K (K $M) ∀k (k ∈ N) ∃N (N ∈ N) ∀ (f ∈ C∞(M))∀ξ1, . . . , ξk (ξ1, . . . , ξk ∈ X(M))[∥∥Lwξ1 . . . Lwξk uε∥∥ = ( sup f∈C∞(M) ∣∣Lwξ1 . . . Lwξkuε(f)∣∣ ) = O(εq), ε→ 0 ]} , (2.16) where Lwξk denoting the weak Lie derivative in L.Schwartz sense.In the definition the Landau symbol aε = O (ψ (ε)) appears, having the following meaning: ∃C (C > 0)∃ε0 (ε0 ∈ (0, 1]) ∀ε (ε < ε0) [aε ≤ Cψ (ε)] . Definition 2.3.2. Elements of G(M,Σ) are denoted by: u = cl[(uε)ε] , (uε)ε +N (M,Σ). (2.17) Remark 2.3.1. With componentwise operations (*,± ) G(M,Σ) is a fine sheaf of differential algebras with respect to the Lie derivative defined by Lξu , cl[(Lξuε)ε]. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., u ∈ G(M,Σ) iff u ◦ ψα ∈ G(ψα(Vα)) for all charts (Vα, ψα), where on the open set ψα(Vα) ⊂ Rn in the respective estimates Lie derivatives are replaced by partial derivatives. Remark 2.6. Smooth functions f ∈ C∞(M\Σ) are embedded into G(M,Σ) simply by the "constant" embedding σ, i.e., σ(f) = cl[(f)ε], hence C∞(M\Σ) is a faithful subalgebra of G(M,Σ). 2.4 Point Values of 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 2.4.1. Inserting p ∈ M into u ∈ G(M,Σ) yields a well defined element of the ring of constants (also called generalized numbers) K (corresponding to K = R resp. C), defined as the set of moderate nets of numbers ((rε)ε ∈ K(0,1] with |rε| = O(ε−N ) for some N) modulo negligible 48 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance nets (|rε| = O(εm) for each m); componentwise insertion of points of M into elements of G(M,Σ) yields well-defined generalized numbers, i.e.,elements of the ring of constants: KΣ= Ec (M,Σ) /Nc (M,Σ) (2.18) (with KΣ = RΣ or K = CΣ for K = R or K = C), where Ec (M,Σ) = { (rε)ε ∈ K I |∃n (n ∈ N) [ |rε| = O ( ε−n ) as ε→ 0 ]} , Nc (M,Σ) = { (rε)ε ∈ K I |∀m (m ∈ N) [|rε| = O (εm) as ε→ 0] } I = (0, 1]. (2.19) Supergeneralized 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 compactly supported nets (pε)ε ∈ (M\Σ)(0,1] with respect to the relation pε ∼ p′ε :⇔ dh(pε, p′ε) = O(εm) for all m, where dh denotes the distance on M\Σ induced by any Riemannian metric. Definition 2.4.2. For u ∈ G(M,Σ) 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 2.4.3. We say that an element r ∈ K is strictly nonzero if there exists a representative (rε)ε and a q ∈ N such that |rε| > εq 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 2.4.5. Let Ω be an open subset of Rn\Σ. On a set ΩΣ :  ΩΣ = { (xε)ε ∈ (Ω\Σ) I |∃p (p > 0) [|xε| = O (εp)] } ={ (xε)ε ∈ (Ω\Σ) I |∃p (p > 0)∃ε0 (ε0 > 0) [|xε| ≤ εp, for 0 < ε < ε0] } (2.20) we introduce an equivalence relation: (xε)ε ∼ (yε)ε ⇐⇒ ∀q (q > 0) ∀ε (ε > 0) [|xε − yε| ≤ ε q, for 0 < ε < ε0] (2.21) and denote by ΩΣ = ΩΣ/ ∼ the set of supergeneralized points. The set of points with compact support is ΩΣ,c = { x = cl[(xε)ε] ∈ ΩΣ|∃K (K ⊂ Ω\Σ) ∃ε0 (ε0 > 0) [xε ∈ K for 0 < ε < ε0] } (2.22) Definition 2.4.6. A supergeneralized function u ∈ G(M,Σ) is called associated to zero, u ≈ 0 on Ω ⊆M in L. Schwartz's sense if one (hence any) representative (uε)ε converges to zero weakly,i.e. w limε→0 uε = 0 (2.23) We shall often write: u ≈ Sch 0. (2.24) 49 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Definition 2.4.7. The G(M,Σ)-module of supergeneralized sections in vector bundlesespecially the space of generalized tensor fields T rs (M\Σ)-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 Grs(M,Σ) = G(M,Σ)⊗ T rs (M\Σ) , (2.25) where T rs (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, 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 Grs(M,Σ) ∼= L(M)(G01(M,Σ)r, G10(M,Σ)s; G(M,Σ)). (2.26) In particular a supergeneralized metric is defined to be a symmetric, supergeneralized (0, 2)-tensor field gab = [((gε)ab )ε] (with its index independent of ε and) whose determinant det(gab) 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(gab) we have ∀K ⊂ M\Σ ∃m ∈ N [infp∈K |det(gab (ε))| ≥ εq] 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: gab is a generalized metric iff (on every relatively compact open subset V ofM) there exists a representative ((gε)ab )ε of gab such that for fixed ε (small enough)(gε)ab = gab (ε) (resp. (gε)ab |V ) is a classical pseudo-Riemannian metric and det(gab) is invertible in the algebra of generalized functions. A generalized metric induces a G(M,Σ)-linear isomorphism from G10(M,Σ) to G01(M,Σ) and the inverse metric gab , [(g−1ab (ε))ε] is a well defined element of G20(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. 2.5 Distributional General Relativity We briefly summarize the basics of distributional 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. In standard general relativity, the space-time is assumed to be a four-dimensional differenti-able manifoldM endowed with the Lorentzian metric ds2 = gμνdx μdxν (μ, ν = 0, 1, 2, 3). At each point p of space-timeM , the metric can be diagonalized as ds2p = ημν(dX μ)p(dX ν)p with ημν , (−1, 1, 1, 1), by choosing the coordinate system {Xμ;μ = 0, 1, 2, 3} appropriately. In superdistributional general relativity the space-time is assumed to be a four dimensional differentiable manifoldM\Σ, where dim (M) = 4,dim (Σ) 6 3 endowed with the Lorentzian supergeneralized metric( ds2ε ) ε = (gμν (ε) dx μdxν)ε ;μ, ν = 0, 1, 2, 3). (2.27) 50 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance at each point p ∈M\Σ, the metric can be diagonalized as( ds2p (ε) ) ε = (ημν(dX μ ε )p(dX ν ε )p)ε with ημν , (−1, 1, 1, 1), (2.28) by choosing the generalized coordinate system {(Xμε )ε ;μ = 0, 1, 2, 3} appropriately. The classical smooth curvature tensor is given by Rρσμν , ∂μ { ρ σ ν } − ∂ν { ρ σ μ } + { ρ λ μ }{ λ σ ν } − { ρ λ ν }{ λ σ μ } (2.29) with { ρ σ ν } being the smooth Christoffel symbol.The supergeneralized nonsmooth curvature tensor is given by (R ρ σμν (ε))ε , ∂μ ({ ρ σ ν } ε ) ε − ∂ν ({ ρ σ μ } ε ) ε + ({ ρ λ μ } ε ) ε ({ λ σ ν } ε ) ε − − ({ ρ λ ν } ε ) ε ({ λ σ μ } ε ) ε (2.30) with ({ ρ σ ν } ε ) ε being the supergeneralized Christoffel symbol.The fundamental classical action integral I is I = 1 c ∫ (LG + LM )d4x, (2.31) where LM is the Lagrangian density of a gravitational source and LG is the gravitational Lagrangian density given by LG = 1 2κ G . (2.32) Here κ is the Einstein gravitational constant κ = 8πG/c4 and G is defined by G = √ −ggμν ({ λ μ ρ }{ ρ ν λ } − { λ μ ν }{ ρ λ ρ }) (2.33) with g = det(gμν). There exists the relation √ −gR = G+ ∂μDμ , (2.34) with Dμ = − √ −g ( gμν { λ ν λ } − gνλ { μ ν λ }) . (2.35) Thus the supergeneralized fundamental action integral (Iε)ε is (Iε)ε = 1 c ∫ ( ( LG (ε) ) ε + (LM (ε))ε)d 4x , (2.36) where (LM (ε))ε is the supergeneralized Lagrangian density of a gravitational source and ( LG (ε) ) ε is the supergeneralized gravitational Lagrangian density given by( LG (ε) ) ε = 1 2κ (Gε)ε . (2.37) Here κ is the Einstein gravitational constant κ = 8πG/c4 and (Gε)ε is defined by (Gε)ε = √ − (gε)ε (g μν ε )ε (({ λ μ ρ } ε ) ({ ρ ν λ } ε ) − ({ λ μ ν } ε )({ ρ λ ρ } ε )) (2.38) 51 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance with gε = det [(gμν (ε))ε]. There exists the relation √ − (gε)ε (Rε)ε = (Gε)ε + ∂μ (D μ ε )ε , (2.39) with (Dμε )ε = − √ − (gε)ε ( (gμνε )ε ({ λ ν λ } ε ) ε − ( gνλε ) ε ({ μ ν λ } ε ) ε ) . (2.40) Also, we have defined the classical scalar curvature by R = Rμμ (2.41) with the smooth Ricci tensor Rμν = R λ μλν . (2.42) from the action I, the classical Einstein equation Gμ ν = Rμ ν − 1 2 δμ νR = κTμ ν , (2.43) follows, where Tμ ν is defined by Tμ ν = Tμ ν √ −g (2.44) with T νμ , 2gμλ δLM δgλν (2.45) being the energy-momentum density of the classical gravity source. Thus we have defined the supergeneralized scalar curvature by (Rε)ε = (R μ μ (ε))ε (2.46) with the supergeneralized Ricci tensor (Rμν (ε))ε = ( Rλμλν (ε) ) ε . (2.47) from the action (Iε)ε , the generalized Einstein equation (Gμ ν (ε))ε = (Rμ ν (ε))ε − 1 2 δμ ν (Rε)ε = κ (Tμ ν (ε))ε , (2.48) follows, where (Tμ ν (ε))ε is defined by (Tμ ν (ε))ε = ( Tμ ν (ε) ) ε√ − (gε)ε (2.49) 52 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance with( T νμ (ε) ) ε , 2 (gμλ (ε))ε δ (LM (ε))ε δ (gλν (ε))ε (2.50) being the supergeneralized energy-momentum density of the supergeneralized gravity source. The classical energy-momentum pseudo-tensor density t νμ of the gravitational field is defined by t νμ = δμ ν LG − ∂LG ∂gστ,ν gστ,μ (2.51) with gστ,ν = ∂gστ/∂x ν . The supergeneralized energy-momentum pseudo-tensor density t νμ of the gravitational field is defined by ( t νμ (ε) ) ε = δμ ν ( LG (ε) ) ε − ( ∂LG (ε) ∂gστ,ν (ε) ) ε (gστ,μ (ε))ε (2.52) with (gστ,ν (ε))ε = (∂gστ (ε) /∂x ν)ε. 3 Distributional Schwarzschild Geometry from Nonsmooth Regularization via Horizon 3.1 Calculation of the Stress-tensor by Using Nonsmooth Regularization Via Horizon In this 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, θ, φ), r = 2m the Schwarzschild metric (1.12) takes the form above horizon r > 2m and below horizon r < 2m correspondingly  above horizon r > 2m : ds+2 = h+(r)dt2 − [ h+(r) ]−1 dr2 + r2dΩ2, h+(r) = −1 + 2m r = −r − 2m r below horizon r < 2m : ds−2 = h−(r)dt2 − h−(r)−1dr2 + r2dΩ2, h−(r) = −1 + 2m r = 2m− r r (3.1) Remark 3.1.1. Following the above discussion we consider the metric coefficients h+ (r) , [ h+(r) ]−1 h− (r) ,and [ h−(r) ]−1 as an element of D′(R3) and embed it into (G(R3)) by replacements above horizon r > 2m and below horizon r 6 2m correspondingly 53 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance r > 2m : r − 2m 7−→ √ (r − 2m)2 + ε2, r 6 2m : 2m− r 7−→ √ (2m− r)2 + ε2. Remark 3.1.2. Note that, accordingly, we have fixed the differentiable structure of the manifold: the usual Schwarzschild coordinates and the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1) are extended on r = 2m through the horizon. Therefore we have above horizon r > 2m and below horizon r 6 2m correspondingly h(r) = { −r − 2m r if r > 2m 0 if r 6 2m } 7−→ ( h+ε (r) ) ε = − √ (r − 2m)2 + ε2 r  ε , where ( h+ε (r) ) ε ∈ G(R3, B+ (2m,R)), B+ (2m,R) = { x ∈ R3|2m 6 ∥x∥ 6 R } . h−1(r) = { − r r − 2m, r > 2m ∞, r = 2m } 7−→ ( h+ε )−1 (r) = h−(r) = { −r − 2m r if r 6 2m 0 if r ≥ 2m } 7−→ h−ε (r) = =  √ (2m− r)2 + ε2 r  ε ∈ G(R3, B− (0, 2m)) , where B− (0, 2m) = { x ∈ R3|0 < ∥x∥ 6 2m }{ − r r − 2m, r < 2m ∞, r = 2m } 7−→ ( h−ε )−1 (r) = =  r√ (r − 2m)2 + ε2  ε ∈ G(R3, B− (0, 2m)) (3.2) Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) gorizon, i.e.,  ( ds+2ε ) ε = ( h+ε (r)dt 2 ) ε − ([ h+ε (r) ]−1 dr2 ) ε + r2dΩ2 ,( ds−2ε ) ε = ( h−ε (r)dt 2 ) ε − ([ h−ε (r) ]−1 dr2 ) ε + r2dΩ2 (3.3) The generalized Ricci tensor above horizon [ R+ ]β α may now be calculated componentwise using the classical formulae  ([ R+ε ]0 0 ) ε = ([ R+ε ]1 1 ) ε = 1 2 (( h+′′ε ) ε + 2 r ( h+′ε ) ε ) ([ R+ε ]2 2 ) ε = ([ R+ε ]3 3 ) ε = ( h+′ε ) ε r + 1 + ( h+ε ) ε r2 . (3.4) From (3.2) we obtain 54 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance h+′ε (r) = − r − 2m r [ (r − 2m)2 + ε2 ]1/2 + [ (r − 2m)2 + ε2 ]1/2 r2 , r ( h+′ε ) ε + 1 + ( h+ε ) ε = r − r − 2mr [(r − 2m)2 + ε2]1/2 + [ (r − 2m)2 + ε2 ]1/2 r2  + 1 − √ (r − 2m)2 + ε2 r = − r − 2m[ (r − 2m)2 + ε2 ]1/2 + [ (r − 2m)2 + ε2 ]1/2 r + 1 − √ (r − 2m)2 + ε2 r = − r − 2m[ (r − 2m)2 + ε2 ]1/2 + 1. h′′ε (r) = − ( r − 2m r [ (r − 2m)2 + ε2 ]1/2 )′ +  [ (r − 2m)2 + ε2 ]1/2 r2  ′ = = − 1 r [ (r − 2m)2 + ε2 ]1/2 + (r − 2m)2r [(r − 2m)2 + ε2]3/2 + r − 2mr2 [(r − 2m)2 + ε2]1/2 + + r − 2m r2 [ (r − 2m)2 + ε2 ]1/2 − 2 [ (r − 2m)2 + ε2 ]1/2 r3 . r2 ( h+′′ε ) ε + 2r ( h+′ε ) ε = r2 { − 1 r [ (r − 2m)2 + ε2 ]1/2 + (r − 2m)2r [(r − 2m)2 + ε2]3/2 + r − 2mr2 [(r − 2m)2 + ε2]1/2 + + r − 2m r2 [ (r − 2m)2 + ε2 ]1/2 − 2 [ (r − 2m)2 + ε2 ]1/2 r3 + +2r − r − 2mr [(r − 2m)2 + ε2]1/2 + [ (r − 2m)2 + ε2 ]1/2 r2  = − r[ (r − 2m)2 + ε2 ]1/2 + r (r − 2m)2[(r − 2m)2 + ε2]3/2 + r − 2m[(r − 2m)2 + ε2]1/2 + + r − 2m[ (r − 2m)2 + ε2 ]1/2 − 2 [ (r − 2m)2 + ε2 ]1/2 r + − 2 (r − 2m)[ (r − 2m)2 + ε2 ]1/2 + 2 [ (r − 2m)2 + ε2 ]1/2 r = − r[ (r − 2m)2 + ε2 ]1/2 + r (r − 2m)2[(r − 2m)2 + ε2]3/2 . (3.5) Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation) Φ(r) = π∫ 0 sin θdθ 2π∫ 0 dφΦ(x) and let Φ(x) be the function Φ(x) ∈ S+2m(R3), where by S + 2m(R 3) we denote the class of all functions Φ(x) with compact support such that: (i) supp(Φ(x)) ⊂ {x| ∥x∥ ≥ 2m} (ii) Φ(r) ∈ C∞ (R) . Then for any function Φ(x) ∈ S2m(R3) we get:∫ K ([ R+ε ]2 2 ) ε Φ(x⃗) d3x = ∫ K ([ R+ε ]3 3 ) ε Φ(x⃗) d3x = R∫ 2m ( r ( h+′ε ) ε + 1 + ( h+ε ) ε ) Φ(r)dr = R∫ 2m { − r − 2m[ (r − 2m)2 + ε2 ]1/2 } Φ(r)dr + R∫ 2m Φ(r)dr. (3.6) 55 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance By replacement r − 2m = u, from (3.6) we obtain∫ K ([ R+ε ]2 2 ) ε Φ(x) d3x = ∫ K ([ R+ε ]3 3 ) ε Φ(x) d3x = − R−2m∫ 0 uΦ(u+ 2m)du (u2 + ε2)1/2 + R−2m∫ 0 Φ(u+ 2m)du. (3.7) By replacement u = εη, from (3.7) we obtain the expression I+3 (ε) = ∫ K ([ R+ε ]3 3 ) ε Φ(x) d3x = I+2 (ε) = ∫ K ([ R+ε ]2 2 ) ε Φ(x⃗) d3x = −ε×  R−2mε∫ 0 ηΦ(εη + 2m)dη (η2 + 1)1/2 − R−2m ε∫ 0 Φ(εη + 2m)dη  . (3.8) From Eq.