# Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance

Jaykov Foukzon, Alexander Alexeevich Potapov, Menkova Elena Romanovna

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Jaykov Foukzon, Alexander Alexeevich Potapov, Menkova Elena Romanovna. Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance: Distributional SAdS BH Spacetime-Induced Vacuum Dominance. Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance, Book Publisher International (a part of SCIENCEDOMAIN International), 2019, ISBN: 978-93-89562-22-4, ISBN 978-93-89562-23-1. 10.9734/bpi/mono/978-93-89562-22-4. hal-04004572

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Singular semi-Riemannian geometry using Colombeau approach.

Jaykov Foukzon¹, Elena Men'kova² Alexander Potapov³

**Abstract** This book is an exposition of "Singular Semi-Riemannian Geometry"- the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. This book also dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry. 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 and not classically. In this paper essentially new class Colombeau solutions to Einstein field 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 $E_T$ of a static and asymptotically flat spacetime, gives $E_T = m$, as it should be. The vacuum

energy density of free scalar quantum field $\Phi$ 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 $\langle\Phi^2\rangle$ has a singular behavior on BHs horizon $r_+$: $\langle\Phi^2(r)\rangle\sim|r-r_+|^{-2}$. 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

on curved spacetime, vacuum fluctuations, vacuum dominance

**Mathematics Subject Classification** : Primary: 83C15; secondary: 83C15, 83C75, 46F30, 46T30.

**Physics and Astronomy Classification Scheme** : Primary: 04.20.Cv; secondary: 04.20.Jb, 04.20.Dw, 02.30.Sa, 02.40.Vh
**Introduction.**

Classical semiclassical and and nonclassical semi-Riemannian manifolds (*M*,*g*).

1.Singular semi-Riemannian geometry using Colombeau approush.

1.1.Paragraph on linear and nonlinear distributional geometry in general relativity.

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.

1.2.2.The ring of Colombeau generalized numbers $\tilde{\mathbb{R}}_\delta$.
Point values of Colombeau generalized functions.

1.3.The point free classical Colombeau geometry.

1.4.The Point-Free Loop Quantum Gravity.

**1.4.1. Classical point-free phase space.**

**1.4.2. Quantization.**

1.5. Schwarzschild Black Hole.

**1.5.1. Classical point-like phase space variables.**

1.6. Classical point-like Loop Quantum Gravity contradict with a linear
Colombeau distributionel geometry. The point free quantum Schwarzschild geometry.

1.7. Generalized Stokes' theorem.

1.7.1.The Colombeau generalized curvilinear coordinates.

**1.7.2. Generalized Stokes' theorem.**

1.8.The Colombeau Generalized Curvature Tensor.

1.9.Generalized Einstein's field equations.

1.10.The densitized Einstein field equations revisited.

1.10.1.Paragraph on the A. Einstein and N. Rosen paper from 1935.

1.10.2.Paragraph on Möller abnormal famous paper from 1943

1.10.3.The densitized Einstein field equations revisited by using the linear
distributional geometry.

1.11.The nonlinear distributional Möller geometry using the full algebra of the
Colombeau generalized functions. A.Einstein as the Copernicus of nonlinear
distributional geometry.

1.12.The Distributional Schwarzschild Geometry by using the linear L.Schwartz distributions and by using the full algebra of the Colombeau generalized functions.

1.12.1. The Distributional Schwarzschild Geometry at the Origin. The Smooth Regularization of the Singularity at the Origin.

1.12.2. The Nonsmooth Regularization of the Singularity at the Origin.

1.12.3. The Smooth Regularization Via Horizon in Schwarzschild coordinates and the smooth regularization at horizon in isotropic coordinates.
1.12.3.1. The smooth regularization via Horizon in Schwarzschild coordinates.

1.12.3.2. The smooth regularization at horizon in isotropic coordinates.

1.12.4. The Nonsmooth Regularization Via Gorizon.

1.12.5. Distributional Eddington-Finkelstein Space-time.

## 2. Nonlinear Distributional General Relativity.

2.1 Notation and Basic Notions from Standard Colombeau Theory.

2.2 Point Values of Generalized Functions on smooth manifold *M*. Generalized Numbers.

2.3. The nonlinear distributional Schwarzschild geometry.
The truncated distributional Schwarzschild geometry. Gravitational singularity.

2.4. The nonlinear distributional Schwarzschild geometry in isotropic coordinates.

2.5. Generalized Colombeau Calculus.

2.5.1. Super Generalized Functions.

2.5.2 Point Values of Supergeneralized Functions on smooth manifold *M*.
Supergeneralized Numbers.

2.5.3. Remarks on distributional derivatives in $D'(\mathbb{R})$.

2.5.4. The super distributional geometry of the Schwarzschild space-time.

2.6. Nonlinear Superistributional General Relativity.

3. Distributional Schwarzschild Geometry from Nonsmooth Regularization via Horizon

3.1 Calculation of the Stress-tensor by Using Nonsmooth Regularization Via Horizon.

3.2 Examples of Distributional Geometries. Calculation of the Distributional Quadratic Scalars by Using Nonsmooth Regularization via Horizon.

4 Quantum Scalar Field in Curved Distributional Space-time.

4.1. Canonical Quantization in Curved Distributional Space-time.

4.2. Defining Distributional Outgoing Modes.

4.2.1. Boundary Conditions at the Horizon of the Distributional SAdS BH Geometry.

5. Energy-Momentum Tensor Calculation by using Colombeau Distributional Modes.

6. Distributional SAdS BH Space-time-induced Vacuum Dominance

6.1. Adiabatic Expansion of Green Functions in Curved Distributional Space-time.

6.2. Effective Action for the Quantum Matter Fields in Curved Distributional Space-time.

6.3. Stress-tensor Renormalization.

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.

7.2 The Colombeau Distributional Kerr Space-time in Boyer - Lindquist form.

8. Conclusions and Remarks

References A.

References B.

Appendix A
This paper is divided into six sections.

**Sh chapter 1** contains a brief introduction in contemporary theory of the degenerate and
singular semi Riemannian manifolds using the linear distributional geometry based on
L.Schwartz distributions $D'(M)$ on smooth manifold $M$ and Colombeau distributional
geometry based on full algebra $G(M)$ of the Colombeau generalized functions on
smooth
manifold $M$.

<u>Subsection 1.1</u> contains a brief review the results in General Relativity obtained by
using
the linear and nonlinear distributional approach in order to resolve problems with
canonical Schwarzschild geometry.

Subsection 1.2 contains the elementary explanation of the basic notions of
Colombeau
generalized functions and Colombeau generalized numbers.

<u>Subsection1.3</u> contains a brief introduction in point free classical Colombeau
geometry.

Subsection1.4 contains a brief introduction in point-free loop quantum gravity.

<u>Subsection1.5</u> contains description of the classical point-like phase space variables
corresponding to the classical Schwarzschild metric.

<u>Subsection1.6</u> contains explanation why classical point-like Loop Quantum Gravity
contradict with a well established the linear Colombeau geometry.

Subsection1.6.1-1.6.2 contains a brief introduction in point free quantum

Schwarzschild
geometry.

<u>Subsection1.7</u> contains Generalized Stokes' theorem corresponding to nonlinear
<u>Colombeau geometry.</u>

<u>Subsection1.8</u> contains derivation of the Colombeau Generalized Curvature Tensor.

<u>Subsection1.9</u> contains derivation of the Generalized Einstein's field equations
obtained
in [37].

<u>Subsection1.10</u> contains explanation why the canonical
densitized Einstein field equations from contemporary literature wrong and revision is
needed,etc.

<u>Subsection1.10.1</u> contains some important remarks on the A. Einstein and N. Rosen
<u>paper from 1935.</u>

<u>Subsection1.10.2</u> contains some important remarks on Möller abnormal famous paper
from 1943

<u>Subsection1.10.3</u> contains derivation of the <u>revisited</u> densitized Einstein field
equations
by using the <u>special kind</u> of the linear distributional geometry.

<u>Subsection1.11</u> contains the nonlinear distributional Möller geometry using the
full algebra of the Colombeau generalized functions originally derived in [5] in order to
avoid the Polchinski's paradox.

**Subsection 1.12** contains a brief review of the distributional Schwarzschild geometry by using the linear L.Schwartz distributions and by using the full algebra of the Colombeau generalized functions.

**Subsection 1.12.1** contains a brief review of the distributional Schwarzschild geometry at the origin by using the smooth regularization of the singularity at the origin.

**Subsection 1.12.2** contains a brief review of the nonsmooth regularization of the singularity at the origin.

**Subsection 1.12.3** contains a brief review of the smooth regularization via Horizon in Schwarzschild coordinates and the smooth regularization at horizon in isotropic coordinates.

**Subsection 1.12.4** contains a brief review of the nonsmooth regularization via Horizon.

**Subsection 1.12.5** Distributional Eddington-Finkelstein Space-time is considered.

**Chapter 2** contains a brief introduction in contemporary generalized Colombeau calculus.

**Subsection 2.1-2.2** contains basic notions of the canonical Colombeau theory [1],[2]

**Subsection 2.3** contains the full derivation of the nonlinear distributional Schwarzschild geometry. The truncated distributional Schwarzschild geometry and Gravitational singularity is considered.

**Subsection 2.4** contains the full derivation of the nonlinear distributional Schwarzschild geometry in isotropic coordinates.

**Subsection 2.5** contains a brief introduction in generalized Colombeau calculus.

**Subsections 2.5.1-2.5.3** contains basic definitions of the super generalized functions.

**Subsection 2.5.4** contains the full derivation of the super distributional geometry of the Schwarzschild space-time.

**Chapter 3** contains the full derivation of the linear distributional Schwarzschild geometry from nonsmooth regularization via horizon

**Subsection 3.1** contains the full calculation of the Stress-tensor by using nonsmooth regularization via horizon.

**Subsection 3.2** contains examples of distributional geometries.

**Chapter 4** contains quantum field theory of the scalar field in curved distributional space-time.

**Subsection 4.1** contains a brief introduction in canonical quantization in curved distributional Space-time.

**Subsection 4.2** contains an example of the distributional outgoing modes.

**Subsection 4.2.1** contains a boundary conditions at the horizon of the distributional SAdS BH geometry.

**Chapter 5** contains the full calculation of the energy-momentum tensor by using Colombeau Distributional Modes.

**Chapter 6** contains the main theoretical result of this paper: Distributional SAdS
BH Space-time-induced Vacuum Dominance.

<u>Subsection 6.1</u> contains the full derivation of the adiabatic expansion of Green
functions in curved distributional space-time.

<u>Subsection 6.2</u> contains effective action for the scalar quantum matter fields in
curved distributional space-time.

Subsection 6.3 contains stress-tensor renormalization.

**_Shapter 7_** contains novel explanation of the active galactic nuclei.

<u>Subsection 7.1</u> contains a brief introduction in the current paradigm for active galactic nuclei and results of the observations of the high energy emission from galactic jets.

<u>Subsection 7.1</u> contains Novel Explanation of the Active Galactic Nuclei based on the
Colombeau distributional Kerr Space-time in Boyer - Lindquist form.

1.Introduction. Classical, semiclassical and nonclassical
semi-Riemannian manifolds (M, g).

The classical Cartan's structural equations show in a compact way the relation
between a connection and its curvature, and reveals their geometric interpretation in
terms of moving frames. In order to study the mathematical properties of singularities,
we need to study the geometry of manifolds endowed on the tangent bundle with a
symmetric bilinear form which is allowed to become _degenerate_ or _singular_ (or both
_degenerate_ and _singular_ ) on semi Riemannian manifold (M, g) or on submanifolds of
semi Riemannian manifold (M, g). But if the fundamental tensor is allowed to be
degenerate or singular, there are some obstructions in constructing the geometric
objects normally associated to the fundamental tensor. Also, local orthonormal frames
and coframes no longer exist, as well as the metric connection and its curvature
operator.

**Definition 1.1.** (i) Semi Riemannian manifold (M, g) is nonclassical if the fundamental tensor g is allowed to be degenerate or singular, (ii) semi Riemannian manifold (M, g) is internally nonclassical if the fundamental tensor g is not allowed to be degenerate or singular but there exists semi Riemannian submanifold (*M*,*g*'), *M*' ⊊ *M*, *g*' = *g*|_{*M*'} such that the fundamental tensor *g*′ is allowed to be degenerate or singular, (iii) otherwise we will be say that (M,g) is classical.

will be say that (M,g) is classical.

**Remark 1.1.** In the nonclassical case the main problem arises from the degeneracy of the $\det(g_{ij}(\hat{x}))$ on some isolated points: $\det(g_{ij}(\hat{x}^0)) = 0, \hat{x}^0 \in M$ or some submanifold $\det(g_{ij}(\hat{x})) = 0$ for all $\hat{x} \in M' \subsetneq M$ and consequantly the corresponding Christoffel symbols bcome infinity.

bcome infinity.

In mathematical literature more than 50 yers accepted that a nonclassical semi
Riemannian manifold mentioned above **impossible treated classically**, i.e. by using
canonical apparatus of the Riemannian geometry. However in the contemporary
mathematical literature, manifolds with degenerate metric tensors have been studied
only fore some special case called a Reinhart manifold (see for example [3] References
B).
**Remark 1.1.** In this paper we studied nonclassical semi Riemannian manifold $(M,g)$ extrinsically, i.e. as degenerate submanifolds of Colombeau generalized semi Riemannian manifolds furnished with non degenerate Colombeau generalized or super generalized fundamental tensor $(g_{ij,\epsilon})_\epsilon \in \mathcal{G}_0^2(M,\Sigma)$, see subsect.2.5, Definition 2.5.8.

**Definition 1.2.** We shall say that Colombeau generalized semi Riemannian manifold $(M, (g_{i,j,\epsilon}(\hat{x}))_\epsilon)$ is the Colombeau extension of the nonclassical semi Riemannian manifold $(M, g_{i,j,0}(\hat{x}))$ if there exists the canonical imbedding $(M, g_{i,j,0}(\hat{x})) \hookrightarrow (M, (g_{i,j,\epsilon}(\hat{x}))_\epsilon)$, see subsect.1.1, where such imbedding is obtained for Schwarzschild space-time with metric

(1.6) in canonical Schwarzschild coordinates $(t, r, \theta, \phi)$.

**Remark 1.2.** The canonical imbedding $(M, g_{i,j,0}(\hat{x})) \hookrightarrow (M, (g_{i,j,\epsilon}(\hat{x}))_\epsilon)$ easily obtained by apropiate $\epsilon$-regularization of the tensor $g_{i,j,0}(\hat{x})$, see subsections 1.11.1-1.11.5.

**Remark 1.3.** (i) Note that the notion of the *classical* Riemannian curvature comes from the study of parallel transport on a *classical* Riemannian manifold, see Fig.1.1. For instance, if

a vector $A_i(\hat{x})$ is moved around closed contour (or a loop) on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as *holonomy*.

The classical holonomy presented by the classical formula (1.1) for the change $\Delta A_k$ in a smooth vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$.

The classical formula for the change in a smooth vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$ or integral measure of the *classical holonomy* of the surface $\Sigma_\Gamma$ spanning by $\Gamma$ reads [4]:

$$ \Delta A_k(\Gamma) = \oint_{\Gamma} \delta A_k = \oint_{\Gamma} \Gamma_{kl}^{i}(\hat{x}) A_k dx^l. \quad (1.1) $$

Now applying *classical Stokes' theorem* (see Theorem 1.10.1) to the integral (1.1) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f^{im})$, one

obtains, (see subsection 1.10.3) [4]:

$$ \Delta A_k = \frac{1}{2} R_{klm}^{i}(x) A_i(x) \Delta f^{im}, \quad (1.2) $$

where $R_{klm}^{i}(x)$ is a tensor field of the fourth rank:

$$ R_{klm}^{i}(x) = \frac{\partial(\Gamma_{km}^{i}(x))}{\partial x^{l}} - \frac{\partial(\Gamma_{kl}^{i}(x))}{\partial x^{m}} + \Gamma_{ni}^{i}(x)\Gamma_{km}^{n}(x) - \Gamma_{nm}^{i}(x)\Gamma_{kl}^{n}(x). \quad (1.3) $$

The tensor field $R_{klm}^{i}(x)$ is called the classical curvature tensor or the classical Riemann tensor.

The classical Riemann tensor that is a tensorial *measure* of holonomy
(ii) Various generalizations capture in an abstract form this idea of *curvature* as a
*measure of holonomy*.

<figure><img src="image_2.png" /><figcaption>Fig.1.1.Parallel transporting a vector from
$A \rightarrow N \rightarrow B \rightarrow A$ yields a different vector.
This failure to return to the initial vector
is measured by the change in as smooth
vector $A_i(\hat{x})$ after parallel displacement around
infinitesimal closed contour $\Gamma$ or
by the *classical* integral measure $\Delta A_k(\Gamma)$ of the
*holonomy* of the surface $\Sigma_{\Gamma}$ spanning by contour $\Gamma$.</figcaption></figure>

(iv) Note that in classical case the change $\Delta A_k(\Gamma)$ always finite,i.e. $\Delta A_k(\Gamma) < \infty$

(v) If semi Riemannian manifold (M, g) is nonclassical the condition ΔA<sub>k</sub>(Γ) < ∞ is not always holds and we shall say that contour Γ is <u>regular</u> if ΔA<sub>k</sub>(Γ) < ∞, see Fig.1.2 if ΔA<sub>k</sub>(Γ) = ∞ we shall say that contour Γ is a <u>singular</u> contour or singular loop, see Fig.1.3.

<figure><img src="image_3.png" /><figcaption>Fig.1.2.Regular contour Γ
beyond BH horizon.</figcaption></figure>
**Definition 1.3.** We shall say that a point $\hat{x}_0 \in \Sigma_\Gamma$ is a singular point of the surface $\Sigma_\Gamma \subset M$
if the Levi-Civitá connection is not available at point $\hat{x}_0$, i.e. some of the Christoffel symbols become infinity at point $\hat{x}_0$.

<figure><img src="image_4.png" /><figcaption>Fig.1.3. Singular point $\hat{x}_0$ at BH horizon and corresponding singular contour $\Gamma_{\text{sing}}$.</figcaption></figure>

**Remark 1.4.** Note that any singular contour $\Gamma_{\text{sing}}$ contains at least one singular point $\hat{x}_0 \in \Gamma_{\text{sing}}$, see Fig.1.3 and we shall abbreviate $\Gamma_{\text{sing}}^{\hat{x}_0}$ if $\hat{x}_0 \in \Gamma_{\text{sing}}^{\hat{x}_0}$.

**Remark 1.5.** (i) Note that the classical formula (1.1) holds only for regular loops but obviously breaks down on singular loops $\Gamma_{\text{sing}}$, since $\Delta A_k(\Gamma_{\text{sing}}) = \infty$.

(ii) Note that Eqs.(1.2)-(1.3) again no longer hold for singular loops $\Gamma_{\text{sing}}$ (see sect.1.8). and therefore the classical Ricci scalar and the classical Kretschman scalar is not

holds at BH horizon. However in classical literature (see [3],[4]) it was just pulled to the BH horizon by the ears.

**Remark 1.6.** (i) Note that for trunketed singular contour $\Gamma_{\text{sing}}^{\#} = \Gamma_{\text{sing}}^{\hat{x}_0} \setminus \{\hat{x}_0\}$ see Fig.1.4, the

Levi-Civit'a connection is available at whole contour $\Gamma_{\text{sing}}^{\#}$ and classical formula (1.1) reads:

$$ \Delta A_k(\Gamma_{\text{sing}}^{\#}) = \oint_{\Gamma_{\text{sing}}^{\hat{x}_0} \setminus \{\hat{x}_0\}} \delta A_k = \oint_{\Gamma_{\text{sing}}^{\hat{x}_0} \setminus \{\hat{x}_0\}} \Gamma_{kl}^i(\hat{x}) A_k dx^l. $$

Obviously for trunketed singular contour $\Gamma_{\text{sing}}^{\#} = \Gamma_{\text{sing}}^{\hat{x}_0} \setminus \{\hat{x}_0\}$ again we get

$\Delta A_k(\Gamma_{\hat{x}_0} \setminus \{\hat{x}_0\}) = \infty$ and therefore the Eqs.(1.2)-(1.3) again no longer hold for trunketed singular loops $\Gamma_{\text{sing}}^{\#}$.

(ii) The semi Riemannian manifold $(M,g)$ which contain trunketed singular loops $\Gamma_{\text{sing}}^{\hat{x}_0}$ such that $\Gamma_{\text{sing}}^{\hat{x}_0} \subset M, \hat{x}_0 \notin M$, seems as classical since the Levi-Civit'a connection is available at whole
<figure><img src="image_5.png" /><figcaption>Fig.1.4. Trunketed singular contour (loop) $\Gamma^{\#}_{\text{sing}} = \Gamma_{\hat{x}_0} \setminus \{\hat{x}_0\}$ with deleted singular point $\hat{x}_0$.</figcaption></figure>

(iii) Note that in order to obtain the full geometrical properties of BH horizon one needs non classical definition of the integral measure $\widetilde{\Delta A}_k(\Gamma_{\text{sing}})$ of the holonomy of the surface

$\Sigma_{\Gamma_{\text{sing}}} \text{ spanning by singular loop } \Gamma_{\text{sing}} \text{ such that } \widetilde{\Delta A}_k(\Gamma_{\text{sing}}) \text{ is well defined quantity.}$

(iv) By using contemporary Colombeau approach [1] one obtains appropriate definition of the non classical integral measure $\widetilde{\Delta A}_k(\Gamma_{\text{sing}})$ of the holonomy as direct generalization in

natural way of the Eq.(1.1) (see subsection 1.8 Remark 1.8.8,Eq.(1.8.10))

$$ \widetilde{\Delta A}_k(\Gamma_{\text{sing}}) \triangleq (\Delta A_{k,\varepsilon})_{\varepsilon} = \left( \oint_{\Gamma_{\text{sing}}^{\hat{x}^0}} \delta A_{k,\varepsilon} \right)_{\varepsilon} = \left( \oint_{\Gamma_{\text{sing}}^{\hat{x}^0}} \Gamma_{kl,\varepsilon}^{i}(\hat{x}) A_{k,\varepsilon} dx^l \right)_{\varepsilon}. \quad (1.4) $$

where point $\hat{x}^0$ belongs to BH horizon and $\hat{x}^0 \in \Gamma_{\text{sing}}^{\hat{x}^0}$, $\widetilde{\Delta A}_k(\Gamma_{\text{sing}}) \in \tilde{\mathbb{R}}$,

**Definition 1.4.** We shall say that a surface $\Sigma_{\Gamma}^{\text{sing}} \subset M$ is a singular surface if $\Sigma_{\Gamma}^{\text{sing}}$ it has at least one singular point point $\hat{x}_0 \in \Sigma_{\Gamma}^{\text{sing}}$, see Fig.1.5.
<figure><img src="image_6.png" /><figcaption>Fig.1.5. Singular surface $\Sigma_{\Gamma}^{\text{sing}}$ with singular point $r = 0$, spanning by regular contour $\Gamma$.</figcaption></figure>

**Remark 1.7.** Note that for the case of the surface $\Sigma_{\Gamma}^{\text{sing}}$ again problem arises, (see sect.1.8. Remark 1.8.3) even if it spanned by regular contour $\Gamma$, see Fig.1.5. In this case the non classical integral measure $\widetilde{\Delta} A_k(\Gamma, \Sigma_{\Gamma}^{\text{sing}})$ of the holonomy reads:

$$ \widetilde{\Delta} A_k(\Gamma, \Sigma_{\Gamma}^{\text{sing}}) \triangleq (\Delta A_{k,\varepsilon})_{\varepsilon} = \left( \oint_{\Gamma} \delta A_{k,\varepsilon} \right)_{\varepsilon} = \left( \oint_{\Gamma} \Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}_0) A_{k,\varepsilon} dx^l \right)_{\varepsilon}. \quad (1.5) $$

**Definition 1.5.** Let $M$ be a differentiable manifold equipped with it canonical topology $\mathfrak{I}$.

Let $M^{\#}$ be a submanifold $M^{\#} \subsetneq M$. The closure $M_{\mathfrak{I}}^{\#}$ of a submanifold $M^{\#} \subsetneq M$ of points in a topological space $(M, \mathfrak{I})$ consists of all points in $M^{\#}$ together with all limit points of $M^{\#}$.

The closure of $M_{\mathfrak{I}}^{\#}$ may equivalently be defined as the union of $M^{\#}$ and its boundary $\partial M^{\#}$,

and also as the intersection of all closed sets containing $M^{\#}$. Intuitively, the closure can be

thought of as all the points that are either in $M^{\#}$ or "near" $M^{\#}$. A point which is in the closure of $M^{\#}$ is a point of closure of $M^{\#}$.

**Definition 1.6.** Let $(M,g)$ and $(M^{\#}, g^{\#})$ semi Riemannian manifolds such that (i) $M^{\#} \subsetneq M$,

(ii) $M_{\mathfrak{I}}^{\#} = M$ and (iii) $g^{\#} = g|_{M^{\#}}$. Assume that: (1) the Levi-Civit'a connection

$\Gamma_{kj}^{il} = \frac{1}{2}[g^{lm}][(g_{mk,j}^{\#} + g_{mj,k}^{\#} - g_{kj,m}^{\#})]$ corresponding to metric tensor $g^{\#}$ is available at

whole semi Riemannian manifold $(M^{\#}, g^{\#})$ and (2) the Levi-Civit'a connection

$\Gamma_{kj}^{l} = \frac{1}{2}[g^{lm}][(g_{mk,j} + g_{mj,k} - g_{kj,m})]$ corresponding to metric tensor $g$ is not available at its

boundary $\partial M^{\#}$, i.e. $(M,g)$ is a nonclassical semi Riemannian manifold. Then we shall say

that $(M^{\#}, g^{\#})$ is a *semiclassical* semi Riemannian manifold.

**Remark 1.8.** It follows from consideration above that semiclassical semi Riemannian manifolds obviously *impossible* treated *classically* as nonclassical semi Riemannian
manifolds mentioned above.

**Example 1.1.** Obviously the Levi-Civita connection is available at whole Schwarzschild spacetime (see Remark 1.10.1) (Sch, $g_{ij}^{Sch}(t, r, \theta, \phi)$),

$$ \mathbf{Sch} = (S^2 \times \{r > 2m\} \cup \{0 < r < 2m\}) \times \mathbb{R}, $$

but spacetime Sch contains truncated singular loops $\Gamma_{\text{sing}}^{\#} = \Gamma_{\hat{x}_0} \setminus \{\hat{x}_0\} \subset \text{Sch}$, see Fig.1.4,

and in particular the Levi-Civita connection is not available at whole its topological closure $(\overline{\mathbf{Sch}}, g_{ij}^{\text{Sch}}(t, r, \theta, \phi))$, where

$$ \hat{x}_0 \in \overline{\mathbf{Sch}} = (S^2 \times (\{r \ge 2m\} \cup \{0 \le r \le 2m\})) \times \mathbb{R}. $$

Thus Schwarzschild spacetime is not classical but exactly is a semiclassical semi Riemannian manifold. Similarly the Levi-Civita connection is available at the open semi

Riemannian manifold $(\mathbf{Sch}^+, g_{ij}^{\text{Sch}^+}(t, r, \theta, \phi))$ above Schwarzschild horizon: $\mathbf{Sh}^+ = (S^2 \times \{r > 2m\}) \times \mathbb{R}$ but is not available at whole its topological closure $(\overline{\mathbf{Sch}}^+, g_{ij}^{\overline{\text{Sch}}^+}(t, r, \theta, \phi))$, $\overline{\mathbf{Sch}}^+ = (S^2 \times \{r \ge 2m\}) \times \mathbb{R}$.