(3.8) we obtain I+3 (ε) = I + 2 (ε) = −ε Φ(2m) 0! R−2m ε∫ 0 [ η (η2 + 1)1/2 − 1 ] dη− − ε 2 1! R−2m ε∫ 0 [ η (η2 + 1)1/2 − 1 ] Φ(1)(ξ)ηdη = −εΦ(2m) √(R− 2m ε )2 + 1− ( R− 2m ε − 1 )− − ε 2 1 R−2m ε∫ 0 [ η (η2 + 1)1/2 − 1 ] Φ(1)(ξ)ηdη, (3.9) Where we have expressed the function Φ(εη + 2m) as Φ(εη + 2m) =∑n−1l=0 Φ (l)(2m) l! (εη)l + 1 n! (εη)nΦ(n)(ξ) , ξ , θεη + 2m , 1 > θ > 0 , n = 1 (3.10) with Φ(l)(ξ) , dlΦ/dξl. Equations (3.9)-(3.10) gives  lim ε→0 I+3 (ε) = lim ε→0 I+2 (ε) = lim ε→0 −εΦ(2m) √(R− 2m ε )2 + 1 + 1− R− 2m ε + +lim ε→0 − ε21 R−2m ε∫ 0 [ η (η2 + 1)1/2 − 1 ] Φ(1)(ξ)ηdη  = 0 (3.11) Thus in S ′2m ( B+R (2m) ) ⊂ S ′2m ( R3 ) ⊂ D′(R3), where B+ (2m,R) = { x ∈ R3|2m 6 ∥x∥ 6 R } from Eq.(3.11) we obtain w − limε→0 [ R+ε ]3 3 = lim ε→0 I+3 (ε) = 0, w − lim ε→0 [ R+ε ]2 2 = lim ε→0 I+2 (ε) = 0. (3.12) For ([ R+ε ]1 1 ) ε , ([ R+ε ]0 0 ) ε we get: 56 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance  2K ([ R+ε ]1 1 ) ε Φ(x) d3x = 2K ([ R+ε ]0 0 ) ε Φ(x) d3x = R∫ 2m ( r2 ( h+′′ε ) ε + 2r ( h+′ε ) ε ) Φ(r)dr = = R∫ 2m { − r[ (r − 2m)2 + ε2 ]1/2 + r (r − 2m)2[ (r − 2m)2 + ε2 ]3/2 } Φ(r)dr. (3.13) By replacement r − 2m = u, from (3.13) we obtain I+1 (ε) = 2K ([ R+ε ]1 1 ) ε Φ(x) d3x = I+2 (ε) = 2K ([ R+ε ]0 0 ) ε Φ(x) d3x = R∫ 2m ( r2 ( h+′′ε ) ε + 2r ( h+′ε ) ε ) Φ(r)dr = = R−2m∫ 0 { − u+ 2m (u2 + ε2)1/2 + u2 (u+ 2m) (u2 + ε2)3/2 } Φ(u+ 2m)du. (3.14) By replacement u = εη, from (3.14) we obtain 2K ([ R+ε ]1 1 ) ε Φ(x) d3x = 2K ([ R+ε ]0 0 ) ε Φ(x) d3x = = R∫ 2m ( r2 ( h+′′ε ) ε + 2r ( h+′ε ) ε ) Φ(r)dr = = ε R−2m ε∫ 0 { − εη + 2m (ε2η2 + ε2)1/2 + ε2η2 (εη + 2m) (ε2η2 + ε2)3/2 } Φ(εη + 2m)dη = − R−2m ε∫ 0 ε2ηΦ(εη + 2m)dη (ε2η2 + ε2)1/2 − 2m R−2m ε 0 εΦ(εη + 2m)dη (ε2η2 + ε2)1/2 + R−2m ε∫ 0 ε4η3Φ(εη + 2m)dη (ε2η2 + ε2)3/2 + 2m R−2m ε 0 ε3η2Φ(εη + 2m)dη (ε2η2 + ε2)3/2 = ε − R−2mε∫ 0 ηΦ(εη + 2m)dη (η2 + 1)1/2 + R−2m ε∫ 0 η3Φ(εη + 2m)dη (η2 + 1)3/2 + 2m − R−2mε∫ 0 Φ(εη + 2m)dη (η2 + 1)1/2 + R−2m ε∫ 0 η2Φ(εη + 2m)dη (η2 + 1)3/2  . (3.15) From Eq.(3.15) we obtain I+0 (ε) = I + 1 (ε) = 2m Φ(2m) 0! R−2m ε∫ 0 [ − 1 (η2 + 1)1/2 + η2 (η2 + 1)3/2 ] dη+ + ε 1! R−2m ε∫ 0 Φ(1)(ξ) [ − 1 (η2 + 1)1/2 + η2 (η2 + 1)3/2 ] ηdη+ + εΦ(2m) 0! R−2m ε∫ −2m ε [ − 1 (η2 + 1)1/2 + η2 (η2 + 1)3/2 ] dη+ + ε2 1! R−2m ε∫ 0 Φ(1)(ξ) [ − 1 (η2 + 1)1/2 + η2 (η2 + 1)3/2 ] ηdη, (3.16) Where we have expressed the function Φ(εη + 2m) as 57 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance  Φ(εη + 2m) =∑n−1l=0 Φ αβ(l)(2m) l! (εη)l + 1 n! (εη)nΦαβ(n)(ξ) , ξ , θεη + 2m , 1 > θ > 0 , n = 1 (3.17) With Φ(l)(ξ) , dlΦ/dξl.Equation (3.17) gives w -lim ε→0 I+0 (ε) = w -lim ε→0 I+1 (ε) = 2mΦ(2m)lim ε→0  R−2m ε∫ 0 [ − 1 (η2 + 1)1/2 + η2 (η2 + 1)3/2 ] dη  = 2mΦ(2m) lim s→∞ [∫ s 0 η2dη (η2 + 1)3/2 − ∫ s 0 dη (η2 + 1)1/2 ] = = −2mΦ(2m). (3.18) Where use is made of the relation lim s→∞ [ s∫ 0 η2dη (η2 + 1)3/2 − s∫ 0 dη (u2 + 1)1/2 ] = −1 (3.19) Thus in S ′2m ( B+ (2m,R) ) ⊂ S ′2m(R3) We obtain w -lim ε→0 [ R+ε ]1 1 = w -lim ε→0 [ R+ε ]0 0 = −mΦ(2m). (3.20) The supergeneralized Ricci tensor below horizon [ R−ε ]β α = [ R−ε ]β α May now be calculated componentwise using the classical formulae ([ R−ε ]0 0 ) ε = ([ R−ε ]1 1 ) ε = 1 2 (( h−′′ε ) ε + 2 r ( h−′ε ) ε ) ,([ R−ε ]2 2 ) ε = ([ R−ε ]3 3 ) ε = ( h−′ε ) ε r + 1 + ( h−ε ) ε r2 . (3.21) From Eq.(3.21) we obtain h−ε (r) = − r − 2m r 7−→ h−ε (r) =  √ (2m− r)2 + ε2 r  = −h+ε (r), r < 2m. h−′ε (r) = −h+′ε (r) = r − 2m r [ (r − 2m)2 + ε2 ]1/2 − [ (r − 2m)2 + ε2 ]1/2 r2 , r ( h−′ε ) ε + 1 + ( h−ε ) ε = −r ( h+′ε ) ε + 1− ( h+ε ) ε = r − 2m[ (r − 2m)2 + ε2 ]1/2 + 1. h−′′ε (r) = −h+′′ε (r) = − r − 2m r2 [ (r − 2m)2 + ε2 ]1/2 + 2 [ (r − 2m)2 + ε2 ]1/2 r3 . r2 ( h−′′ε ) ε + 2r ( h−′ε ) ε = −r2 ( h+′′ε ) ε − 2r ( h+′ε ) ε = r[ (r − 2m)2 + ε2 ]1/2 − r (r − 2m)2[ (r − 2m)2 + ε2 ]3/2 . (3.22) Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation 58 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Φ(r) = π∫ 0 sin θdθ 2π∫ 0 dφΦ(x) where Φ(x⃗) ∈ C∞ ( R3 ) , Φ(x) is a function with compact support K such that K ⊆ B− (0, 2m) = { x ∈ R3|0 6 ∥x∥ 6 2m } We get:∫ K ([ R−ε ]2 2 ) ε Φ(x⃗) d3x = ∫ K ([ R−ε ]3 3 ) ε Φ(x⃗) d3x = 2m∫ 0 ( r ( h−′ε ) ε + 1 + ( h−ε ) ε ) Φ(r)dr = 2m∫ 0 { r − 2m[ (r − 2m)2 + ε2 ]1/2 } Φ(r)dr + 2m∫ 0 Φ(r)dr. (3.23) By replacement r − 2m = u, from Eq.(3.23) we obtain  ∫ K ([ R−ε ]2 2 ) ε Φ(x) d3x = ∫ K ([ R−ε ]3 3 ) ε Φ(x) d3x = 0∫ −2m uΦ(u+ 2m)du (u2 + ε2)1/2 + 0∫ −2m Φ(u+ 2m)du. (3.24) By replacement u = εη, from (3.23) we obtain I−3 (ε) = ∫ K ([ R−ε ]3 3 ) ε Φ(x) d3x = I−2 (ε) = ∫ K ([ R−ε ]2 2 ) ε Φ(x⃗) d3x = ε×  0∫ − 2m ε ηΦ(εη + 2m)dη (η2 + 1)1/2 + 0∫ − 2m ε Φ(εη + 2m)dη  , (3.25) Which is calculated to give I−3 (ε) = I − 2 (ε) = ε Φ(2m) 0! 0∫ − 2m ε [ η (η2 + 1)1/2 + 1 ] dη+ + ε2 1! 0∫ − 2m ε [ η (η2 + 1)1/2 + 1 ] Φ(1)(ξ)ηdη = εΦ(2m) 1−√(2m ε )2 + 1 + 2m ε + + ε2 1 0∫ − 2m ε [ η (η2 + 1)1/2 + 1 ] Φ(1)(ξ)ηdη, (3.26) Where we have expressed the function Φ(εη + 2m) as  Φ(εη + 2m) =∑n−1l=0 Φ (l)(2m) l! (εη)l + 1 n! (εη)nΦ(n)(ξ) , ξ , θεη + 2m , 1 > θ > 0 , n = 1 (3.27) With Φ(l) , dlΦ/drl. Equation (3.27) gives 59 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance  lim ε→0 I−3 (ε) = lim ε→0 I−2 (ε) = lim ε→0 εΦ(2m) 1−√(2m ε )2 + 1 + 2m ε + +lim ε→0  ε22 0∫− 2m ε [ η (η2 + 1)1/2 + 1 ] Φ(1)(ξ)ηdη  = 0. (3.28) Thus in S ′2m ( B−R (2m) ) ⊂ S ′2m(R3), where B− (0, 2m) = { x ∈ R3|0 6 ∥x∥ 6 2m } from Eq.(3.28) we obtain w − limε→0 [ R−ε ]3 3 = lim ε→0 I−3 (ε) = 0. w − lim ε→0 [ R−ε ]2 2 = lim ε→0 I−2 (ε) = 0. (3.29) For ([ R−ε ]1 1 ) ε , ([ R−ε ]0 0 ) ε we get: 2K ([ R−ε ]1 1 ) ε Φ(x) d3x = 2K ([ R−ε ]0 0 ) ε Φ(x) d3x = 2m∫ 0 ( r2 ( h−′′ε ) ε + 2r ( h−′ε ) ε ) Φ(r)dr = = 2m∫ 0 { r[ (r − 2m)2 + ε2 ]1/2 − r (r − 2m)2[ (r − 2m)2 + ε2 ]3/2 } Φ(r)dr. (3.30) By replacement r − 2m = u, from (3.30) we obtain I+1 (ε) = 2 ([ R−ε ]1 1 ) ε Φ(x) d3x = I+2 (ε) = 2 ([ R−ε ]0 0 ) ε Φ(x) d3x = 2m∫ 0 ( r2 ( h−′′ε ) ε + 2r ( h−′ε ) ε ) Φ(r)dr = = 0∫ −2m { u+ 2m (u2 + ε2)1/2 − u 2 (u+ 2m) (u2 + ε2)3/2 } Φ(u+ 2m)du. (3.31) By replacement u = εη, from (3.31) we obtain 2K ([ R−ε ]1 1 ) ε Φ(x) d3x = 2K ([ R−ε ]0 0 ) ε Φ(x) d3x = 0∫ − 2m ε ( r2 ( h−′′ε ) ε + 2r ( h−′ε ) ε ) Φ(r)dr = = ε 0∫ − 2m ε { εη + 2m (ε2η2 + ε2)1/2 − ε 2η2 (εη + 2m) (ε2η2 + ε2)3/2 } Φ(εη + 2m)dη = 0∫ − 2m ε ε2ηΦ(εη + 2m)dη (ε2η2 + ε2)1/2 + 2m0− 2m ε εΦ(εη + 2m)dη (ε2η2 + ε2)1/2 − − 0∫ − 2m ε ε4η3Φ(εη + 2m)dη (ε2η2 + ε2)3/2 − 2m0− 2m ε ε3η2Φ(εη + 2m)dη (ε2η2 + ε2)3/2 = ε 0∫ − 2m ε ηΦ(εη + 2m)dη (η2 + 1)1/2 − 0∫ − 2m ε η3Φ(εη + 2m)dη (η2 + 1)3/2 + +2m  0∫ − 2m ε Φ(εη + 2m)dη (η2 + 1)1/2 − 0∫ − 2m ε η2Φ(εη + 2m)dη (η2 + 1)3/2  . (3.32) 60 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance which is calculated to give I−0 (ε) = I − 1 (ε) = 2m Φ(2m) 0! εl 0∫ − 2m ε [ 1 (η2 + 1)1/2 − η 2 (η2 + 1)3/2 ] dη+ + ε 1! 2m ε∫ 0 Φ(1)(ξ) [ 1 (η2 + 1)1/2 − η 2 (η2 + 1)3/2 ] ηdη +O ( ε2 ) . (3.33) where we have expressed the function Φ(εη + 2m) as Φ(εη + 2m) = ∑n−1 l=0 Φαβ(l)(2m) l! (εη)l + 1 n! (εη)nΦαβ(n)(ξ) , ξ , θεη + 2m , 1 > θ > 0 , n = 1 (3.34) with Φ(l)(ξ) , dlΦ/dξl.Equation (3.34) gives lim ε→0 I−0 (ε) = lim ε→0 I−1 (ε) = 2mlim ε→0  Φ(2m)0! 0∫− 2m ε [ 1 (η2 + 1)1/2 − η 2 (η2 + 1)3/2 ] dη  = 2mΦ(2m)lim s→0 [∫ 0 −s dη (η2 + 1)1/2 − ∫ 0 −s η2dη (η2 + 1)3/2 ] = = 2mΦ(2m). (3.35) where use is made of the relation lim s→∞ [ 0∫ −s dη (u2 + 1)1/2 − 0∫ −s η2dη (η2 + 1)3/2 ] = 1. (3.36) Thus in S ′ ( B− (0, 2m) ) ⊂ S ′(R3) we obtain w -lim ε→0 [ R−ε ]1 1 = w -lim ε→0 [ R−ε ]0 0 = mΦ(2m). (3.37) Using Egs. (3.12),(3.20),(3.29),(3.37) we obtain ∫ [( T+rr +T +θ θ +T +φ φ +T +t t ) + ( T−rr +T −θ θ +T −φ φ +T −t t )]√ −gd3x = 0 (3.38) 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 d3x the coordinate volume element, gives ET = ∫ ( Trr +T θ θ +T φ φ +T t t )√ −gd3x = m, (3.39) We revrite now the Schwarzschild metric (3.3) in the form 61 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance { ( ds±2ε ) ε = ( h±ε (r)dt 2 ) ε − (( 1 + C±ε (r) ) dr2 ) ε + r2dΩ2 C±ε (r) = −1 + [ h±ε (r) ]−1 . (3.40) Using Eq.(A.5) from Eq.(3.40) one obtains for r ≍ 2m( R±μν (ε)R±μν (ε) ) ε =(( 1 2 ( h±ε )′′ + 1 r ( h±ε )′)2) ε + 2 ([ − ( h±ε )′ r + 1 r2 ]2) ε + ([ 1 2 ( h±ε )′′ + ( h±ε )′ r ]2) ε ≍ ≍ 1 4 [ (r − 2m)2 + ε2 ] + 2 (2m)4 , (3.41) and (3.40) (Rρσμν (ε)Rρσμν (ε))ε =((( h±ε )′′)2 + 2 (( h±ε )′ r )2 + 4 1 r4 + 2 (( h±ε )′ r )2 ) ε ≍ ≍ 1[ (r − 2m)2 + ε2 ] + 4 (2m)4 . (3.42) 3.2 Examples of Distributional Geometries. Calculation of the Distributional Quadratic Scalars by Using Nonsmooth Regularization via Horizon Let us consider again the Schwarzschild metric (3.1) ds2 = h(r)dt2 − h(r)−1dr2 + r2dΩ2, h(r) = −1 + 2m r = −r − 2m r , h−1(r) = − r r − 2m. (3.43) We revrite now the Schwarzschild metric (3.43) above Horizon (r ≥ 2m) in the form ds+2 = −A+(r)dt2 + ( A+(r) )−1 (r)dr2 + r2dΩ2, A+(r) = r − 2m r ,( A+(r) )−1 = r r − 2m. (3.44) Following the above discussion we consider the singular metric coefficient A−1(r) as an element of D′(R3) and embed it into (G(R3)) by replacement r − 2m 7−→ √ r2 + ε2 − 2m. (3.45) Thus above Horizon (r ≥ 2m) the corresponding distributional metric ( ds+2ε ) ε takes the form  ( ds+2ε ) ε = ( −A+ε (r)dt2 + ( A+ε (r) )−1 dr2 ) ε + r2dΩ2,( A+ε (r) ) ε = (√ r2 + ε2 − 2m r ) ε ,(( A+ε (r) )−1) ε = ( r√ r2 + ε2 − 2m ) ε . (3.46) We revrite now the Schwarzschild metric (3.43) below Horizon (r < 2m) in the form 62 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance  ds −2 = A−(r)dt2 − ( A−(r) )−1 dr2 + r2dΩ2, A−(r) = 2m− r r , ( A−(r) )−1 = r 2m− r . (3.47) Following the above discussion we consider the singular metric coefficient A−1(r) as an element of D′(R3) and embed it into (G(R3)) by replacement 2m− r 7−→ 2m− √ r2 + ε2. (3.48) Thus belov Horizon (r < 2m) the corresponding distributional metric ( ds−2ε ) ε takes the form  ( ds−2ε ) ε = ( A−ε (r)dt 2 − ( A−ε (r) )−1 dr2 ) ε + r2dΩ2,( A−ε (r) ) ε = ( 2m− √ r2 + ε2 r ) ε , (( A−ε (r) )−1) ε = ( r 2m− √ r2 + ε2 ) ε . (3.49) From Eq.(3.46) one obtains ( A+ε )′ = ( − √ r2 + ε2 − 2m r )′ = − 1√ r2 + ε2 + √ r2 + ε2 − 2m r2( A+ε )′′ = r (r2 + ε2)3/2 − 2 √ r2 + ε2 − 2m r3 + 1 r √ r2 + ε2 (3.50) From Eq.(3.46) using Eq.(A.5) one obtains (R (ε))ε = ( −4A ′ ε r + 2AεCε r2 −A′′ε ) ε = −4 r ( − 1√ r2 + ε2 + √ r2 + ε2 − 2m r2 ) ε + 2 r2 − − ( r (r2 + ε2)3/2 − 2 √ r2 + ε2 − 2m r3 + 1 r √ r2 + ε2 ) ε . (3.51) From Eq.