Thus semi Riemannian manifold $\mathbf{Sh}^+$ is not classical but exactly is a semiclassical semi

Riemannian manifold.

**Remark 1.9.** Note that in physical literature the spacetime $\mathbf{Sch}^+ = (S^2 \times \{r > 2m\}) \times \mathbb{R}$ mistakenly considered as classical semi Riemannian manifold. Note that only spacetimes $(\mathbf{Sch}_\delta^+, g_{ij}^{\overline{\mathbf{Sch}}_\delta^+})$, $\mathbf{Sch}_\delta^+ = (S^2 \times \{r > 2m + \delta\}) \times \mathbb{R}$, $\delta > 0$ is a classical semi Riemannian manifolds.

<figure><img src="image_7.png" /><figcaption>Fig.1.6. Region of the classical semi Riemannian geometry above Schwarzschild horizon</figcaption></figure>

$$ \mathbf{Sh}_\delta^+ = (S^2 \times \{r > 2m + \delta\}) \times \mathbb{R}, \delta > 0. $$

**Remark 1.10.** Note that in physical literature
<figure><img src="image_8.png" /><figcaption>Fig.1.7. The classical semi Riemannian Geometry presented by Eddington-Finkelstein spacetime</figcaption></figure>

**Remark 1.10.** There exist examples of the nonclassical semi Riemannian manifolds. However in physical literature these nonclassical semi Riemannian manifolds mistakenly were considered as classical semi Riemannian manifolds.

1. The Schwarzschild metric represented by Schwarzschild coordinates $(t, r, \theta, \phi)$ reads

$$ds^2 = -(1 - \frac{r_s}{r})(dt)^2 + \left(1 - \frac{r_s}{r}\right)^{-1}(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2]. \quad (1.6)$$

The Christoffel symbols in Schwarzschild coordinates $(t, r, \theta, \phi)$ reads

$$\begin{align}
\Gamma^r_{tt} &= \frac{c^2 r_s (r-r_s)}{2r^3}, \quad \Gamma^t_{tr} = \frac{r_s}{2r(r-r_s)}, \quad \Gamma^r_{rr} = -\frac{r_s}{2r(r-r_s)}, \nonumber \\
\Gamma^\theta_{r\theta} &= \frac{1}{r}, \quad \Gamma^\phi_{r\phi} = \frac{1}{r}, \quad \Gamma^r_{\theta\theta} = -(r-r_s), \quad \Gamma^\phi_{\theta\phi} = \cot\theta, \tag{1.7} \\
\Gamma^r_{\phi\phi} &= -(r-r_s)\sin 2\theta, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta. \nonumber
\end{align}$$

**Remark 1.3.** Note that: (i) the Schwarzschild metric is allowed to become _degenerate_ and _singular_ at Schwarzschild horizon {$r = r_s$}, (ii) the Schwarzschild metric is allowed to become _degenerate_ and _singular_ at Schwarzschild singularity {$r = 0$}, (iii) the Christoffel symbols $\Gamma^t_{tr}$ and $\Gamma^r_{rr}$ become infinity at Schwarzschild horizon {$r = r_s$}.

Thus the Schwarzschild space-time is nonclassical semi Riemannian manifold. The full nonlinear distributional Schwarzschild geometry at horizon is obtained in subsect.

2.3.

2. The _singular_ transformation of the Schwarzschild metric (1.6) from the usual Schwarzschild time coordinate $t$ to the advanced null coordinate $v$ with

$$cv = ct + r + r_s \ln(r - r_s) \quad (1.8)$$

leads to the ingoing Eddington-Finkelstein metric (1.9) with coordinates $(v, r, \theta, \phi)$. The Schwarzschild metric (1.6) in Eddington-Finkelstein coordinates $(v, r, \theta, \phi)$, reads

$$ds^2 = -(1 - \frac{r_s}{r})c^2 dv^2 + 2cdvdr + r^2 d\Omega^2. \quad (1.9)$$

The Christoffel symbols in Eddington-Finkelstein coordinates $(v, r, \theta, \phi)$, reads
$$
\begin{align}
\Gamma^v_{vv} &= \frac{cr_s}{2r^2}, \quad \Gamma^r_{vv} = \frac{c^2r_s(r-r_s)}{2r^3}, \quad \Gamma^r_{vr} = -\frac{cr_s}{2r^2}, \quad \Gamma^\theta_{r\theta} = \frac{1}{r}, \nonumber \\
\Gamma^\phi_{r\phi} &= \frac{1}{r}, \quad \Gamma^v_{\theta\theta} = -\frac{r}{c}, \quad \Gamma^r_{\theta\theta} = -(r-r_s), \quad \Gamma^\phi_{\theta\phi} = \cot\theta, \tag{1.10}
\end{align}
$$

$$
\Gamma^v_{\phi\phi} = -\frac{r\sin^2\theta}{c}, \quad \Gamma^r_{\phi\phi} = -(r-r_s)\sin^2\theta, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta.
$$

**Remark 1.9.** The Eddington-Finkelstein space-time is classical semi Riemannian manifold except any finite neighborhood of singularity {$r=0$}.

**Remark 1.10.** Note that Eddington-Finkelstein space-time is not equivalent with Schwarzschild space-time, since the transform (1.8) is singular at Schwarzschild horizon {$r = r_s$}.

3. The Schwarzschild metric (1.6) in isotropic coordinates (t, ρ, θ, φ) reads,

$$
ds^2 = -\left(\frac{1-r_s/4\rho}{1+r_s/4\rho}\right)^2 c^2 dt^2 + \left(1+\frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\phi^2), \qquad (1.11)
$$

where

$$
r = \rho \left( 1 + \frac{r_s}{4\rho} \right)^2 \qquad (1.12)
$$

is the coordinate transformation between the Schwarzschild radial coordinate $r$ and the
isotropic radial coordinate $\rho$. The Christoffel symbols reads

$$
\begin{align}
\Gamma^\rho_{tt} &= 2048\rho^4 r_s c^2 \frac{(4\rho - r_s)}{(4\rho + r_s)^7}, \quad \Gamma^t_{t\rho} = \frac{8r_s}{16\rho^2 - r_s^2}, \quad \Gamma^\rho_{\rho\rho} = -\frac{2r_s}{(4\rho + r_s)\rho}, \nonumber \\
\Gamma^\theta_{\rho\theta} &= \frac{4\rho - r_s}{(4\rho + r_s)\rho}, \quad \Gamma^\phi_{\rho\phi} = \frac{4\rho - r_s}{(4\rho + r_s)\rho}, \quad \Gamma^\rho_{\theta\theta} = -\rho \frac{4\rho - r_s}{4\rho + r_s}, \tag{1.13} \\
\Gamma^\phi_{\theta\phi} &= \cot\theta, \quad \Gamma^\rho_{\phi\phi} = -\frac{(4\rho - rs)\rho \sin^2\theta}{4\rho + rs}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta \cos\theta. \nonumber
\end{align}
$$

**Remark 1.11.** (i) Note that "Schwarzschild space-time in isotropic coordinates" is nonclassical semi Riemannian manifold, since the metric (1.6) is allowed to become _degenerate_ at horizon $\rho^* = r_s/4$, (ii) note that "Schwarzschild space-time in isotropic coordinates" is not equivalent with Schwarzschild space-time in Schwarzschild coordinates $(t, r, \theta, \phi)$. The full nonlinear distributional "Schwarzschild geometry in isotropic coordinates" at horizon is obtained in subsect. 2.4.

coordinates" at horizon is obtained in subsect. 2.4.

**Remark 1.10.** Note that the space-times mentioned above in physical literature mistakenly

were considered as the same geometrical object in different coordinates only.

1.1.Remarks on linear and nonlinear distributional
geometry in general relativity.Why

A degenerate (singular) semi-Riemannian manifold (M; g) is a differentiable manifold
M endowed with a symmetric bilinear form g ∈ T₂M 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 [26],[30].

During last 30 years the applications classical linear distributional geometry in general relativity was many developed [5]-[31].

**Remark 1.1.1.** Let $(R_{bcd,\epsilon}^a)_\epsilon \in \mathcal{G}_\delta(\mathbb{R}^4)$ be Colombeau generalized function obtained using

the standard definition of the Riemann curvature in a coordinate basis,i.e.

$$ (R_{bcd,\epsilon}^a)_\epsilon = (\Gamma_{db,c,\epsilon}^a)_\epsilon - (\Gamma_{cb,d,\epsilon}^a)_\epsilon + (\Gamma_{cf,\epsilon}^a \Gamma_{ab,\epsilon}^f)_\epsilon - (\Gamma_{df,\epsilon}^a \Gamma_{cb,\epsilon}^f)_\epsilon, \quad (1.1.1) $$

where $(\Gamma_{bc,\epsilon}^a)_\epsilon \in \mathcal{G}_\delta(\mathbb{R}^n)$ and $\Gamma_{bc,\epsilon}^a, \epsilon \in (0,\delta], \delta \le 1$ is the regularized Levi-Civita connection

coefficients in terms of the regularized metric $g_{ab,\epsilon}, \epsilon \in (0,1]$ such that $(g_{ab,\epsilon})_\epsilon, (g_\epsilon^{ab})_\epsilon \in \mathcal{G}(\mathbb{R}^4), (\det(g_{ab,\epsilon}))_\epsilon \neq 0_{\mathbb{R}}$. 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 $D'(\mathbb{R}^n)$, [18],[19].

**Remark 1.1.2.** This is the case even for the well-known Schwarzschild spacetime, which

is given in the Schwarzschild coordinates $(\hat{x}^0, \hat{r}, \theta, \phi)$, by the metric

$$ ds^2 = -(1 - \frac{a}{\hat{r}})(d\hat{x}^0)^2 + (1 - \frac{a}{\hat{r}})^{-1}(d\hat{r})^2 + \hat{r}^2[(d\theta)^2 + \sin^2\theta(d\phi)^2]. \quad (1.1.2) $$

Here, $a$ is the Schwarzschild radius $a = 2GM/c^2$ with $G, M$ and $c$ being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. Obviously the fundamental tensor corresponding to $ds^2$ has the components which is degenerate or singular: (i) at $\hat{r} = 0$ and (ii) at $\hat{r} = a$.

**Remark 1.1.3.** Note that in classical papers [5]-[31],etc. (i) the Colombeau distributional metric tensor $(g_{ab,\epsilon})_\epsilon \in \mathcal{G}_\delta(\mathbb{R}^4)$ related to $(R_{bcd,\epsilon}^a)_\epsilon \in \mathcal{G}_\delta(\mathbb{R}^4)$ 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 $(R_{bcd,\epsilon}^a)_\epsilon \in \mathcal{G}_\delta(\mathbb{R}^4)$ classical Schwartz distributions in $D'(\mathbb{R}^n)$, (iii) there is no any important physical applications of the classical linear distributional geometry were obtained.

**Remark 1.1.4.** Originally fundamental physical applications of the Colombeau nonlinear distributional geometry has been obtained in author paper [33]-[37].

By using now the Cartesian coordinates $(\hat{x}^0, \hat{x}^1, \hat{x}^2, \hat{x}^3)$, which are related to $(\hat{x}^0, \hat{r}, \theta, \phi)$ through the canonical relations: $\hat{x}^1 = \hat{r}\cos\phi\sin\theta$, $\hat{x}^2 = \hat{r}\sin\phi\sin\theta$, $\hat{x}^3 = \hat{r}\cos\theta$, the metric (1.1.2) reads $ds^2 = \hat{g}_{\mu\nu}d\hat{x}^\mu d\hat{x}^\nu$, where at points $\hat{r} \neq 0, \hat{r} \neq a$ the metric $\hat{g}_{\mu\nu}$ is given by
[29]:

$$
\begin{align}
\hat{g}_{00} &= -(1-h), \quad \hat{g}_{0\alpha} = 0, \tag{1.1.3} \\
\hat{g}_{\alpha\beta} &= \delta^{\alpha\beta} + h(1-h)^{-1} \frac{\hat{x}^a \hat{x}^\beta}{\hat{r}^2}, \quad \alpha, \beta = 1,2,3 \nonumber
\end{align}
$$

with $h \triangleq a/\hat{r}$. Well known that at points $\hat{r} \neq 0, \hat{r} \neq a$:

$$
\begin{align}
\kappa \hat{T}_0^0 &= -\frac{h'}{\hat{r}} - \frac{h}{\hat{r}^2}, \\
\kappa \hat{T}_0^\alpha &= 0, \quad \kappa \hat{T}_\alpha^0 = 0, \tag{1.1.4} \\
\kappa \hat{T}_\alpha^\beta &= \delta_\alpha^\beta \left(-\frac{h''}{2} - \frac{h'}{\hat{r}}\right) + \frac{\hat{x}^\alpha \hat{x}^\beta}{\hat{r}^2} \left(\frac{h''}{2} - \frac{h}{\hat{r}^2}\right),
\end{align}
$$

where the hatted symbols $\hat{T}_{\mu}^{\nu}$ represent the quantity $\tilde{T}_{\mu}^{\nu}$ in the coordinate system $\{\hat{x}^{\mu}; \mu = 0, 1, 2, 3\}$. Also, we have defined $h' \triangleq dh/d\hat{r}$ and $h'' \triangleq d^2h/d\hat{r}^2$.

**Remark 1.1.5.** We extend now the quantity (1.1.3)-(1.1.4) in point $\hat{r} = 0$ as Colombeau generalized functions from Colombeau algebra $G_{\delta}(\mathbb{R}^3)$. Regularizing now the function

$h = a/\hat{r}$ as $(h_{\varepsilon})_{\varepsilon} = a/(\sqrt{\hat{r}^2 + \varepsilon^2})_{\varepsilon}$ and the function $\frac{\hat{x}^a \hat{x}^b}{\hat{r}^2}$ as $\frac{\hat{x}^a \hat{x}^b}{(\hat{r}^2 + \varepsilon^2)_{\varepsilon}}$ with $\varepsilon \in (0,1]$,

we

replace now the singular metric (1.1.3) by the Colombeau generalized metric

$$
ds^2 = (\hat{g}_{\mu\nu,\varepsilon} d\hat{x}^{\mu} d\hat{x}^{\nu})_{\varepsilon}, \qquad (1.1.5)
$$

where

$$
\begin{align}
(\hat{g}_{00,\epsilon})_{\epsilon} &= -(1-h_{\epsilon}), \quad (\hat{g}_{0a,\epsilon})_{\epsilon} = 0_{\mathbb{R}}, \tag{1.1.6} \\
(\hat{g}_{\alpha\beta,\epsilon})_{\epsilon} &= \delta^{\alpha\beta} + ((h_{\epsilon}(1-h_{\epsilon})^{-1})_{\epsilon}) \left( \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2 + \epsilon^2} \right)_{\epsilon}, \quad \alpha, \beta = 1, 2, 3 \nonumber
\end{align}
$$

and therefore

$$
\begin{align}
\kappa \left( \hat{T}_{0,\epsilon}^{0}(\hat{x}) \right)_{\epsilon} &= - \left( \frac{a\epsilon^2}{(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon}, \quad \kappa \left( \hat{T}_{0}^{\alpha}(\hat{x}; \epsilon) \right)_{\epsilon} = 0_{\mathbb{R}}, \quad \kappa \left( \hat{T}_{\alpha}^{0}(\hat{x}; \epsilon) \right)_{\epsilon} = 0_{\mathbb{R}}, \nonumber \\
\kappa \left( \hat{T}_{\alpha}^{\beta}(\hat{x}; \epsilon) \right)_{\epsilon} &= \delta_{\alpha}^{\beta} \left( \frac{3a\epsilon^2}{2(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon} - \tag{1.1.7} \\
& \quad \left( \left( \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2 + \epsilon^2} \right)_{\epsilon} \right) \left( \frac{a\epsilon^2}{(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon} \left( \frac{5}{2} + \left( \frac{\epsilon^2}{\hat{r}^2 + \epsilon^2} \right)_{\epsilon} \right). \nonumber
\end{align}
$$

Note that from Eq. (1.1.7) one obtains

$$
\hat{T}_{\mu}^{\nu}(\hat{x}) \triangleq w\lim_{\epsilon \to 0} \hat{T}_{\mu,\epsilon}^{\nu}(\hat{x}) \sim -Mc^2 \delta_{\mu}^{0} \delta_{0}^{\nu} \delta^{(3)}(\hat{x}). \quad (1.1.8)
$$

**Remark 1.1.6.** Note that $(\hat{T}_0^0(\hat{x}; \varepsilon))_\varepsilon, (\hat{T}_\alpha^\beta(\hat{x}; \varepsilon))_\varepsilon \in \mathcal{G}(\mathbb{R}^4)$. Thus the generalized

Einstein

equation [37] related to Eq.(1.1.6)-Eq.(1.1.7) in Colombeau notations reads:

$$
(\hat{G}_{\mu,\epsilon})^{\nu}_{\epsilon} = (\hat{R}_{\mu,\epsilon})^{\nu}_{\epsilon} - \frac{1}{2} \delta_{\mu}^{\nu} (\hat{R}_{\epsilon})_{\epsilon} = \kappa (\tilde{T}_{\mu,\epsilon})^{\nu}_{\epsilon}, \quad (1.1.9)
$$

where
$$ (\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon} = (\hat{R}^{\mu}_{\mu,\varepsilon}(\hat{r}))_{\varepsilon} = -\left(\frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} + \left(\frac{2a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} \quad (1.1.10) $$

**Remark 1.1.7.** Note that the regularized scalar curvature $\hat{R}_{\varepsilon}$ has the well-defined weak limit $\hat{R}_{w}$ in $D'(\mathbb{R}^n)$

$$ \hat{R}_{w} \triangleq w\lim_{\epsilon \to 0} \hat{R}_{\epsilon} = -\frac{4}{3}\pi a \delta^{(3)}(\hat{x}). \quad (1.1.11) $$

**Remark 1.1.8.** Note that: (i) for any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}$ such that $\text{st}((\hat{r}_{\varepsilon})_{\varepsilon}) = \hat{r}_{\text{fin}} \neq 0$, (see Definition 1.2.5) where $\hat{r}_{\text{fin}} \in \mathbb{R}$ from Eq.(1.1.10) it follows that

$$ \text{st}((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}) = -\text{st}\left(\left(\frac{a\varepsilon^2}{(\hat{r}_{\varepsilon}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon}\right) = -\text{st}\left(\left(\frac{a\varepsilon^2}{(\hat{r}_{\text{fin}}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon}\right) = 0, \quad (1.1.12) $$

(ii) for any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}$ such that $(\hat{r}_{\varepsilon}) \le \tilde{\mathbb{R}}(\varepsilon)$ (see Definition 1.2.4) from Eq.(1.1.10) it follows that

$$ ((\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon}) \approx \tilde{\mathbb{R}} - \infty, \quad (1.1.13) $$

(iii) at origin $(\hat{r}_{\varepsilon}^O)_{\varepsilon} = \tilde{\mathbb{R}} 0_{\tilde{\mathbb{R}}}$ (see Definition 1.2.4) one obtains

$$ ((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^O))_{\varepsilon}) = \tilde{\mathbb{R}} - \left(\frac{a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right)_{\varepsilon} = \tilde{\mathbb{R}} - \frac{a}{(\varepsilon^3)_{\varepsilon}}, \quad (1.1.14) $$

where $\varepsilon \in (0, \delta]$.

**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 $\delta$-function. For the regularized quadratic scalars one obtains [29]:

$$ \hat{R}_{\varepsilon}^{\mu\nu}(\hat{r})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}) = \frac{1}{2}\left[\frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right]^2 + 2\left[\frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right]^2 = \frac{13}{2}\left[\frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right]^2 \quad (1.1.15) $$

$$ \hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}) = \frac{4a^2}{\hat{r}^2 + \varepsilon^2}\left[\frac{3}{(\hat{r}^2 + \varepsilon^2)^2}\right] - \frac{12a^2\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^4} + \frac{9a^2\varepsilon^4}{(\hat{r}^2 + \varepsilon^2)^5}. $$

**Remark 1.1.10.** Note that in contrast with the regularized scalar curvature $\hat{R}_{\varepsilon}$ the regularized quadratic scalars do not have the weak limits, which can be symbolically written as

$$ \begin{aligned} \hat{R}^{\mu\nu}(\hat{x})\hat{R}_{\mu\nu}(\hat{x}) &\triangleq \lim_{\epsilon \to 0} \hat{R}^{\mu\nu}(\hat{x}; \epsilon)\hat{R}_{\mu\nu}(\hat{x}; \epsilon) \sim 40\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2, \\ \hat{R}^{\rho\sigma\mu\nu}(\hat{x})\hat{R}_{\rho\sigma\mu\nu}(\hat{x}) &\triangleq \lim_{\epsilon \to 0} \hat{R}^{\rho\sigma\mu\nu}(\hat{x}; \epsilon)\hat{R}_{\rho\sigma\mu\nu}(\hat{x}; \epsilon) \\ &\sim \frac{12a^2}{\hat{r}^6} + \frac{16\pi a^2}{3}\frac{1}{\hat{r}^3}\delta^{(3)}(\hat{x}) + 16\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2. \end{aligned} \quad (1.1.16) $$

**Remark 1.1.11.** However Colombeau quadratic scalars $(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ and
($\hat{R}_\epsilon^{\rho\sigma\mu\nu}(\hat{r}_\epsilon)\hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}_\epsilon)$) is well defined as Colombeau generalized functions in $G_\delta(\tilde{\mathbb{R}}^3)$.

$$ (\hat{R}_\epsilon^{\mu\nu}(\hat{r}_\epsilon)\hat{R}_{\mu\nu,\epsilon}(\hat{r}_\epsilon))_\epsilon = \frac{1}{2}\left[\left(\frac{3a\epsilon^2}{(\hat{r}_\epsilon^2 + \epsilon^2)^{5/2}}\right)_\epsilon\right]^2 + 2\left[\left(\frac{a\epsilon^2}{(\hat{r}_\epsilon^2 + \epsilon^2)^{5/2}}\right)_\epsilon\right]^2, \\ (\hat{R}_\epsilon^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}))_\epsilon = \frac{12a^2}{((\hat{r}_\epsilon^2 + \epsilon^2)^3)_\epsilon} - \left(\frac{12a^2\epsilon^2}{(\hat{r}_\epsilon^2 + \epsilon^2)^4}\right)_\epsilon + \qquad (1.1.17) \\ +\left(\frac{9a^2\epsilon^4}{(\hat{r}_\epsilon^2 + \epsilon^2)^5}\right)_\epsilon. $$

**Remark 1.1.12.** Note that Colombeau quadratic scalars $(\hat{R}_\epsilon^{\mu\nu}(\hat{r})\hat{R}_{\mu\nu,\epsilon}(\hat{r}))_\epsilon$ and $(\hat{R}_\epsilon^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}))_\epsilon$ can be triating only nonclassically as Colombeau generalized functions extended on $\tilde{\mathbb{R}}^3 = \tilde{\mathbb{R}} \times \tilde{\mathbb{R}} \times \tilde{\mathbb{R}}$, since at origin $(\hat{r}_\epsilon^O)_\epsilon = 0_{\tilde{\mathbb{R}}}$ we get

$$ (\hat{R}_\epsilon^{\mu\nu}(\hat{r}_\epsilon^O)\hat{R}_{\mu\nu,\epsilon}(\hat{r}_\epsilon^O))_\epsilon = \frac{1}{2}\left[\left(\frac{3a\epsilon^2}{(\epsilon^2)^{5/2}}\right)_\epsilon\right]^2 + 2\left[\left(\frac{a\epsilon^2}{(\epsilon^2)^{5/2}}\right)_\epsilon\right]^2, \\ (\hat{R}_\epsilon^{\rho\sigma\mu\nu}(\hat{r}_\epsilon^O)\hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}_\epsilon^O))_\epsilon = \frac{12a^2}{(\epsilon^6)_\epsilon} - \left(\frac{12a^2\epsilon^2}{(\epsilon^8)}\right)_\epsilon + \qquad (1.1.18) \\ +\left(\frac{9a^2\epsilon^4}{(\epsilon^{10})}\right)_\epsilon. $$

**Remark 1.1.13.** In the usual Schwarzschild coordinates $(t, r, \theta, \phi)$, $r \neq a$ the Schwarzschild metric (1.1.2) takes the form above horizon $r > a$ and below horizon $r < a$ correspondingly

$$ \left\{ \begin{array}{l} \text{above horizon } r > 2m: ds^{+2} = h^{+}(r)dt^2 - [h^{+}(r)]^{-1}dr^2 + r^2d\Omega^2, \\ h^{+}(r) = -1 + \frac{a}{r} = -\frac{r-a}{r} \qquad (1.1.19) \\ \text{below horizon } r < 2m: ds^{-2} = h^{-}(r)dt^2 - h^{-}(r)^{-1}dr^2 + r^2d\Omega^2, \\ h^{-}(r) = -1 + \frac{a}{r} = \frac{a-r}{r} \end{array} \right. $$

Following the above discussion we consider the metric coefficients $h^+(r)$, $[h^+(r)]^{-1}$, $h^-(r)$, and $[h^-(r)]^{-1}$ as an element of $\mathcal{D}'(\mathbb{R}^3)$ and embed it into $G_\delta(\mathbb{R}^3)$ by replacements above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly

$$ r \ge 2m: r - 2m \mapsto \sqrt{(r - 2m)^2 + \epsilon^2}; r < 2m: 2m - r \mapsto \sqrt{(2m - r)^2 + \epsilon^2}. \qquad (1.1.20) $$

Inserting (1.1.20) into (1.1.2) we obtain Colombeau generalized object modeling the singular Schwarzschild metric above (below) horizon, i.e.,

$$ (ds_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \qquad (1.1.21) \\ (ds_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2} $$

The generalized Ricci tensor above horizon $([\mathbb{R}_\epsilon^+]_\alpha)_\epsilon$ may now be calculated componentwise using the classical formulae
$$ 
\begin{aligned}
([\mathbf{R}_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} &= ([\mathbf{R}_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2}((h_{\varepsilon}^{++})_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}^{+\prime})_{\varepsilon}) \\
([\mathbf{R}_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} &= ([\mathbf{R}_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{(h_{\varepsilon}^{+\prime})_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+})_{\varepsilon}}{r^2},
\end{aligned}
\quad (1.1.22) $$

where

$$ 
\begin{aligned}
(h_{\varepsilon}^{+\prime}(r))_{\varepsilon} &= -\frac{r-2m}{r([([r-2m)^2 + \varepsilon^2]]^{1/2})_{\varepsilon}} + \frac{([([r-2m)^2 + \varepsilon^2]]^{1/2})_{\varepsilon}}{r^2}, \\
(h_{\varepsilon}^{\prime\prime}(r))_{\varepsilon} &= -\frac{1}{r([([r-2m)^2 + \varepsilon^2]]^{1/2})_{\varepsilon}} + \frac{(r-2m)^2}{r([([r-2m)^2 + \varepsilon^2]]^{3/2})_{\varepsilon}} + \\
&\quad + \frac{r-2m}{r^2([([r-2m)^2 + \varepsilon^2]]^{1/2})_{\varepsilon}} + \frac{r-2m}{r^2([([r-2m)^2 + \varepsilon^2]]^{1/2})_{\varepsilon}} - \\
&\quad - \frac{2([([r-2m)^2 + \varepsilon^2]]^{1/2})_{\varepsilon}}{r^3}.
\end{aligned}
\quad (1.1.23) $$

From Eq.(1.1.22)- Eq.(1.1.23) we obtain (see sect.3)

$$ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}(r)]_{1}^{1} = w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}(r)]_{0}^{0} = -2m\delta(r-2m). \quad (1.1.24) $$

**Remark 1.1.14.** Note that the $\epsilon$-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, geophysics, microlocal analysis and general relativity so far.