(3.51) for r = 2m one obtains (R (ε))ε = 6 (2m)2 − 1 (2m)2 + 1 (2m)3 = 1 (2m)3 + 5 (2m)2 (3.52) Remark 3.2.1. Note that curvature scalar (R (ε))ε again nonzero but nonsingular. Let us introduce now the general metric which has the form [11]:{ ds2 = −A (r) (dx0)2 − 2D (r) dx0dr + (B (r) + C (r) (dr)2 +B (r) r2 [ (dθ)2 + sin2 θ(dφ)2 ] , (3.53) Where A (r) = Ω2 ( 1− a K (r) ) , B (r) = K2 (r) ρ2 (r) , C (r) = ( 1− a K (r) )−1 (K′ (r))2 − K 2 (r) ρ2 (r) − ( 1− a K (r) ) (f ′)2 , D (r) = Ω ( 1− a K (r) ) f ′,K′ (r) , dK (r) /dr, f ′ (r) , df (r) /dr, K (r) = ρ (r)− |a| , a < 0. (3.54) 63 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 3.2.2. Note that the coordinates t = x0/c and r are time and space coordinates, respectively, only if 1− a K > 0 , ( 1− a K )−1 (K′)2 − ( 1− a K ) (f ′)2 > 0 . (3.55) In the Cartesian coordinate system {xμ;μ = 0, 1, 2, 3} with x1 = r cosφ sin θ, x2 = r sinφ sin θ, x3 = r cos θ, (3.56) The metric (3.53)-(3.55) takes the form ds2 = gμνdx μdxν (3.57) With gμν given by g00 = −A , g0α = −D xα r , gαβ = Bδ αβ + C xαxβ r2 . (3.58) From Eq.(3.54) one obtain A (r) = Ω2 ( ρ (r) ρ (r)− |a| ) , B = (ρ (r)− |a|)2 ρ2 (r) , C (r) = ( ρ (r)− |a| ρ (r) ) − (ρ (r)− |a|) 2 ρ2 (r) − ( ρ (r) ρ (r)− |a| ) (f ′ (r))2 , D (r) = Ω ( ρ (r) ρ (r)− |a| ) f ′ (r) , f ′ (r) , df (r) /dr. (3.59) Regularizing the function (ρ (r)− |a|)−1 above gorizon (under condition ρ (r)− |a| ≥ 0) such as ρ (r)− |a| ≥ 0 : ( ρ (r)− |a|)−1 7→ (ρε (r)− |a|)−1 = (√ ρ2 (r) + ε2 − |a| )−1 (3.60) with ε ∈ (0, 1] from Eq.(3.59)-Eq.(3.60) one obtains A+ε (r) = Ω 2 ( ρε (r) ρε (r)− |a| ) , B+ε (r) = (ρε (r)− |a|)2 ρ2ε (r) , C+ε (r) = ( ρε (r)− |a| ρε (r) ) − (ρε (r)− |a|) 2 ρ2ε (r) − ( ρε (r) ρε (r)− |a| ) (f ′ (r))2 , D+ε (r) = Ω ( ρε (r) ρε (r)− |a| ) f ′ (r) , f ′ (r) , df (r) /dr. (3.61) Regularizing the function (|a| − ρ (r))−1 below gorizon (under condition |a| − ρ (r) ≥ 0) such as |a| − ρ (r) ≥ 0 : (|a| − ρ (r))−1 7→ (|a| − ρε (r)) = ( |a| − √ r2 + ε2 )−1 (3.62) with ε ∈ (0, 1] from Eq.(3.59),Eq.(3.62) one obtains A−ε (r) = −Ω2 ( ρε (r) |a| − ρε (r) ) , B−ε (r) = (|a| − ρε (r))2 ρ2ε (r) , C−ε (r) = − ( |a| − ρε (r) ρε (r) ) − (|a| − ρε (r)) 2 ρ2ε (r) + ( ρε (r) |a| − ρε (r) ) (f ′ (r))2, D−ε (r) = −Ω ( ρε (r) |a| − ρε (r) ) f ′ (r) , f ′ (r) , df (r) /dr. (3.63) 64 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 3.2.3. Finally the metric (3.57) becomes the Colombeau object of the form( ds2ε ) ε = ( g±μν (ε) dx μdxν ) ε (3.64) with gμν (ε) given by g±00 (ε) = −A±ε (r) , g ± 0α (ε) = −D±ε (r) xα r , g±αβ (ε) = B ± ε (r) δ αβ + C±ε (r) xαxβ r2 . (3.65) Using now Eq. A2 one obtains that the Colombeau curvature scalars ( R± (ε) ) ε in terms of Colombeau generalized functions ( A±ε (r) ) ε , ( B±ε (r) ) ε , ( C±ε (r) ) ε , ( D±ε (r) ) ε is expressed as ( R+ (ε) ) ε = ( r2 + ε2(√ r2 + ε2 − |a| )2 [ 9aε2 (r2 + ε2) 5 2 − 2aε 2 r2 (r2 + ε2) 3 2 ]) ε( R− (ε) ) ε = − ( r2 + ε2(√ r2 + ε2 − |a| )2 [ 9aε2 (r2 + ε2) 5 2 − 2aε 2 r2 (r2 + ε2) 3 2 ]) ε (3.66) Remark 3.2.4. Note that (i) on horizon r = a Colombeau scalars ( R± (ε) ) ε well defined and becomes to infinite large Colombeau generalized numbers ( R+ (ε) ) ε =( a2 + ε2(√ a2 + ε2 − |a| )2 [ 9aε2 (a2 + ε2) 5 2 − 2aε 2 a2 (a2 + ε2) 3 2 ]) ε = 7a−2 ( ε−2 ) ε ∈ R,( R− (ε) ) ε = − ( a2 + ε2(√ a2 + ε2 − |a| )2 [ 9aε2 (a2 + ε2) 5 2 − 2aε 2 r2 (r2 + ε2) 3 2 ]) ε = −7a−2 ( ε−2 ) ε ∈ R (3.67) (ii) for r = a Colombeau scalars ( R± (ε) ) ε well defined and becomes to infinite small Colombeau generalized numbers ( R± (ε) ) ε ≈ ± ( ε2 ) ε . Using now Eq. A2 one obtains that the Colombeau scalars ( R±μν (ε)R±μν (ε) ) ε in terms of Colombeau generalized functions ( A±ε (r) ) ε , ( B±ε (r) ) ε , ( C±ε (r) ) ε , ( D±ε (r) ) ε is expressed as ( R±μν (ε)R±μν (ε) ) ε = ± ( ( r2 + ε2 )2(√ r2 + ε2 − |a| )4 {[ 5 2 [ 3aε2 (r2 + ε2) 5 2 ]2 − 2aε 2 r2 (r2 + ε2) 3 2 ] + 2 [ 3aε2 (r2 + ε2) 5 2 + aε2 r2 (r2 + ε2) 3 2 ]2}) ε (3.68) Remark 3.2.5. Note that (i) on horizon r = a Colombeau scalars ( R±μν (ε)R±μν (ε) ) ε well defined and becomes to infinite large Colombeau generalized numbers, (ii) for r = a Colombeau scalars ( R± (ε) ) ε well defined and becomes to infinite small Colombeau generalized numbers. Using now Eq. A2 one obtains that the Colombeau scalars ( R±ρσμν (ε)R±ρσμν (ε) ) ε in terms of Colombeau generalized functions ( A±ε (r) ) ε , ( B±ε (r) ) ε , ( C±ε (r) ) ε , ( D±ε (r) ) ε is expressed as 65 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ( R±ρσμν (ε)R±ρσμν (ε) ) ε = ( 12a2(√ r2 + ε2 − |a| )6 [ 1 + aε2 (r2 + ε2) 3 2 ]2 ∓ ∓ 4a 2(√ r2 + ε2 − |a| )5 [ 1 + aε2 (r2 + ε2) 3 2 ]2 [ 2ε2 r2 (r2 + ε2) + 9ε2 (r2 + ε2) 5 2 ] + a2(√ r2 + ε2 − |a| )4 [ 4ε4r4 (r2 + ε2) + 81ε4(r2 + ε2)3 ] . (3.69) Remark 3.2.6. Note that (i) on horizon r = a Colombeau scalars ( R±ρσμν (ε)R±ρσμν (ε) ) ε well defined and becomes to infinite large Colombeau generalized numbers, (ii) for r = a Colombeau scalars finite( R±ρσμν (ε)R±ρσμν (ε) ) ε = 12a2 (r2 − |a|)6 (3.70) and tends to zero in the limit r → ∞. Remark 3.2.7. Note that under generalized transformatios such as dt = ( d [√ r2 + ε2 − 2m r ] v+ε ) ε + (√ r2 + ε2 − 2m r dv+ε ) ε , (3.71) and dt = ( d [ 2m− √ r2 + ε2 r ] v−ε ) ε + ( 2m− √ r2 + ε2 r dv−ε ) ε , (3.72) The metric given by Eq.(3.61)-Eq.(3.64) becomes to Colombeau metric of the form{ ds±2ε = ∓A± (r, ε) (dv±ε )2 − 2v±ε D±2 (r, ε) dv±ε dr + [ B± (r, ε) + C±1 ( v±ε , r, ε )] (dr)2+ +B± (r, ε) r2 [ (dθ)2 + sin2 θ(dφ)2 ] . (3.73) 4 Quantum Scalar Field in Curved Distributional Space-time 4.1 Canonical Quantization in Curved Distributional Space-time Much of formalism can be explained with Colombeau generalized scalar field.The basic concepts and methods extend straightforwardly to distributional tensor and distributional spinor fields. To being with let's take a spacetime of arbitrary dimension D, with a metric gμν of signature(+− ...−). The action for the Colombeau generalized scalar field (φε)ε ∈ G(M) is (Sε)ε = (∫ M dDx 1 2 √ |gε| (gμνε ∂μφε∂νφε)− ( m2 +Rε ) φ2ε ) ε . (4.1) The corresponding equation of motion is([ ε +m2 + ξRε ] φε ) ε , ε ∈ (0, 1] . (4.2) Here (εφε)ε = ( |gε|−1/2 ∂μ |gε|1/2 gμνε ∂μφε ) ε . (4.3) 66 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance With } explicit, the mass m should be replaced by m/}.Separating out a time coordinate x0, xμ = (x0, xi), i = 1, 2, 3 we can write the action as (Sε)ε = (∫ dx0Lε ) ε , (Lε)ε = (∫ dD−1xε ) ε . (4.4) The canonical momentum at a time x0 is given by (πε (x))ε = (δLε/ δ (∂0φε (x)))ε = ( |hε|1/2 nμ∂μφε (x) ) ε , (4.5) where x labels a point on a surface of constant x0, the x0 argument of (φε)ε is suppressed, n μ is the unit normal to the surface, and (|hε|)ε is the determinant of the induced spatial metric (hij (ε))ε. To quantize, the Colombeau generalized field (φε)ε and its conjugate momentum (πε (x))ε are now promoted to hermitian operators and required to satisfy the canonical commutation relation,([ φε (x) , πε ( y )]) ε = i~δD−1 ( x, y ) , ε ∈ (0, 1] . (4.6) Here ∫ dD−1yδD−1 ( x, y ) f ( y ) = f (x) for any scalar function f ∈ D ( R3 ) , without the use of a metric volume element. We form now a conserved bracket from two complex Colombeau solutions to the scalar wave equation (4.2) by (⟨φε, φε⟩)ε = (∫ Σ dΣμj μ ε ) ε , ε ∈ [0, 1] , (4.7) where (jμε (φε, φε))ε = (i/~) ( |gε|1/2 gμνε ( φε∂νφε − φε∂νφε )) ε . (4.8) This bracket is called the generalized Klein-Gordon inner product, and (⟨φε, φε⟩)ε the generalized Klein Gordon norm of (φε)ε . The generalized current density (j μ ε (φε, φε))ε is divergenceless,i.e. (∂μj μ ε (φε, φε))ε = 0 when the Colombeau generalized functions (φε)ε and (φε)ε satisfy the KG equation (4.2), hence the value of the integral in (4.7) is independent of the space-like surface Σ over which it is evaluated, provided the functions vanish at spatial infinity. The generalized KG inner product satisfies the relations( ⟨φε, φε⟩ ) ε = − (⟨ φε, φε ⟩) ε = (⟨φε, φε⟩)ε , ε ∈ [0, 1] . (4.9) We define now the annihilation operator associated with a complex Colombeau solution (φε)ε by the bracket of (φε)ε with the generalized field operator (φε)ε : (a (φε))ε = (⟨φε, φε⟩)ε . (4.10) It follows from the hermiticity of (φε)ε that the hermitian conjugate of (a (φε))ε is given by( a† (φε) ) ε = − ( a ( φε )) ε . (4.11) From Eq.(4.5) and CCR (4.6) one obtains([ a (φε) , a † (φε) ]) ε = (⟨φε, φε⟩)ε . (4.12) Note that from Eq.(4.11) it follows ([a (φε) , a (φε)])ε = − (⟨ φε, φε ⟩) ε , ([ a† (φε) , a † (φε) ]) ε = − (⟨φε, φε⟩) (4.13) Note that if (φε)ε is a positive norm solution with unit norm and with, then (a (φε))ε and a † (φε) 67 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance satisfy the commutation relation ([ a† (φε) , a (φε) ]) ε = 1. Suppose now that |Ψ⟩ is a normalized quantum state satisfying (a (φε) |Ψ⟩)ε = 1, then for each n, the state |n,Ψ⟩ = ((1/ √ n!)(a (φε)) n |Ψ⟩)ε is a normalized eigenstate of the number operator (N [(φε)])ε = ( a† (φε) a (φε) ) ε with eigenvalue n. The span of all these states defines a Fock space of the distributional (φε)εwavepacket "n-particle excitations" above the state |Ψ⟩ .If we want to construct the full Hilbert space of the field theory in curved distributional spacetime,how can we proceed? We should find a decomposition of the space of complex Colombeau solutions to the wave equation (4.2) S into a direct sum of a positive norm subspace Sp and its complex conjugate Sp, such that all brackets between solutions from the two subspaces vanish. That is, we must find a direct sum decomposition: S = Sp⊕Sp (4.14) Such that (⟨φε, φε⟩)ε > 0,∀ (φε)ε ∈ Sp (4.15) and (⟨φε, φε⟩)ε > 0, ∀ (φε)ε , (φε)ε ∈ Sp. (4.16) The condition (4.15) implies that each (φε)ε in Sp can be scaled to define its own harmonic oscillator sub-albegra. The second condition implies, according to (4.13), that the annihilators and creators for (φε)ε and (φε)ε in the subspace Sp commute amongst themselves: ([a (φε) , a (φε)])ε = ([ a† (φε) , a † (φε) ]) ε = 0. (4.17) Given such a decompostion a total Hilbert space H for the field theory can be defined as the space of finite norm sums of possibly infinitely many states of the form( a† (φ1,ε) ...a † (φn,ε) |0⟩ ) ε , (4.18) Where |0⟩ is a state such that (a (φn,ε) |0⟩)ε = 0 for all (φε)ε in Sp. The state |0⟩ , as in classical case, is called a Fock vacuum and Hilbert space H is called a Fock space. The representation of the field operator on this Fock space is hermitian and satisfies the canonical commutation relations in sense of Colombeau generalized function. 4.2 Defining Distributional Outgoing Modes For illustration we consider the non-rotating, uncharged d-dimensional SAdS BH with a distributional line element( ds2ε ) ε = ( −fεdt2 + f −1ε dr2 ) ε + r2dΩ2d−2, ε ∈ (0, 1] , (4.19) Where fε = 0, ε ∈ (0, 1] ,fε=0 = 1 + r2 L2 − r d−3 0 rd−3 , (4.20) Where dΩ2d−2 is the metric of the (d − 2)-sphere, and the AdS curvature radius squared L2 is related to the cosmological constant by L2 = −(d− 2)(d− 1)/2Λ. The parameter r0 is proportional to the mass M of the spacetime: M = (d− 2)Ad−2rd−30 /16π, where Ad−2 = 2π(d−1)/2/Γ[(d− 1)/2]. The distributional Schwarzschild geometry corresponds to L → ∞.The corresponding equation of motion (4.2) for massless case are 68 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (∇μ∇μφε)ε = (d− 2) γ 4 (d− 1) (Rε)ε , (Gε,μν)ε + Λ(gε,μν)ε = 8πG (Tε,μν)ε , (Tε,μν)ε  δ (x) . (4.21) The time-independence and the spherical symmetry of the metric imply the canonical decomposition (φε (t, r, θ))ε = e −iωt ( Ψlm,ε (r)Ylm (θ) r(d−2)/2 + ε ) ε , (4.22) Where Ylm (θ) denotes the d-dimensional scalar spherical harmonics, satisfying ∆Ωd−2Ylm (θ) = −l (l + d− 3)Ylm (θ) , (4.23) Where∆Ωd−2 the Laplace-Beltrami operator.Substituting the decomposition (4.22) into Eq. (6) one get a radial wave equation( f2ε d2Ψlm,ε (r) dr2 + fεf ′ ε dΨlm,ε (r) dr + ( ω2 − Vlm,ε (r) ) Ψlm,ε (r) ) ε = 0. (4.24) We define now a "tortoise" distributional coordinate (r∗ε)ε = (r ∗ ε (r))ε by the relation( dr∗ε dr ) ε = ( f −1ε (r) ) ε . (4.25) By using a "tortoise" distributional coordinate the Eq.