### Basic idea.

**Definition 1.2.1.** The algebra moderate functions $C_M^\infty(\mathbb{R}^n)$ on $\mathbb{R}^n$ is the algebra of families of smooth functions $(f_\epsilon(x))_\epsilon \triangleq (f_\epsilon(x))_{\epsilon, x \in \mathbb{R}^n, \epsilon \in (0, \delta], \delta \le 1}$ (smooth $\epsilon$-regularisations, where $\epsilon$ is the regularization parameter), such that: (i) for all compact subsets $K$ of $\mathbb{R}^n$ and all multiindices $\alpha$, there is an $N > 0$ such that

$$ \sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_{\epsilon}(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\epsilon^{-N}), \epsilon \to 0, \quad (1.2.1) $$

with addition and multiplication defined by natural way:

$$ (f_{\epsilon}(x))_{\epsilon} + (g_{\epsilon}(x))_{\epsilon} = (f_{\epsilon}(x) + g_{\epsilon}(x))_{\epsilon} \quad (1.2.2) $$

and

$$ (f_{\epsilon}(x))_{\epsilon} \times (g_{\epsilon}(x))_{\epsilon} = (f_{\epsilon}(x) \times g_{\epsilon}(x))_{\epsilon}. \quad (1.2.3) $$

**Definition 1.2.2.** The ideal $\mathcal{N}_\delta(\mathbb{R}^n)$ of negligible functions is defined in the same way but with the partial derivatives instead bounded by $O(\epsilon^N)$ for all $N > 0$, i.e.

$$ \sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_{\epsilon}(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\epsilon^N), \epsilon \to 0. \quad (1.2.4) $$

**Definition 1.2.3.** The Colombeau Algebra $\mathcal{G}_\delta(\mathbb{R}^n)$ [1],[2] is defined as the quotient algebra

$$ \mathcal{G}_{\delta}(\mathbb{R}^n) = C_M^{\infty}(\mathbb{R}^n)/\mathcal{N}_{\delta}(\mathbb{R}^n). \quad (1.2.5) $$

Elements of $\mathcal{G}_\delta(\mathbb{R}^n)$ are denoted by:

$$ u = \mathbf{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}_\delta(\mathbb{R}^n). \quad (1.2.6) $$

### Embedding of distributions

The space of Schwartz distributions $D'(\mathbb{R}^n)$ can be embedded into the Colombeau algebra $\mathcal{G}_\delta(\mathbb{R}^n)$ by (component-wise) convolution with any element $(\varphi_\epsilon)_\epsilon$ of the algebra $\mathcal{G}_\delta(\mathbb{R}^n)$ having as representative a $\delta$-net, i.e. a family of smooth functions $\varphi_\epsilon$ such that $\varphi_\epsilon \to \delta$ in $D'(\mathbb{R}^n)$ as $\epsilon \to 0$.

**Remark 1.2.2.** Note that the embedding $\iota: D'(\mathbb{R}^n) \hookrightarrow \mathcal{G}_\delta(\mathbb{R}^n)$ is non-canonical, because it

depends on the choice of the $\delta$-net.

**Example 1.2.1.** Delta function $\delta(x) \in D'(\mathbb{R})$ for example has the following different representatives in Colombeau algebra $\mathcal{G}_\delta(\mathbb{R})$:

$$ \begin{aligned} \frac{1}{2} \left( \frac{1}{\sqrt{\pi \epsilon}} \exp \left( -\frac{x^2}{4\epsilon} \right) \right)_\epsilon & \in \mathcal{G}_\delta(\mathbb{R}), \quad \frac{1}{\pi} \left( \frac{1}{x} \sin \left( \frac{x}{\epsilon} \right) \right)_\epsilon \in \mathcal{G}_\delta(\mathbb{R}), \\ \frac{1}{\pi} \left( \frac{\epsilon}{x^2 + \epsilon^2} \right)_\epsilon & \in \mathcal{G}_\delta(\mathbb{R}), \quad \frac{1}{\pi} \left( \frac{1}{x^2} \sin^2 \left( \frac{x}{\epsilon} \right) \right)_\epsilon \in \mathcal{G}_\delta(\mathbb{R}), \end{aligned} \quad (1.2.7) $$

since
$$ 
\begin{aligned}
& \frac{1}{2} \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{x^2}{4\epsilon}\right) \rightarrow \delta(x), \quad \frac{1}{\pi} \frac{1}{x} \sin\left(\frac{x}{\epsilon}\right) \rightarrow \delta(x), \\
& \frac{1}{\pi} \frac{\epsilon}{x^2 + \epsilon^2} \rightarrow \delta(x), \quad \frac{1}{\pi} \frac{1}{x^2} \sin^2\left(\frac{x}{\epsilon}\right) \rightarrow \delta(x)
\end{aligned}
\qquad (1.2.8) $$

in $D'$ as $\epsilon \to 0$.

**Remark 1.2.2.** However note that embedding $D'(\mathbb{R}^n) \hookrightarrow \mathcal{G}(\mathbb{R}^n)$ does not mean 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 value at any point $x \in \mathbb{R}^n$ these point values of the Colombeau generalized functions is the Colombeau generalized numbers.

**Example 1.2.2.** Delta function $\delta(x)$ ill defined at point $x = 0$ since $\delta(0) = \infty$. However any Colombeau generalized function defined by Eq.(1.2.7) has well defined point value at point $x = 0$. For example

$$ 
\begin{aligned}
\frac{1}{2} \left( \left. \frac{1}{\sqrt{\pi \epsilon}} \exp\left(-\frac{x^2}{4\epsilon}\right) \right|_{\epsilon=0} \right) &= \frac{1}{2\sqrt{\pi \epsilon}} \left( \left. \frac{1}{\sqrt{\epsilon}} \right|_{\epsilon=0} \right) \in \tilde{\mathbb{R}}_{\delta}, \\
\frac{1}{\pi} \left( \left. \frac{\epsilon}{x^2 + \epsilon^2} \right|_{\epsilon=0} \right) &= \frac{1}{\pi} \left( \left. \frac{1}{\epsilon} \right|_{\epsilon=0} \right) \in \tilde{\mathbb{R}}_{\delta}.
\end{aligned}
\qquad (1.2.9) $$

Here $\tilde{\mathbb{R}}$ is the ring of real Colombeau generalized numbers [34].

### 1.2.2. The ring of Colombeau generalized numbers $\tilde{\mathbb{R}}_{\delta}$. Point values of Colombeau generalized functions.

**Designation 1.2.1.** (I) We denote by $\tilde{\mathbb{R}}_{\delta}$, $\delta \le 1$ the ring of real Colombeau generalized numbers. Recall that by definition $\tilde{\mathbb{R}}_{\delta} = \mathbb{E}_{\mathbb{R},\delta}(\mathbb{R})/\mathbb{N}_{\delta}(\mathbb{R})$ where [34],[36],[37]:

$$ 
\begin{aligned}
\mathbb{E}_{\mathbb{R},\delta}(\mathbb{R}) &= \{(x_{\epsilon})_{\epsilon} \in \mathbb{R}^{(0,\delta)} | (\exists a \in \mathbb{R}_+) (\exists \epsilon_0 \in (0,1)) (\forall \epsilon \le \epsilon_0) [|x_{\epsilon}| \le \epsilon^{-a}]\}, \\
\mathbb{N}_{\delta}(\mathbb{R}) &= \{(x_{\epsilon})_{\epsilon} \in \mathbb{R}^{(0,\delta)} | (\forall a \in \mathbb{R}_+) (\exists \epsilon_0 \in (0,1)) (\forall \epsilon \le \epsilon_0) [|x_{\epsilon}| \le \epsilon^a]\}.
\end{aligned}
\qquad (1.2.10) $$

(II) In this subsection we will be write for short $\tilde{\mathbb{R}}$ instead $\tilde{\mathbb{R}}_{\delta}$.

Notice that the ring $\tilde{\mathbb{R}}$ arises naturally as the ring of constants of the Colombeau algebras $\mathcal{G}_{\delta}(\Omega)$. Recall that there exists natural embedding $\tilde{r} : \mathbb{R} \hookrightarrow \tilde{\mathbb{R}}$ such that for all $r \in \mathbb{R}$, $\tilde{r} = (r_{\epsilon})_{\epsilon}$ where $r_{\epsilon} \equiv r$ for all $\epsilon \in (0,1]$. We say that $r$ is standard number and abbreviate $r \in \mathbb{R}$ for short. The ring $\tilde{\mathbb{R}}$ can be endowed with the structure of a partially ordered ring: for $r,s \in \tilde{\mathbb{R}}$ $\eta \in \mathbb{R}_+$, $\eta \le \delta$ we abbreviate $r \le_{\tilde{\mathbb{R},\eta}} s$ or simply $r \le_{\tilde{\mathbb{R}}} s$ if and only if there are representatives $(r_{\epsilon})_{\epsilon}$ and $(s_{\epsilon})_{\epsilon}$ with $r_{\epsilon} \le s_{\epsilon}$ for all $\epsilon \in (0,\eta]$. Colombeau generalized number $r \in \tilde{\mathbb{R}}$ with representative $(r_{\epsilon})_{\epsilon}$ we abbreviate $\mathbb{cl}[(r_{\epsilon})_{\epsilon}]$.

**Definition 1.2.4.** (i) Let $\tilde{\delta} = \mathbb{cl}[(\delta_{\epsilon})_{\epsilon}] \in \tilde{\mathbb{R}}$. We say that $\tilde{\delta}$ is infinite small Colombeau generalized number and abbreviate $\tilde{\delta} \approx_{\tilde{\mathbb{R}}} \emptyset$ if there exists representative $(\delta_{\epsilon})_{\epsilon}$ and some $q \in \mathbb{N}$ such that $|\delta_{\epsilon}| = O(\epsilon^q)$ as $\epsilon \to 0$. (ii) Let $\Delta \in \tilde{\mathbb{R}}$. We say that $\Delta$ is infinite large
Colombeau generalized number and abbreviate $\Delta = \tilde{\mathbb{R}} \approx \infty$ if $\Delta_{\tilde{\mathbb{R}}}^{-1} \approx \tilde{\mathbb{R}} \approx \tilde{0}$. (iii) Let $\mathbb{R}_{\pm\infty}$ be $\mathbb{R} \cup \{\pm\infty\}$ We say that $\Theta \in \tilde{\mathbb{R}}_{\pm\infty}$ is infinite Colombeau generalized number and abbreviate $\Theta = \tilde{\mathbb{R}} \pm \infty_{\tilde{\mathbb{R}}} \text{ if there exists representative } (\Theta_{\varepsilon})_{\varepsilon} \text{ where } |\Theta_{\varepsilon}| = \infty \text{ for all } \varepsilon \in (0,1]. \text{ Here we abbreviate } \mathcal{E}_M(\mathbb{R}_{\pm\infty}) = \mathcal{E}_M(\mathbb{R} \cup \{\pm\infty\}), \mathcal{N}(\mathbb{R}_{\pm\infty}) = \mathcal{N}(\mathbb{R} \cup \{\pm\infty\}) \text{ and } \tilde{\mathbb{R}}_{\pm\infty} = \mathcal{E}_M(\mathbb{R}_{\pm\infty})/\mathcal{N}(\mathbb{R}_{\pm\infty})$ **Definition 1.2.5.** (Standard Part Mapping). (i) The standard part mapping $\text{st} : \tilde{\mathbb{R}} \to \mathbb{R}$ is defined by the formula:

$$ \text{st}(x) = \sup\{r \in \mathbb{R} | r \le \tilde{\mathbb{R}} x\}. \qquad (1.2.11) $$

If $x \in \tilde{\mathbb{R}}$, then $\text{st}(x)$ is called the standard part of $x$.

(ii) The mapping $\text{st} : \tilde{\mathbb{R}} \to \mathbb{R} \cup \{\pm\infty\}$ is defined by (i) and by $\text{st}(x) = \pm\infty$ for $x \in \tilde{\mathbb{R}}$ and for $x \in \tilde{\mathbb{R}}_{\pm\infty}$, respectively.

**Definition 1.2.6.** [37]. Let $(f_{\varepsilon}(x))_{\varepsilon} \in \mathcal{G}(\mathbb{R})$ and $\tilde{x} \in \tilde{\mathbb{R}}$, then $\text{cl}[(f_{\varepsilon}(\tilde{x}))_{\varepsilon}] \in \tilde{\mathbb{R}}$. We will say that Colombeau generalized number $\text{cl}[(f_{\varepsilon}(\tilde{x}))_{\varepsilon}]$ is a point values of Colombeau generalized function $(f_{\varepsilon}(x))_{\varepsilon}$ at point $\tilde{x} \in \tilde{\mathbb{R}}$.

**Definition 1.2.7.** (Principal value mapping) The principal value mapping $\text{p.v.} : \tilde{\mathbb{R}} \to \mathbb{R}$ of Colombeau generalized function $(f_{\varepsilon}(x))_{\varepsilon}$ at point $\tilde{x} \in \tilde{\mathbb{R}}$ is defined by the formula:

$$ \text{p.v.} \{ \text{cl}[(f_{\varepsilon}(\tilde{x}))_{\varepsilon}] \} = \sup_{\varepsilon \in (0,1]} |f_{\varepsilon}(\tilde{x})|. \qquad (1.2.12) $$

We will be write for short $\text{p.v.} [(f_{\varepsilon}(\tilde{x}))_{\varepsilon}]$.

**Example 1.2.3.** The principal value of the curvature scalar $(\hat{R}_{\varepsilon}(r, a))_{\varepsilon}$ (1.1.10) at point $\tilde{r} \in \tilde{\mathbb{R}}$ reads

$$ \text{p.v.} \left[ \left( \hat{R}_{\varepsilon}(\tilde{r}, a) \right)_{\varepsilon} \right] = \sup_{\varepsilon \in (0,1]} \frac{a\varepsilon^2}{(\tilde{r}^2 + \varepsilon^2)^{5/2}}. \qquad (1.2.13) $$

<figure><img src="image_9.png" /><figcaption>Fig.1.Plot of the function</figcaption></figure>

$$ \begin{gathered} R(a, \tilde{r}, \varepsilon) = \frac{a\varepsilon^2}{(\tilde{r}^2 + \varepsilon^2)^{5/2}}, \\ a = 1, \tilde{r} = 10^{-3}, z = \varepsilon \in (0, 0.01]. \\ R(10^{-3}, 7 \times 10^{-4}) = 1.808 \times 10^8. \end{gathered} $$

$$ \text{p.v.} \left[ \left( \hat{R}_{\varepsilon}(\tilde{r}, a) \right)_{\varepsilon} \right] \approx 1.808 \times 10^8. $$
### 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 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 $(f_{\varepsilon}(x))_{\varepsilon} \in \mathcal{G}_{\delta}(\mathbb{R})$ and $\mathrm{cl}[(\tilde{x}_{\varepsilon}))_{\varepsilon}] \in \tilde{\mathbb{R}}_{\delta}$. Assume that $\mathrm{cl}[(f_{\varepsilon}(\tilde{x}_{\varepsilon}))_{\varepsilon}] \in \tilde{\mathbb{R}}_{\delta}$. We will say that Colombeau generalized number $\mathrm{cl}[(f_{\varepsilon}(\tilde{x}_{\varepsilon}))_{\varepsilon}]$ is a point value of Colombeau generalized function $(f_{\varepsilon}(x))_{\varepsilon}$ at point $(\tilde{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$.

**Example 1.3.1.** For any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$, $\varepsilon \in (0, \delta]$ the point values $(\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ of Colombeau generalized function $(\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon}$ (see Eq.(1.1.10) reads

$$ (\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon} = (\hat{R}^{\mu}_{\mu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}_{\varepsilon}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} = -a\frac{(\varepsilon^2)_{\varepsilon}}{[(\hat{r}_{\varepsilon}^2)_{\varepsilon} + (\varepsilon^2)_{\varepsilon}]^{5/2}} = -a\frac{\delta^2(\varepsilon_1^2)_{\varepsilon_1}}{[(\hat{r}_{\varepsilon}^2)_{\varepsilon} + \delta^2(\varepsilon_1^2)_{\varepsilon_1}]^{5/2}}, \quad (1.3.1) $$

where $\varepsilon_1 \in (0, 1]$.

**Remark 1.3.2.** We choose now $(\hat{r}_{\varepsilon})_{\varepsilon} = (\hat{r}_{\varepsilon}^{*})_{\varepsilon} = \eta(\varepsilon)_{\varepsilon} = \eta\delta(\varepsilon_1)_{\varepsilon_1}$ and from Eq.(1.3.1) we get

$$ \begin{aligned} (\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^{*}))_{\varepsilon} = -a \frac{\delta^2(\varepsilon_1^2)_{\varepsilon_1}}{[(\hat{r}_{\varepsilon}^{*2})_{\varepsilon} + \delta^2(\varepsilon_1^2)_{\varepsilon_1}]^{5/2}} = -a \frac{\delta^2(\varepsilon_1^2)_{\varepsilon_1}}{[\eta\delta(\varepsilon_1^2)_{\varepsilon_1} + \delta^2(\varepsilon_1^2)_{\varepsilon_1}]^{5/2}} = & \\ -a \frac{\delta^2(\varepsilon_1^2)_{\varepsilon_1}}{[\eta^2\delta^2(\varepsilon_1^2)_{\varepsilon_1} + \delta^2(\varepsilon_1^2)_{\varepsilon_1}]^{5/2}} = -a \frac{\delta^2(\varepsilon_1^2)_{\varepsilon_1}}{\delta^5(\varepsilon_1^5)_{\varepsilon_1}[\eta^2 + 1]^{5/2}} = & \\ \frac{-a}{\delta^3(\varepsilon_1^3)_{\varepsilon_1}[\eta^2 + 1]^{5/2}} & \end{aligned} \quad (1.3.2) $$

Thus in "point limit" $\delta \to 0$ the curvature scalar $(\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^{*}))_{\varepsilon}$ diverges as $\delta^{-3}(\varepsilon_1^{-3})_{\varepsilon_1}$.

**Remark 1.3.2.** In order to prevent the divergence mentioned above, we assume now that there exist fundamental generalized length $l^* = \mathrm{cl}[(l_{\varepsilon_1}^*)_{\varepsilon_1}] = \eta \mathrm{cl}[(\varepsilon_1)_{\varepsilon_1}]$, $\varepsilon_1 \in (0, 1]$,
$\eta \in \mathbb{R}$, $\delta \ll \eta \ll 1$, such that: $(\hat{r}_{\varepsilon_1})_{\varepsilon_1} \ge l^*$, see [36] sec.2. Thus from Eq.(1.3.1) we get now instead Eq.(1.3.2)

$$ \begin{gathered} \left| (\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon} \right| \le \left( \frac{a\varepsilon^2}{(l_{\varepsilon_1}^{*2} + \varepsilon^2)^{5/2}} \right)_{\varepsilon} = a \frac{a(\varepsilon^2)_{\varepsilon}}{\left[ (l_{\varepsilon_1}^{*2})_{\varepsilon} + (\varepsilon^2)_{\varepsilon} \right]^{5/2}} = \\ a \frac{\delta^2(\varepsilon_1^2)_{\varepsilon_1}}{\left[ \eta^2(\varepsilon_1^2)_{\varepsilon_1} + \delta^2(\varepsilon_1^2)_{\varepsilon_1} \right]^{1/2}} \frac{1}{\left[ \eta^2(\varepsilon_1^2)_{\varepsilon_1} + \delta^2(\varepsilon_1^2)_{\varepsilon_1} \right]^{1/2}} \frac{1}{\left[ (\hat{r}_{\varepsilon_1})_{\varepsilon_1} + (\varepsilon^2)_{\varepsilon} \right]^{3/2}} = \tag{1.3.3} \\ \frac{a\delta^2}{\left[ \eta^2 + \delta^2 \right]} \frac{1}{\left[ (\hat{r}_{\varepsilon_1}^2)_{\varepsilon_1} + (\varepsilon^2)_{\varepsilon} \right]^{3/2}} = \frac{a\delta^2}{\left[ \eta^2 + \delta^2 \right]} \frac{1}{\left[ (\hat{r}_{\varepsilon}^2)_{\varepsilon} \right]^{3/2}}. \end{gathered} $$

## 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 $\Sigma$ furnished by canonical point-like geometry, where a topology of space-time M of the form $M \cong \mathbb{R} \times \Sigma$ 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 $A_a^i(x)$ is an SU(2) connection and $E_i^a(x)$ is the densitised triad vector field, conjugate to A :

$$ \{A_a^i(x), E_j^b(x')\} = 8\pi\gamma\delta^i_j\delta^b_a\delta(x-x'), \qquad (1.4.1) $$

with G the gravitational constant and $\gamma$ the Immirzi parameter. The conjugate pair are constraint to satisfy the system

$$ \begin{gathered} G_i = D_a E_i^a = 0, H_b = E_i^a F_{ab}^i = 0, \tag{1.4.2} \\ H = \epsilon_{ijk} F_{ab}^i E_j^a E_k^b - 2(1 + \gamma^2) K_{[a}^i K_{b]}^j E_i^a E_j^b = 0, \end{gathered} $$

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 $A_i^a$ along a curve $\gamma$, defined by $h_{\gamma}[A] = P[\exp(\int_{\gamma} A)]$, and fluxes of $E_i^a$ through the surface $S$, defined by $E_i(S) = \int_S d^2\sigma n_a E_i^a$. 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 $(\Gamma, j_l, i_n)$ consisting of a graph $\Gamma$ with nodes in $\Sigma$, labeled by intertwiners $i_n$, and links connecting different nodes, labeled by SU(2) representations $j_l$.

**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.

Canonical quantization of nonlinear distributional GRT can be expressed as an $\tilde{SU}(2)$ gauge theory on the 3 dimensional Colombeau distributional manifold $\tilde{\Sigma}$ furnished by Colombeau point-free geometry, where a topology of space-time $\tilde{M}$ of the form $\tilde{M} \cong \tilde{\mathbb{R}}_\delta \times \tilde{\Sigma}$ is assumed, in a background independent manner. In such formulation of GRT, the gravitational field is described by a pair of conjugate variables $((A_\epsilon)_\epsilon, (E_\epsilon)_\epsilon)$, where $(A_{a,\epsilon}^i(x_\epsilon))_\epsilon$ is an $\tilde{SU}(2)$ Colombeau distributional connection and $(E_{i,\epsilon}^a(x_\epsilon))_\epsilon$ is the distributional densitised triad vector field, conjugate to $(A_\epsilon)_\epsilon$:

$$ \{ (A_{a,\epsilon}^i(x_\epsilon))_\epsilon, (E_j^b(x'_\epsilon))_\epsilon \} = 8\pi\gamma\delta_j^i\delta_a^b(\delta(x_\epsilon - x'_\epsilon))_\epsilon, \quad (1.4.3) $$

with $G$ the gravitational constant and $\gamma$ the Immirzi parameter. The conjugate pair are constraint to satisfy the system

$$ \begin{gathered} (G_{i,\epsilon})_{\epsilon} = \tilde{\mathbb{R}} (D_{a,\epsilon} E_{i,\epsilon}^{a})_{\epsilon} = 0, (H_{b,\epsilon})_{\epsilon} = (E_{i,\epsilon}^{a} F_{ab,\epsilon})_{\epsilon} = \tilde{\mathbb{R}} 0_{\tilde{\mathbb{R}}}, \\ (H_{\epsilon})_{\epsilon} = \tilde{\mathbb{R}} \epsilon_{ijk} (F_{ab,\epsilon}^{i} E_{j,\epsilon}^{a} E_{k,\epsilon}^{b})_{\epsilon} - 2(1 + \gamma^2) (K_{[a,\epsilon}^{i} K_{b,\epsilon]}^{j} E_{i,\epsilon}^{a} E_{j,\epsilon}^{b})_{\epsilon} = \tilde{\mathbb{R}} 0_{\tilde{\mathbb{R}}}, \end{gathered} \quad (1.4.4) $$

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 $((A_\epsilon)_\epsilon, (E_\epsilon)_\epsilon)$ satisfying the constraint system (1.4.3). On quantization, one smears the basic Colombeau generalized fields $((A_\epsilon)_\epsilon, (E_\epsilon)_\epsilon)$ to holonomies of $(A_{i,\epsilon}^a)_\epsilon$ along a curve $\gamma = (\gamma_\epsilon)_\epsilon$, defined by

$$ \begin{gathered} (h_\gamma[A_\epsilon])_\epsilon = P\left[\exp\left(\int_\gamma A_\epsilon\right)_\epsilon\right], \quad \text{and fluxes of } (E_{i,\epsilon}^a)_\epsilon \text{ through the surface } \tilde{S}, \text{ defined by} \\ (E_{i,\epsilon}(\tilde{S}))_\epsilon = \left(\int_{\tilde{S}} d^2\sigma n_a E_{i,\epsilon}^a\right)_\epsilon. \end{gathered} $$

A spin network is a triplet $(\tilde{\Gamma}, j_l, i_n)$ consisting of a graph $\tilde{\Gamma}$ with nodes in $\tilde{\Sigma}$, labeled by intertwiners $i_n$, and links connecting different nodes, labeled by $\tilde{SU}(2)$ representations $j_l$.

### 1.4.1. Classical point-free phase space

**Definition 1.4.1.** (1) The general linear group over Colombeau algebras $\tilde{\mathbb{R}}, \tilde{\mathbb{C}}$ (the set of real, complex Colombeau numbers) is the group of $n \times n$ invertible matrices of real (complex) Colombeau numbers, and is denoted by $GL_n(\tilde{\mathbb{R}}), GL_n(\tilde{\mathbb{C}})$ or $GL(n, \tilde{\mathbb{R}}), GL(n, \tilde{\mathbb{C}})$.

(2) The unitary group of degree $n$ over Colombeau algebra $\tilde{\mathbb{C}}$, denoted $\tilde{U}(n)$, or $U(n, \tilde{\mathbb{C}})$ is the group of $n \times n$ unitary matrices over $\tilde{\mathbb{C}}$.

(3) The unitary group is a subgroup of the general linear group $GL(n, \tilde{\mathbb{C}})$.

(4) In the simple case $n = 1$, the group $U(n, \tilde{\mathbb{C}})$ corresponds to the circle group $\tilde{T}$, consisting of all Colombeau complex numbers with absolute value 1 under
multiplication,i.e. $\tilde{T} = \{z \in \tilde{\mathbb{C}} | |z| = 1\}$.

(5) The special unitary group of degree $n$, denoted $\tilde{SU}(n)$, is the Lie group of $n \times n$ unitary matrices over Colombeau algebra $\tilde{\mathbb{R}}$ with determinant 1.

The Colombeau distributional manifold $\tilde{\Sigma}$ over Colombeau algebra $\tilde{\mathbb{R}}$ having the symmetry group $\tilde{S}$ with an isotropy subgroup $\tilde{F}$, can be decomposed as $\tilde{\Sigma} \cong \tilde{\Sigma}/\tilde{S} \times \tilde{S}/\tilde{F}$.
The connection can generally be written as $(A_{\varepsilon})_{\varepsilon} = (A_{\tilde{\Sigma}/\tilde{S},\varepsilon})_{\varepsilon} + (A_{\tilde{S}/\tilde{F},\varepsilon})_{\varepsilon}$. Then $(A_{\tilde{\Sigma}/\tilde{S},\varepsilon})_{\varepsilon}$ can be considered as the connection of the reduced theory and its holonomies along curves in $\tilde{\Sigma}/\tilde{S}$ can be quantized. For the spherically symmetric case, $\tilde{\Sigma} \cong \tilde{\mathbb{R}}_{\delta} \times \tilde{S}^2$, and the symmetry group is $\tilde{S} = \tilde{SU}(2)$. This implies identifying $\tilde{\Sigma}/\tilde{S}$ with $\tilde{\mathbb{R}}_{\delta}$ and the gauge group of the reduced theory $F$ with $U(1)$. Therefore, reduced connections are $\tilde{U}(1)$ gauge fields on $\tilde{\mathbb{R}}$. Roughly speaking, spherical symmetry implies that our basic fields, in the spherical coordinate $((x_{\varepsilon}), \theta, \phi)$, are independent of angular variables. Thus, the Colombeau generalized connection $(A_{\varepsilon}(\tilde{x}_{\varepsilon}))_{\varepsilon}$ is just a function of the radial coordinate; $(A_{\varepsilon})_{\varepsilon} = (A_{\varepsilon}(x_{\varepsilon}))_{\varepsilon}$. These connections and triads of the reduced spherically symmetric phase space have the general form:

$$ (A_{\varepsilon})_{\varepsilon} = [(A_{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon}\tau_3[(dx_{\varepsilon})_{\varepsilon}] + (([A_1(x_{\varepsilon}))_{\varepsilon}\tau_1 + [(A_2(x_{\varepsilon}))_{\varepsilon}\tau_2][d\theta_{\varepsilon})_{\varepsilon}] + \\ (((A_{1,\varepsilon}(x_{\varepsilon})\tau_2)_{\varepsilon} - (A_{2,\varepsilon}(x_{\varepsilon})\tau_1 \sin\theta_{\varepsilon})_{\varepsilon}) + \tau_3(\cos\theta_{\varepsilon})_{\varepsilon})[(d\phi_{\varepsilon})_{\varepsilon}] \quad (1.4.5) $$

and

$$ (E_{\varepsilon})_{\varepsilon} = (E_{\varepsilon}^{x_{\varepsilon}}(x_{\varepsilon})\tau_3 \sin\theta_{\varepsilon}\partial_{x_{\varepsilon}})_{\varepsilon} + ((E_{\varepsilon}^{1}(x_{\varepsilon})\tau_1)_{\varepsilon} + \\ (E_{\varepsilon}^{2}(x_{\varepsilon})\tau_2)\sin\theta_{\varepsilon}\partial_{\theta_{\varepsilon}})_{\varepsilon} + ((E_{\varepsilon}^{1}(x_{\varepsilon})\tau_2)_{\varepsilon} - (E_{\varepsilon}^{2}(x_{\varepsilon})\tau_1)\partial_{\phi_{\varepsilon}})_{\varepsilon} \quad (1.4.6) $$

correspondingly, where $\tau_i = -\frac{i}{2}\sigma_i$ are the generators of $\tilde{su}(2)$ algebra. They define the Colombeau generalized symplectic structure:

$$ (\Omega_{\varepsilon})_{\varepsilon} = \frac{1}{2\gamma G} \left( \int dx_{\varepsilon} (dA_{x_{\varepsilon},\varepsilon} \wedge dE_{\varepsilon}^{x_{\varepsilon}} + 2dA_{1,\varepsilon} \wedge dE_{\varepsilon}^{1} + 2dA_{2,\varepsilon} \wedge dE_{\varepsilon}^{2}) \right)_{\varepsilon} \quad (1.4.7) $$

However, a suitable canonical transformation can be made resulting in Colombeau generalized canonical variables $((A_{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon}, (E_{\varepsilon}^{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon}), (\gamma(K_{\phi_{\varepsilon},\varepsilon}(x_{\varepsilon}))_{\varepsilon}, (E_{\varepsilon}^{\phi_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon})$ and $((\eta_{\varepsilon}(x_{\varepsilon}))_{\varepsilon}, (P_{\varepsilon}^{\eta_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon})$:

$$ (\Omega_{\varepsilon})_{\varepsilon} = \frac{1}{2\gamma G} \left( \int dx_{\varepsilon} (dA_{x_{\varepsilon}} \wedge dE_{\varepsilon}^{x_{\varepsilon}} + d(\gamma K_{\phi_{\varepsilon},\varepsilon}) \wedge dE_{\varepsilon}^{\phi_{\varepsilon}} + 2d\eta_{\varepsilon} \wedge dP_{\varepsilon}^{\eta_{\varepsilon}}) \right)_{\varepsilon} \quad (1.4.8) $$

with $(K_{\phi_{\varepsilon},\varepsilon})_{\varepsilon}$ being the $(\phi_{\varepsilon})_{\varepsilon}$ component of the extrinsic Colombeau generalized curvature. The Gauss constraint, generating $\tilde{U}(1)$ gauge transformations, takes the form:

$$ (G_{\varepsilon}[\lambda_{\varepsilon}])_{\varepsilon} = \left( \int dx_{\varepsilon} \lambda_{\varepsilon} (E_{\varepsilon}^{x_{\varepsilon}} + P_{\varepsilon}^{\eta_{\varepsilon}}) \right)_{\varepsilon} = \tilde{\mathbb{R}} \quad 0_{\tilde{\mathbb{R}}}, \quad (1.4.9) $$

where prime denotes differentiation with respect to $x_{\varepsilon}$.

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
$$ (H_{\varepsilon}[N_{\varepsilon}])_{\varepsilon} = -\frac{1}{2G} \times \left( \int dx_{\varepsilon} N_{\varepsilon}(x_{\varepsilon}) \frac{1}{\sqrt{|E_{\varepsilon}^{x_{\varepsilon}}|}} \left( (1 - \Gamma_{\phi_{\varepsilon}, \varepsilon}^{2} + K_{\phi_{\varepsilon}, \varepsilon}^{2}) E_{\varepsilon}^{\phi_{\varepsilon}} + \frac{2}{\gamma} K_{\phi_{\varepsilon}, \varepsilon} E_{\varepsilon}^{x_{\varepsilon}} (A_{x_{\varepsilon}, \varepsilon} + \eta'_{\varepsilon}) + 2E_{\varepsilon}^{x_{\varepsilon}} \Gamma'_{\phi_{\varepsilon}, \varepsilon} \right) \right)_{\varepsilon} \quad (1.4.10) $$

### 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 $(A_{x,\varepsilon})_{\varepsilon}$ along curves $(\gamma_{\varepsilon})_{\varepsilon}$ in $\tilde{\mathbb{R}}$ are defined as $(h_{\varepsilon}^{(\gamma_{\varepsilon})})_{\varepsilon} \equiv \exp\left[\frac{i}{2}\left(\int_{\gamma_{\varepsilon}} A_{x_{\varepsilon}}(x_{\varepsilon})\right)_{\varepsilon}\right]$ which are elements in $\tilde{U}_{\delta}(1) = \tilde{\mathbb{R}}_{\delta}/\tilde{\mathbb{Z}}$. For $(A_{\phi_{\varepsilon},\varepsilon})_{\varepsilon}$ point holonomies $\exp[i\mu(A_{\phi_{\varepsilon},\varepsilon}(x_{\varepsilon}))_{\varepsilon}]$ are used which belongs to the space of continuous almost periodic functions on the Bohr compactification of real line $\tilde{\mathbb{R}}_{\delta}$, and point holonomies of $(\eta_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{S}}^{1}$, have the form $\exp[(i\eta_{\varepsilon}(x_{\varepsilon}))_{\varepsilon}]$ which are elements of $\tilde{U}(1)$.

The kinematical Hilbert space of the present reduced theory is the space spanned by spin network state $(T_{g,k,\mu,\varepsilon})_{\varepsilon}$:

$$ (T_{g,k,\mu,\varepsilon})_{\varepsilon} = \prod_{e \in g} \exp\left(\frac{i}{2} k_e \left(\int_e dx_{\varepsilon} A_{x_{\varepsilon}}(x_{\varepsilon})\right)_{\varepsilon}\right) \left(\prod_{v \in V(g)} \exp(i\mu_{v,\varepsilon}\gamma K_{\phi_{\varepsilon},\varepsilon}(v)) \exp(ik_v \eta_{\varepsilon}(v))\right)_{\varepsilon} \quad (1.4.11) $$

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 $\tilde{\mathbb{R}}_{Bohr}$ representations $(\mu_{v,\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ and irreducible $\tilde{\mathbb{S}}^{1}$ representation $k_v \in \tilde{\mathbb{Z}}$, while edges are labeled by irreducible representations of $\tilde{U}_{\delta}(1)$.

Holonomies act on spin network states by multiplication. Their corresponding momenta, on the other hand, act by differentiation

$$ (\hat{E}_{\varepsilon}^{x_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon})_{\varepsilon} = \gamma \frac{\ell_p^2}{2} \left( (k_{e^{+}(x_{\varepsilon})} + k_{e^{-}(x_{\varepsilon})}) T_{g,k,\mu,\varepsilon} \right)_{\varepsilon}, \quad (1.4.12) $$

$$ \int dx_{\varepsilon} \hat{E}_{\varepsilon}^{\phi_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu} = \gamma \ell_p^2 \sum_{v} \mu_{v,\varepsilon} T_{g,k,\mu,\varepsilon}, \quad (1.4.13) $$

$$ \int dx_{\varepsilon} \hat{P}_{\varepsilon}^{\eta_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon} = 2\gamma \ell_p^2 \sum_{v} k_{v,\varepsilon} T_{g,k,\mu,\varepsilon}. \quad (1.4.14) $$

The generalized volume operator can be expressed as

$$ (\hat{V}_{\varepsilon})_{\varepsilon} = 4\pi \int dx_{\varepsilon} |\hat{E}_{\varepsilon}^{\phi_{\varepsilon}}(x_{\varepsilon})| \sqrt{|\hat{E}_{\varepsilon}^{x_{\varepsilon}}(x_{\varepsilon})|} \quad (1.4.15) $$

which is diagonal in spin network representation

$$ (\hat{V}_{\varepsilon} T_{g,k,\mu,\varepsilon})_{\varepsilon} = (V_{k,m,\varepsilon} T_{g,k,\mu,\varepsilon})_{\varepsilon}, \quad (1.4.16) $$

where

$$ (V_{k,m,\varepsilon})_{\varepsilon} = 4\pi\gamma^{3/2}\ell_p^3 \left( \sum_v |\mu_v| \sqrt{\frac{1}{2}|k_{e^+(x_v)} + k_{e^-(x_v)}|} \right)_{\varepsilon}. \quad (1.4.17) $$

Implementing the Gauss constraint as an operator on spin networks to select the gauge invariant states, leads to a restriction on labels
$$ (\hat{G}_{\varepsilon}[\lambda_{\varepsilon}] T_{g,k,\mu,\varepsilon})_{\varepsilon} = \gamma l_p^2 (\sum_{\nu} \lambda_{\varepsilon}(\nu) (k_{e^+(x_{\varepsilon})} - k_{e^-(x_{\varepsilon})} + 2k_{\nu}) T_{g,k,\mu,\varepsilon})_{\varepsilon} \quad (1.4.18) $$

$$ (\hat{G}_{\varepsilon}[\lambda_{\varepsilon}] T_{g,k,\mu})_{\varepsilon} = 0_{\mathbb{R}} \Rightarrow k_{\nu} = -\frac{1}{2} (k_{e^+(x_{\varepsilon})} - k_{e^-(x_{\varepsilon})})_{\varepsilon}. \quad (1.4.19) $$

Imposing now this on (1.4.11) results in the gauge invariant states

$$ (T_{g,k,\mu,\varepsilon})_{\varepsilon} = \prod_{e \in g} \exp \left[ \frac{i}{2} k_e \left( \int_e dx_{\varepsilon} (A_{x_{\varepsilon}}(x_{\varepsilon}) + \eta'_{\varepsilon}) \right)_{\varepsilon} \right] \left( \prod_{v \in V(g)} \exp(i\mu_v \gamma K_{\phi_{\varepsilon},\varepsilon}(v)) \right)_{\varepsilon}. \quad (1.4.20) $$

## 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 mass m in coordinate system $(x, \theta, \phi)$ reads

$$ ds^2 = -\left(1 - \frac{2m}{x}\right) dt^2 + \left(1 - \frac{2m}{x}\right)^{-1} dx^2 + x^2 d\Omega^2. \quad (1.5.1) $$

Horizon $x_+$ appear where $g_{00} = 0$:

$$ x_+ - 2m = 0. \quad (1.5.2) $$

The event horizons partition space-time into 2 regions: I ($x > x_+$), and II ($0 < x < x +$). By inspecting the sign of $g_{00}$, one observes that in region II, $x$ and $t$ interchange their roles and becomes time-like and space-like respectively.

### Classical point-like phase space variables

In region II, the metric of space-time takes the form

$$ ds^2 = -\left(\frac{2m}{t} - 1\right)^{-1} dt^2 + \left(\frac{2m}{t} - 1\right) dx^2 + t^2 d\Omega^2. \quad (1.5.3) $$

According to definition of tetrad (frame) fields $g_{\mu\nu} = \eta_{IJ}e_{\mu}^{I}e_{\nu}^{J}$, 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 $g_{\mu\nu} = \eta_{IJ}e_{\mu}^{I}e_{\nu}^{J}$ 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

$$ e^0 = \pm \left(\frac{2m}{t} - 1\right)^{-1/2} dt; \quad e^1 = \pm t \sin\theta d\phi; \quad e^2 = \pm t d\theta; \quad e^3 = \pm \left(\frac{2m}{t} - 1\right)^{1/2} dx, \quad (1.5.4) $$

which gives the compatible spin connection components

$$ \begin{aligned} \omega^{30} &= -\omega^{03} = \left(-\frac{m}{t^2}\right)dx; \quad \omega^{20} = -\omega^{02} = \left(\frac{2m}{t} - 1\right)^{1/2}d\theta, \\ \omega^{10} &= -\omega^{01} = \left(\frac{2m}{t} - 1\right)^{1/2}\sin\theta d\phi; \quad \omega^{12} = -\omega^{21} = \cos\theta d\phi. \end{aligned} \quad (1.5.5) $$

The A field can be constructed using spin connections:
$$
\begin{align}
A^3 &= \pm \gamma \left(-\frac{m}{t^2}\right) dx, A^2 = \pm \gamma \left(\frac{2m}{t} - 1\right)^{1/2} d\theta, \tag{1.5.6} \\
A^1 &= \pm \gamma \left(\frac{2m}{t} - 1\right)^{1/2} \sin\theta d\phi, A^3 = \pm \cos\theta d\phi. \nonumber
\end{align}
$$

E field on Σ<sub>in</sub> we choose a gauge in which e<sub>μ</sub><sup>0</sup> = n<sub>μ</sub>, 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 × S<sup>2</sup>. The 3 triad fields become:

$$
e^1 = \pm t \sin\theta d\phi; e^2 = \pm t d\theta; e^3 = \pm \left(\frac{2m}{t} - 1\right)^{1/2} dx, \quad (1.5.7)
$$

with determinant

$$
\det(e) = t^2 \sin\theta \left(\frac{2m}{t} - 1\right)^{1/2}, \quad (1.5.8)
$$

and inverse triad

$$
e_1 = \pm \frac{1}{t \sin\theta} \partial_{\phi}; \quad e_2 = \pm \frac{1}{t} \partial_{\theta}; \quad e_3 = \pm \left(\frac{2m}{t} - 1\right)^{-1/2} \partial_x, \qquad (1.5.9)
$$

The *E* fields become

$$
E_1 = \pm t \left( \frac{2m}{t} - 1 \right)^{1/2} \partial_{\phi}, E_2 = \pm t \left( \frac{2m}{t} - 1 \right)^{1/2} \sin\theta \partial_{\theta}, E_3 = \pm t^2 \sin\theta \partial_x. \quad (1.5.10)
$$

The 3 triad fields (1.5.7) define their compatible spin connection, Γ<sup>ij</sup> ∧ e<sup>j</sup> + de<sup>i</sup> = 0 :

$$
\Gamma^{12} = -\Gamma^{21} = \cos\theta d\phi, \qquad (1.5.11)
$$

and

$$
\Gamma^3 = \frac{1}{2} (\epsilon^{312} \Gamma^{12} + \epsilon^{321} \Gamma^{21}) = \cos\theta d\phi. \quad (1.5.12)
$$

Extrinsic curvature is related to A via γK = A − Γ reads

$$
K_r^3 = \frac{1}{\gamma} A_r^3 = \pm \left(-\frac{m}{t^2}\right) dx, K_{\theta}^2 = \frac{1}{\gamma} A_{\theta}^2 = \pm \left(\frac{2m}{t} - 1\right)^{1/2}, \qquad (1.5.13)
$$

$$
K_{\phi}^{1} = \frac{1}{\gamma} A_{\phi}^{1} = \pm \left(\frac{2m}{t} - 1\right)^{1/2} \sin\theta. \qquad (1.5.14)
$$

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 {$A_a^i(x), E_j^b(x_j')$} =
$\delta_i^j \delta_a^b \delta(x - x')$.

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

$$
H_{\theta} = \gamma t \left( \frac{2m}{t} - 1 \right) \cos\theta \{ \mathbf{sgn}(A_{\phi}^{1}) + \mathbf{sgn}(A_{\theta}^{2} A_{\phi}^{3}) \}, \quad (1.5.15)
$$

$$
G_2 = t \left( \frac{2m}{t} - 1 \right)^{1/2} \cos\theta \{ \mathbf{sign}(E_2^\theta) + \mathbf{sign}(A_\phi^3 E^\phi) \}, \quad (1.5.16)
$$

and Hamiltonian constraint gives:

$$
C = t \left( \frac{2m}{t} - 1 \right) \sin^2 \theta \{ \mathbf{sign}(E_2^\theta) + \mathbf{sign}(E_1^\phi) \}. \quad (1.5.17)
$$

For the above constraints to be zero we must have

$$
\operatorname{sign}(E_2^\theta) = -\operatorname{sign}(E_1^\phi), \quad \operatorname{sign}(A_\phi^3) = +1, \quad \operatorname{sign}(A_\phi^1) = -\operatorname{sign}(A_\theta^2). \qquad (1.5.18)
$$

This leaves us with two alternatives corresponding to the residual gauge freedom
$$ (b, p_b) \rightarrow (-b, -p_b). $$

$$
\begin{align}
A_a^i &= c\tau_3 dr + b\tau_2 d\theta + (\cos\theta\tau_3 - b\sin\theta\tau_1)d\phi \tag{1.5.19} \\
E_i^a &= p_c\tau_3 \sin\theta\partial_r + p_b\tau_2 \sin\theta\partial_\theta - p_b\tau_1\partial_\phi, \nonumber
\end{align}
$$

and

$$
\begin{align}
A_a^i &= c\tau_3 dr - b\tau_2 d\theta + (\cos\theta\tau_3 + b\sin\theta\tau_1)d\phi \tag{1.5.20} \\
E_i^a &= p_c\tau_3 \sin\theta\partial_r - p_b\tau_2 \sin\theta\partial_\theta + p_b\tau_1\partial_\phi, \nonumber
\end{align}
$$

where,

$$ b = \pm\gamma \left( \frac{2m}{t} - 1 \right)^{1/2} ; \quad c = \pm\gamma \left( -\frac{m}{t^2} \right), \qquad (1.5.21) $$

$$ p_c = \pm t^2 ; \quad p_b = t \left( \frac{2m}{t} - 1 \right)^{1/2} . \qquad (1.5.22) $$

The momentum $p_c = \pm t^2$ 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 Schwarzschild black hole).

Region I.

The analoguos calculations for region I with line element (1.5.1) leads to the following phase space coordinates

$$
\begin{align}
\tilde{A}_a^i &= \tilde{c}\tau_3 dr + \tilde{b}\tau_2 d\theta + (\cos\theta\tau_3 - \tilde{b}\sin\theta\tau_1)d\phi \tag{1.5.23} \\
\tilde{E}_i^a &= \tilde{p}_c\tau_3 \sin\theta\partial_r + \tilde{p}_b\tau_2 \sin\theta\partial_\theta - \tilde{p}_b\tau_1\partial_\phi, \nonumber
\end{align}
$$

$$
\begin{align}
\tilde{A}_a^i &= \tilde{c}\tau_3 dr - \tilde{b}\tau_2 d\theta + (\cos\theta\tau_3 + \tilde{b}\sin\theta\tau_1)d\phi \tag{1.5.24} \\
\tilde{E}_i^a &= \tilde{p}_c\tau_3 \sin\theta\partial_r - \tilde{p}_b\tau_2 \sin\theta\partial_\theta + \tilde{p}_b\tau_1\partial_\phi, \nonumber
\end{align}
$$

where,

$$ \tilde{b} = \pm\gamma \left(1 - \frac{2m}{x}\right)^{1/2} ; \quad \tilde{c} = \mp\gamma \left(\frac{m}{x^2}\right) ; \quad \tilde{p}_c = \pm x^2 ; \quad \tilde{p}_b = x \left(1 - \frac{2m}{x}\right)^{1/2} . \qquad (1.5.25) $$

This defines variables (1.4.5)-(1.4.6) introduced above in subsection 1.4 as

$$ A_x = \tilde{c}, E^x = \tilde{p}_c; \gamma K_\phi = \tilde{b}, E^\phi = \tilde{p}_b; \eta = (2n+1)\pi, P^\eta = 0 \qquad (1.5.27) $$

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 free quantum Schwarzschild geometry.

We remind that in accordance with a linear Colombeau geometry approach [30], the Schwarzschild black hole, etc. has a distributional source ~ $\delta(\hat{x}) \in D'(\mathbb{R}^3)$, 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.
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 $\Delta \sim l_{Pl}$.

Remind that under naive formal calculation the Kretschmann scalar curvature of the Schwarzschild black hole reads [43]:

$$ \hat{R}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu}(\hat{r}) = \frac{48M^2}{t^6}. \qquad (1.6.1) $$

Obviously (1.6.1) indicates that in this case the singularity of space-time lies at $\hat{r} = 0$ as well. The classical phase space variables calculated in subsect.1.5 $\{c, p_c\}$ used in this section are given by Eq.(1.5.21)-Eq.(1.5.22) and therefore

$$ b = \pm\gamma \left(\frac{2m}{t} - 1\right)^{1/2} ; c = \mp\gamma \left(\frac{m}{t^2}\right); p_c = \pm t^2 ; p_b = t \left(\frac{2m}{t} - 1\right)^{1/2}. \qquad (1.6.2) $$

Let us consider the following quantity on the classical point-like phase space [43]:

$$ \mathcal{R} = \frac{1}{2\pi\gamma G} \{c, \sqrt{|p_c|}\} = \frac{\operatorname{sgn}(p_c)}{\sqrt{|p_c|}} = \frac{1}{t}. \qquad (1.6.3) $$

Following the methods presented in [44], we expand now the holonomy along $x$ direction of $\Sigma = \mathbb{R} \times S^2$ with oriented length $\tau$ as

$$ h_x^{(\tau)} = 1 + \epsilon \int_0^\tau dxc \tau_3 + \mathcal{O}(\epsilon^2), \qquad (1.6.4) $$

and rewrite $\mathcal{R}$ as

$$ \mathcal{R} = \frac{1}{2\pi\gamma G} \text{tr} \left( \tau_3 h_x^{(\tau)} \{h_x^{(\tau)-1}, \sqrt{|p_c|}\} \right). \qquad (1.6.5) $$

Now, quantization would be straightforward:

$$ \begin{aligned} \hat{R} &= \frac{1}{2\pi\gamma\ell_{Pl}^2} \text{tr} \left( \tau_3 \hat{h}_x^{(\tau)} \left[ \hat{h}_x^{(\tau)-1}, \sqrt{|\hat{p}_c|} \right] \right) \\ &= \frac{1}{2\pi\gamma\ell_{Pl}^2} \left( \cos\left(\frac{\tau c}{2}\right) \sqrt{|\hat{p}_c|} \sin\left(\frac{\tau c}{2}\right) - \sin\left(\frac{\tau c}{2}\right) \sqrt{|\hat{p}_c|} \cos\left(\frac{\tau c}{2}\right) \right). \end{aligned} \qquad (1.6.6) $$

Its action on $|\tau, \mu\rangle$ which are the simplified version of the spin network states in this reduced model (with $\mu$ being the oriented length along the equator of $S^2$), then becomes:

$$ \hat{R}|\tau, \mu\rangle = \frac{1}{2\pi\sqrt{\gamma\ell_{Pl}}} \left( \sqrt{|\tau+1|} - \sqrt{|\tau-1|} \right) |\tau, \mu\rangle. \qquad (1.6.7) $$

Such operator $\hat{R}|\tau, \mu\rangle$ has a bounded spectrum with maximum value of $(\sqrt{2} \pi \sqrt{\gamma} \ell_{Pl})^{-1}$. Thus the Kretschmann scalar curvature, which is classically divergent, at quantum level has a maximum value of [43]:

$$ \hat{R}^{\rho\sigma\mu\nu}(\hat{r}) \hat{R}_{\rho\sigma\mu\nu}(\hat{r}) \Big|_{\max} = \frac{48M^2}{\hat{r}^6} \Big|_{\max} = \frac{48M^2}{\gamma^3 \pi^6 l_{Pl}^6}. \qquad (1.6.8) $$

**Remark 1.6.2.** Note that a quantity $(\hat{R}_\epsilon(\hat{r}))_\epsilon$ which is classically has a weak distributional

limit, at quantum level obtained by canonical LQG has a maximum value of
$$ \left| \hat{R}_{\varepsilon}(\hat{r}) \right| \Big|_{\max} \sim \frac{M \varepsilon^2}{l_{Pl}^5} \Big|_{\max} . \qquad (1.6.9) $$

Thus $\lim_{\varepsilon \to 0} \hat{R}_{\varepsilon}(\hat{r}) = 0$ since RHS of the Eq.(1.6.9) vanishes in the limit $\varepsilon \to 0$.