(4.24) can be written in the form of a Schrödinger equation with the potential Vlm,ε (r)( dΨε (r ∗ ε) dr∗ε ) ε + (( ω2 − Vε (r∗ε) ) Ψε (r ∗ ε) ) ε = 0. (4.26) Note that the tortoise distributional coordinate (r∗ε (r))ε becomes to infinite Colombeau constant[ (r∗ε (r+))ε ] = [ (ln ε)ε ] at the horizon, i.e. as r → r+, but its behavior at infinity is strongly dependent on the cosmological constant: [ (r∗ε (r+))ε ] = +∞ for asymptotically-flat spacetimes, and [ (r∗ε (r+))ε ] = finite Colombeau constant for the SAdSd geometry. 4.2.1 Boundary Conditions at the Horizon of the Distributional SAdS BH Geometry For most spacetimes of interest the potential (Vε (r ∗ ε (r)))ε = 0 as r = r+, i.e. (|r∗ε (r)|)ε = +∞, and in this limit solutions to the wave equation (4.26) behave as (Ψε (t, r ∗ ε))ε   (exp [−iω (t± r ∗ ε (r))])ε , as r r+. (4.27) Note that classically nothing should leave the horizon and thus classically only ingoing modes (corresponding to a plus sign) should be present,i.e. (Ψε (t, r ∗ ε))ε   (exp [−iω (t+ r ∗ ε (r))])ε , as r r+. (4.28) Note that for non-extremal spacetimes, the tortoise coordinate tends to (r∗ε (r))ε = (∫ f −1ε (r) dr ) ε   [ (f ′ε (r+))ε ]−1 (ln (|r − r+|+ ε))ε as r r+, (4.29) 69 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Where (f ′ε (r+))ε > 0. Therefore near the horizon, outgoing modes behave as{ (exp [−iω (t− r∗ε (r))])ε = { (exp [−iωυ∗ε (t, r)])ε } (exp [−2iωr∗ε (r)])ε = = { (exp [−iωυ∗ε (t, r)])ε }{( [|r − r+|+ ε]2iω/f ′ ε (r+) ) ε } , (4.30) Where (υ∗ε (t, r))ε = t + (r ∗ ε (r))ε . Now Eq. (4.30) shows that outgoing modes is Colombeau generalized function of class G(R). 5 Energy-Momentum Tensor Calculation by using Colombeau Distributional Modes We shall assume now any distributional spacetime which is conformally static in both the asymptotic past and future. We will be considered distributional spacetime which is conformally flat in the asymptotic past,i.e. { ds2ε ∼ ( f2ε,in(−dt2 + dx⃗2) ) ε asymp. past ds2ε ∼ ( f2ε,out(−dt2 + hε,ijdxidxj) ) ε , asymp. future (5.1) Where ε ∈ (0, 1] (fε,J)ε = (fε,J(t, x⃗))ε > 0,J ∈ {in, out}, are smooth functions and hε,ij = hε,ij(x⃗), i, j = 1, 2, 3, are the components of an arbitrary distributional spatial metric. Note that we use the same labels t and x⃗ = (x1, x2, x3) for coordinates in the asymptotic past and future only for simplicity; they are obviously defined on non-intersecting regions of the spacetime.) In each of these asymptotic regions the distributional field (Φε)ε can be written as (Φε)ε = ( Φε/fε,J ) ε , where( Φε ) ε satisfies − ( ∂2 ∂t2 Φ ) ε = − ( ∆ε,JΦε ) ε + ( Vε,JΦε ) ε , (5.2) Where (∆ε,in)ε is the flat Laplace operator, (∆ε,out)ε is the Laplace operator associated with the spatial metric (hεij)ε, and the effective potential VJ is given by  (Vε,J)ε = ( ∆εJfε,J fε,J ) ε + ( f 2ε,J(m 2 + ξRε) ) ε = (1− 6ξ) ( ∆εJfε,J fε,J ) ε +m2 ( f 2ε,J ) ε + ξ (Kε,J)ε , (5.3) With (Kε,in)ε = 0, Kε,out = Kε,out(x⃗) the scalar curvature associated with the spatial distributional metric (hε,ij)ε. We assume now this condition: (i) the massless (m = 0) field with arbitrary coupling ξ in spacetimes which are asymptotically flat in the past and asymptotically static in the future,i.e. fin = 1 and fε,out = fε,out(x⃗), as those describing the formation of a static BH from matter initially scattered throughout space, and (ii) the massless, conformally coupled field (m = 0 and ξ = 1/6). With this assumptions for the potential, two different sets of positive-norm distributional modes, ( u (+) ε,k⃗ ) ε and 70 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ( v (+) ε,α ) ε , can be naturally defined by the requirement that they are the solutions of Eq.(5.2) which satisfy the asymptotic conditions: ( u (+) ε,k⃗ ) ε ≍ past (16π3ωk⃗) −1/2 e−i(ωk⃗t−k⃗*x⃗) ( f −1ε,in ) ε (5.4) and( v (+) ε,α ) ε ≍ future (2πα) −1/2e−iπαt ( f −1ε,out Fε,α(x⃗) ) ε , (5.5) Where k⃗ ∈ R3, ωk⃗ := ∥k⃗∥, πα > 0, and (Fε,α(x⃗))ε are Colombeau solutions of ([−∆ε,out + Vε,out(x⃗)]Fε,α(x⃗))ε = π 2 α (Fε,α(x⃗))ε , (Fε,α(x⃗))ε ∣∣ ε=0 ∈ C∞ ( R3 ) (5.6) Satisfying the normalization (∫ Σout d3x √ h Fε,α(x⃗) ∗Fε,β(x⃗) ) ε = δα,β (5.7) On a Cauchy surface Σout in the asymptotic future. Note that each Fε,α, ε ∈ [0, 1] can be chosen to be real without loss of generality. There are reasonable situations where the distributional modes( v (+) ε,α ) ε , given in Eq. (5.5), together with distributional modes ( v (−) ε,α ) ε fail to form a complete set of distributional normal modes. This happens whenever the operator ([−∆ε,out + Vε,out(x⃗)])ε in Eq. (5.2) happens to possess normalizable i.e., satisfying Eq. (5.7) eigenfunctions with negative eigenvalues, π2α = −Ω2α < 0. In this case, additional positive-norm modes ( w (+) ε,α ) ε with the asymptotic behavior ( w (+) ε,α ) ε ≍ future ( eΩαt−iπ/12 + e−Ωαt+iπ/12 )( Fε,α(x⃗)√ 2Ωα fε,out(x⃗) ) ε (5.8) and their complex conjugates ( w (−) ε,α ) ε are necessary in order to expand an arbitrary Colombeau solution of Eq.(5.1) As a direct consequence, at least some of the in-modes ( u (±) ε,k⃗ ) ε (typically those with low ωk⃗) eventually undergo an exponential growth.This asymptotic divergence is reflected on the unbounded increase of the vacuum fluctuations, (⟨ Φ2ε(x⃗) ⟩) ε ≍ future κ e2Ωt 2Ω [( Fε(x⃗) fε,out(x⃗) ) ε ]2 [ 1 +O(e−εt) ] , (5.9) Where F (x⃗) is the eigenfunction of Eq. (5.6) associated with the lowest negative eigenvalue allowed, π2α = −Ω2, ε is some positive constant, and κ is a dimensionless constant (typically of order unity) whose exact value depends globally on the spacetime structure (since it crucially depends on the projection of each ( u (±) ε,k⃗ ) ε on the mode ( w (±) ε,α ) ε whose π2α = −Ω2; κ also depends on the initial state, here assumed to be the vacuum |0⟩in). As one would expect, these wild quantum fluctuations give an important contribution to the vacuum energy stored in the field. In fact, the expectation 71 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance value of its distributional energy-momentum tensor, (⟨Tε,μν(x⃗)⟩)ε , ε ∈ (0, 1] , in the asymptotic future is found to be dominated by this exponential growth:  (⟨Tε,00(x⃗)⟩)ε ≍future {(⟨ Φ2ε(x⃗) ⟩) ε }{ (1− 4ξ) 2 ( Ω2 + (DFε) 2 F 2 +m2f 2ε + ξKε ) ε +(1− 6ξ) ( 2ξD2fε fε + (Dfε) 2 2f 2ε − DifεD iFε fεFε ) ε +O(e−εt) } , (5.10)  (⟨Tε,0i(x⃗)⟩)ε ≍future{(⟨ Φ2ε(x⃗) ⟩) ε }{ (1− 4ξ) ( ΩDiFε Fε ) ε − (1− 6ξ) ( ΩDifε fε ) ε +O(e−εt) } , (5.11) (⟨Tε,ij(x⃗)⟩)ε ≍future{(⟨ Φ2ε(x⃗) ⟩) ε }{ (1− 2ξ) ( DiFεDjFε F 2ε ) ε − 2ξ ( DiDjFε Fε ) ε + ξ ( Rε,ij ) ε + (1− 4ξ)hij 2 ( Ω2 − ( (DFε) 2 F 2ε −m2f2ε − ξKε ) ε ) +(1− 6ξ) [( DifεDjfε f2ε − DifDjF fεFε − DjfεDiFε fεFε ) ε + +(hε,ij)ε ( 2ξD2fε fε − (Dfε) 2 2f2ε + DkfεD kFε fεFε ) ε ] +O(e−εt), (5.12) Where Di is the derivative operator compatible with the distributional metric (hε,ij)ε (so that ∆out = D 2), ( Rε,ij ) ε is the associated distributional Ricci tensor so that (Kε,out)ε = ( hijε Rε,ij ) ε , and we have omitted the subscript out in (fε,out)ε and (Kε,out)ε for simplicity. The Eqs. (5.10-5.12), together with Eq.(5.9), imply that on time scales determined by Ω−1, the vacuum fluctuations of the field should overcome any other classical source of energy, therefore taking control over the evolution of the background geometry through the semiclassical Einstein equations (in which (⟨Tε,μν⟩)ε is included as a source term for the distributional Einstein tensor). We are then confronted with a startling situation where the quantum fluctuations of a field, whose energy is usually negligible in comparison with classical energy components, are forced by the distributional background spacetime to play a dominant role. We are still left with the task of showing that there exist indeed well-behaved distributional background spacetimes in which the operator [ (−∆ε,out + Vε,out(x⃗))ε ] possesses negative eigenvalues π2α < 0,condition on which depends Eq(5.9). Experience from quantum mechanics tells us that this typically occurs when (Vε,out)ε gets sufficiently negative over a sufficiently large region. It is easy to see from Eq. (5.3) that, except for very special geometries (as the flat one), one can generally find appropriate values of ξ ∈ R which make (Vε,out)ε as negative as would be necessary in order to guarantee the existence of negative eigenvalues. For distributional BH spacetime using Eq.(5.9)-Eq.(5.12) one obtains (⟨ Φ2ε(r) ⟩) ε ≍ future κ e2Ωt 2Ω ( r1/2Fε(r)( (r − r+)2 + ε2 )1/4 ) ε 2 , r → r+ (5.13) 72 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (⟨Tε,00(r⟩)ε ≍future{(⟨ Φ2ε(r) ⟩) ε }{ (1− 4ξ) 2 ( Ω2 + (DFε (r+)) 2 F 2 (r+) +m2 ( (r − r+)2 + ε2 )1/2 + ξKε ) ε +(1− 6ξ) ( − 2ξD 2fε (r)( (r − r+)2 + ε2 )1/4 + (Dfε (r))2( (r − r+)2 + ε2 )1/4 − Difε (r)DiFε( (r − r+)2 + ε2 )1/4 Fε ) ε  , r → r+, fε (r) = ( (r − r+)2 + ε2 )1/4 (5.14)  (⟨Tε,0i(r)⟩)ε ≍future{(⟨ Φ2ε(r) ⟩) ε }(1− 4ξ) ( ΩDiFε (r+) Fε (r+) ) ε − (1− 6ξ) ( ΩDifε (r)( (r − r+)2 + ε2 )1/4 ) ε  , r → r+, fε (r) = ( (r − r+)2 + ε2 )1/4 (5.15) (⟨ Tε,ij(r) ⟩) ε ≍ future{(⟨ Φ2ε(r) ⟩) ε }{ (1 − 2ξ) ( DiFεDj Fε F2ε ) ε − 2ξ ( DiDj Fε Fε ) ε + ξ ( Rε,ij ) ε + (1 − 4ξ)hij 2 ( Ω2 − ( (DFε ( r+ ) )2 F2ε ( r+ ) − m2 ((r − r+)2 + ε2)1/2 − ξKε ) ε ) +(1 − 6ξ)   DifεDjfε(( r − r+ )2 + ε2)1/2 − DifDj F(( r − r+ )2 + ε2)1/4 Fε − DjfεDiFε(( r − r+ )2 + ε2)1/2 Fε  ε + + ( hε,ij ) ε  2ξD2fε(( r − r+ )2 + ε2)1/4 − (Dfε) 2 2 (( r − r+ )2 + ε2)1/2 + DkfεD kFε(( r − r+ )2 + ε2)1/4 Fε  ε  , r → r+. (5.16) Remark 5.1. Note that in spite of the unbounded growth at r → r+ in Eq.(5.13)-Eq.(5.16), (⟨Tε,μν⟩)ε is covariantly conserved: ( ∇μ ⟨ Tμε,ν ⟩) ε = 0. In the static case (fε,out)ε = (fε,out(x⃗))ε, for instance for distributional BH geometry, this implies that the total vacuum energy is kept constant, although it continuously flows from spatial regions where its density is negative to spatial regions where it is positive. Remark 5.2. Note that the singular behavior at r → r+ appearing in Eq.(5.13)-Eq.(5.16) leads only to asymptotic divergences, i.e. all the quantities remain finite everywhere except horizon. 6 Distributional SAdS BH Space-time-induced Vacuum Dominance 6.1 Adiabatic Expansion of Green Functions Using equation of motion Eq.