### 1.6.2. The point free quantum Schwarzschild geometry.
Classical point-free phase space variables

In region II, the Colombeau metric of point-free Schwarzschild space-time takes the form

$$ (ds_{\varepsilon}^2)_{\varepsilon} = - \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{-1} (dt_{\varepsilon})_{\varepsilon} + \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right] (dx_{\varepsilon}^2)_{\varepsilon} + (t_{\varepsilon}^2 d\Omega_{\varepsilon}^2)_{\varepsilon}, \qquad (1.6.10) $$

where $cl[(t_{\varepsilon})_{\varepsilon}] \in \tilde{\mathbb{R}}$. According to definition of Colombeau tetrad (frame) fields $(g_{\mu\nu,\varepsilon})_{\varepsilon} = \eta_{IJ}(e_{\mu,\varepsilon}^I e_{\nu,\varepsilon}^J)_{\varepsilon}$, 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 $(g_{\mu\nu,\varepsilon})_{\varepsilon} = \eta_{IJ}(e_{\mu,\varepsilon}^I e_{\nu,\varepsilon}^J)_{\varepsilon}$ 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 $((A_{\varepsilon})_{\varepsilon}, (E_{\varepsilon})_{\varepsilon})$, a particular labeling must be chosen which will be clear below. The suitable choice for labeling 4 orthogonal Colombeau generalized frame fields reads

$$ \begin{aligned} (e_{\varepsilon}^0)_{\varepsilon} &= \pm \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{-1/2} (dt_{\varepsilon})_{\varepsilon} ; (e_{\varepsilon}^1)_{\varepsilon} = \pm [(t_{\varepsilon})_{\varepsilon}] [(\sin \theta_{\varepsilon})_{\varepsilon}] [(d\phi_{\varepsilon})_{\varepsilon}]; \\ (e_{\varepsilon}^2)_{\varepsilon} &= \pm [(t_{\varepsilon})_{\varepsilon}] [(d\theta_{\varepsilon})_{\varepsilon}] ; (e_{\varepsilon}^3)_{\varepsilon} = \pm \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{1/2} [(dx_{\varepsilon})_{\varepsilon}], \end{aligned} \qquad (1.6.11) $$

which gives the compatible Colombeau generalized spin connection components

$$ \begin{aligned} (\omega_{\varepsilon}^{30})_{\varepsilon} &= -(\omega_{\varepsilon}^{03})_{\varepsilon} = -\frac{m}{[(t_{\varepsilon}^2)_{\varepsilon}]} [(dx_{\varepsilon})_{\varepsilon}]; \\ (\omega_{\varepsilon}^{20})_{\varepsilon} &= -(\omega_{\varepsilon}^{02})_{\varepsilon} = \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{1/2} [(d\theta_{\varepsilon})_{\varepsilon}], \qquad (1.6.12) \\ (\omega_{\varepsilon}^{10})_{\varepsilon} &= -(\omega_{\varepsilon}^{01})_{\varepsilon} = \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{1/2} [(\sin \theta_{\varepsilon})_{\varepsilon}] [(d\phi_{\varepsilon})_{\varepsilon}]; \\ (\omega_{\varepsilon}^{12})_{\varepsilon} &= -(\omega_{\varepsilon}^{21})_{\varepsilon} = [(\cos \theta_{\varepsilon})_{\varepsilon}] [(d\phi_{\varepsilon})_{\varepsilon}]. \end{aligned} $$

The $(A_{\varepsilon})_{\varepsilon}$ field can be constructed using spin connections:

$$ \begin{aligned} (A_{\varepsilon}^3)_{\varepsilon} &= \pm \gamma \left(-\frac{m}{(t_{\varepsilon}^2)_{\varepsilon}}\right) [(dx_{\varepsilon})_{\varepsilon}], (A_{\varepsilon}^2)_{\varepsilon} = \pm \gamma \left[\frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1\right]^{1/2} [(d\theta_{\varepsilon})_{\varepsilon}], \\ (A_{\varepsilon}^1)_{\varepsilon} &= \pm \gamma \left[\frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1\right]^{1/2} [(\sin \theta_{\varepsilon})_{\varepsilon}] [(d\phi_{\varepsilon})_{\varepsilon}], (A_{\varepsilon}^3)_{\varepsilon} = \pm [(\cos \theta_{\varepsilon})_{\varepsilon}] [(d\phi_{\varepsilon})_{\varepsilon}]. \end{aligned} \qquad (1.6.13) $$

To construct Colombeau generalized field $(E_{\varepsilon})_{\varepsilon}$ on $\tilde{\Sigma}$ we choose a gauge in which $e_{\mu}^{0} = n_{\mu}$, 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 $\tilde{\Sigma} = \tilde{\mathbb{R}} \times \tilde{S}^2$. The
Colombeau generalized 3 triad fields become:

$$ (e_{\epsilon}^{1})_{\epsilon} = \pm [(t_{\epsilon}^{1/2})_{\epsilon}] [(\sin\theta_{\epsilon})_{\epsilon}] [(d\phi_{\epsilon})_{\epsilon}]; (e_{\epsilon}^{2})_{\epsilon} = \pm [(t_{\epsilon})_{\epsilon}] [(d\theta_{\epsilon})_{\epsilon}]; \\ (e_{\epsilon}^{3})_{\epsilon} = \pm \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} [(dx_{\epsilon})_{\epsilon}], \qquad (1.6.14) $$

with determinant

$$ (\det(e_{\epsilon}))_{\epsilon} = [(t_{\epsilon}^{2})_{\epsilon}] [(\sin\theta_{\epsilon})_{\epsilon}] \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2}, \qquad (1.6.15) $$

and inverse triad

$$ (e_{1,\epsilon})_{\epsilon} = \pm \frac{1}{[(t_{\epsilon})_{\epsilon}] [(\sin\theta_{\epsilon})_{\epsilon}]} [(\partial_{\phi_{\epsilon}})_{\epsilon}]; (e_{2,\epsilon})_{\epsilon} = \pm \frac{1}{[(t_{\epsilon})_{\epsilon}]} [(\partial_{\theta_{\epsilon}})_{\epsilon}]; \\ (e_{3,\epsilon})_{\epsilon} = \pm \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{-1/2} [(\partial_{x,\epsilon})_{\epsilon}]. \qquad (1.6.16) $$

The $(E_{\epsilon})_{\epsilon}$ fields become

$$ (E_{1,\epsilon})_{\epsilon} = \pm [(t_{\epsilon})_{\epsilon}] \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} (\partial_{\phi_{\epsilon}})_{\epsilon}, \\ (E_{2,\epsilon})_{\epsilon} = \pm [(t_{\epsilon})_{\epsilon}] \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} [(\sin\theta_{\epsilon})_{\epsilon}] [(\partial_{\theta_{\epsilon}})_{\epsilon}], \\ (E_{3,\epsilon})_{\epsilon} = \pm [(t_{\epsilon}^{2})_{\epsilon}] [(\sin\theta_{\epsilon})_{\epsilon}] [(\partial_{x,\epsilon})_{\epsilon}]. \qquad (1.6.17) $$

The Colombeau generalized 3 triad fields (1.6.14) define their generalized compatible spin connection, $(\Gamma_{\epsilon}^{ij} \wedge e_{\epsilon}^{j}) + (de_{\epsilon}^{i})_{\epsilon} = 0_{\mathbb{R}}$:

$$ (\Gamma_{\epsilon}^{12})_{\epsilon} = -(\Gamma_{\epsilon}^{21})_{\epsilon} = [(\cos\theta_{\epsilon})_{\epsilon}] [(d\phi_{\epsilon})_{\epsilon}], \qquad (1.6.18) $$

and

$$ (\Gamma_{\epsilon}^{3})_{\epsilon} = \frac{1}{2} (\epsilon^{312} \Gamma_{\epsilon}^{12} + \epsilon^{321} \Gamma_{\epsilon}^{21})_{\epsilon} = [(\cos\theta_{\epsilon})_{\epsilon}] [(d\phi_{\epsilon})_{\epsilon}]. \qquad (1.6.19) $$

Extrinsic distributional curvature is related to $(A_{\epsilon})_{\epsilon}$ via $\gamma(K_{\epsilon})_{\epsilon} = (A_{\epsilon})_{\epsilon} - (\Gamma_{\epsilon})_{\epsilon}$ reads

$$ (K_{r_{\epsilon}, \epsilon}^{3})_{\epsilon} = \frac{1}{\gamma} (A_{r_{\epsilon}, \epsilon}^{3})_{\epsilon} = \pm \left( -\frac{m}{(t_{\epsilon}^{2})_{\epsilon}} \right) [(dx_{\epsilon})_{\epsilon}], \\ (K_{\theta_{\epsilon}, \epsilon}^{2})_{\epsilon} = \frac{1}{\gamma} (A_{\theta_{\epsilon}, \epsilon}^{2})_{\epsilon} = \pm \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2}, \qquad (1.6.20) $$

$$ (K_{\phi_{\epsilon}, \epsilon}^{1})_{\epsilon} = \frac{1}{\gamma} (A_{\phi_{\epsilon}, \epsilon}^{1})_{\epsilon} = \pm \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} [(\sin\theta_{\epsilon})_{\epsilon}]. \qquad (1.6.21) $$

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

$$ (\{A_{a,\epsilon}^{i}(x_{\epsilon}), E_{j}^{b}(x'_{\epsilon})\})_{\epsilon} = \delta_{j}^{i} (\delta_{a}^{b} \delta(x_{\epsilon} - x'_{\epsilon}))_{\epsilon}. $$

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
$$ (H_{\theta_{\epsilon}, \epsilon})_{\epsilon} = \gamma[(t_{\epsilon})_{\epsilon}] \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right] [(\cos \theta_{\epsilon})_{\epsilon}] \{ \mathbf{sign}[(A_{\phi_{\epsilon}, \epsilon}^{1})_{\epsilon}] + \mathbf{sign}[(A_{\theta_{\epsilon}, \epsilon}^{2} A_{\phi_{\epsilon}, \epsilon}^{3})_{\epsilon}] \} \quad (1.6.22) $$

$$ (G_{2,\epsilon})_{\epsilon} = [(t_{\epsilon})_{\epsilon}] \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} [(\cos \theta_{\epsilon})_{\epsilon}] \{ \mathbf{sign}[(E_{2,\epsilon}^{\theta_{\epsilon}})_{\epsilon}] + \mathbf{sign}[(A_{\phi_{\epsilon}}^{3} E_{\epsilon}^{\phi_{\epsilon}1})_{\epsilon}] \} \quad (1.6.23) $$

and Hamiltonian constraint gives:

$$ (C_{\epsilon})_{\epsilon} = [(t_{\epsilon})_{\epsilon}] \left( \frac{2m}{[(t_{\epsilon})_{\epsilon}]} - 1 \right) [(\sin^2 \theta_{\epsilon})_{\epsilon}] \{ \mathbf{sign}[(E_{2,\epsilon}^{\theta_{\epsilon}})_{\epsilon}] + \mathbf{sign}[(E_{1,\epsilon}^{\phi_{\epsilon}})_{\epsilon}] \} \quad (1.6.24) $$

For the above constraints to be zero we must have

$$ \begin{gathered} \mathbf{sign}[(E_{2,\epsilon}^{\theta_{\epsilon}})_{\epsilon}] = -\mathbf{sign}[(E_{1,\epsilon}^{\phi_{\epsilon}})_{\epsilon}], \quad (1.6.25) \\ \mathbf{sign}[(A_{\phi_{\epsilon},\epsilon}^{3})_{\epsilon}] = +1, \quad \mathbf{sign}[(A_{\phi_{\epsilon},\epsilon}^{1})_{\epsilon}] = -\mathbf{sign}[(A_{\theta_{\epsilon},\epsilon}^{2})_{\epsilon}]. \end{gathered} $$

This leaves us with two alternatives corresponding to the residual gauge freedom
$$ ((b_{\epsilon})_{\epsilon}, (p_{b_{\epsilon},\epsilon})_{\epsilon}) \rightarrow -(b_{\epsilon})_{\epsilon}, -(p_{b_{\epsilon},\epsilon})_{\epsilon}). $$

$$ \begin{aligned} (A_{a,\epsilon}^{i})_{\epsilon} &= [(c_{\epsilon})_{\epsilon}]\tau_{3}[(dr_{\epsilon})_{\epsilon}] + [(b_{\epsilon})_{\epsilon}]\tau_{2}[(d\theta_{\epsilon})_{\epsilon}] + \\ &\quad [((\cos\theta_{\epsilon})_{\epsilon})\tau_{3} - [(b_{\epsilon})_{\epsilon}][(\sin\theta_{\epsilon})_{\epsilon}]\tau_{1}][d\phi_{\epsilon})_{\epsilon}] \end{aligned} \quad (1.6.26) $$

$$ \begin{aligned} (E_{i,\epsilon}^{a})_{\epsilon} &= [(p_{c_{\epsilon},\epsilon})_{\epsilon}]\tau_{3}[(\sin\theta_{\epsilon})_{\epsilon}][\partial_{r_{\epsilon}})_{\epsilon}] + \\ &\quad [(p_{b_{\epsilon},\epsilon})_{\epsilon}]\tau_{2}[(\sin\theta_{\epsilon})_{\epsilon}][\partial_{\theta_{\epsilon}})_{\epsilon}] - [(p_{b_{\epsilon},\epsilon})_{\epsilon}]\tau_{1}[(\partial_{\phi_{\epsilon}})_{\epsilon}], \end{aligned} $$

and

$$ \begin{aligned} (A_{a,\epsilon}^{i})_{\epsilon} &= [(c_{\epsilon})_{\epsilon}]\tau_{3}[(dr_{\epsilon})_{\epsilon}] - [(b_{\epsilon})_{\epsilon}]\tau_{2}[(d\theta_{\epsilon})_{\epsilon}] + \\ &\quad [((\cos\theta_{\epsilon})_{\epsilon})\tau_{3} + [(b_{\epsilon})_{\epsilon}][(\sin\theta_{\epsilon})_{\epsilon}]\tau_{1}]d\phi_{\epsilon} \end{aligned} \quad (1.6.27) \\ (E_{i,\epsilon}^{a})_{\epsilon} = [(p_{c_{\epsilon},\epsilon})_{\epsilon}]\tau_{3}[(\sin\theta_{\epsilon})_{\epsilon}][\partial_{r_{\epsilon}})_{\epsilon}] - [(p_{c_{\epsilon},\epsilon})_{\epsilon}]\tau_{2}[(\sin\theta_{\epsilon})_{\epsilon}][\partial_{\theta_{\epsilon}})_{\epsilon}] + \\ +[(p_{b_{\epsilon},\epsilon})_{\epsilon}]\tau_{1}[(\partial_{\phi_{\epsilon}})_{\epsilon}], \end{aligned} $$

where,

$$ (b_{\epsilon})_{\epsilon} = \pm\gamma \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} ; \quad (c_{\epsilon})_{\epsilon} = \mp\gamma \frac{m}{[(t_{\epsilon})_{\epsilon}^2]}, \quad (1.6.28) $$

$$ (p_{c_{\epsilon},\epsilon})_{\epsilon} = \pm(t_{\epsilon}^2)_{\epsilon} ; \quad (p_{b_{\epsilon},\epsilon})_{\epsilon} = [(t_{\epsilon})_{\epsilon}] \left( \left[ \left( \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right) \right]^{1/2} \right). \quad (1.6.29) $$

The momentum $(p_{c_{\epsilon},\epsilon})_{\epsilon} = \pm(t_{\epsilon}^2)$ is a monotonic generalized function on $\tilde{\mathbb{R}}$ 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

$$ \begin{aligned} (\tilde{A}_{a,\epsilon})_{\epsilon} &= [ (\tilde{c}_{\epsilon})_{\epsilon} ] \tau_3 [(dr_{\epsilon})_{\epsilon}] + [ (\tilde{b}_{\epsilon})_{\epsilon} ] \tau_2 [(d\theta_{\epsilon})_{\epsilon}] + \\ &\quad [ ((\cos\theta_{\epsilon})_{\epsilon}) \tau_3 - [ (\tilde{b}_{\epsilon})_{\epsilon} ] [ (\sin\theta_{\epsilon})_{\epsilon} ] \tau_1 ] [(d\phi_{\epsilon})_{\epsilon}], \end{aligned} \quad (1.6.30) $$

$$ \begin{aligned} (\tilde{E}_{i,\epsilon}^{a})_{\epsilon} &= [ (\tilde{p}_{c_{\epsilon},\epsilon})_{\epsilon} ] \tau_3 [(\sin\theta_{\epsilon})_{\epsilon}] [ (\partial_{r_{\epsilon}})_{\epsilon} ] + \\ &\quad [ (\tilde{p}_{b_{\epsilon},\epsilon})_{\epsilon} ] \tau_2 [(\sin\theta_{\epsilon})_{\epsilon}] [ (\partial_{\theta_{\epsilon}})_{\epsilon} ] - [ (\tilde{p}_{b_{\epsilon},\epsilon})_{\epsilon} ] \tau_1 [(\partial_{\phi_{\epsilon}})_{\epsilon} ], \end{aligned} $$
$$
\begin{align}
(\tilde{A}_{a,\epsilon}^i)_\epsilon &= [(\tilde{c}_\epsilon)_\epsilon] \tau_3 [(dr_\epsilon)_\epsilon] - [(\tilde{b}_\epsilon)_\epsilon] \tau_2 [(d\theta_\epsilon)_\epsilon] + \nonumber \\
&\quad [((\cos\theta_\epsilon)_\epsilon] \tau_3 + [(\tilde{b}_\epsilon)_\epsilon] [((\sin\theta_\epsilon)_\epsilon] \tau_1) [(d\phi_\epsilon)_\epsilon], \tag{1.6.31} \\
(\tilde{E}_{i,\epsilon}^a)_\epsilon &= [(\tilde{p}_{c_\epsilon,\epsilon})_\epsilon] \tau_3 [((\sin\theta_\epsilon)_\epsilon] [((\partial_{r_\epsilon})_\epsilon] - \nonumber \\
&\quad [((\tilde{p}_{b_\epsilon,\epsilon})_\epsilon] \tau_2 [((\sin\theta_\epsilon)_\epsilon] [((\partial_{\theta_\epsilon})_\epsilon] + [((\tilde{p}_{b_\epsilon,\epsilon})_\epsilon] \tau_1 [((\partial_{\phi_\epsilon})_\epsilon], \nonumber
\end{align}
$$

where,

$$
\begin{align}
(\tilde{b}_{\epsilon})_{\epsilon} &= \pm \gamma \left[ 1 - \frac{2m}{(x_{\epsilon})_{\epsilon}} \right]^{1/2} ; \quad (\tilde{c}_{\epsilon})_{\epsilon} = \mp \gamma \frac{m}{[(x_{\epsilon})_{\epsilon}]} ; \quad (\tilde{p}_{c_{\epsilon},\epsilon})_{\epsilon} = \pm (x_{\epsilon}^2)_{\epsilon} ; \tag{1.6.32} \\
(\tilde{p}_{b_{\epsilon}})_{\epsilon} &= [(x_{\epsilon})_{\epsilon}] \left[ 1 - \frac{2m}{(x_{\epsilon})_{\epsilon}} \right]^{1/2} . \nonumber
\end{align}
$$

This defines variables (1.4.5)-(1.4.6) introduced above as

$$
\begin{align}
(A_{x_\epsilon, \epsilon})_\epsilon &= (\tilde{c}_\epsilon)_\epsilon, (E_\epsilon^{x_\epsilon})_\epsilon = (\tilde{p}_{c_\epsilon, \epsilon})_\epsilon; \gamma(K_{\phi_\epsilon, \epsilon})_\epsilon = (\tilde{b}_\epsilon)_\epsilon, \tag{1.6.33} \\
(E_\epsilon^{\phi_\epsilon})_\epsilon &= (\tilde{p}_{b_\epsilon, \epsilon})_\epsilon; \eta = (2n+1)\pi, (P_\epsilon^{\eta_\epsilon})_\epsilon = 0 \notag \\
&\sim \mathbb{R} \nonumber
\end{align}
$$

which constitute a 4 dimensional phase space.

Let us consider the following quantity on the point-free phase space mentioned above

$$
(\mathcal{R}_{\epsilon})_{\epsilon} \equiv \frac{1}{2\pi\gamma G} \left\{ [(c_{\epsilon})_{\epsilon}], \left[ \sqrt{[(|p_{c_{\epsilon},\epsilon}|)_{\epsilon}]} \right] \right\} = \frac{\mathbf{sign}[(p_{c_{\epsilon},\epsilon})_{\epsilon}]}{\sqrt{[(|p_{c_{\epsilon},\epsilon}|)_{\epsilon}]}}, = \frac{1}{[(t_{\epsilon})_{\epsilon}]} \quad (1.6.33)
$$

Following the canonical methods presented in [44], we expand now the holonomy
along $x$

direction of $\tilde{\Sigma} = \tilde{\mathbb{R}} \times \tilde{S}^2$ with oriented generalized length $(\tau_{\epsilon})_{\epsilon}$ as

$$
(h_{x_{\epsilon}, \epsilon}^{(\tau_{\epsilon})})_{\epsilon} = 1 + \epsilon \left( \int_{0}^{\tau_{\epsilon}} dx_{\epsilon} c_{\epsilon} \tau_{3} \right)_{\epsilon} + (\mathcal{O}_{\epsilon}(\epsilon^2))_{\epsilon}, \quad (1.6.34)
$$

and rewrite $(\mathcal{R}_\epsilon)_\epsilon$ as

$$
(\mathcal{R}_\epsilon)_\epsilon = \frac{1}{2\pi\gamma G} \mathrm{tr} \left( \tau_3 \left[ \left( h_{x_\epsilon, \epsilon}^{(\tau_\epsilon)} \right)_\epsilon \right] \left\{ \left[ \left( h_{x_\epsilon, \epsilon}^{(\tau_\epsilon)-1} \right)_\epsilon \right], \sqrt{[(|p_{c_\epsilon, \epsilon}|)_\epsilon]} \right\} \right). \quad (1.6.35)
$$

Now, quantization would be straightforward:

$$
\begin{align}
(\mathcal{R}_\epsilon)_\epsilon &= \frac{1}{2\pi\gamma l_{Pl}^2} \mathrm{tr} \left( \tau_3 \left[ \left( \hat{h}_{x_\epsilon, \epsilon}^{(\tau_\epsilon)} \right)_\epsilon \right] \left\{ \left[ \left( \hat{h}_{x_\epsilon, \epsilon}^{(\tau_\epsilon)-1} \right)_\epsilon \right], \sqrt{[(|\hat{p}_{c_\epsilon, \epsilon}|)_\epsilon]} \right\} \right) \nonumber \\
&= \frac{1}{2\pi\gamma l_{Pl}^2} \left( \left[ \cos\left(\frac{\tau_\epsilon c_\epsilon}{2}\right)_\epsilon \right] \sqrt{[(|\hat{p}_{c_\epsilon, \epsilon}|)_\epsilon]} \left[ \sin\left(\frac{\tau_\epsilon c_\epsilon}{2}\right)_\epsilon \right] - \right. \tag{1.6.36} \\
&\qquad \left. \left[ \sin\left(\frac{\tau_\epsilon c_\epsilon}{2}\right)_\epsilon \right] \sqrt{[(|\hat{p}_{c_\epsilon, \epsilon}|)_\epsilon]} \left[ \cos\left(\frac{\tau_\epsilon c_\epsilon}{2}\right)_\epsilon \right] \right). \nonumber
\end{align}
$$

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 S²), then
becomes:

$$
(\mathcal{R}_\epsilon | \tau_\epsilon, \mu_\epsilon \rangle)_\epsilon = \frac{1}{2\pi\sqrt{\gamma l_{Pl}}} \left( \sqrt{|(\tau_\epsilon)_\epsilon + 1|} - \sqrt{|(\tau_\epsilon)_\epsilon - 1|} \right) |(\tau_\epsilon)_\epsilon, (\mu_\epsilon)_\epsilon\rangle. \quad (1.6.37)
$$

Such operator $(\mathcal{R}_\epsilon | \tau_\epsilon, \mu_\epsilon \rangle)_\epsilon$ has a bounded spectrum with maximum value of
$\left( \sqrt{2} \pi \sqrt{\gamma} l_{Pl} \right)^{-1}$.
**Remark 1.6.3.** Thus the Colombeau generalized Kretschmann scalar curvature $(\hat{R}_{\epsilon}^{\rho\sigma\mu\nu}(t_{\epsilon})\hat{R}_{\rho\sigma\mu\nu,\epsilon}(t_{\epsilon}))_{\epsilon}$, which is classically has infinite large point value $\sim \text{cl}[(\epsilon^{-6})_{\epsilon}] \in \tilde{\mathbb{R}}$ (see Eq.(1.1.18)), at quantum level has a maximum value of:

$$ \frac{\text{cl}[(\hat{R}_{\epsilon}^{\rho\sigma\mu\nu}(t_{\epsilon})\hat{R}_{\rho\sigma\mu\nu,\epsilon}(t_{\epsilon}))_{\epsilon}]}{\text{cl}[(t_{\epsilon}^{6})_{\epsilon}]} \Big|_{\max} \sim \frac{M^2}{\text{cl}[(t_{\epsilon}^{6})_{\epsilon}]} \Big|_{\max} \le \tilde{\mathbb{R}} \text{ st} \left( \frac{M^2}{\text{cl}[(t_{\epsilon}^{6})_{\epsilon}]} \Big|_{\max} \right) \le \frac{M^2}{\gamma^3 \pi^6 l_{Pl}^6}. \quad (1.6.38) $$

**Remark 1.6.4.** Note that the Colombeau generalized curvature scalar $(\hat{R}_{\epsilon}(t_{\epsilon}))_{\epsilon}$ obtained at quantum level by point-free LQG by using similarly calculation as it has been applied above, has nonzero maximum value

$$ (|\hat{R}_{\epsilon}(t_{\epsilon})|)_{\epsilon}|_{\max} \sim \frac{M}{l_{Pl}^3}|_{\max}. \quad (1.6.39) $$

**Remark 1.6.5.** We emphasize that in contrast with trivial (zero valued) result obtained at quantum level for Colombeau generalized curvature scalar $(\hat{R}_{\epsilon}(t_{\epsilon}))_{\epsilon}$ by using canonical LQG, see Remark 1.6.2, Colombeau generalized curvature scalar $(\hat{R}_{\epsilon}(t_{\epsilon}))_{\epsilon}$ 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, $(x_{\epsilon}^{0})_{\epsilon}$, $(x_{\epsilon}^{1})_{\epsilon}$, $(x_{\epsilon}^{2})_{\epsilon}$, $(x_{\epsilon}^{3})_{\epsilon}$, to another generalized coordinate system $(x_{\epsilon}^{\prime 0})_{\epsilon}$, $(x_{\epsilon}^{\prime 1})_{\epsilon}$, $(x_{\epsilon}^{\prime 2})_{\epsilon}$, $(x_{\epsilon}^{\prime 3})_{\epsilon}$: transform according to the relation

$$ (x_{\epsilon}^{i})_{\epsilon} = (f_{\epsilon}^{i}(x_{\epsilon}^{0}, x_{\epsilon}^{\prime 1}, x_{\epsilon}^{\prime 2}, x_{\epsilon}^{\prime 3}))_{\epsilon}, \quad (1.7.1) $$

where the $(f_{\epsilon}^{i})_{\epsilon}$ are certain Colombeau generalized functions and where $(\mathbf{J}_{\epsilon}(x^{\prime 0}, x^{\prime 1}, x^{\prime 2}, x^{\prime 3}))_{\epsilon}$

$$ (\mathbf{J}_{\epsilon}(x_{\epsilon}^{\prime 0}, x_{\epsilon}^{\prime 1}, x_{\epsilon}^{\prime 2}, x_{\epsilon}^{\prime 3}))_{\epsilon} = \left( \frac{\partial(x_{\epsilon}^{0}, x_{\epsilon}^{1}, x_{\epsilon}^{2}, x_{\epsilon}^{3})}{\partial(x_{\epsilon}^{\prime 0}, x_{\epsilon}^{\prime 1}, x_{\epsilon}^{\prime 2}, x_{\epsilon}^{\prime 3})} \right)_{\epsilon} \neq 0_{\tilde{\mathbb{R}}} \quad (1.7.2) $$

is the Jacobian of the Colombeau generalized transformation (1.7.1).

**Remark 1.7.1.** When we transform the coordinates, their Colombeau differentials $(dx_{\epsilon}^{i})_{\epsilon}$ transform according to the relation

$$ (dx_{\epsilon}^{i})_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{\prime k}} dx_{\epsilon}^{\prime k} \right)_{\epsilon} = \left[ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{\prime k}} \right)_{\epsilon} \right] (dx_{\epsilon}^{\prime k})_{\epsilon}. \quad (1.7.3) $$

**Definition 1.7.1.** Every tuple of four Colombeau quantities $(A_{\epsilon}^{i})_{\epsilon}$, $i = 0, 1, 2, 3$, which under a transformation (1.7.1) of coordinates, transform like the Colombeau coordinate
differentials (1.7.2), is called Colombeau contravariant four-vector:

$$ (A_{\epsilon}^{i})_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}} A_{\epsilon}^{k} \right)_{\epsilon} = \left[ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} \right] (A_{\epsilon}^{k})_{\epsilon}. \quad (1.7.4) $$

Let $(\varphi_{\epsilon})_{\epsilon}$ be the Colombeau scalar. Under a coordinate transformation (1.7.1), the four Colombeau quantities $(\frac{\partial \varphi_{\epsilon}}{\partial x_{\epsilon}^{i}})_{\epsilon}$, $i = 0, 1, 2, 3$ transform according to the formula

$$ \left( \frac{\partial \varphi_{\epsilon}}{\partial x_{\epsilon}^{i}} \right)_{\epsilon} = \left( \frac{\partial \varphi_{\epsilon}}{\partial x_{\epsilon}^{k}} \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} = \left( \frac{\partial \varphi_{\epsilon}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon}. \quad (1.7.5) $$

**Definition 1.7.2.** Every tuple of four Colombeau generalized functions $(A_{i,\epsilon})_{\epsilon}$ which, under

a coordinate transformation (1.7.1), transform like the Colombeau derivatives of a scalar,

is called Colombeau generalized covariant four-vector

$$ (A_{i,\epsilon})_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} A'_{k,\epsilon} \right)_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} \right)_{\epsilon} (A'_{k,\epsilon})_{\epsilon}. \quad (1.7.6) $$

**Definition 1.7.3.** We call the Colombeau generalized contravariant tensor of the second

rank, $(A_{\epsilon}^{ik})_{\epsilon}$, 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

$$ (A_{\epsilon}^{ik})_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} \frac{\partial x_{\epsilon}^{l}}{\partial x_{\epsilon}^{i}} A'_{im,\epsilon} \right)_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} \frac{\partial x_{\epsilon}^{l}}{\partial x_{\epsilon}^{i}} \right)_{\epsilon} (A'_{im,\epsilon})_{\epsilon} \quad (1.7.7) $$

and a mixed Colombeau generalized tensor transforms as follows

$$ (A_{k,\epsilon}^{i})_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{l}}{\partial x_{\epsilon}^{k}} A_{m,\epsilon}^{l} \right)_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{l}}{\partial x_{\epsilon}^{k}} A_{m,\epsilon}^{l} \right)_{\epsilon} \quad (1.7.8) $$

**Remark 1.7.2.** Note that the scalar product of two four-vectors $(A_{\epsilon}^{i} B_{i,\epsilon})$ is invariant since

$$ (A_{\epsilon}^{i} B_{i,\epsilon})_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{l}}{\partial x_{\epsilon}^{i}} A_{\epsilon}^{l} B'_{m,\epsilon} \right)_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{l}}{\partial x_{\epsilon}^{i}} A_{\epsilon}^{l} B'_{m,\epsilon} \right)_{\epsilon} = (A_{\epsilon}^{l} B'_{l,\epsilon})_{\epsilon}. \quad (1.7.9) $$

The unit four-tensor $\delta_{k}^{i}$ is defined the same as in classical case: $\delta_{k}^{i} = 0$ for $i \neq k$ and $\delta_{k}^{i} = 1$ for $i = k$. If $(A_{\epsilon}^{k})_{\epsilon}$ is a Colombeau generalized four-vector, then multiplying by $\delta_{k}^{i}$ we obtain

$$ (A_{\epsilon}^{k} \delta_{k}^{i})_{\epsilon} = (A_{\epsilon}^{i})_{\epsilon}, \quad (1.7.10) $$

i.e. again Colombeau generalized four-vector; this proves that $\delta_{k}^{i}$ is a tensor.

**Remark 1.7.3.** The square of the Colombeau generalized line element $(ds_{\epsilon}^{2})_{\epsilon}$ in curvilinear

coordinates is a quadratic form in the differentials $dx^{i}$, $i = 0, 1, 2, 3$:

$$ (ds_{\epsilon}^{2})_{\epsilon} = (g_{ik,\epsilon} dx^{i} dx^{k})_{\epsilon} = [(g_{ik,\epsilon})_{\epsilon}] dx^{i} dx^{k}. \quad (1.7.11) $$

where the $(g_{ik,\epsilon})_{\epsilon}$ are Colombeau generalized functions of the coordinates; $(g_{ik,\epsilon})_{\epsilon}$ is symmetric in the indices $i$ and $k$:
$$ (g_{ik,\epsilon})_{\epsilon} = (g_{ki,\epsilon})_{\epsilon}. \qquad (1.7.12) $$

**Definition 1.7.4.** Since the (contracted) product of $(g_{ik,\epsilon})_{\epsilon}$ and the contravariant tensor $dx^i dx^k$ is a scalar, the $(g_{ik,\epsilon})_{\epsilon}$ form a covariant tensor; it is called the Colombeau generalized metric tensor.

**Definition 1.7.5.** Two tensors $(A_{ik,\epsilon})_{\epsilon}$ and $(B_{\epsilon}^{ik})_{\epsilon}$ are said to be reciprocal to each other if

$$ (A_{ik,\epsilon} B_{\epsilon}^{ik})_{\epsilon} = [(A_{ik,\epsilon})_{\epsilon}] [(B_{\epsilon}^{ik})_{\epsilon}] = \delta_{k}^{i}. \qquad (1.7.13) $$

In particular the contravariant metric tensor is the tensor $(g_{ik,\epsilon})_{\epsilon}$ reciprocal to the tensor $(g_{\epsilon}^{ik})_{\epsilon}$, that is,

$$ \{(g_{ik,\epsilon})_{\epsilon}\} \{(g_{\epsilon}^{ik})_{\epsilon}\} = \delta_{k}^{i}. \qquad (1.7.14) $$

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:

$$ (A_{\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{ik} A_{k,\epsilon})_{\epsilon}, (A_{i,\epsilon})_{\epsilon} = (g_{ik,\epsilon} A_{\epsilon}^{k})_{\epsilon}. \qquad (1.7.15) $$

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:

$$ (A_{k,\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{il} A_{lk,\epsilon})_{\epsilon}, (A_{i,\epsilon}^{k})_{\epsilon} = (g_{\epsilon}^{il} g_{\epsilon}^{km} A_{lm,\epsilon})_{\epsilon}, \text{etc.} \qquad (1.7.16) $$

The completely antisymmetric unit pseudotensor in galilean coordinates we denote by $e^{iklm}$. Let us transform it to an arbitrary system of Colombeau generalized coordinates, and now denote it by $(E_{\epsilon}^{iklm})_{\epsilon}$. We keep the notation $e^{iklm}$ for the quantities defined as before by

$e^{0123} = 1$ (or $e_{0123} = -1$). Let the $x^{i}$, $i = 0, 1, 2, 3$ be galilean, and the $(x_{\epsilon}^{i})_{\epsilon}$, $i = 0, 1, 2, 3$ be arbitrary Colombeau generalized curvilinear coordinates. According to the general rules for transformation of Colombeau generalized tensors, we have

$$ (E_{\epsilon}^{iklm})_{\epsilon} = \left[ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x^{l}} \frac{\partial x_{\epsilon}^{k}}{\partial x^{l}} \frac{\partial x_{\epsilon}^{m}}{\partial x^{l}} \right)_{\epsilon} \right] e^{prst}, \qquad (1.7.17) $$

or

$$ (E_{\epsilon}^{iklm})_{\epsilon} = \{(\mathbf{J}_{\epsilon}(x^{0}, x^{1}, x^{2}, x^{3}))_{\epsilon}\} e^{prst}, \qquad (1.7.18) $$

where $(\mathbf{J}_{\epsilon}(x^{0}, x^{1}, x^{2}, x^{3}))_{\epsilon} \neq 0$ is the determinant formed from the derivatives $\partial x^{i} / \partial x^{l}$, i.e. it is just the Colombeau generalized Jacobian of the Colombeau generalized transformation from the galilean to the Colombeau generalized curvilinear coordinates:

$$ (\mathbf{J}_{\epsilon}(x^{0}, x^{1}, x^{2}, x^{3}))_{\epsilon} = \left( \frac{\partial(x_{\epsilon}^{0}, x_{\epsilon}^{1}, x_{\epsilon}^{2}, x_{\epsilon}^{3})}{\partial(x^{0}, x^{1}, x^{2}, x^{3})} \right)_{\epsilon}. \qquad (1.7.19) $$

This Colombeau generalized Jacobian can be expressed in terms of the determinant
of the Colombeau generalized metric tensor $(g_{ik,\epsilon})_{\epsilon}$ (in the system $(x_{\epsilon}^i)_{\epsilon}$). To do this we write the formula for the transformation of the metric tensor:

$$ (g_{\epsilon}^{ik})_{\epsilon} = \left[ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x'^{i}} \frac{\partial x_{\epsilon}^{k}}{\partial x'^{k}} \right)_{\epsilon} \right] g^{(0)im}, \qquad (1.7.20) $$

where

$$ g^{(0)im} = g_{im}^{(0)} = \begin{cases} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \\ 1 & 1 & 1 & -1 \end{cases}, \qquad (1.7.21) $$

and equate the determinants of the two sides of this equation. The determinant of the

reciprocal tensor $\det|(g_{\epsilon}^{ik})_{\epsilon}| = 1/(g_{\epsilon})_{\epsilon}$. The determinant $\det|g^{(0)im}| = -1$. Thus we have $1/(g_{\epsilon})_{\epsilon} = -(\mathbf{J}_{\epsilon}^2(x'^0, x'^1, x'^2, x'^3))_{\epsilon}$, and so

$$ (\mathbf{J}_{\epsilon}^2(x'^0, x'^1, x'^2, x'^3))_{\epsilon} = 1/\sqrt{(g_{\epsilon})_{\epsilon}}. \qquad (1.7.22) $$

Thus, in curvilinear coordinates the antisymmetric unit tensor of rank four must be defined as

$$ (E_{\epsilon}^{iklm})_{\epsilon} = \frac{1}{\sqrt{-(g_{\epsilon})_{\epsilon}}} e^{iklm} \qquad (1.7.23) $$

and its covariant components are

$$ (E_{iklm,\epsilon})_{\epsilon} = \sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm}. \qquad (1.7.24) $$

In a galilean coordinate system $x'^i, i = 0, 1, 2, 3$ the integral of a scalar with respect to $d\Omega' = dx'^0 dx'^1 dx'^2 dx'^3$ is also a scalar, i.e. the element $d\Omega'$ behaves like a scalar in the integration. On transforming to Colombeau generalized curvilinear coordinates $(x_{\epsilon}^i)_{\epsilon}, i = 0, 1, 2, 3$, the element of integration $d\Omega'$ goes over into

$$ d\Omega' := \{(\mathbf{J}_{\epsilon}^{-1})_{\epsilon}\} d\Omega = \sqrt{-(g_{\epsilon})_{\epsilon}} (d\Omega_{\epsilon})_{\epsilon}, \qquad (1.7.25) $$

where $(d\Omega_{\epsilon})_{\epsilon} = \{(dx_{\epsilon}^0)_{\epsilon}\} \{(dx_{\epsilon}^1)_{\epsilon}\} \{(dx_{\epsilon}^2)_{\epsilon}\} \{(dx_{\epsilon}^3)_{\epsilon}\}$.

Thus, in Colombeau generalized curvilinear coordinates, when integrating over a four-volume the quantity $\sqrt{-(g_{\epsilon})_{\epsilon}} (d\Omega_{\epsilon})_{\epsilon}$ 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 $(dS_{\epsilon}^{ikl})_{\epsilon}$; the vector dual to it is gotten by multiplying by the tensor $\sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm}$, so it is equal to

$$ \sqrt{-(g_{\epsilon})_{\epsilon}} (dS_{\epsilon,i})_{\epsilon} = -\frac{1}{6} \sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm} (dS_{\epsilon}^{kim})_{\epsilon}. \qquad (1.7.26) $$

**Remark 1.7.5.** Let $(df_{\epsilon}^{ik})_{\epsilon}$ be the element of two-dimensional Colombeau generalized surface spanned by two infinitesimal Colombeau generalized displacements, the dual Colombeau generalized tensor is defined as
$$ \sqrt{-(g_{\epsilon})_{\epsilon}} (df_{ik,\epsilon}^{*})_{\epsilon} = \frac{1}{2} \sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm} (df_{\epsilon}^{lm})_{\epsilon}. \quad (1.7.27) $$

We will be use the designations $(dS_{\epsilon,i})$ and $(df_{ki,\epsilon}^*)_{\epsilon}$ for $e_{iklm}(dS_{\epsilon}^{kim})_{\epsilon}$ and $e_{iklm}(df_{\epsilon}^{lm})_{\epsilon}$ (and not for their products by $\sqrt{-(g_{\epsilon})_{\epsilon}}$).

### 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

$$ (dS_{i,\epsilon})_{\epsilon} := [(d\Omega_{\epsilon})_{\epsilon}] \left(\frac{\partial}{\partial x_{\epsilon}^{i}}\right)_{\epsilon}. \quad (1.7.28) $$

**Remark 1.7.7.** (Generalized Stokes' theorem) Note that for the integral of Colombeau generalized vector $(A_{\epsilon}^i)_{\epsilon}$ we have

$$ \left(\oint A_{\epsilon}^{i} dS_{i,\epsilon}\right)_{\epsilon} = \left(\int \frac{\partial A_{\epsilon}^{i}}{\partial x_{\epsilon}^{i}} d\Omega_{\epsilon}\right)_{\epsilon} = \int \left[ \left(\frac{\partial A_{\epsilon}^{i}}{\partial x_{\epsilon}^{i}}\right)_{\epsilon} \right] [(d\Omega_{\epsilon})_{\epsilon}]. \quad (1.7.29) $$

which is the generalization of Stokes' theorem.

Note that in galilean coordinates the Colombeau generalized differentials $(dA_{i,\epsilon})_{\epsilon}$ of a vector $(A_{i,\epsilon})_{\epsilon}$ form the Colombeau generalized vector, and the derivatives $(\partial A_{i,\epsilon}/\partial x_{\epsilon}^k)_{\epsilon}$ 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; $(dA_{i,\epsilon})_{\epsilon}$ is not a vector, and $(\partial A_{i,\epsilon}/\partial x_{\epsilon}^k)_{\epsilon}$ is not the Colombeau generalized tensor. This is due to the fact that $(dA_{i,\epsilon})_{\epsilon}$ 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 infinitesimally 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 $x^i$ is $(A_{\epsilon}^i)_{\epsilon}$, then at the neighboring point $x^i + dx^i$ it is equal to $(A_{\epsilon}^i)_{\epsilon} + (dA_{\epsilon}^i)_{\epsilon} = (A_{\epsilon}^i + dA_{\epsilon}^i)_{\epsilon}$. We subject the vector $(A_{\epsilon}^i)_{\epsilon}$ to an infinitesimal parallel displacement to the point $x^i + dx^i$; the change in the vector which results from this we denote by $(\delta A_{\epsilon}^i)_{\epsilon}$. Then the difference $(DA_{\epsilon}^i)_{\epsilon}$ between the two Colombeau generalized vectors which are now located at the same point is

$$ (DA_{\epsilon}^{i})_{\epsilon} = (dA_{\epsilon}^{i})_{\epsilon} - (\delta A_{\epsilon}^{i})_{\epsilon}. \quad (1.7.30) $$

The change $(\delta A_{\epsilon}^i)_{\epsilon}$ 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 $(\delta A_{\varepsilon}^{i})_{\varepsilon}$ has the form

$$ (\delta A_{\varepsilon}^{i})_{\varepsilon} = -(\Gamma_{kl,\varepsilon}^{i} A_{\varepsilon}^{k} dx^{l})_{\varepsilon}, \qquad (1.7.31) $$

where $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon}$ are certain Colombeau generalized functions of the coordinates. Their form depends, of course, on the coordinate system; for a galilean coordinate system $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon} = 0_{\mathbb{R}}$. From this it is already clear that the quantities $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon}$ 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 $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon}$ vanish over all of space. But we can choose a coordinate system for which the $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon}$ become $0_{\mathbb{R}}$ over a given infinitesimal region. The quantities $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon}$, are called generalized Christoffel symbols. In addition to the quantities $(\Gamma_{kl,\varepsilon}^{i})_{\varepsilon}$ we shall later also use Colombeau generalized quantities $(\Gamma_{i,kl,\varepsilon})_{\varepsilon}$ defined as follows

$$ (\Gamma_{i,kl,\varepsilon})_{\varepsilon} = (g_{im,\varepsilon} \Gamma_{km,\varepsilon}^{m})_{\varepsilon}. \qquad (1.7.32) $$

Conversely,

$$ (\Gamma_{kl,\varepsilon}^{i})_{\varepsilon} = (g_{\varepsilon}^{im} \Gamma_{m,kl,\varepsilon})_{\varepsilon}. \qquad (1.7.33) $$

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 parallel displacement. Let $(A_{i,\varepsilon})_{\varepsilon}$ and $(B_{\varepsilon}^{i})_{\varepsilon}$ be any covariant and contravariant vectors. Then from $\delta(A_{i,\varepsilon}B_{\varepsilon}^{i})_{\varepsilon} = 0_{\mathbb{R}}$, we have

$$ (B_{\varepsilon}^{i} \delta A_{i,\varepsilon}) = -(A_{i,\varepsilon} \delta B_{\varepsilon}^{i})_{\varepsilon} = (\Gamma_{kl,\varepsilon}^{i} B_{\varepsilon}^{k} A_{i,\varepsilon} dx^{l})_{\varepsilon} \qquad (1.7.34) $$

or, changing the indices,

$$ (B_{\varepsilon}^{i} \delta A_{i,\varepsilon}) = (\Gamma_{il,\varepsilon}^{k} B_{\varepsilon}^{i} A_{k,\varepsilon} dx^{l})_{\varepsilon} \qquad (1.7.35) $$

From this, by the arbitrariness of the $(B_{\varepsilon}^{i})_{\varepsilon}$ one obtains

$$ (\delta A_{i,\varepsilon})_{\varepsilon} = ((\Gamma_{il,\varepsilon}^{k} A_{k,\varepsilon})_{\varepsilon}) dx^{l} \qquad (1.7.36) $$

which determines the change in a covariant vector under a parallel displacement.

Substituting (1.7.31) and $(dA_{\varepsilon}^{i})_{\varepsilon} = ((\partial A_{\varepsilon}^{i}/\partial x^{l})_{\varepsilon})dx^{l}$ in (1.7.30), we obtain

$$ (DA_{\varepsilon}^{i})_{\varepsilon} = \left[ \left( \frac{\partial A_{\varepsilon}^{i}}{\partial x^{l}} \right)_{\varepsilon} + (\Gamma_{kl,\varepsilon}^{i} A_{\varepsilon}^{k})_{\varepsilon} \right] dx^{l}. \qquad (1.7.37) $$

## 1.8. The Colombeau Generalized Curvature Tensor.

**Remark 1.8.1.** (i) Note that the notion of the classical Rimannian curvature comes from

the study of parallel transport on a classical Rimannian manifold $(M,g)$, see Fig.1.8.1.

For

instance, if a vector is moved around a loop $\Gamma$ on the surface $\Sigma_{\Gamma} \subset M$ of a sphere keeping
parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as holonomy.

(ii) Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy.

<figure><img src="image_10.png" /><figcaption>Fig.1.8.1. Parallel transporting a vector from $A \to N \to B \to A$ yields a different vector. This failure to return to the initial vector is measured by the classical holonomy of the surface $\Sigma_\Gamma$ spanning by $\Gamma$.</figcaption></figure>

**Remark 1.8.2.** Let $(M, g)$ be a semirimannian manifold. Let $\Gamma_{\bar{x}^0}$ be infinitesimal closed contour and let $\Sigma_\Gamma \subset M$ be the corresponding surface spanning by $\Gamma$, see Fig.1.8.1. We assume now that christoffel symbols $\Gamma^i_{kl}(\hat{x})$ are smooth on $\Sigma_\Gamma \cup \Gamma$. The classical formula for the change in a smooth vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$ reads [4]:

$$ \Delta A_k(\Gamma) = \oint_{\Gamma} \delta A_k = \oint_{\Gamma} \Gamma^i_{kl}(\hat{x}) A_k dx^l. \qquad (1.8.1) $$

We remind now the classical Stokes' theorem.

**Theorem.1.8.1.** (Stokes' theorem) If $\omega$ is a smooth $(n-1)$-form with compact support on smooth $n$-dimensional manifold $\Omega$ with-boundary $\partial\Omega$ of $\Omega$ given the induced orientation, and $i: \partial\Omega \hookrightarrow \Omega$ is the inclusion map, then

$$ \int_{\Omega} d\omega = \int_{\partial\Omega} i^*\omega. \qquad (1.8.2) $$

**Remark 1.8.2.** Conventionally, $\int_{\partial\Omega} i^*\omega$ is abbreviated as $\int_{\partial\Omega} \omega$, since the pullback of a differential form by the inclusion map is simply its restriction to its domain $i^*\omega = \omega|_{\partial\Omega}$. Here $d$ is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as $\oint_{\partial\Omega} \omega$ to stress the fact that the $(n-1)$-manifold $\partial\Omega$ has no boundary.
For the further transformation of the integral (1.8.1), we must note the following. The values of the vector $A_i$ 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 $A_i$ at points inside the infinitesimal contour $\Gamma$ as being uniquely determined by their values on the contour itself by the formulas

$$ \delta A_i(x) = \Gamma_{il}^n(x) A_n(x) dx^l, \qquad (1.8.3) $$

i.e.,by the derivatives

$$ \frac{\partial A_i(x)}{\partial x^l} = (\Gamma_{il}^n(x) A_n(x)). \qquad (1.8.4) $$

Now applying **classical Stokes' theorem** (see Theorem 1.8.1) to the integral (1.8.1) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f^{im})$, we get:

$$ \begin{aligned} \Delta A_k &= \frac{1}{2} \left[ \frac{\partial(\Gamma_{km}^i(x)A_i(x))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(x)A_i(x))}{\partial x^m} \right] \Delta f^{im} = \\ &= \frac{1}{2} \left[ A_i(x) \left( \frac{\partial(\Gamma_{km}^i(x))}{\partial x^l} \right)_\epsilon - A_i(x) \frac{\partial(\Gamma_{kl}^i(x))}{\partial x^m} + \left( \frac{\partial A_i(x)}{\partial x^l} \right) \Gamma_{km}^i(x) - \left( \frac{\partial A_i(x)}{\partial x^m} \right) \Gamma_{kl}^i(x) \right] \Delta f^{im}. \end{aligned} \qquad (1.8.5) $$

**Remark 1.8.3.** Note that: (i) the regularity condition in Stokes' theorem, i.e. $\omega$ is a **smooth** $(n-1)$-form, **essentially important** and without this condition the Stokes' theorem is no longer holds. However in physical literature the regularity condition usually missing in formulation of the Stokes' theorem, see for example [4].

(ii) Obviously without Stokes' theorem impossible to derive the Eq.(1.8.5) and therefore

the expression in the RHS of the Eq.(1.8.5) without the regularity condition of the functions

$\Gamma_{km}^n(x)$, does not make any rigorous mathematical sense, i.e. the Eq.(1.8.5) becomes to absurdum.

Substituting now the values of the derivatives (1.8.4) into Eq.(1.8.5), we get

$$ \Delta A_k = \frac{1}{2} R_{klm}^i(x) A_i(x) \Delta f^{im}, \qquad (1.8.6) $$

where $R_{klm}^i(x)$ is a tensor field of the fourth rank:

$$ R_{klm}^i(x) = \frac{\partial(\Gamma_{km}^i(x))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(x))}{\partial x^m} + \Gamma_{ni}^i(x)\Gamma_{km}^n(x) - \Gamma_{nm}^i(x)\Gamma_{kl}^n(x). \qquad (1.8.7) $$

**Definition 1.8.1.** The tensor field $R_{klm}^i(x)$ is called the classical curvature tensor or the classical Riemann tensor.

The classical Riemann tensor that is a tensorial **measure** of the classical holonomy.

**Definition 1.8.2.** Let $(M,g)$ be a nonclassical semirimannian manifold, i.e. the manifold endowed on the tangent bundle with a symmetric bilinear form which is allowed to become **degenerate (singular)**. Let $\Sigma_\Gamma \subset M$ be the surface spanning by $\Gamma$ and let $\hat{x}^0$ be
a

point such that $\hat{x}^0 \in \Sigma_\Gamma, \hat{x}^0 \notin \Gamma$. Assume that the classical Levi-Civitá connection is
available on $\Sigma_\Gamma \setminus \{\hat{x}^0\}$.

(i) We will say that $\hat{x}^0$ is a singular point if the classical Levi-Civitá connection is not available on $\{\hat{x}^0\}$, see Fig.1.8.2.

(ii) We will say that a surface Σ<sub>Γ</sub> is a singular surface if Σ<sub>Γ</sub> contains at least one singular

point, see Fig.1.8.2.

(iii) We will say that the surface Σ<sub>Γ</sub> admit the *classical tensorial measure of holonomy*
(or admit the classical Riemann tensor) iff the Eq.(1.8.6) and Eq.(1.8.7) holds.

<figure><img src="image_11.png" /><figcaption>Fig.1.8.2.Infinitesimal closed contour Γ
and corresponding singular surface Σ<sub>Γ</sub> ⊃ ˆx<sup>0</sup>
spanning by Γ.</figcaption></figure>

Due to the degeneracy of the metric (1.10.12)

at $r = 2m, r = 0,$

the classical Levi-Civitá connection $\Gamma_{kj}^{+l}(\{\}) =$
$= \frac{1}{2}[g^{lm}(\{\})][(g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})]$
is available only on $\mathbb{R}_{+}^{3}\{r = 2m, r = 0\}$.
<figure><img src="image_12.png" /><figcaption>Fig.1.8.3. Infinitesimal closed contour Γ and corresponding singular surface $\Sigma^*=\Sigma_\Gamma \ni \hat{x}^0$ with singular point $\hat{x}^0$ (Rindler horizon) in Rindler space-time:
$$ds^2 = -\alpha R^2 d\eta^2 + dR^2.$$</figcaption></figure>

**Remark 1.8.4.** Obviously the surface $\Sigma_\Gamma$ admit the classical tensorial measure of holonomy if there is no singular points $\hat{x}^0 \in \Sigma_\Gamma$. Thus in order to avoid difficulty which

arises from singular points one needs to replace the definition of holonomy (1.8.1) by the definition appropriate for the case of the singular surfaces

**Remark 1.8.5.** Let $M$ be a separable, smooth orientable Hausdorff manifold of dimension

$n$ endowed with Colombeau generalized metric tensor $((g_{i,j,\varepsilon}(\hat{x}))_{ij})_\varepsilon \in \mathcal{G}_0^2(M)$ whose determinant $(\det(g_{i,j,\varepsilon}))_\varepsilon$ is invertible in $\mathcal{G}(M)$, see subsection 2.1 and [15],[16],[17],[35].

**Remark 1.8.6.** (i) Let $(M, g_{i,j,0}(\hat{x}))$ be a semirimannian manifold endowed with $\mathbb{R}$-valued metric tensor defined by the formula: $g_{i,j,0}(\hat{x}) = ((g_{i,j,\varepsilon}(\hat{x}))_{ij})_\varepsilon|_{\varepsilon=0}$.

(ii) We assume now that $(M, g_{i,j,0}(\hat{x}))$ is a nonclassical semirimannian manifold, i.e. the manifold endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate (singular).

**Example 1.8.1.** For instance the christoffel symbols $\Gamma^i_{kl,0}(\hat{x}, \hat{x}^0)$ corresponding to the metric tensor $g_{i,j,0}(\hat{x})$ become infinite at some singular point $\hat{x}^0$ by formulae

$$ \left\{ \begin{array}{l} \Gamma^i_{kl,0}(\hat{x}, \hat{x}^0) \propto \Xi_{kl}(\hat{x})(x_i - x_i^0)^{-\delta}, \delta \ge 1 \\ \Xi_{kl}(\hat{x}) \in C^\infty(\Sigma_{\hat{x}^0}). \end{array} \right. \qquad (1.8.8) $$

It follows from Eq. (1.8.8) the Levi-Civita connection is not available at point $\hat{x}^0$.