(5.2) one can obtain corresponding distributional generalization of the canonical Green functions equations. In particular for the distributional propagator iG±ε (x, x ′) = ⟨ 0 ∣∣T (φ±ε (x)φ±ε (x′))∣∣ 0⟩ , ε ∈ (0, 1] (6.1) One obtains directly([ ε,x +m2 + ξR± (x, ε) ] G±ε (x, x ′) ) ε = − [ −g± (x, ε) ]−1/2 δn (x− x′) . (6.2) Special interest attaches to the short distance behaviour of the Green functions, such as ( G±ε (x, x ′) ) ε in the limit ∥x− x′∥ → 0 with a fixed ε ∈ (0, 1] . We obtan now an adiabatic expansion of 73 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ( G±ε (x, x ′) ) ε .Introducing Riemann normal coordinates yμ for the point x, with origin at the point x′ we have expanding( g±μυ (x, ε) ) ε = ημυ + 1 3 [( R±μαυβ (ε) ) ε ] yαyβ − 1 6 [( R±μαυβ;γ (ε) ) ε ] yαyβyγ+ + [ 1 20 ( R±μαυβ;γδ (ε) ) ε + 2 45 [( R±αμβλ (ε) ) ε ] ( R±λγvδ (ε) ) ε ] yαyβyγyδ + ... (6.3) Where ημυ is the Minkowski metric tensor, and the coefficients are all evaluated at y = 0. Defining now( L ±ε (x, x ′) ) ε = [(( −g±μυ (x, ε) )1/4) ε ] ( G±ε (x, x ′) ) ε (6.4) and its Colombeau-Fourier transform by( L ±ε (x, x ′) ) ε = (2π)−n (∫ dnke−ikyL ±ε (k) ) ε (6.5) Where ky = ηαβkαyβ , one can work in a sort of localized momentum space. Expanding (6.2) in normal coordinates and converting to k-space, ( L ±ε (k) ) ε can readily be solved by iteration to any adiabatic order. The result to adiabatic order four (i.e., four derivatives of the metric) is ( L±ε (k) ) ε = ( k2 − m2 )−1 − ( 1 6 − ξ )( k2 − m2 )−2 ( R± (ε) ) ε + + i 2 ( 1 6 − ξ ) ∂α ( k2 − m2 )−2 ( R±;α (ε) ) ε − − 1 3 [( a± αβ (ε) ) ε ] ∂α∂β ( k2 − m2 )−2 + [( 1 6 − ξ )2 ( R±2 (ε) ) ε + 2 3 ( a±λ λ (ε) ) ε ] ( k2 − m2 )−3 (6.6) Where ∂α = ∂/∂k α,( a±αβ (ε) ) ε ≍ ( 1 2 − ξ )( R±;αβ (ε) ) ε + 1 120 ( R±;αβ (ε) ) ε − 1 140 ( R± λαβ;λ (ε) ) ε − − 1 30 [( R± λα (ε) ) ε ] ( R±λβ (ε) ) ε + 1 60 [( R±κ λα β (ε) ) ε ] ( R±κλ (ε) ) ε + + 1 60 [( R± λμκα (ε) ) ε ] ( R±λμκβ (ε) ) ε , (6.7) and we are using the symbol ≍ to indicate that this is an asymptotic expansion. One ensures that Eq.(6.5) represents a time-ordered product by performing the k0 integral along the appropriate contour in Fig.4. This is equivalent to replacing m2 by m2 − iε. Similarly, the adiabatic expansions of other Green functions can be obtained by using the other contours in Fig.3. Substituting Eq.(6.6) into Eq.(6.5) gives  ( L±ε ( x, x′ )) ε = (2π)−n ×(∫ dnke−iky [ a±0 ( x, x′; ε ) + a±1 ( x, x′; ε ) ( − ∂ ∂m2 ) + a±2 ( x, x′; ε ) ( ∂ ∂m2 )2] ( k2 − m2 )−1) ε (6.8) where( a±0 (x, x ′; ε) ) ε = 1 (6.9) and, to adiabatic order 4, ( a±1 (x, x ′; ε) ) ε =( 1 6 − ξ )( R± (ε) ) ε − i 2 ( 1 6 − ξ )[( R±;α (ε) ) ε ] yα − 1 3 [( a±αβ (ε) ) ε ] yαyβ( a±2 (x, x ′; ε) ) ε = 1 2 ( 1 6 − ξ )( R±2 (ε) ) ε + 1 3 ( a±λλ (ε) ) ε (6.10) 74 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Fig. 4. The contour in the complex k0plane C to be used in the evaluation of the integral giving L +. The cross indicates the pole at k0 = ( |k|2 +m2 )1/2 . With all geometric quantities on the right-hand side of Eq.(6.10) evaluated at x′. ( k2 −m2 + iε )−1 = −i ∞∫ 0 dseis(k 2−m2+iε) (6.11) in Eq.(6.8), then the dnk integration may be interchanged with the ds integration, and performed explicitly to yield (dropping the iε) ( L ±ε (x, x ′) ) ε = −i (4π)−n/2 ( ∞∫ 0 ids (is)−n/2 exp [ −im2s+ σ (x, x ′) 2is ]± ε (x, x′; is) ) ε σ (x, x′) = 1 2 yαy α. (6.12) The function σ (x, x′) which is one-half of the square of the proper distance between x and x′, while the function (Lε (x, x ′; is))ε has the following asymptotic adiabatic expansion( L ±ε (x, x ′; is) ) ε ≍ ( a±0 (x, x ′; ε) ) ε + is ( a±1 (x, x ′; ε) ) ε + (is)2 ( a±2 (x, x ′; ε) ) ε + ... (6.13) Using Eq.(6.4), equation (6.12) gives a representation of ( G±ε (x, x ′) ) ε :  ( G±ε ( x, x′ )) ε = −i (4π)−n/2 ([( ∆ 1/2 ± ( x, x′; ε )) ε ]∞∫ 0 ids (is)−n/2 exp [ −im2s + σ ( x, x′ ) 2is ] Fε ( x, x′; is )) ε (6.14) Where (∆± (x, x ′; ε))ε is the distributional Van Vleck determinant (∆± (x, x ′; ε))ε = −det [∂μ∂νσ (x, x ′)] ([ g± (x, ε) g± (x′, ε) ]−1/2) ε (6.15) In the normal coordinates about x′ that we are currently using, (∆± (x, x ′; ε))ε reduces to 75 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ([ −g± (x, ε) ]−1/2) ε .The full asymptotic expansion of ( F±ε (x, x′; is) ) ε to all adiabatic orders are ( F±ε (x, x′; is) ) ε ≍∞j=0 (is)j ( a±2 (x, x ′; ε) ) ε (6.16) With ( a±0 (x, x ′; ε) ) ε = 1, the other ( a±j (x, x ′; ε) ) ε being given by canonical recursion relations which enable their adiabatic expansions to be obtained. The expansions (6.13) and (6.16) are, however, only asymptotic approximations in the limit of large adiabatic parameter T. If (6.16) is substituted into (6.14) the integral can be performed to give the adiabatic expansion of the Feynman propagator in coordinate space: ( G±ε (x, x ′) ) ε ≍ − (4πi)−n/2 ( ∆ 1/2 ± (x, x ′; ε) ∞ j=0 a ± j (x, x ′; ε) ( − ∂ ∂m2 )j × × [( −2m 2 σ )n−2 4 H (2) (n−2)/2 (( 2m2σ ) 1 2 )]) ε (6.17) In which, strictly, a small imaginary part iε should be subtracted from σ. Since we have not imposed global boundary conditions on the distributional Green function Colombeau solution of (6.2), the expansion (6.17) does not determine the particular vacuum state in (6.1). In particular, the "iε" in the expansion of ( G±ε (x, x ′) ) ε only ensures that (6.17) represents the expectation value, in some set of states, of a time-ordered product of fields. Under some circumstances the use of "iε" in the exact representation (6.14) may give additional information concerning the global nature of the states 6.2 Effective Action for the Quantum Matter Fields in Curved Distributional Space-time As in classical case one can obtain Colombeau generalized quantity (Wε)ε , called the effective action for the quantum matter fields in curved distributional spcetime, which, when functionally differentiated, yields( 2 (−g (ε)) 1 2 δWε δgμν (ε) ) ε = (⟨Tμν (ε)⟩)ε (6.18) To discover the structure of (Wε)ε, let us return to first principles, recalling the Colombeau pathintegral quantization procedure such as developed in []. Our notation will imply a treatment for the scalar field, but the formal manipulations are identical for fields of higher spins. Note that the generating functional (Zε [Jε])ε = (∫ D [φε] exp { iSm (ε) + i ∫ Jε (x)φε (x) d nx }) ε (6.19) was interpreted physically as the vacuum persistence amplitude (⟨outε, 0|0, inε⟩)ε . The presence of the external distributional current density (Jε)ε can cause the initial vacuum state ( |0, inε⟩)ε to be unstable, i.e., it can bring about the production of particles. In flat space, in the limit (Jε)ε = 0, no particles are produced, and one have the normalization condition (Zε [0])ε = (∫ D [φε] exp { iSm (ε) + i ∫ Jε (x)φε (x) d nx }) ε ∣∣ J=0 = (⟨0ε|0ε⟩)ε = 1. (6.20) However, when distributional spacetime is curved, we have seen that, in general, (|0,outε⟩)ε = (|0, inε⟩)ε , (6.21) even in the absence of source currents J. Hence (6.19) will no longer apply. Path-integral quantization 76 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance still works in curved distributional spacetime; one simply treats (Sm (ε))ε in (6.19) as the curved distributional spacetime matter action, and (Jε (x))ε as a current density (a scalar density in the case of scalar fields). One can thus set Jε = 0 in (6.19) and examine the variation of (Zε[0])ε : (δZε[0])ε = i ∫ D [φε] δSm (ε) exp [iSm (φε; ε)] = i (⟨outε, 0|δSm (ε) |0, inε⟩)ε . (6.22) Note that( 2 (−g (ε)) 1 2 δSm (ε) δgμν (ε) ) ε = (Tμν (ε))ε . (6.23) From (6.22) and (6.23) one obtains directly( 2 (−g (ε)) 1 2 δZε[0] δgμν (ε) ) ε = i (⟨outε, 0|Tμν (ε) |0, inε⟩)ε (6.24) Noting that the matter action Sm (ε) appears exponentiated in (6.19), one obtains directly Zε[0] = (exp (iWε))ε (6.25) and (exp (Wε))ε = −i (ln ⟨outε, 0|0, inε⟩)ε . (6.26) Following canonical calculation one obtains( Z±ε [0] ) ε ∝ ([ det ( −G±ε (x, x′) )] 1 2 ) ε (6.27) Where the proportionality constant is metric-independent and can be ignored. Thus we obtain( W±ε ) ε = −i ( lnZ±ε [0] ) ε = − i 2 ( tr [ ln ( −Ĝ±ε )]) ε . (6.28) In(6.28) ( Ĝ±ε ) ε is to be interpreted as an Colombeau generalized operator which acts on an linear space I of generalized vectors |x⟩ ,normalized by (⟨x |x′⟩)ε = δ (x− x ′) ([ −g± (x, ε) ]− 1 2 ) ε (6.29) in such a way that( G±ε (x, x ′) ) ε = ( ⟨x| Ĝ±ε |x′⟩ ) ε . (6.30) Remark 6.1. Note that the trace tr [] of an Colombeau generalized operator (Rε)ε which acts on a linear space I, is defined by (tr [Rε])ε = (∫ dnx [ −g± (x, ε) ] 1 2 Rxx;ε ) ε = (∫ dnx [ −g± (x, ε) ] 1 2 ⟨x| Rxx;ε |x′⟩ ) ε . (6.31) Writing now the Colombeau generalized operator ( Ĝ±ε ) ε as ( Ĝ±ε ) ε = − (±−1 ε ) ε = −i (∞∫ 0 ds exp [ −s±ε ]) ε , (6.32) 77 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance By Eq.(6.14) one obtains ( ⟨x| exp [ −s±ε ] |x′⟩ ) = i (4π)−n/2 [( ∆ 1/2 ± (x, x ′; ε) ) ε ] exp [ −im2s+ σ (x, x ′) 2is ]± ε (x, x′; is) (is)−n/2 (6.33) Now, assuming (ε)ε to have a small negative imaginary part, we obtains(∞∫ Λ ds (is)−1 i exp [ −s±ε ]) ε = ( Ei ( −iΛ±ε )) ε (6.34) Where Ei (x) is the exponential integral function. Remark 6.2. Note that for x→ 0 Ei (x) = γ + ln (−x) +O (x) (6.35) Where γ is the Euler's constant. Substituting now (6.35) into (6.34) and letting Λ → 0 we obtain( ln ( −Ĝ±ε )) ε = − (ln (ε))ε = (∞∫ 0 ds exp [ −s±ε ] (is)−1 ) ε , (6.36) Which is correct up to the addition of a metric-independent infinite large Colombeau constant Ω ∈ R that can be ignored in what follows. Thus, in the generalized De Witt-Schwinger representation (6.33) or (6.14) we have ( ⟨x| ln ( −Ĝ±ε ) |x′⟩ ) ε = ( ∞∫ m2 G±ε ( x, x′;m2 ) dm2 ) ε , (6.37) Where the integral with respect to m2 brings down the extra power of (is)−1 that appears in Eq.(6.36). Returning now to the expression (6.28) for ( W±ε ) ε using Eq.(6.37) and Eq.(6.31) we get ( W±ε ) ε = i 2 [(∫ dnx [ −g± (x, ε) ] 1 2 ) ε ]( lim x→x′ ∞∫ m2 G±ε ( x, x′;m2 ) dm2 ) ε (6.38) Interchanging the order of integration and taking the limit x→ x′ one obtains ( W±ε ) ε = i 2 ( ∞∫ m2 dm2 ∫ dnx [ −g± (x, ε) ] 1 2 G±ε ( x, x;m2 )) ε . (6.39) Colombeau quantity ( W±ε ) ε is colled as the one-loop effective action. In the case of fermion effective actions, there would be a remaining trace over spinorial indices. From Eq.(6.39) we may define an effective Lagrangian density ( L±ε;eff (x) ) ε by ( W±ε ) ε = (∫ dnx [ −g± (x, ε) ] 1 2 L±ε;eff (x) ) ε (6.40) Whence one get ( L±ε (x) ) ε = [ −g± (x, ε) ] 1 2 ε;eff ± (x) = i 2 ( lim x→x′ ∞∫ m2 dm2G±ε ( x, x′;m2 )) ε . (6.41) 78 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance 6.3 Stress-tensor Renormalization Note that ( L±ε (x) ) ε diverges at the lower end of the s integral because the σ/2s damping factor in the exponent vanishes in the limit x → x′. (Convergence at the upper end is guaranteed by the −iε that is implicitly added to m2 in the De Witt-Schwinger representation of ( L±ε (x) ) ε . In four dimensions, the potentially divergent terms in the DeWittSchwinger expansion of ( L±ε (x) ) ε are  ( L±ε;div (x) ) ε = − ( 32π2 )−1 ( lim x→x′ [( ∆ 1/2 ± (x, x ′; ε) ) ε ] ∞∫ 0 ds s3 exp [ −im2s+ σ (x, x ′) 2is ] × × [ a±0 (x, x ′; ε) + isa±1 (x, x ′; ε) + (is)2 a±2 (x, x ′; ε) ]) ε (6.42) Where the coefficients a±0 , a ± 1 and a ± 2 are given by Eq.(6.9)-Eq.(6.10). The remaining terms in this asymptotic expansion, involving a±3 and higher, are finite in the limit x→ x′. Let us determine now the precise form of the geometrical ( L±ε;div (x) ) ε terms, to compare them with the conventional gravitational Lagrangian that appears in (2.38). This is a delicate matter because (6.48) is, of course, infinite. What we require is to display the divergent terms in the form ∞× [geometrical object]. This can be done in a variety of ways. For example, in n dimensions, the asymptotic (adiabatic) expansion of ( L±ε;eff (x) ) ε is ( L±ε;eff (x) ) ε ≍ 2−1 (4π)−n/2 ( lim x→x′ [( ∆ 1/2 ± (x, x ′; ε) ) ε ]∞ j=0 aj (x, x ′; ε)× × ∞∫ 0 ids (is)j−1−n/2 exp [ −im2s+ σ (x, x ′) 2is ]) ε (6.43) of which the first n/2 + 1 terms are divergent as σ → 0. If n is treated as a variable which can be analytically continued throughout the complex plane, then we may take the x→ x′ limit ( L±ε;eff (x) ) ε ≍ 2−1 (4π)−n/2 ( ∞ j=0aj (x; ε) ∞∫ 0 ids (is)j−1−n/2 exp [ −im2s ]) ε = 2−1 (4π)−n/2j=0 ∞aj (x; ε) ( m2 )n/2−j Γ ( j − n 2 ) , aj (x; ε) = aj (x, x; ε) . (6.44) From Eq.(6.44) follows we shall wish to retain the units of L±ε;eff (x) as (length) −4, even when n = 4. It is therefore necessary to introduce an arbitrary mass scale μ and to rewrite Eq.(6.44) as( L±ε;eff (x) ) ε ≍ 2−1 (4π)−n/2 ( m μ )n−4 ( ∞ j=0aj (x; ε) ( m2 )4−2j Γ ( j − n 2 )) ε . (6.45) If n→ 4, the first three terms of Eq.(6.45) diverge because of poles in the Γfunctions: Γ ( −n 4 ) = 4 n (n− 2) ( 2 4− n − γ ) +O (n− 4) , Γ ( 1− n 2 ) = 4 (2− n) ( 2 4− n − γ ) +O (n− 4) , Γ ( 2− n 2 ) = 2 4− n − γ +O (n− 4) . (6.46) 79 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Denoting these first three terms by ( L±ε;div (x) ) ε , we have( L±ε;div (x) ) ε = (4π)−n/2 { 1 n− 4 + 1 2 [ γ + ln ( m2 μ2 )]}([ 4m4a0 (x; ε) n (n− 2) − 2m2a1 (x; ε) n− 2 + a2 (x; ε) ]) ε . (6.47) The functions a0 (x; ε) , a1 (x; ε) and a2 (x; ε) are given by taking the coincidence limits of (6.9)- (6.10) ( a±0 (x; ε) ) ε = 1,( a±1 (x; ε) ) ε = ( 1 6 − ξ )( R± (ε) ) ε ,( a±2 (x; ε) ) ε = 1 180 ( R±αβγδ (x, ε)R ±αβγδ (x, ε) ) ε − 1 180 ( R±αβ (x, ε)R±αβ (x, ε) ) ε − −1 6 ( 1 5 − ξ )( R± (x, ε) ) ε + 1 2 ( 1 6 − ξ )( R±2 (x, ε) ) ε . (6.48) Finally one obtains ( L±ε;ren (x) ) ε ≍ − 1 64π2 (∞∫ 0 ids ln (is) ∂3 ∂ (is)3 [ ± ε (x, x; is) e −ism2 ]) ε . (6.49) Special interest attaches to field theories in distributional spasetime in which the classical action (Sε)ε is invariant under distributional conformal transformations,i.e., (gμν (x, ε))ε → ( Ω2ε (x) gμν (x, ε) ) ε , ( ḡ±μν (x, ε) ) ε . (6.50) From the definitions one has ( Sε [ ḡ±μν (x, ε) ]) ε = ( Sε [ g±μν (x, ε) ]) ε + (∫ dnx ( δSε [ ḡ±μν (x, ε) ] δḡ±ρσ (x, ε) ) δḡ±ρσ (x, ε) ) ε . (6.51) From Eq.(6.51) one obtains T± ρρ [ g±μν (x, ε) , ε ] = − ( Ω2ε (x) [−g (x, ε)] 1 2 δSε [ ḡ±μν (x, ε) ] δΩε (x) ) ε ∣∣∣∣∣ Ωε=1 , (6.52) and it is clear that if the classical action is invariant under the conformal transformations (6.50), then the classical stress-tensor is traceless.Because conformal transformations are essentially a rescaling of lengths at each spacetime point x, the presence of a mass and hence a fixed length scale in the theory will always break the conformal invariance. Therefore we are led to the massless limit of the regularization and renormalization procedures used in the previous section. Although all the higher order (j > 2) terms in the DeWitt-Schwinger expansion of the effective Lagrangian (6.45) are infrared divergent at n = 4 as m → 0, we can still use this expansion to yield the ultraviolet divergent terms arising from j = 0, 1, and 2 in the four-dimensional case. We may put m = 0 immediately in the j = 0 and 1 terms in the expansion, because they are of positive power for n ∼ 4. These terms therefore vanish. The only nonvanishing potentially ultraviolet divergent term is therefore j = 2 : 2−1 (4π)−n/2 ( m μ )n−4 a2 (x, ε) Γ ( 2− n 2 ) , (6.53) Which must be handled carefully. Substituting for a2(x) with ξ = ξ(n) from (6.48), and rearranging 80 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance terms, we may write the divergent term in the effective action arising from (6.53) as follows ( W± ε,div ) ε = 2−1 (4π)−n/2 ( m μ )n−4 Γ ( 2 − n 2 )(∫ dnx [ −g± (x, ε) ] 1 2 a2 (x, ε) ) ε = 2−1 (4π)−n/2 ( m μ )n−4 Γ ( 2 − n 2 )(∫ dnx [ −g± (x, ε) ] 1 2 [ αz±ε (x) + βG ± ε (x) ]) ε + O (n − 4) (6.54) where (zε (x))ε = ( R±αβγδ (x, ε)R± αβγδ (x, ε) ) ε − 2 ( R±αβ (x, ε)R± αβ (x, ε) ) ε + 1 3 ( R±2 (x, ε) ) ε ,( G±ε (x) ) ε = ( R±αβγδ (x, ε)R± αβγδ (x, ε) ) ε (6.55) and α = 1 120 , β = − 1 360 . (6.56) Finally one obtains ⟨ Tμμ (x, ε) ⟩ ren = − ( 1/2880π2 ) [ α ( zε (x) − 23R ± (x, ε) ) ε + β ( G±ε (x) ) ε ] = − ( 1/2880π2 ) [( R± αβγδ (x, ε)R±αβγδ (x, ε) ) ε − ( R± αβ (x, ε)R±αβ (x, ε) ) ε − R± (x, ε) ] . (6.57) Note that from Eq.(3.42) for r → 2m follows that (Rρσμν (ε)Rρσμν (ε))ε ≍ ([ (r − 2m)2 + ε2 ]−1) ε + 4 (2m)4 . (6.58) Thus for the case of the distributional Schwarzchild spesetime given by the distributional metric (3.40) using Eq.(6.57) and Eq.(6.58) for r → 2m one obtains⟨ Tμμ (x, ε) ⟩ ren ≍ − ( 2880π2 )−1 [([ (r − 2m)2 + ε2 ]−1) ε + 4 (2m)4 ] . (6.59) This result in a good agreement with Eq.(5.14)-Eq.(5.16). 7 Novel Explanation of the Active Galactic Nuclei. The Power Source of Quasars as a Result of Vacuum Polarization by the Gravitational Singularities on BHs Horizon 7.1 The Current Paradigm for Active Galactic Nuclei. High Energy Emission from Galactic Jets Accretion of gas onto the supermassive Kerr black holes lurking at the center of active galactic nuclei (AGN) gives rise to powerful relativistic jets. We remind that in the standard theory of an accretion disk around a black hole it is assumed that there is no coupling between the disk and the central black hole [51]. However, in the presence of a magnetic field, a magnetic coupling between the disk and the black hole could exist and play an important role in the balance and transportation of energy and angular momentum.In the absence of the magnetic coupling, the energy source for the radiation of the disk is the gravitational energy of the disk (i.e., the gravitational binding energy between the disk and the black hole). But, if 81 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance the magnetic coupling exists and the black hole is rotating, the rotational energy of the black hole provides an additional energy source for the radiation of the disk.With the magnetic coupling, the black hole exerts a torque on the disk, which transfers energy and angular momentum between the black hole and the disk. If the black hole rotates faster than the disk, energy and angular momentum are extracted from the black hole and transfered to the disk. The energy deposited into the disk is eventually radiated away by the disk,which will increase the efficiency of the disk and make the disk brighter than usual. If the black hole rotates slower than the disk, energy and angular momentum are transfered from the disk to the black hole, which will lower the efficiency of the disk and make the disk dimmer than usual. Therefore, the magnetic coupling between the black hole and the disk has important effects on the radiation properties of the disk [52]-[53]. Fig. 5. Jet from Black Hole in a Galaxy Pictor A. The active galaxy Pictor A lies nearly 500 million light-years from Earth and contains a supermassive black hole at its centre. This is a composite radio and X-ray image. Fig. 6. Fourier coverage (uv-coverage) of the fringe fitted data (i.e.,reliable fringe detections) of the Radio Astron observations of BL Lac on 2013 November 10-11at 22 GHz. Color marks the lower limit of observed brightness temperature obtained from visibility amplitudes. Adapted from [54]. 82 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance The current paradigm for AGN is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed 1011 to 1012 K [55]. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10. The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than 5.3Gλ, must indeed exceed 2 × 1013K and can reach as high as   3 × 1014K [55]. As follows from Fig. 7.2, these visibilities correspond to the structural scales of 30 − 40 μas oriented along position angles of 25◦−30◦.These values are indeed close to the width of the inner jet and the normal to its direction.The observed, Tb,obs, and intrinsic, Tb,int, brightness temperatures are related by Tb,obs = δ(1 + z) −1Tb,int (7.1.1) With δ = 7.2.The estimeted by (4.1.1) a lower limit of the intrinsic brightness temperature in the core component of our Radio Astron observations of Tb,int > 2.910 12 K. It is commonly considered that inverse Compton losses limit the intrinsic brightness temperature for incoherent synchrotron sources, such as AGN, to about 1012K [53].In case of a strong flare, the "Compton catastrophe" is calculated to take about one day to drive the brightness temperature below 1012K [53]. The estimated lower limit for the intrinsic brightness temperature of the core in the Radio Astron image of Tb,int > 2.910 12K is therefore more than an order of magnitude larger than the equipartition brightness temperature limit established in [55] and at least several times larger than the limit established by inverse Compton cooling. Remark 4.1.1. Note that if the estimate of the maximum brightness temperature given in [53], is closer to actual values, it would imply Tb;int5× 1013K. This is difficult to reconcile with current incoherent synchrotron emission models from relativistic electrons, requiring alternative models such as emission from relativistic protons. Remark 4.1.2. However the proton, as we know, is 1836 times heavier than an electron and absolutely huge energy is required to accelerated it to sublight speed. Remark 4.1.3.These alternative models such as emission from relativistic protons can be suported by semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes presented by rotating gravitational singularities. 7.2 The Colombeau Distributional Kerr Space-time in BoyerLindquist form The classical Kerr metric in Boyer-Lindquist form reads ds2 = −Ξ (r, θ) dt2 − 4mra sin 2 θ ρ2 dtdφ+ ρ2 ∆a dr2 + ρ2dθ2+( r2 + a2 + 2mra2 sin2 θ ρ2 ) sin2 θdφ2, (7.2.1) Where ρ2 = r2 + a2 cos2 θ,∆a = r 2 − 2mr + a2, Ξ (r, θ) = 1− 2mr ρ2 = r2 − 2mr + a2 cos2 θ ρ2 . (7.2.2) Remark 7.2.1. Note that For small values of the parameter a, where we may neglect terms of the order of a2, we get from (7.2.1) the Lense-Thirring metric with Jz = ma 83 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance ds2 = − ( 1− 2m r ) dt2 + ( 1− 2m r )−1 dr2 + r2 ( dθ2 + sin2 θdφ2 ) − 4ma sin 2 θ r dtdφ. (7.2.3) I. Slow Kerr gravitational singularity: a < m. Note that Ξ (r, θ) = r2 − 2mr + a2 cos2 θ ρ2 = ( r − rE+ (θ) ) (r − rE (θ)) ρ2 , (7.2.4) Where rE± (θ) = m ± √ m2 − a2 cos2 θ and ∆a = r2 − 2mr + a2 = (r − r+) (r − r−) , where r± = m± √ m2 − a2. Remark 7.2.2. Let Ξ (θ) be a submanifold given by equation φ = const, then metric (7.2.1) restricted on submanifold ∪θΞ (θ) reads ds2 = −Ξ (r, θ) dt2 + ρ 2 ∆a dr2 + ρ2dθ2. (7.2.5) Note that: (i) the metric (7.2.5) is degenerate on outer ergosurfaces r = rE+ (θ) and inner ergosurfaces r − rE (θ) ,(ii) the metric (7.2.5) is singular on horizon r = r+,(iii) the metric (7.2.5) is singular on submanifold given by equation r = r−. Remark 7.2.3. Note that we will be consider the distributional Kerr spacetime not as full distributionel BH with Colombeau generalized metric (7.2.7), but only as gravitational singularity located on submanifold ∪θΞ (θ) which coincide with outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11), Eq.(7.2.19) and Eq.(7.2.20) submanifold ∪θΞ (θ) presented the singular boundary of distributional spacetime presented by Colombeau generalized metric (7.2.7). We introduce now the following regularized (above ergosurface r = rE+ (θ)) quantity Ξ+ε (rε, θ) = (rε − rE (θ)) √( rε − rE+ (θ) )2 + ε2 ρ2ε (rε) , ∆a,ε = r 2 ε − 2mrε + a2, (7.2.6) Where ρ2ε = ρ 2 ε (rε) = r 2 ε + a 2 cos2 θ, ε ∈ (0, δ] , rε ≥ rE+ (θ) ≥ r+.Thus Colombeau generalized metric (above ergosurface r = rE+ (θ)) corresponding to classical Kerr metric (7.2.1) reads( ds+2ε ) ε = − [ (Ξε (rε, θ))ε ] dt2 − [( 4mrεa sin 2 θ ρ2ε ) ε ] dtdφ+[( ρ2ε ∆a,ε ) ε ] [( dr2ε ) ε ] + ( ρ2ε ) ε dθ2+( r2ε + a 2 + 2mra2 sin2 θ ρ2ε ) ε sin2 θdφ2. (7.2.7) Remark 7.2.4. Let Ξ (θ) be a submanifold given by equation φ = const, then Colombeau generalized metric (7.2.7) restricted on Ξ (θ) reads ( ds+2ε ) ε = − [ (Ξε (rε, θ))ε ] dt2 + [( ρ2ε ∆a,ε ) ε ] [( dr2ε ) ε ] + [( ρ2ε ) ε ] dθ2. (7.2.8) Note that Colombeau generalized metric (7.2.7) nondegenerate on outer ergosurfaces (rε)ε = rE+ (θ) ,see Pic.7.1. 