Let $\Gamma_{\hat{x}^0} \ni \hat{x}^0$ be infinitesimal closed contour and let $\Sigma_{\Gamma_{\hat{x}^0}} \subset M$ be the corresponding surface spanning by $\Gamma_{\hat{x}^0}$, see Fig.1.8.4.
<figure><img src="image_13.png" /><figcaption>Fig.1.8.4. Singular point at BH horizon
{r = 2m} and corresponding singular point.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig.1.8.4. Singular point at Rindler horizon</figcaption></figure>

$$ds^2 = -\alpha R^2 d\eta^2 + dR^2.$$

**Remark 1.8.7.** The classical formula (1.8.1) for the change in a smooth vector $A_i(\hat{x})$ after

parallel displacement around infinitesimal closed contour $\Gamma_{\hat{x}^0}$ (see Fig.1.8.2) reads:

$$\Delta A_k(\Gamma_{\hat{x}^0}) = \oint_{\Gamma_{\hat{x}^0}} \delta A_k = \oint_{\Gamma_{\hat{x}^0}} \Gamma_{kl}^i(\hat{x}, \hat{x}^0) A_k dx^l. \qquad (1.8.9)$$

Obviously the differential form $\Gamma_{kl}^i(\hat{x})A_k dx^l$ is not locally integrable in neighborhood of the point $\hat{x}^0 \in \Sigma_\Gamma$ and therefore $\Delta A_k(\Gamma_{\hat{x}^0}) = \infty$.

**Remark 1.8.8.** In order to avoid these difficulties with divergence $\Delta A_k(\Gamma_{\hat{x}^0}) = \infty$, etc. we

consider the canonical imbedding $(M, g_{i,j,0}(\hat{x})) \hookrightarrow (M, (g_{i,j,\varepsilon}(\hat{x}))_\varepsilon)$, and we extend now the

classical formula (1.8.1) from a nonclassical semirimannian manifold $(M, g_{i,j,0}(\hat{x}))$ up to Colombeau manifold $(M, (\det(g_{i,j,\varepsilon}))_\varepsilon)$ in natural way and obtain the formula for the Colombeau generalized change in a vector after parallel displacement around any infinitesimal closed contour $\Gamma$. This generalized change $(\Delta A_{k,\varepsilon})_\varepsilon \in \tilde{\mathbb{R}}$ can clearly be
written

in the following form

$$ (\Delta A_{k,\varepsilon})_{\varepsilon} = \left( \oint_{\Gamma_{\hat{x}^0}} \delta A_{k,\varepsilon} \right)_{\varepsilon}, \qquad (1.8.10) $$

where the Colombeau integral is taken over the given contour $\Gamma_{\hat{x}^0}$.

**Definition 1.8.3. (i)** Let $(M, (g_{\varepsilon})_{\varepsilon})$ be the Colombeau generalized semirimannian manifold,

and let $\Gamma_{\hat{x}^0}$ be infinitesimal closed contour such that $\hat{x}^0 \in \Gamma_{\hat{x}^0} \subset M$. Let $\Theta(\hat{x}^0)$ be a closed

infinitesimal neighborhood of $\hat{x}^0$, then we abbreviate $\delta_{\hat{x}^0} \triangleq \Gamma_{\hat{x}^0} \cap \Theta(\hat{x}^0)$. We will be say

that a point $\hat{x}^0 \in \Gamma_{\hat{x}^0}$ is a singular pont of the Colombeau generalized manifold $(M, (g_{\varepsilon})_{\varepsilon})$ if

$$ (\Delta A_{k,\varepsilon}(\Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}))_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{inf}} = \tilde{\mathbb{R}} \setminus \tilde{\mathbb{R}}_{\text{fin}} \quad \text{and} \quad \Delta A_k(\Gamma_{\hat{x}^0} \setminus \delta_{\hat{x}^0}) \in \mathbb{R}, \text{i.e. the quantity} \\ (\Delta A_{k,\varepsilon}(\Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}))_{\varepsilon} $$

is infinite large Colombeau generalized number and the quantity $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \delta_{\hat{x}^0})$ is finite.

(ii) We will be say that a closed contour is a singular contour $\Gamma_{\hat{x}^0}$ if it contains at least one

singular pont $\hat{x}^0 \in \Gamma_{\hat{x}^0}$, see Fig.1.8.2.-Fig.1.8.3.

**Definition 1.8.4. (i)** We will be say that a semirimannian manifold is a singular manifold

if there exists at least one singular (isolated) pont $\hat{x}^0 \in M$.

Substituting now in place of $(\delta A_{k,\varepsilon})_{\varepsilon}$ the expression (1.7.36), we get

$$ (\Delta A_{k,\varepsilon}(\hat{x}, \hat{x}^0))_{\varepsilon} = \left( \oint_{\Gamma} \Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}^0) A_i(\hat{x}) dx^l \right)_{\varepsilon} \in \tilde{\mathbb{R}}, \qquad (1.8.11) $$

where for any $i, k, l = 0, 1, 2, 3 : (\Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}^0))_{\varepsilon} \in \mathcal{G}(\mathbb{R}^4), \hat{x} = x = (x^0, x^1, x^2, x^3), A_i(x) \in \mathcal{D}(G)$ and where $\hat{x}^0 \notin \Gamma, \hat{x}^0 \in \Sigma_{\Gamma} \subset G \subset \mathbb{R}^4$. Note that the vector $A_i$ which appears in the integrand obviously changes as we move along the contour $\Gamma$.

**Definition 1.8.5.** We will be say that generalized change $(\Delta A_{k,\varepsilon}(\hat{x}, \hat{x}^0))_{\varepsilon}$ exists in the sense

of the Schwartz distributions with compact support if for any $A_i(\hat{x}) \in \mathcal{D}(G)$ the limit: $\lim_{\varepsilon \to 0} \Delta A_{k,\varepsilon}(\hat{x}, \hat{x}^0)$ exists in $\mathcal{D}'(G)$, i.e. for any $g(\hat{x}) \in \mathcal{D}(G)$, where $\hat{x} \in G$ the following limit

exists

$$ \lim_{\varepsilon \to 0} \Delta A_{k,\varepsilon}(\hat{x}, \hat{x}^0) = \lim_{\varepsilon \to 0} d^4 y \left( \oint_{\Gamma} \Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}^0) A_i(\hat{x}) g(\hat{x}) dx^l \right) \qquad (1.8.12) $$

Of course in this case obviously $\text{cl}[ (\Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}^0))_{\varepsilon} ] \in \iota(\mathcal{D}'(G)) \cap \mathcal{G}(\mathbb{R}^4)$ where $\iota$ is an imbedding the Schwartz distributions $\mathcal{D}'(G)$ into the full Colombeau algebra $\mathcal{G}(\mathbb{R}^4)$ : $\iota : \mathcal{D}'(G) \hookrightarrow \mathcal{G}(\mathbb{R}^4)$.

For the further transformation of this Colombeau integral (1.8.11), we must note the following. The values of the vector $A_i$ 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 $(A_{i,\epsilon})_{\epsilon}$ at points inside the infinitesimal contour $\Gamma$ as being uniquely determined by their values on the contour itself by the formulas

$$ (\delta A_{i,\epsilon}(\hat{x}))_{\epsilon} = (\Gamma_{il,\epsilon}^{n}(\hat{x}, \hat{x}^0) A_{n,\epsilon}(\hat{x}) dx^l)_{\epsilon}, \qquad (1.8.13) $$

i.e., by the Colombeau derivatives

$$ \left( \frac{\partial A_{i,\epsilon}(\hat{x})}{\partial x^l} \right)_{\epsilon} = (\Gamma_{il,\epsilon}^{n}(\hat{x}, \hat{x}^0) A_{n,\epsilon}(\hat{x}))_{\epsilon}. \qquad (1.8.14) $$

Now applying generalized Stokes' theorem (see Theorem 1.8.2 below) to the Colombeau integral (1.8.11) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f_{\epsilon}^{im})_{\epsilon}$, we get:

$$ \begin{aligned} (\Delta A_{k,\epsilon})_{\epsilon} &= \frac{1}{2} \left[ \left( \frac{\partial(\Gamma_{km,\epsilon}^i(\hat{x}, \hat{x}^0) A_i(\hat{x}))}{\partial x^l} \right)_{\epsilon} - \left( \frac{\partial(\Gamma_{kl,\epsilon}^i(\hat{x}, \hat{x}^0) A_i(\hat{x}))}{\partial x^m} \right)_{\epsilon} \right] (\Delta f_{\epsilon}^{im})_{\epsilon} \\ &= \frac{1}{2} \left[ A_i(\hat{x}) \left( \frac{\partial(\Gamma_{km,\epsilon}^i(\hat{x}, \hat{x}^0))}{\partial x^l} \right)_{\epsilon} - A_i(\hat{x}) \left( \frac{\partial(\Gamma_{kl,\epsilon}^i(\hat{x}, \hat{x}^0))}{\partial x^m} \right)_{\epsilon} + \right. \\ & \qquad \left. \left( \frac{\partial A_i(\hat{x})}{\partial x^l} \right)_{\epsilon} (\Gamma_{km,\epsilon}^i(\hat{x}, \hat{x}^0))_{\epsilon} - \left( \frac{\partial A_i(\hat{x})}{\partial x^m} \right)_{\epsilon} (\Gamma_{kl,\epsilon}^i(\hat{x}, \hat{x}^0))_{\epsilon} \right] (\Delta f_{\epsilon}^{im})_{\epsilon}. \end{aligned} \qquad (1.8.15) $$

**Definition 1.8.6.** Colombeau generalized $k$-form $(\omega_{\epsilon})_{\epsilon}$ on a differentiable manifold $M$ is a

smooth section of the bundle of alternating Colombeau generalized $k$-tensors on $M$. Equivalently, $(\omega_{\epsilon})_{\epsilon}$ associates to each $x \in M$ an alternating Colombeau generalized $k$-tensor $(\omega_{x,\epsilon})_{\epsilon}$, in such a way that in any chart for $M$, the coefficients $(\omega_{i_1...i_k,\epsilon})_{\epsilon}$ are Colombeau generalized functions.

**Theorem 1.8.2.** (Generalized Stokes' Theorem) Let $(\omega_{\epsilon})_{\epsilon}$ be Colombeau generalized differential form. Then the Colombeau integral of a differential form $(\omega_{\epsilon})_{\epsilon}$ over the boundary of some orientable manifold $\Sigma \subset M$ is equal to the integral of its exterior Colombeau derivative $(d\omega_{\epsilon})_{\epsilon}$ over the whole of $\Sigma$, i.e.,

$$ \int_{\partial\Sigma} (\omega_{\epsilon})_{\epsilon} = \left( \int_{\partial\Sigma} \omega_{\epsilon} \right)_{\epsilon} = \left( \int_{\Sigma} d\omega_{\epsilon} \right)_{\epsilon} = \int_{\Sigma} (d\omega_{\epsilon})_{\epsilon}. \qquad (1.8.16) $$

Proof. Immediately from the classical Stokes' Theorem and definitions.

**Example 1.8.2.** For example, for the integral of Colombeau generalized vector $(A_{i,\epsilon}(x))_{\epsilon}$ we have

$$ \begin{aligned} \left( \oint_{\Gamma} A_{i,\epsilon} dx^i \right)_{\epsilon} &= \left( \int_{\Sigma} df^{ki} \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_{\epsilon} = \frac{1}{2} \left( \iint \left[ \left( df_{\epsilon}^{ki} \right)_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} - \frac{\partial A_{i,\epsilon}}{\partial x^k} \right) \right)_{\epsilon} = \\ & \frac{1}{2} \iint \left[ \left( df_{\epsilon}^{ki} \right)_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} - \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_{\epsilon} = \frac{1}{2} \iint \left[ \left( df_{\epsilon}^{ki} \right)_{\epsilon} \right] \left[ \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} \right)_{\epsilon} - \left( \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_{\epsilon} \right], \end{aligned} \qquad (1.8.17) $$

where $\Gamma = \partial\Sigma$ and $(df_{\epsilon}^{ki})_{\epsilon} = (dx_{\epsilon}^{i} dx_{\epsilon}^{k})_{\epsilon} - (dx_{\epsilon}^{k} dx_{\epsilon}^{i})_{\epsilon}$ is the infinitesimal element of surface which is given by the antisymmetric Colombeau generalized tensor of second
rank $(df_{\varepsilon}^{ki})_{\varepsilon}$.

Substituting the values of the derivatives (1.4.3) into Eq.(1.4.4), we get

$$ (\Delta A_{k,\varepsilon})_{\varepsilon} = \frac{1}{2} (R_{klm,\varepsilon}^{i}(x) A_{i,\varepsilon}(x) \Delta f_{\varepsilon}^{lm})_{\varepsilon}, \quad (1.8.18) $$

where $(R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon}$ is a Colombeau generalized tensor field of the fourth rank:

$$ (R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon} = \left( \frac{\partial(\Gamma_{km,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))}{\partial x^{l}} \right)_{\varepsilon} - \left( \frac{\partial(\Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))}{\partial x^{m}} \right)_{\varepsilon} + (\Gamma_{ni,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}) \Gamma_{km,\varepsilon}^{n}(\hat{x}, \hat{x}^{0}))_{\varepsilon} - (\Gamma_{nm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}) \Gamma_{kl,\varepsilon}^{n}(\hat{x}, \hat{x}^{0}))_{\varepsilon}. \quad (1.8.19) $$

**Definition 1.8.7.** The tensor field $(R_{kim,\varepsilon}^{l}(\hat{x}, \hat{x}^{0}))_{\varepsilon}$ is called the distributional curvature tensor

or the distributional Riemann tensor.

**Remark 1.8.9.** Note that in general case for any $i, k, l = 0, 1, 2, 3$ : $cl[(R_{klm,\varepsilon}^{i}(x))_{\varepsilon}] \in G(\mathbb{R}^{4})$.

**Definition 1.8.8.** We will say that the distributional Riemann tensor $(R_{klm,\varepsilon}^{i}(x, \hat{x}^{0}))_{\varepsilon}$ exists in

the sense of the Schwartz distributions if for any $i, k, l = 0, 1, 2, 3$ and for any $A_{i}(x) \in D(G)$

the limit exists

$$ \lim_{\varepsilon \to 0} \int_G R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}) A_{i}(x) d^{4}x. \quad (1.8.20) $$

**Definition 1.8.9.** We will say that the distributional Riemann tensor $(R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon}$ exists in

the classical sense at point $\tilde{x} \in \mathbb{R}^4$ if there exists standard part of point value of Colombeau generalized function $(R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon}$ at point $\tilde{x} \in \mathbb{R}^4$, i.e.

$$ st(cl[(R_{klm,\varepsilon}^{i}(\tilde{x}, \tilde{x}^{0}))_{\varepsilon}]) \in \mathbb{R}. $$

From the expression (1.8.19) it follows directly that $\forall x \in \mathbb{R}$ the distributional curvature tensor is antisymmetric in the indices $l$ and $m$:

$$ (R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon} = \tilde{\mathbb{R}} - (R_{kml,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon} \quad (1.8.21) $$

and therefore for any Colombeau generalized vector $(\hat{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{fin}}^4$ the following identity holds

$$ (R_{klm,\varepsilon}^{i}(\hat{x}_{\varepsilon}, \hat{x}^{0}))_{\varepsilon} = \tilde{\mathbb{R}} - (R_{kml,\varepsilon}^{i}(\hat{x}_{\varepsilon}, \hat{x}^{0}))_{\varepsilon}. \quad (1.8.22) $$

Obviously the following identity holds

$$ (R_{kim,\varepsilon}^{l}(\hat{x}, \hat{x}^{0}))_{\varepsilon} + (R_{mkl,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon} + (R_{lmk,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon} = \tilde{\mathbb{R}} \quad 0_{\tilde{\mathbb{R}}} \quad (1.8.23) $$

and therefore for any Colombeau generalized vector $(\hat{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{fin}}^4$ the following identity holds

$$ (R_{kim,\varepsilon}^{l}(\hat{x}_{\varepsilon}, \hat{x}^{0}))_{\varepsilon} + (R_{mkl,\varepsilon}^{i}(\hat{x}_{\varepsilon}, \hat{x}^{0}))_{\varepsilon} + (R_{lmk,\varepsilon}^{i}(\hat{x}_{\varepsilon}, \hat{x}^{0}))_{\varepsilon} = \tilde{\mathbb{R}} \quad 0_{\tilde{\mathbb{R}}} \quad (1.8.24) $$

In addition to the mixed distributional curvature tensor $(R_{klm,\varepsilon}^{i}(\hat{x}, \hat{x}^{0}))_{\varepsilon}$, one also uses the covariant distributional curvature tensor

$$ (R_{iklm,\varepsilon}(\hat{x}, \hat{x}^{0}))_{\varepsilon} = (g_{in,\varepsilon}(\hat{x}, \hat{x}^{0}) R_{klm,\varepsilon}^{n}(\hat{x}, \hat{x}^{0}))_{\varepsilon} = \tilde{\mathbb{R}} ((g_{in,\varepsilon}(\hat{x}, \hat{x}^{0}))_{\varepsilon}) ((R_{klm,\varepsilon}^{n}(\hat{x}, \hat{x}^{0}))_{\varepsilon}) \quad (1.8.25) $$
and therefore for any Colombeau generalized vector $(\hat{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{fin}}^4$ the following identity holds

$$ (R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = (g_{in,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0) R_{klm,\varepsilon}^n(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} \qquad (1.8.26) \\ ((g_{in,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon})((R_{klm,\varepsilon}^n(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}) $$

Obviously by means of simple calculation the following expressions for $(R_{iklm,\varepsilon}((x, \hat{x}^0)))_{\varepsilon}$ holds

$$ \frac{1}{2} \left( \left( \left( \frac{\partial^2 g_{im,\varepsilon}(\hat{x}, \hat{x}^0)}{\partial x^k \partial x^l} \right)_\varepsilon + \left( \frac{\partial^2 g_{kl,\varepsilon}(\hat{x}, \hat{x}^0)}{\partial x^i \partial x^m} \right)_\varepsilon - \left( \frac{\partial^2 g_{il,\varepsilon}(\hat{x}, \hat{x}^0)}{\partial x^k \partial x^m} \right)_\varepsilon - \right. \right. \\ \left. \left. - \left( \frac{\partial^2 g_{km,\varepsilon}(\hat{x}, \hat{x}^0)}{\partial x^i \partial x^l} \right)_\varepsilon + \right. \right. \\ \left. \left. + ((g_{np,\varepsilon}(\hat{x}, \hat{x}^0))_{\varepsilon}) [ (\Gamma_{kl,\varepsilon}^n(\hat{x}, \hat{x}^0) \Gamma_{im,\varepsilon}^p(\hat{x}, \hat{x}^0))_{\varepsilon} - (\Gamma_{km,\varepsilon}^n(\hat{x}, \hat{x}^0) \Gamma_{il,\varepsilon}^p(\hat{x}, \hat{x}^0))_{\varepsilon} ] \right) \right) \right) \qquad (1.8.27) $$

and therefore for any Colombeau generalized vector $(\hat{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{fin}}^4$ the following identity holds

$$ \frac{1}{2} \left( \left( \left( \frac{\partial^2 g_{im,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0)}{\partial x^k \partial x^l} \right)_\varepsilon + \left( \frac{\partial^2 g_{kl,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0)}{\partial x^i \partial x^m} \right)_\varepsilon - \left( \frac{\partial^2 g_{il,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0)}{\partial x^k \partial x^m} \right)_\varepsilon \right. \right. \\ \left. \left. - \left( \frac{\partial^2 g_{km,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0)}{\partial x^i \partial x^l} \right)_\varepsilon \right) + \right. \\ \left. ((g_{np,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}) [ (\Gamma_{kl,\varepsilon}^n(\hat{x}_{\varepsilon}, \hat{x}^0) \Gamma_{im,\varepsilon}^p(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} - (\Gamma_{km,\varepsilon}^n(\hat{x}_{\varepsilon}, \hat{x}^0) \Gamma_{il,\varepsilon}^p(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} ] \right) \right) \qquad (1.8.28) $$

From this expressions (1.8.28) it follows

$$ (R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} - (R_{kilm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} - (R_{ikml,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}, \qquad (1.8.28) \\ (R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} (R_{lmik,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}. $$

For $(R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}$ and for any Colombeau generalized vector $(\hat{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{fin}}^4$ the following identities holds

$$ (R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} (R_{lmik,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}. \qquad (1.8.29) $$

For $(R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}$ the following identities holds

$$ (R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} + (R_{imkl,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} + (R_{ilmk,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} 0_{\tilde{\mathbb{R}}} \qquad (1.8.30) $$

The generalized Bianchi identity holds

$$ (R_{ikl;m,\varepsilon}^n(\hat{x}, \hat{x}^0))_{\varepsilon} + (R_{imkl;l,\varepsilon}^n(\hat{x}, \hat{x}^0))_{\varepsilon} + (R_{ilm;k,\varepsilon}(\hat{x}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} 0_{\tilde{\mathbb{R}}} \qquad (1.8.31) $$

and for any Colombeau generalized vector $(\hat{x}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\text{fin}}^4$ the following identities holds

$$ (R_{ikl;m,\varepsilon}^n(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} + (R_{imkl;l,\varepsilon}^n(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} + (R_{ilm;k,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon} = \tilde{\mathbb{R}} 0_{\tilde{\mathbb{R}}}. \qquad (1.8.32) $$

From the Colombeau generalized curvature tensor we can, by contraction, construct Colombeau generalized tensor of the second rank. This contraction can be carried out in only one way: contraction of the Colombeau generalized tensor $(R_{iklm,\varepsilon}(\hat{x}_{\varepsilon}, \hat{x}^0))_{\varepsilon}$ on the
indices i and k or l and m gives zero because of the antisymmetry in these indices, while contraction on any other pair always gives the same result, except for sign. We define the Colombeau generalized tensor $(R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε$ (the generalized Ricci tensor) as

$$ (R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε = (g^l_ε R_{ilmk,ε}(\hat{x}_ε, \hat{x}^0))_ε = (R^l_{ilk,ε}(\hat{x}_ε, \hat{x}^0))_ε. \qquad (1.8.33) $$

From Eq.(1.8.19) we get

$$ (R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε = \left( \frac{\partial(\Gamma^l_{ik,ε}(\hat{x}_ε, \hat{x}^0))}{\partial x^l} \right)_ε - \left( \frac{\partial(\Gamma^l_{il,ε}(\hat{x}_ε, \hat{x}^0))}{\partial x^m} \right)_ε + (\Gamma^l_{ik,ε}(\hat{x}_ε, \hat{x}^0)\Gamma^m_{lm,ε}(x, \hat{x}^0))_ε - (\Gamma^m_{il,ε}(\hat{x}_ε, \hat{x}^0)\Gamma^l_{km,ε}(\hat{x}_ε, \hat{x}^0))_ε. \qquad (1.8.33) $$

This Colombeau generalized tensor $(R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε$ is symmetric:

$$ (R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε = (R_{ki,ε}(\hat{x}_ε, \hat{x}^0))_ε. \qquad (1.8.34) $$

Contracting $(R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε$, we obtain the invariant

$$ (R(\hat{x}_{\epsilon}, \hat{x}^0))_{\epsilon} = (g^{ik}_{\epsilon}(\hat{x}_{\epsilon}, \hat{x}^0) R_{ik,\epsilon}(\hat{x}_{\epsilon}, \hat{x}^0))_{\epsilon} \qquad (1.8.35) $$

which is called the Colombeau generalized scalar curvature. Finally, contracting $(R_{ik,ε}(\hat{x}_ε, \hat{x}^0))_ε$, we obtain the invariant

$$ (R(\hat{x}_{\epsilon}, \hat{x}^0))_{\epsilon} = (g^{ik}_{\epsilon}(\hat{x}_{\epsilon}, \hat{x}^0) R_{ik,\epsilon}(\hat{x}_{\epsilon}, \hat{x}^0))_{\epsilon} \qquad (1.8.36) $$

which is the point value at point $(\hat{x}_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_{\text{fin}}^4$ of the Colombeau generalized scalar curvature $(R(\hat{x}, \hat{x}^0))_\epsilon$. We remind that the point value at point $(\hat{x}_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_{\text{fin}}^4$ of the Colombeau generalized scalar curvature $(R(\hat{x}_\epsilon, \hat{x}^0))_\epsilon$ is $\text{cl}[(R(\hat{x}_\epsilon, \hat{x}^0))_\epsilon] \in \tilde{\mathbb{R}}$.

## 1.9. Generalized Einstein's field equations

The action functional for the gravitational field reads [37]:

$$ \left( \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon}. \qquad (1.9.1) $$

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

$$ \left( \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon} = \left( \int G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon} + \left( \int \frac{\partial(\sqrt{-g_{\epsilon}} w_{\epsilon}^i)}{\partial x^i} d\Omega \right)_{\epsilon}, \qquad (1.9.2) $$

where $(G_\epsilon)_\epsilon$ contains only the tensor $(g_{ik,\epsilon})_\epsilon$ and its first derivatives, and the integrand of the second integral has the form of a divergence of a certain quantity $(w_\epsilon^i)_\epsilon$. 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

$$ \delta\left(\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega\right)_{\epsilon} = \left(\delta\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega\right)_{\epsilon} = \left(\delta\int G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega\right)_{\epsilon}. \qquad (1.9.3) $$
The left side is Colombeau scalar; therefore the expression on the right is also

Colombeau scalar (the quantity $(G_\epsilon)_\epsilon$ itself is, of course, not Colombeau scalar). The quantity $(G_\epsilon)_\epsilon$ satisfies the condition imposed above, since it contains only the $(g_{ik,\epsilon})_\epsilon$ and its Colombeau derivatives. Thus finally we obtain

$$ \delta S[(g_\epsilon)_\epsilon] = -\frac{c^3}{16\pi\kappa} \left( \delta \int G_\epsilon \sqrt{-g_\epsilon} d\Omega \right)_\epsilon = -\frac{c^3}{16\pi k} \left( \delta \int R_\epsilon \sqrt{-g_\epsilon} d\Omega \right)_\epsilon. \quad (1.9.4) $$

The constant $\kappa$ is called the gravitational constant. The dimensions of $\kappa$ follow from (1.9.4). Its numerical value is $\kappa = 6.67 \times 10^{-8} \text{sm}^3 \times \text{gr}^{-1} \times \text{sec}^{-2}$.