84 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 7.2.4. Note that for small values of the parameter a, where we may neglect terms of the order of a2, we get from (7.2.7) effectively the following Colombeau generalized metric ( ds+2ε ) ε = − ( 1− rE+ (θ) (rε)ε ) dt2 + ( 1− rE+ (θ) (rε)ε )−1 [( dr2ε ) ε ] +[( r2ε ) ε ] ( dθ2 + sin2 θdφ2 ) − 4ma sin 2 θ (rε)ε dtdφ, (7.2.9) Where rE+ (θ) = m+ √ m2 − a2 cos2 θ. Remark 7.2.5. Note that Colombeau generalized metric (7.2.9) restricted on Ξ (θ) reads ( ds+2ε ) ε = − ( 1− rE+ (θ) (rε)ε ) dt2 + ( 1− rE+ (θ) (rε)ε )−1 [( dr2ε ) ε ] + [( r2ε ) ε ] dθ2 (7.2.10) (I) Let ( Ra≪1 (rε, ε) ) ε be Colombeau generalized curvature scalar (R (rε, ε))ε corresponding to the metric (7.2.10) with a≪ 1. Main singular part sing [( Ra≪1 (rε, ε) ) ε ] of the Colombeau generalized curvature scalar ( Ra≪1 (rε, ε) ) ε corresponding to the metric (7.2.10) reads sing [( Ra≪1 (rε, ε) ) ε ] =R  ε2 rE+ (θ) [( rε − rE+ (θ) )2 + ε2 ]3/2  ε , (7.2.11) Where cl [ (rε)ε ] ≈R rE+ (θ) ,see Appendix Eq.(A.12). (II) Let ( Rμν(a≪1) (rε, ε)R (a≪1) μν (rε, ε) ) ε be Colombeau generalized quadratic scalar (Rμν (rε, ε)Rμν (rε, ε))ε corresponding to the metric (7.2.10) with a ≪ 1. From Eq.(7.2.10) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Rμν(a≪1) (rε, ε)R (a≪1) μν (rε, ε) ) ε ] of the quadratic scalar ( Rμν(a≪1) (rε, ε)R (a≪1) μν (rε, ε) ) ε reads: sing [( Rμν(a≪1) (rε, ε)R (a≪1) μν (rε, ε) ) ε ] =R ( ε4 4 ( rE+ (θ) )4 [ ε2 + (rε − 2m)2 ]3 ) ε . (7.2.12) (III) Let ( Rρσμν(a≪1) (rε, ε)R (a≪1) ρσμν (rε, ε) ) ε be Colombeau generalized quadratic scalar (Rρσμν (rε, ε)Rρσμν (rε, ε))ε corresponding to the metric (7.2.10) with a ≪ 1. From Eq.(7.2.10) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Rρσμν(a≪1) (rε, ε)R (a≪1) ρσμν (rε, ε) ) ε ] of the Colombeau generalized quadratic scalar( Rρσμν(a≪1) (rε, ε)R (a≪1) ρσμν (rε, ε) ) ε reads sing [( Rρσμν(a≪1) (rε, ε)R (a≪1) ρσμν (rε, ε) ) ε ] =R ( ε4 4 ( rE+ (θ) )4 [ ε2 + (rε − 2m)2 ]3 ) ε . (7.2.13) 85 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 7.2.6. Note that from Eq.(7.2.11)-Eq.(7.2.13) at outer ergosurfaces (rε)ε = rE+ (θ) ,(see Pic.7.1) follows that sing [( Ra≪1 (rε, ε) ) ε ]∣∣ (rε)ε=rE+ (θ) =R r −1 E+ (θ) ( ε−1 ) ε ≈R ∞. (7.2.14) and sing [( Rμν(a≪1) (rε, ε)R (a≪1) μν (rε, ε) ) ε ]∣∣∣ (rε)ε=rE+ (θ) =R r −4 E+ (θ) ( ε−2 ) ε ≈R ∞, sing [( Rρσμν(a≪1) (rε, ε)R (a≪1) ρσμν (rε, ε) ) ε ]∣∣∣ (rε)ε=rE+ (θ) =R r −4 E+ (θ) ( ε−2 ) ε ≈R ∞. (7.2.15) Let ( Ra<m (rε, ε) ) ε be Colombeau generalized curvature scalar ( Ra<m (rε, ε) ) ε corresponding to the metric (7.2.8) with a < m. We let now that ∆ε = Aε (Bε + Cε) = Aερ 2 ε∆ −1 a,ε, Bεr 2 ε = ρ 2 ε, Bε + Cε = ρ 2 ε∆ −1 a,ε, Aε = Ξ + ε (r, θ) , Dε = 0. (7.2.16) From Eq.(7.2.8),Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) we obtain ( Ra<m (rε, ε) ) ε = ( Aε ∆ε [ 2 rε ( −2A ′ ε Aε − 3B ′ ε Bε + ∆′ε ∆ε ) + 2 r2 Cε Bε − A ′′ ε Aε − 2B ′′ ε Bε + 1 2 ( B′ε Bε )2 − 2A ′ εB ′ ε AεBε + ( 1 2 A′ε Aε + B′ε Bε ) ∆′ε ∆ε ]) ε =( ∆a,ε ρ2ε [ 2 rε ( −2A ′ ε Aε − 3B ′ ε Bε + ∆′ε ∆ε ) + 2 r2 Cε Bε − A ′′ ε Aε − 2B ′′ ε Bε + 1 2 ( B′ε Bε )2 − 2A ′ εB ′ ε AεBε + ( 1 2 A′ε Aε + B′ε Bε ) ∆′ε ∆ε ]) ε . (7.2.17) Note that ∂ ∂rε √( rε − rE+ (θ) )2 + ε2 = rε − rE+ (θ)√( rε − rE+ (θ) )2 + ε2 ∂2 ∂r2ε √( rε − rE+ )2 + ε2 = ε2[( rε − rE+ )2 + ε2 ] 3 2 . (7.2.18) From Eq.(7.2.17) and Eq.(7.2.18) one obtains that main singular part sing [( Ra<m (rε, ε) ) ε ] of the Colombeau generalized curvature scalar ( Ra<m (rε, ε) ) ε corresponding to the metric (7.2.8) (mod nonsingular multiplier) reads sing [( Ra<m (rε, ε) ) ε ] =R  ε2[( rε − rE+ (θ) )2 + ε2 ]2  ε . (7.2.19) Remark 7.2.7. (I) Let ( Rμν(a<m) (rε, ε)R (a<m) μν (rε, ε) ) ε be Colombeau generalized quadratic scalar (Rμν (rε, ε)Rμν (rε, ε))ε corresponding to the metric (7.2.8) with a < m. From Eq.(7.2.8)-Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Rμν(a<m) (rε, ε)R (a<m) μν (rε, ε) ) ε ] 86 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance of the Colombeau generalized quadratic scalar ( Rμν(a<m) (rε, ε)R (a<m) μν (rε, ε) ) ε reads sing [( Rμν(a<m) (rε, ε)R (a<m) μν (rε, ε) ) ε ] =R ( ε4 4 ( rE+ (θ) )4 [ ε2 + (rε − 2m)2 ]3 ) ε . (7.2.20) (II) Let ( Rρσμν(a<m) (rε, ε)R (a<m) ρσμν (rε, ε) ) ε be Colombeau generalized quadratic scalar (Rρσμν (rε, ε)Rρσμν (rε, ε))ε corresponding to the metric (7.2.8) with a < m. From Eq.(7.2.8), Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Rρσμν(a<m) (rε, ε)R (a<m) ρσμν (rε, ε) ) ε ] of the Colombeau generalized quadratic scalar ( Rρσμν(a<m) (rε, ε)R (a<m) ρσμν (rε, ε) ) ε reads sing [( Rρσμν(a<m) (rε, ε)R (a<m) ρσμν (rε, ε) ) ε ] =R ( ε4 4 ( rE+ (θ) )4 [ ε2 + (rε − 2m)2 ]3 ) ε . (7.2.21) II. Critical Kerr gravitational singularity: a=m. Note that in contrast with full distributional Kerr spacetime the case of the critical Kerr gravitational singularity considered in this subsection (see Remark 7.2.3) not principal different in comparison with a case of the slow Kerr gravitational singularity considered above. In particular the Eqs.(7.2.19)- (7.2.21) holds with rE+ (θ) given by Eq.(7.2.22) rE+ (θ) = m+ √ m2 − a2 cos2 θ = m ( 1 + sin2 θ ) . (7.2.22) III. Fast Kerr gravitational singularity: a > m. LetΞη (θ) be a submanifold given by equations (i) φ = const and (ii) m 2 − a2 cos2 θ ≥ 0,i.e. cos2 θ ≤ m 2 a2 = η. (7.2.23) Let Θη be a set Θη = { θ| cos2 θ ≤ η } and let χ (θ, η) be the indicator function of the set Θη,i.e. χ (θ, η) is the function defined to be identically 1 on Θη, and is 0 elsewhere. We introduce now the following regularized (above ergosurface r = rE+ (θ) , θ ∈ Θη) quantity Ξ+ε (rε, θ, η) = χ (θ, η) (rε − rE (θ)) √( rε − rE+ (θ) )2 + ε2 ρ2ε (rε) , (7.2.25) where ρ2ε = ρ 2 ε (rε) = r 2 ε + a 2 cos2 θ, ε ∈ (0, δ] , rε ≥ rE+ (θ) > 0.Thus Colombeau generalized metric (above ergosurface r = rE+ (θ)) corresponding to classical Kerr metric (7.2.1) reads( ds+2ε ) ε = −χ (θ, η) [ (Ξε (rε, θ, η))ε ] dt2 − [( 4mrεa sin 2 θ ρ2ε ) ε ] dtdφ+[( ρ2ε ∆a,ε ) ε ] [( dr2ε ) ε ] + ( ρ2ε ) ε dθ2+( r2ε + a 2 + 2mra2 sin2 θ ρ2ε ) ε sin2 θdφ2. (7.2.26) 87 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Remark 7.2.8. Note that we will be consider the distributional Kerr spacetime not as full distributionel BH with Colombeau generalized metric (7.2.7), but only as gravitational singularity located on submanifold ∪θ∈ΘηΞη (θ) which coincide with an part of the outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11),Eq.(7.2.19) and Eq.(7.2.20) submanifold ∪θ∈ΘηΞη (θ) presented the singular boundary of distributional spacetime with Colombeau generalized metric (7.2.26). Fig. 7. Ergosurface, horizon and singularity for slow Kerr black hole. Fig. 8.Ergosurface,horizon,and singularity for critical Kerr black hole. 88 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Fig. 9. Ergosurface, horizon and singularity for fast Kerr black hole. Remark 7.2.9. Let Ξη (θ) be a submanifold given by equations (i) φ = const and (ii) cos 2 θ ≤ η,then Colombeau generalized metric (7.2.26) restricted on submanifold ∪θ∈Θη Ξη (θ) reads( ds+2ε ) ε = −χ (θ, η) [ (Ξε (rε, θ, η))ε ] dt2 + [( ρ2ε ∆a,ε ) ε ] [( dr2ε ) ε ] + [( ρ2ε ) ε ] dθ2. (7.2.27) Note that Colombeau generalized metric (7.2.27) nondegenerate on outer ergosurfaces (rε)ε = rE+ (θ) ,see Pic.7.3. From Eq.(7.2.27) and and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Ra>m (rε, ε) ) ε ] of the Colombeau generalized curvature scalar ( Ra>m (rε, ε) ) ε corresponding to the metric (7.2.27) (mod nonsingular multiplier) reads sing [( Ra>m (rε, ε) ) ε ] =R  χ (θ, η) ε2[( rε − rE+ (θ) )2 + ε2 ]2  ε . (7.2.28) Remark 7.2.10. (I) Let ( Rμν(a>m) (rε, ε)R (a>m) μν (rε, ε) ) ε be Colombeau generalized quadratic scalar (Rμν (rε, ε)Rμν (rε, ε))ε corresponding to the metric (7.2.27) with a > m. From Eq.(7.2.27) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Rμν(a>m) (rε, ε)R (a>m) μν (rε, ε) ) ε ] of the Colombeau generalized quadratic scalar ( Rμν(a>m) (rε, ε)R (a>m) μν (rε, ε) ) ε reads sing [( Rμν(a>m) (rε, ε)R (a>m) μν (rε, ε) ) ε ] =R ( χ (θ, η) ε4 4 ( rE+ (θ) )4 [ ε2 + (rε − 2m)2 ]3 ) ε . (7.2.29) (II) Let ( Rρσμν(a>m) (rε, ε)R (a>m) ρσμν (rε, ε) ) ε be Colombeau generalized quadratic scalar 89 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (Rρσμν (rε, ε)Rρσμν (rε, ε))ε corresponding to the metric (7.2.27) with a > m. From Eq.(7.2.27) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing [( Rρσμν(a>m) (rε, ε)R (a>m) ρσμν (rε, ε) ) ε ] of the Colombeau generalized quadratic scalar ( Rρσμν(a>m) (rε, ε)R (a>m) ρσμν (rε, ε) ) ε reads sing [( Rρσμν(a>m) (rε, ε)R (a>m) ρσμν (rε, ε) ) ε ] =R ( χ (θ, η) ε4 4 ( rE+ (θ) )4 [ ε2 + (rε − 2m)2 ]3 ) ε . (7.2.30) 8 Conclusions and Remarks This book dealing with an extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry,see for example [1]-[2],[5]-[7] and [14]-[15].The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this book essentially new class Colombeau solutions to Einstein field equations is obtained. We have shown that a succesfull approach for dealing with curvature tensor valued distribution is to first impose admisible the nondegeneracy conditions on the metric tensor, and then take its derivatives in the sense of classical distributions in space S ′2m(R3). 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(R3,Σ)). in order to obtain superdistributional curvatures, may renders a more rigorous setting for discussing situations like the ones considered in this paper. The vacuum energy density of free scalar quantum field Φ with a distributional background spacetime also is considered.It have been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations ⟨ Φ2 ⟩ has a singular behavior on BHs horizon r+: ⟨ Φ2 (r) ⟩   |r − r+|−2 .We argue that this vacuum dominance may bear important astrophysical implications. Acknowledgement To reviewers provided important clarifications. 90 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Competing Interests Authors have declared that no competing interests exist. References [1] Colombeau JF. New generalized functions and multiplication of distributions. North Holland, Amsterdam; 1984. [2] Colombeau JF. Elementary introduction to new generalized functions. North Holland, Amsterdam; 1985. [3] Müller T, Grave F. Catalogue of spacetimes; 58. arXiv:0904.4184 [gr-qc] Available: https://arxiv.org/abs/0904.4184v2 [4] Landau LD, Lifschitz EM. 1975 the classical theory of fields pergamon press. Oxford. 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Astrophysical Journal. 2016;817:96 DOI:10.3847/0004-637X/817/2/96 arXiv:1512.04690 [astro-ph.HE] 93 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Appendix A Expressions for the Colombeau quantities (R({}, (ε)))ε , (R μν({}, (ε))Rμν({}, (ε)))ε and (Rρσμν({}, (ε))Rρσμν({}, (ε)) )ε in terms of (Aε)ε , (Bε)ε , (Cε)ε and (Dε)ε , ε ∈ (0, 1]: Let us introduce now Colombeau generalized metric which has the form{ ( ds2 ) ε = − ( Aε (r) (dx 0)2 ) ε − 2 ( Dε (r) dx 0dr ) ε + ( (Bε (r) + Cε (r))(dr) 2 ) ε + ( Bε (r) r 2 [ (dθ)2 + sin2 θ(dφ)2 ]) ε r = cl [ (rε)ε ] ∈ R. (A.1) The Colombeau scalars (R (ε))ε , (R μν (ε)Rμν (ε))ε and (R ρσμν (ε)Rρσμν (ε))ε , in terms of Colombeau generalized functions (Aε (r))ε , (Bε (r))ε , (Cε (r))ε , (Dε (r))ε is expressed as (R (ε))ε = ( Aε ∆ε [ 2 r ( −2A ′ ε Aε − 3B ′ ε Bε + ∆′ε ∆ε ) + 2 r2 AεCε +D 2 ε AεBε − A ′′ ε Aε − 2B ′′ ε Bε + 1 2 ( B′ε Bε )2 − 2A ′ εB ′ ε AεBε + ( 1 2 A′ε Aε + B′ε Bε ) ∆′ε ∆ε ]) ε , (Rμν (ε)Rμν (ε))ε = ( A2ε ∆2ε ( 1 2 A′′ε Aε − 1 4 A′ε∆ ′ ε Aε∆ε + 1 2 A′εB ′ ε AεBε + 1 r A′ε Aε )2) ε + +2 ( A2ε ∆2ε [ 1 r ( 1 2 ∆′ε ∆ε − A ′ ε Aε − 2B ′ ε Bε ) + 1 r2 AεCε +D 2 ε AεBε − 1 2 A′εB ′ ε AεBε − −1 2 B′′ε Bε + 1 4 B′ε∆ ′ ε Bε∆ε ]2) ε +( A2ε ∆2ε [ 1 2 A′′ε Aε − 1 4 A′ε∆ ′ ε Aε∆ε + 1 2 A′εB ′ ε AεBε + B′′ε Bε − 1 2 ( B′ε Bε )2 −1 2 B′ε∆ ′ ε Bε∆ε + 1 r ( A′ε Aε − ∆ ′ ε ∆ε + 2 B′ε Bε )]2) ε , (Rρσμν (ε)Rρσμν (ε))ε =( A2ε ∆2ε ( A′′ε Aε − 1 2 A′ε∆ ′ ε Aε∆ε )2 + 2 A2ε ∆2ε ( 1 r A′ε Aε + 1 2 A′εB ′ ε AεBε )2 +4 A2ε ∆2ε [ 1 r B′ε Bε − 1 r2 AεCε +D 2 ε AεBε + 1 4 ( B′ε Bε )2]2 + +2 A2 ∆2ε [ 1 r ( A′ε Aε + 2 B′ε Bε − ∆ ′ ε ∆ε ) + 1 2 A′εB ′ ε AεBε + B′′ε Bε −1 2 ( B′ε Bε )2 − 1 2 B′ε∆ ′ ε Bε∆ε ]2 ) ε . (A.2) Here (∆ε)ε = (Aε (r) (Bε (r) + Cε (r)))ε + ( D2ε (r) ) ε . (A.3) Assume that (∆ε (r))ε = 1, (Bε (r))ε = 1, (Dε (r))ε = 0. (A.4) From Eq.(A.2)-Eq.(A.4) one obtains 94 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (R (ε))ε = ( −4A ′ ε r + 2AεCε r2 −A′′ε ) ε , (Rμν (ε)Rμν (ε))ε =(( 1 2 A′′ε + 1 r A′ε )2) ε + 2 ([ −A ′ ε r + AεCε r2 ]2) ε + ([ 1 2 A′′ε + 1 r A′ε ]2) ε , (Rρσμν (ε)Rρσμν (ε))ε =( (A′′ε ) 2 + 2 ( A′ε r )2 + 4 (AεCε) 2 r4 + 2 ( A′ε r )2 ) ε . (A.5) We choose now Bε (rε) = 1, Cε (rε) = −1 +A−1ε (rε) , Dε (rε) = 0, (A.6) and rewrite Colombeau generalized object corresponding to Schwarzschild metric above horizon in the following form( ds2ε ) ε = − ( Aε (rε) dt 2 ) ε + ( A−1ε (rε) dr 2 ε ) ε + r2εdΩ 2, (A.7) Where Aε (r) Aε (rε) = −r−1ε √ (rε − 2m)2 + ε2, rε ≥ 2m. (A.8) By differentiation we obtain ∆ε = Aε (Bε + Cε) = 1,∆ ′ ε = 0, A′ε (rε) = −2m (rε − 2m) r2ε √ (rε − 2m)2 + ε2 , A′′ε (r) = 2m ( −16m3 + 24m2rε − 12mr2ε − 4mε2 + 2r3ε + rεε2 ) r3ε [ (rε − 2m)2 + ε2 ]3/2 = 4m (rε − 2m)3 + (rε − 4m) ε2 r3ε [ (rε − 2m)2 + ε2 ]3/2 . (A.9) From Eqs.(A.2)-(A.5) and Eq.(A.9) we obtain (R (r, ε))ε = ( Aε ∆ε [ 2 rε ( −2A ′ ε Aε − 3B ′ ε Bε + ∆′ε ∆ε ) + 2 r2ε AεCε +D 2 ε AεBε − A ′′ ε Aε − 2B ′′ ε Bε + 1 2 ( B′ε Bε )2 − 2A ′ εB ′ ε AεBε + ( 1 2 A′ε Aε + B′ε Bε ) ∆′ε ∆ε ]) ε =( Aε [ 2 rε ( −2A ′ ε Aε ) − 2Aε ( 1−A−1ε ) r2ε − A ′′ ε Aε ]) ε =( −4A ′ ε rε − 2Aε r2ε + 2 r2ε −A′′ε ) ε = 8m (rε − 2m) r3ε √ (rε − 2m)2 + ε2  ε + 2r−3ε (√ (rε − 2m)2 + ε2 ) ε − 2 (r2ε)ε − − ( 2m ( −16m3 + 24m2rε − 12mr2ε − 4mε2 + 2r3ε + rεε2 ) r3ε [ (rε − 2m)2 + ε2 ]3/2 ) ε . (A.10) Finally from Eq.(A.10) one obtains the following expression for the distributional Colombeau scalar 95 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (R (r, ε))ε (R (rε, ε))ε =  8m (rε − 2m) r3ε √ (r − 2m)2 + ε2  ε + 2 [( r−3ε ) ε ](√ (rε − 2m)2 + ε2 ) ε − 2 (r2ε)ε − ( 4m (rε − 2m)3 + (rε − 4m) ε2 r3ε [ (rε − 2m)2 + ε2 ]3/2 ) ε . (A.11) Remark A.1. Note that from Eq.(A.11) follows that: if st ( (rε)ε ) = 0,i.e.(rε)ε ≈R 2m then (rε)ε ≈R 2m =⇒ (R (rε, ε))ε ∼ ( ε2 ) ε ≈R 0. We assume now that cl [ (rε)ε ] ≈R 2m and therefore from Eq.(A.11) we obtain (R (rε, ε))ε ≈R ( 4m2ε2 8m3 [ (rε − 2m)2 + ε2 ]3/2 ) ε . (A.12) Remark A.2. Note that from Eq.(A.12) at horizon r =R 2m follows that: (R (r, ε))ε = ( 4m2ε2 8m3 [ε2]3/2 ) ε = (4m)−1 ( ε−1 ) ε ≈R ∞. (A.13) Remark A.3. Note that from Eq.(A.11) follows that: w-lim ε→0 R (r, ε)   δ (r − 2m) . (A.14) Remark A.4. Let [ (rε − 2m)ε ] ≈R 0, then from Eq.(A.13) we obtain [ (R (rε, ε))ε ] ≈R ( ε2 2m [ (rε − 2m)2 + ε2 ]3/2 ) ε  . (A.15) From Eqs.(A.2) and Eq.(A.9) we obtain (Rμν (rε, ε)Rμν (rε, ε))ε = +2 ([ 1 rε A′ε + −Aε + 1 r2ε ]2) ε + 2 ([ 1 2 A′′ε + 1 rε A′ε ]2) ε = 2 ( 1 r3ε √ ε2 + (rε − 2m)2 + 1 r2ε ) ε − 2m r3ε rε − 2m√ ε2 + (rε − 2m)2 2 ε + 2 ([ 4m (rε − 2m)3 + (rε − 4m) ε2 r3ε [ (rε − 2m)2 + ε2 ]3/2 −2m r3ε rε − 2m√ ε2 + (rε − 2m)2 2 ε . (A.16) Remark A.5. Note that from Eq.(A.16) follows that:if st ( (rε)ε ) = 0,i.e.(rε)ε ≈R 2m then r ≈R 2m =⇒ (R μν (r, ε)Rμν (r, ε))ε ≈R K (r) , K (r) = 12 r2s r6 , rs = 2m. (A.17) 96 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance We assume now that (rε)ε ≈R 2m and therefore from Eq.(A.16) we obtain (Rμν (rε, ε)Rμν (rε, ε))ε ≈R K (rs) + ( ε4 4m4 [ ε2 + (rε − 2m)2 ]3 ) ε (A.18) Remark A.6. Note that from Eq.(A.18) at horizon r =R 2m follows that: (Rμν (rε, ε)Rμν (rε, ε))ε = ( 1 4m4ε2 ) ε ≈R ∞, (A.19) Remark C.6. Let [ (rε − 2m)ε ] ≈R 0, then from Eq.(A1.3) and Eq.(C.12) we obtain [ (Rμν (rε, ε)Rμν (rε, ε))ε ] ≈R K (rs) + [( ε4 4m4 ( ε2 + (rε − 2m)2 )3 ) ε ] (A.20) From Eqs.(A.2) and Eq.(C.3) we obtain (Rρσμν (r, ε)Rρσμν (r, ε))ε =( (A′′ε ) 2 + 2 ( A′ε r )2 ) ε + 4 ([ 1 r2 (1−Aε) ]2) ε + 2 ([ A′ε r ]2 ) ε = ( A′′2ε + 4 A′2ε r2 ) ε + 4 ( 1 r4 (1−Aε)2 ) ε =([ 4m (r − 2m)3 + (r − 4m) ε2 r3 [ (r − 2m)2 + ε2 ]3/2 ]2) ε − 8m 2 (r − 2m)2 r6 [ (r − 2m)2 + ε2 ]+ 4 r4 ( 1 + r−1 √ (r − 2m)2 + ε2 )2 . (A.21) Remark C.7. Note that from Eq.(C.15) follows that: r ≈R 2m =⇒ (R ρσμν (r, ε)Rρσμν (r, ε))ε ≈R K (r) , (A.22) see Definition 1.5.2.(i). We assume now that (rε)ε ≈R 2m and therefore from Eq.(C.10) we obtain (Rρσμν (rε, ε)Rρσμν (rε, ε))ε ≈R K (rs) + ( ε4 4m4 [ ε2 + (rε − 2m)2 ]3 ) ε . (A.23) Remark C.8. Let [ (rε − 2m)ε ] ≈R 0, then from Eq.(A1.3) and Eq.(C.12) we obtain [ (Rρσμν (r, ε)Rρσμν (r, ε))ε ] = K (rs) + ( ε4 4m4 [ ε2 + (rε − 2m)2 ]3 ) ε . (A.24) Remark C.9. Note that from Eq.(C.15) at horizon r = 2m follows that: (Rρσμν (r, ε)Rρσμν (r, ε))ε ≈R ∞, (A.25) see Definition 1.5.2.(ii). Remark A2.6. We assume now there exist an fundamental generalized lengh (lε)ε 97 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (lε)ε∈(0,η] = a (ε)ε∈(0,η] , η ≪ 1, (lε)ε∈(η,1] = a, (A.26) such that ∣∣(rε − ρ)ε∣∣ ≥ (lε)ε = a (ε)ε It meant there exist a thickness thhor = (lε)ε of BH horizon. We introduce a norm ∥thhor∥ of a thickness thhor by formula ∥thhor∥ = supε∈(0,η] |lε| = η, (A.27) By using (A.20) we get the estimate (Rρσμν (rε, ε)Rρσμν (rε, ε))ε ≈R K (rs) + ( ε4 4m4 [ (rε − 2m)2 + ε2 ]3 ) ε∈(0,η] = K (rs) + ( 1 4m4 [ (rε − 2m)2 + ε2 ]) ε∈(0,η] × ( ε2[ (rε − 2m)2 + ε2 ]) ε∈(0,η] × × ( ε2[ (rε − 2m)2 + ε2 ]) ε∈(0,η] ≤ K (rs) + 1 4m4 [a2 + 1]2 ( 1[ (rε − 2m)2 + ε2 ]) ε∈(0,η] ≤ K (rs) + 1 4m4 [a2 + 1]2 ( 1 (r − 2m)2 ) r−2m∈(0,η] . (A.28) Biography of author(s) Jaykov Foukzon Center for Mathematical Sciences, Technion Israel Institute of Technology, Haifa, Israel. He is currently working in the Center for Mathematical Sciences as a Technion of the Israel Institute of Technology, Haifa, Israel. He was born in 1954 in Soviet Union. He completed his doctoral degree in Physics and Mathematics. He completed his education from the Moscow State University in 1975. His Doctoral supervisor was Lev Pontryagin a Soviet and Russian mathematician and physicist from the Russian Space Research Institute, Institute of Control Sciences of Russian Academy of Sciences, Central Aerohydrodynamic Institute (TsAGI) Russia. His skills and activities include Optimal Control, Nonlinear Dynamics, Quantum Chaos, Control Theory, Advanced Control Theory, Systems Dynamics, Theory of Reference Frames (TR), Waves, Non-Standard Model, Standard Model, Calculations, Quantum Field Theory, Number Theory, Electromagnetic Fields, Partial Differential Equations, Actuarial Mathematics, Geometry, Computational Physics, QuantumMechanics, Theoretical Physics, Foundations of Quantum Mechanics, Mathematical Physics, First-order Logic, Special 98 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance and General Relativity, General Relativity, Theoretical Particle Physics, Fundamental Physics, Probability Theory, Gravitational Physics, Field Theory, Theoretical High Energy Physics, High Energy Physics Theory, Particle Physics, Foundations of Physics, Supersymmetry, Classical Mechanics, Quantum Cosmology, Conformal Field Theory, Algebraic Quantum Field Theory, Renormalizationgroup Theory, Quantum Gravity, Electrodynamics, Set Theory, Logic, IT Systems, Algebra, Analysis, Basic, Numbers, Deterministic Quantum Physics, Mathematical Analysis, Differential Geometry. He has published several book as follows: Jaykov Foukzon, Elena R. Men'kova, Alexander A. Potapov. EPRB Paradox Resolution. Bell inequalities revisited. 03/2019; LAMBERT. ISBN: 978613-9-45511-9: Jaykov Foukzon, Alexander Potapov, Elena Men'kova. Schrdingers Cat Paradox Resolution. A New Quantum Mechanical Formalism Based on the Probability Representation of Continuous Observables. 12/2017; LAMBERT. ISBN: 978-3-659-93543-5: S. A. Podosenov, A. A. Potapov, J. Foukzon, E. R. Menkova. Fields, Fractals, Singularities and Quantum Control. 10/2016; in press: S. A. Podosenov, A. A. Potapov, J. Foukzon, E. R. Menkova. Nonholonomic, Fractal and Bound Structures in Relativistic Continua, Electrodynamics, Quantum Mechanics and Cosmology. In three books. Publishing house URSS, Moscow, LENAND, 03/2016. ISBN 978-59710-2456-9: Jaykov Foukzon, Alexander Potapov, Elena Menkova. Large deviations principles of NonFreidlin-Wentzell type. 07/2015; LAMBERT. ISBN: 978-3-659-66379-6 Alexander Alexeevich Potapov Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia. He is working as the head of laboratory of IRE RAS (Kotelnikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia), Academician of the Academy of Engineering Sciences named after A.M. Prokhorov, Academician of RANS, Chief Editor of the International Journal Nonlinear World, Honorable Radio Operator of the Russian Federation. He was born in 1951and completed his doctoral degree in physics and mathematics. He is the author of the basic researches in the field of fractal approaches to radio physics, radio engineering, radar, electrodynamics, electronics, aerial engineering and control problems and broad spectrum of adjacent scientific and technology paths. He is the initiator of the pioneer researches and development works in the field of fractal theory application in radio physics, scaling effects and fractional operators in the radio systems in Russia. He is the founder of the world well-known Russian scientific school of the fractal methods. He is also serving as a president of the joint Chinese-Russian Laboratory of the information technologies and fractal signal processing. He is also working as the Emeritus Professor of the Jinan University (China). He is the academician Prokhorov Prizewinner. He has authored more than 700 scientific publications, including 20 monographs and 1 textbook. The scientific areas of his interest are radio physics, dispersion and diffraction of electromagnetic waves, radio location, recognition and signal processing and images, fractal antennas, deterministic chaos, modern topology, fractal analysis, fractional operators, scaling effects. 99 Colombeau Solutions to Einstein Field Equations in General Relativity:Gravitational singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance Menkova Elena Romanovna All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia. She was born in 1955. She completed her Ph.D. degree and currently working as a Senior researcher in All-Russian Research Institute for Optical and Physical Measurements, VNIIOFI Laboratory of Generation and Measurement of Electromagnetic Pulse Parameters. She completed her education from the Moscow Mendeleev Institute of Chemical Technology in 1978 and from the Moscow Institute of Electronic Mechanical Engineering, Automated Management Systems, qualification improvement faculty in 1987. She usually done the literature searches and physical interpretation of models in the areas of radiation and measurement of pulse electromagnetic fields and their interaction with the conducting bodies. Her research interest focuses on the works connected with problems of relativistic mechanics and electrodynamics, noninertial frames of reference in special and general relativity. ----------------------------------------------– c⃝ Copyright (2019): Authors. The licensee is the publisher (Book Publisher International). DISCLAIMER This chapter is an extended version of the article published by the same authors in the following journal with CC BY license. British Journal of Mathematics & Computer Science, 13(6): 1-54, 2016. 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