We now proceed to the derivation of the equations of the gravitational field. These equations are obtained from the principle of least action $\delta((S_{m,\epsilon})_\epsilon + (S_{g_\epsilon})_\epsilon) = 0_{\mathbb{R}}$, where $(S_{m,\epsilon})_\epsilon$ and $(S_{g_\epsilon})_\epsilon$ are the distributional actions of the gravitational field and matter respectively. We now subject the gravitational Colombeau metric field, that is, the quantities $g_{ik}$, to variation. Calculating the variation $\delta(S_{g_\epsilon})_\epsilon$, we get

$$ \begin{aligned} \delta \left( \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon} = \left( \delta \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon} = \left( \delta \int g_{\epsilon}^{ik} R_{ik,\epsilon} \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon} = & \\ \left\{ \left( \int R_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega \right)_{\epsilon} + \left( \int R_{ik,\epsilon} g_{\epsilon}^{ik} \delta \sqrt{-g_{\epsilon}} d\Omega \right)_{\epsilon} + \left( \int g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} d\Omega \right)_{\epsilon} \right\} & (1.9.5) \\ \int \left\{ \left( R_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} \right)_{\epsilon} + \left( R_{ik,\epsilon} g_{\epsilon}^{ik} \delta \sqrt{-g_{\epsilon}} \right)_{\epsilon} + \left( g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} \right)_{\epsilon} \right\} d\Omega. & \end{aligned} $$

Thus, the variation $S[(g_\epsilon)_\epsilon]$ is equal to

$$ S[(g_\epsilon)_\epsilon] = -\frac{c^3}{16\pi\kappa} \left( \int \left\{ R_{ik,\epsilon} - \frac{1}{2} g_{ik,\epsilon} R_\epsilon \right\} \sqrt{-g_\epsilon} \delta g_\epsilon^{ik} d\Omega \right)_\epsilon. \quad (1.9.10) $$

**Remark 1.9.1.** We note that if we had started from the expression

$$ \delta S_g[(g_\epsilon)_\epsilon] = -\frac{c^3}{16\pi\kappa} \left( \delta \int G_\epsilon \sqrt{-g_\epsilon} d\Omega \right)_\epsilon \quad (1.9.11) $$

for the action of the field, then we get

$$ \delta S[(g_\epsilon)_\epsilon] = -\frac{c^3}{16\pi\kappa} \int \delta(g_\epsilon^{ik})_\epsilon d\Omega \left\{ \left( \frac{\partial \{G_\epsilon \sqrt{-g_\epsilon}\}}{\partial g_\epsilon^{ik}} \right)_\epsilon - \left( \frac{\partial}{\partial x^l} \frac{\partial \{G_\epsilon \sqrt{-g_\epsilon}\}}{\partial \frac{\partial g_\epsilon^{ik}}{\partial x^l}} \right)_\epsilon \right\}. \quad (1.9.12) $$

Comparing Eq. (1.9.12) with Eq. (1.9.10), we get

$$ (R_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} R_{\epsilon})_{\epsilon} = \left\{ \left( \frac{1}{\sqrt{-g_{\epsilon}}} \right)_{\epsilon} \right\} \left\{ \left( \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial g_{\epsilon}^{ik}} \right)_{\epsilon} - \left( \frac{\partial}{\partial x^l} \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial \frac{\partial g_{\epsilon}^{ik}}{\partial x^l}} \right)_{\epsilon} \right\}. \quad (1.9.13) $$

For the variation of the action of the matter we can write

$$ (\delta S_{m,\epsilon})_{\epsilon} = \frac{1}{2c} \left( \int T_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega \right)_{\epsilon}, \quad (1.9.14) $$

where $(T_{ik,\epsilon})_\epsilon \in \mathcal{G}(\mathbb{R}^4)$ is the generalized energy-momentum tensor of the matter
fields.

Thus, from the principle of least action

$$ \delta\{S[(g_{\epsilon})_{\epsilon}] + (S_{m,\epsilon})_{\epsilon}\} = 0_{\mathbb{R}} \qquad (1.9.15) $$

one obtains

$$ -\frac{c^3}{16\pi\kappa} \left( \int \left\{ R_{ik,\epsilon} - \frac{1}{2}g_{ik,\epsilon}R_{\epsilon} - \frac{8\pi\kappa}{c^4}T_{ik,\epsilon} \right\} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega \right)_{\epsilon} = 0_{\mathbb{R}}. \qquad (1.9.16) $$

From Eq.(1.9.16), since of the arbitrariness of the $(\delta g_{\epsilon}^{ik})_{\epsilon} \in \mathcal{G}(\mathbb{R}^4)$ finally we get

$$ (R_{ik,\epsilon})_{\epsilon} - \frac{1}{2}(g_{ik,\epsilon}R_{\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4}(T_{ik,\epsilon})_{\epsilon} \qquad (1.9.17) $$

or, in mixed components,

$$ (R_{i,\epsilon}^k) - \frac{1}{2} \delta_i^k (R_{\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} (T_{i,\epsilon}^k)_{\epsilon}. \qquad (1.9.18) $$

They are called the generalized Einstein equations.

Contracting (1.9.18) on the indices $i$ and $k$, we get

$$ (R_{\epsilon})_{\epsilon} = -\frac{8\pi\kappa}{c^4} (T_{i,\epsilon}^i)_{\epsilon} = -\frac{8\pi\kappa}{c^4} (T_{\epsilon})_{\epsilon}. \qquad (1.9.19) $$

Therefore the generalized Einstein equations of the field can also be written in the form

[37]

$$ (R_{ik,\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} \left\{ (T_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} T_{\epsilon})_{\epsilon} \right\}. \qquad (1.9.20) $$

Note that the generalized Einstein equations of the gravitational field are nonlinear Colombeau equations.

# 1.10. The densitized Einstein field equations revisited.

## 1.10.1. Remarks on the A. Einstein and N. Rosen paper from 1935.

The densitized Einstein field equations originally considered in A. Einstein and N. Rosen paper [32], see also [46]. As an example of the problem which arises from degeneracy of the metric tensor $g_{ik}$, the metric field is considered (see [32],eq.1):

$$ ds^2 = -dx_1^2 - dx_2^2 - dx_3^2 + \alpha^2 x_1^2 dx_4^2. \qquad (1.10.1) $$

The $g_{\mu\nu}$ of this field satisfy in general the equations $R_{klm}^i = 0$, and hence the equations

$$ R_{kl} = R_{klm}^m = 0. \qquad (1.10.2) $$

A. Einstein emphasized that: "The $g_{\mu\nu}$ corresponding to (1.10.1) are regular for all finite (i.e. nonzero) points of space-time. Nevertheless one cannot assert that Eqs.(1.10.2) are satisfied by (1.10.1) for all finite values of $x_1, \dots, x_4$. This is due to the fact that the determinant $g$ of the $g_{\mu\nu}$ vanishes for $x_1 = 0$. The contravariant $g^{\mu\nu}$ therefore become infinite and the tensors $R_{klm}^i$ and $R_{kl}$ take on the form 0/0. From the standpoint of Eqs.(1.10.2) the hyperplane $x_1 = 0$ then represents a singularity of the field".
We now ask whether the field law of gravitation (and later on the field law of gravitation and electricity) could not be modified in a natural way without essential change so that the solution (1.10.1) would satisfy the field equations for all finite points, i.e., also for $x_1 = 0$.

W. Mayer has called our attention to the fact that one can make $R_{kl}^i$ and $R_{kl}$ into rational functions of the $g_{\mu\nu}$, and their first two derivatives by multiplying them by suitable powers of $g$. It is easy to show that in $g^2 R_{kl}$ there is no longer any denominator. If then we replace (1.10.3) by (see [32],eq.3a):

$$R_{kl}^* = g^2 R_{kl} = 0, \qquad (1.10.3)$$

this system of equations is satisfied by (1.10.1) at all finite points. This amounts to introducing in place of the $g^{\mu\nu}$ the cofactors $[g_{\mu\nu}]$ of the $g_{\mu\nu}$ in $g$ in order to avoid the occurrence of denominators. One is therefore operating with tensor densities of a suitable weight instead of with tensors. In this way one succeeds in avoiding singularities of that special kind which is characterized by the vanishing of $g$.

**Remark 1.10.1.** Note that A. Einstein actually rejected densitized field equations by the following reason: "The solution (1) naturally has no deeper physical significance insofar as it extends into spatial infinity. It allows one to see however to what extent the regularization of the hypersurfaces $g=0$ leads to a theoretical representation of matter, regarded from the standpoint of the original theory. Thus, in the framework of the original theory one has the gravitational equations

$$R_{ik} - \frac{1}{2} g_{ik} R = -T_{ik}, \qquad (1.10.4)$$

where $T_{ik}$ is the tensor of mass or energy density. Nevertheless in physical literature the densitized Einstein field equations holds from A. Einstein time untill nowoday, see for example [46].

**Remark 1.10.2.** Note that obviously the system of equations (1.10.3) is satisfied by (1.10.1) at all finite points. Nevertheless these equations can not solved the problem since

the ancretanty 0/0 holds again in tensors $R_{klm}^i$ and $R_{kl}$ on hypersurface $x_1 = 0$.

**Remark 1.10.3.** Note that if some components of the Riemann curvature tensor $R_{klm}^i(\hat{x})$ become ancretanty 0/0 or infinite at point $\hat{x}^0$ one obtain the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood $\Omega$ of the point $\hat{x}^0 \in \Omega$, i.e. in such neighborhood $\Omega$ Riemann curvature tensor $R_{klm}^i(\hat{x})$ must be changed

by formula (1.10.7) see remark 1.10.2.

**Remark 1.10.4.** Note that in Möller's paper [38] the metric (1.10.1) has been derived in fact under abnormal assumption 0/0 = 1 without respect to Levi-Civita connection.

## 1.10.2. Remarks on Möller abnormal famous paper from 1943

Recall that the classical Cartan's structural equations show in a compact way the relation

between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities,
we

need to study the geometry of manifolds endowed on the tangent bundle with a
symmetric

bilinear form which is allowed to become <u><u>degenerate (singular)</u></u>. But if the fundamental
tensor is allowed to be degenerate (singular), there are some obstructions in
constructing

the geometric objects normally associated to the fundamental tensor. Also, local
orthonormal frames and co-frames no longer exist, as well as the metric connection
and

its curvature operator [46].

As an important example of the geometry with the fundamental tensor which is
allowed to

be degenerate, we consider now Möller's uniformly accelerated frame given by
Möller's

line element (1.10.4).Recall that Möller dealing with the following line element [38]:

$$
ds^2 = -\Delta(x)dt^2 + dx^2 + dy^2 + dz^2, \tag{1.10.4}
$$

where $\Delta(x) = (a + gx)^2$.

**Remark 1.10.3.** Of course Möller's metric (1.10.4) degenerate at Möller horizon

$x_{hor} = -a/g.$

However in contrast with A.Einstein paper [32],in famous but ubnormal paper [38]
Möller

<u>mistakenly argue</u> that metric field (1.10.4) is an global vaccuum solution of the
A.Einstein

field equations (1.10.5), i.e. the $g_{\mu\nu}$ of this field for all values of $t,x,y,z$ satisfy the
equations

$$
R_{ik} - \frac{1}{2} g_{ik} R = 0. \tag{1.10.5}
$$

**Remark 1.10.4.** In physical literature this Möller's abnormal mistake holds from
Möller's

time until nowadays.

**Remark 1.10.5.** Note that formally corresponding to the Möller's metric (1.10.4)
classical

Levi-Civita connection reads

$$
\Gamma_{44}^{1}(x) = a + gx, \Gamma_{14}^{4}(x) = \Gamma_{41}^{4}(x) = g(a + gx)^{-1} \quad (1.10.6)
$$

and therefore classical Levi-Civit'a connection (1.10.6) of course is not available at
Möller

horizon since at horizon formal expressions (1.10.6) becomes infinity:

$$
\Gamma_{14}^{4} \left(-\frac{a}{g}\right) = \Gamma_{41}^{4} \left(-\frac{a}{g}\right) = \infty. \qquad (1.10.7)
$$

**Remark 1.10.6.** Note that Möller dealing with Einstein's field equations in the following
form

$$
G_i^k = R_i^k - \frac{1}{2} \delta_i^k R = 0, \qquad (1.10.8)
$$

where $R_i^k$ is the contracted Riemann-Christoffel tensor, formally calculated by
canonical
way by using classical Levi-Civit'a connection (1.10.6) and where $R = R_i^i$. By using the following ansatz

$$ds^2 = -\Delta(x)dt^2 + dx^2 + dy^2 + dz^2, \qquad (1.10.9)$$

Möller finally obtain

$$G_2^2(x) = G_3^3(x) = -\frac{1}{2\Delta(x)} \left[ \Delta''(x) - \frac{(\Delta'(x))^2}{2\Delta(x)} \right] = -\frac{(\Delta^{1/2}(x))''}{\Delta^{1/2}(x)}. \quad (1.10.10)$$

where $\Delta'(x) = d\Delta(x)/dx$.

**Remark 1.10.7.** From Eq. (1.10.10) Möller obtain the following ordinary differential equation

$$(\Delta^{1/2}(x))'' = 0, \qquad (1.10.11)$$

since it was mistakenly assumed that $G_2^2(x)$ and $G_3^3(x)$ for all values of $x$ satisfy the equations

$$G_2^2(x) = G_3^3(x) \equiv 0. \qquad (1.10.12)$$

The equation (1.10.11) obviously has the following trivial general solution

$$\Delta(x) = (a + gx)^2. \qquad (1.10.13)$$

**Remark 1.10.8.** Note that at Möller horizon $x_{hor} = -a/g$ the functions $G_2^2(x)$ and $G_3^3(x)$ of course is not zero identically but becomes uncertainty, since

$$G_2^2(-a/g) = G_3^3(-a/g) = -\frac{([\Delta(-a/g)]^{1/2})''}{[\Delta(-a/g)]^{1/2}} = \frac{0}{0}. \qquad (1.10.14)$$

**Remark 1.10.9.** Note that at any point $x \neq -a/g$ obviously $G_2^2(x) = G_3^3(x) \equiv 0$ since at these points one obtains

$$G_2^2(x) = G_3^3(x) = -\frac{1}{2\Delta(x)} \left[ \Delta''(x) - \frac{(\Delta'(x))^2}{2\Delta(x)} \right] = -\frac{1}{2(a+gx)^2} \left[ 2g^2 - \frac{4g^2(a+gx)^2}{2(a+gx)^2} \right] = -\frac{1}{2(a+gx)^2} [2g^2 - 2g^2] = 0. \qquad (1.10.15)$$

**Remark 1.10.10.** At point $x = -a/g$ the quantity $G_2^2(-a/g)$ and $G_3^3(-a/g)$ well defined only by formal limit

$$G_2^2(-a/g) = G_3^3(-a/g) = \lim_{x \to -a/g} -\frac{0}{2(a+gx)^2} = 0. \qquad (1.10.16)$$

**Remark 1.10.11.** However in the limit $x \to -a/g$ the christoffel symbols (1.10.6) becomes infinity:
$$ \lim_{x \to -a/g} \Gamma_{14}^{4}(x) = \lim_{x \to -a/g} g(a+gx)^{-1} = \infty, \qquad (1.10.17) \\ \lim_{x \to -a/g} \Gamma_{41}^{4}(x) = \lim_{x \to -a/g} g(a+gx)^{-1} = \infty. $$

It follows from (1.10.17) at horizon $x = -a/g$ the canonical expression for the contracted Riemann-Christoffel tensor $R_i^k$ no longer holds, due to the degeneracy of (1.10.4), the Levi-Civita connection is not available at Möller horizon $x = -a/g$.

In the following subsection we resolve this tension rigorously using linear distributional geometry.

### 1.10.3. The densitized Einstein field equations revisited by using the linear distributional geometry.

In order to derive the densitized Einstein field equations rigorously we apply in this subsection the apparatus of the linear distributional geometry. Note that in linear distributional geometry one dealing exactly with Schwartz distributions with compact support but not with full algebra of Colombeau generalized functions, see for example [30].

**Remark 1.10.12.** (i) Note that the notion of the Rimannian curvature comes from the study of parallel transport on a Rimannian manifold, see Fig. For instance, if a vector is moved around a loop on the surface of a sphere keeping parallel throughout the motion,

then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as holonomy.

(ii) Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy.

(iii) Classical holonomy presented by the classical formula (1.10.18) for the change $\Delta A_k$

in a smooth vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$.

(iv) Note that in classical case the change $\Delta A_k(\Gamma)$ always finite, i.e. $\Delta A_k(\Gamma) < \infty$.
<figure><img src="image_15.png" /><figcaption>Fig.1.10.1.Parallel transporting a vector from $A \to N \to B \to A$ yields a different vector. This failure to return to the initial vector is measured by the holonomy of the surface spanning by $\Gamma$.</figcaption></figure>

**Remark 1.10.13.** Let $(M,g)$ be a semirimannian manifold. Let $\Gamma_{\xi^0}$ be infinitesimal closed contour and let $\Sigma_{\Gamma} \subset M$ be the corresponding surface spanning by $\Gamma$, see Fig.1.10.1. We assume now that christoffel symbols $\Gamma^i_{kl}(\hat{x})$ are smooth on $\Sigma_{\Gamma} \cup \Gamma$. The classical formula for the change in a smooth vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$ reads [4]:

$$ \Delta A_k(\Gamma) = \oint_{\Gamma} \delta A_k = \oint_{\Gamma} \Gamma^i_{kl}(\hat{x}) A_k dx^l. \qquad (1.10.18) $$

We remind now the classical Stokes' theorem.

**Theorem.1.10.1.** (Stokes' theorem) If $\omega$ is a smooth $(n-1)$-form with compact support on smooth $n$-dimensional manifold $\Omega$ with-boundary $\partial\Omega$ of $\Omega$ given the induced orientation, and $i: \partial\Omega \to \Omega$ is the inclusion map, then

$$ \int_{\Omega} d\omega = \int_{\partial\Omega} i^*\omega. \qquad (1.10.19) $$

Conventionally, $\int_{\partial\Omega} i^*\omega$ is abbreviated as $\int_{\partial\Omega} \omega$, since the pullback of a differential form by the inclusion map is simply its restriction to its domain $i^*\omega = \omega|_{\partial\Omega}$. Here $d$ is the exterior derivative, which is defined using the manifold structure only. The right-hand side is sometimes written as $\oint_{\partial\Omega} \omega$ to stress the fact that the $(n-1)$-manifold $\partial\Omega$ has no boundary.

For the further transformation of the integral (1.10.18), we must note the following. The values of the vector $A_i$ 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 $A_i$ at points inside the infinitesimal contour $\Gamma$ as
being uniquely determined by their values on the contour itself by the formulas

$$ \delta A_i(x) = \Gamma_{il}^n(x) A_n(x) dx^l, \qquad (1.10.20) $$

i.e., by the derivatives

$$ \frac{\partial A_i(x)}{\partial x^l} = (\Gamma_{il}^n(x) A_n(x)). \qquad (1.10.21) $$

Now applying **classical Stokes' theorem** (see Theorem 1.10.1) to the integral (1.10.18) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f^{im})$, we get [4]:

$$ \begin{aligned} \Delta A_k &= \frac{1}{2} \left[ \frac{\partial(\Gamma_{km}^i(x) A_i(x))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(x) A_i(x))}{\partial x^m} \right] \Delta f^{im} = \\ &= \frac{1}{2} \left[ A_i(x) \left( \frac{\partial(\Gamma_{km}^i(x))}{\partial x^l} \right)_\epsilon - A_i(x) \frac{\partial(\Gamma_{kl}^i(x))}{\partial x^m} + \left( \frac{\partial A_i(x)}{\partial x^l} \right) \Gamma_{km}^i(x) - \left( \frac{\partial A_i(x)}{\partial x^m} \right) \Gamma_{kl}^i(x) \right] \Delta f^{im}. \end{aligned} \qquad (1.10.22) $$

**Remark 1.10.13.** Note that: (i) the regularity condition in Stokes' theorem, i.e. $\omega$ is a **smooth** $(n-1)$-form is **essentially important** and without this condition this theorem is **no longer holds**.

(ii) Obviously without Stokes' theorem **impossible** to derive the Eq.(1.10.22) and therefore

the expression (1.10.24) without the regularity condition of the functions $\Gamma_{km}^n(x)$ does not

make any sense and becomes to absurdum.

Substituting now the values of the derivatives (1.10.21) into Eq.(1.10.22), we get

$$ \Delta A_k = \frac{1}{2} R_{klm}^i(x) A_i(x) \Delta f^{im}, \qquad (1.10.23) $$

where $R_{klm}^i(x)$ is a tensor field of the fourth rank:

$$ R_{klm}^i(x) = \frac{\partial(\Gamma_{km}^i(x))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(x))}{\partial x^m} + \Gamma_{ni}^i(x)\Gamma_{km}^n(x) - \Gamma_{nm}^i(x)\Gamma_{kl}^n(x). \qquad (1.10.24) $$

**Definition 1.10.3.** The tensor field $R_{klm}^i(x)$ is called the classical curvature tensor or the classical Riemann tensor.

The classical Riemann tensor that is a **tensorial measure of holonomy**.

**Definition 1.10.4.** Let $(M,g)$ be a nonclassical semirimannian manifold. Let $\Sigma_\Gamma \subset M$ be the surface spanning by $\Gamma$. We will say that the surface $\Sigma_\Gamma$ admit the **classical tensorial measure of holonomy** (or admit the classical Riemann tensor) iff the Eq.(1.10.23) and Eq.(1.10.24) holds.

**Remark 1.10.14.** Let $(M,g)$ be a nonclassical semirimannian manifold, i.e. the manifold endowed on the tangent bundle with a symmetric bilinear form which is allowed to become **degenerate (singular)**. Let $\Gamma_{\hat{x}^0} \ni \hat{x}^0$ be infinitesimal closed contour and let $\Sigma_{\Gamma_{\hat{x}^0}} \subset M$ be the corresponding surface spanning by $\Gamma_{\hat{x}^0}$, see Fig.1.10.2.

We assume now that:

(i) christoffel symbols $\Gamma_{kl}^i(\hat{x}, \hat{x}^0)$ become infinite at singular point $\hat{x}^0$ by formulae
$$ \left\{ \begin{array}{l} \Gamma_{kl}^{i}(\hat{x}, \hat{x}^{0}) \propto \Xi_{kl}(\hat{x})(x_i - x_i^{0})^{-\delta}, \delta \ge 1 \\ \Xi_{kl}(\hat{x}) \in C^{\infty}(\Sigma_{\Gamma_{\hat{x}^{0}}}) \end{array} \right. \qquad (1.10.25) $$

and (ii) $\hat{x}^0 \in \Gamma_{\hat{x}^0} \subset \Sigma_{\Gamma_{\hat{x}^0}}$. The classical formula (1.10.18) for the change in a smooth vector

$A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma_{\hat{x}^0}$ reads:

$$ \Delta A_k(\Gamma_{\hat{x}^0}) = \oint_{\Gamma_{\hat{x}^0}} \delta A_k = \oint_{\Gamma_{\hat{x}^0}} \Gamma_{kl}^i(\hat{x}, \hat{x}^0) A_k dx^l. \qquad (1.10.26) $$

Obviously the differential form $\Gamma_{kl}^i(\hat{x})A_k dx^l$ does not locally integrable in neighborhood of

the point $\hat{x}^0 \in \Sigma_{\Gamma}$ and therefore $\Delta A_k(\Gamma_{\hat{x}^0}) = \infty$.

**Remark 1.10.15.** Note that under nonregularity conditions $\Delta A_k(\Gamma_{\hat{x}^0}) = \infty$ the classical formula (1.10.26) can not define correctly the holonomy of the surface $\Sigma_{\Gamma_{\hat{x}^0}}$ spanning by

$\Gamma_{\hat{x}^0}$ since the classical holonomy becomes infinity. In order to avoid this difficultyness

one

needs to replace the classical formula (1.10.26) by the formula appropriate for singular case.

We started now from some definitions.

**Definition 1.10.5.** (i) Let $(M,g)$ be a semirimannian manifold, and let $\Gamma_{\hat{x}^0}$ be infinitesimal

closed contour such that $\hat{x}^0 \in \Gamma_{\hat{x}^0} \subset M$. Let $\Theta(\hat{x}^0)$ be a closed infinitesimal neighborhood

of $\hat{x}^0$, then we we abbreviate $\delta_{\hat{x}^0} \triangleq \Gamma_{\hat{x}^0} \cap \Theta(\hat{x}^0)$. We will be say that a point $\hat{x}^0 \in \Gamma_{\hat{x}^0}$ is a singular pont of the manifold $(M,g)$ if $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}) = \infty$ and $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \delta_{\hat{x}^0}) < \infty$.

(ii) We will be say that a closed contour is a singular contour $\Gamma_{\hat{x}^0}$ if it contains at least one

singular pont $\hat{x}^0 \in \Gamma_{\hat{x}^0}$, see Fig.1.10.2-Fig.1.10.3.

**Definition 1.10.6.** (i) We will be say that a semirimannian manifold is a singular manifold

if there exists at least one singular (isolated) pont $\hat{x}^0 \in M$.

(ii)
<figure><img src="image_16.png" /><figcaption>Fig.1.10.2. Singular point at BH horizon and corresponding singular countur.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig.1.10.2. Singular point (0,0) at Rindler horizon
$R = 0$ and corresponding singular countur.</figcaption></figure>

$$ds^2 = dR^2 - (\alpha R)^2 d\eta^2.$$

**Remark 1.10.15.** Obviously the Schwarzschild singularity $r = 0$ is singular isolated point of the Schwarzschild manifold.

**Definition 1.10.3.** Let $(M,g)$ be a nonclassical semirimannian closed manifold, and let $\Gamma_{\hat{x}^0} \subset M$ be infinitesimal closed contour such that $\hat{x}^0 \in \Gamma_{\hat{x}^0} \subset M$. We will be say that a contour $\Gamma_{\hat{x}^0}^\# = \Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}$ with deleted point $\hat{x}^0$ (see Fig.1.10.3) is a singular truncated contour of the open manifold $(M',g')$, where $M' = M \setminus M$ and $g' = g|_{M'}$ if $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}) = \infty$ and $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \delta_{\hat{x}^0}) < \infty$.

**Remark 1.10.16.** (i) Note that the Levi-Civita connection $\Gamma_{kl}^i(\hat{x}, \hat{x}^0)$ is available on any singular truncated contour $\Gamma_{\hat{x}^0}^\# = \Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}$ of the open manifold $(M', g')$ but despite this again $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}) = \infty$. (ii) Note that the semirimannian submanifold $(M', g')$ of nonclassical semirimannian closed manifold $(M,g)$ impossible treated classically, since the classical holonomy breaks down by divergence $\Delta A_k(\Gamma_{\hat{x}^0} \setminus \{\hat{x}^0\}) = \infty$.
<figure><img src="image_18.png" /><figcaption>Fig.1.10.3. Singular truncated contour</figcaption></figure>

$$ \Gamma_{\hat{x}^0}^{\#} = \Gamma_{\hat{x}^0 \setminus \{\hat{x}^0\}} $$

**Definition 1.10.3.** Let $(M,g)$ be a semirimannian manifold, and let $(M_1,g_1)$ be closed submanifold where $g_1 = g|_M$.

**Remark 1.10.17.** Let $(M,g)$ be a nonclassical semirimannian manifold, i.e. the manifold endowed on the tangent bundle with a symmetric bilinear form which is allowed to become degenerate (singular). Let $\Gamma$ be infinitesimal closed contour and let $\hat{x}^0 \in \Sigma_\Gamma \subset M, \hat{x}^0 \notin \Gamma$ be the corresponding surface spanning by $\Gamma$, see Fig.1.10.4 and Fig.1.10.5. Note that the Eq.(1.10.22) again breaks down (see Remark 1.10.13), since the regularity condition of the functions $\Gamma_{km}^n(\hat{x}, \hat{x}^0)$ are violated at point $\hat{x}^0 \in \Sigma_\Gamma$.

<figure><img src="image_19.png" /><figcaption>Fig.1.10.4. Infinitesimal closed contour $\Gamma$ and corresponding singular surface $\Sigma^*=\Sigma_\Gamma \ni \hat{x}^0$ with singular point $\hat{x}^0$ (Rindler horizon) in Rindler space-time:</figcaption></figure>

$$ ds^2 = -\alpha R^2 d\eta^2 + dR^2. $$

**Remark 1.10.18.** In order to avoid the divergence mentioned above we consider the Christoffel symbols $\Gamma_{kl}^i(\hat{x}, \hat{x}^0)$ as distributions on appropriate space of the test functions.

**Definition 1.10.4.** Schwartz distributions with compact support (Schwartz generalized functions with compact support) are a class of linear functionals that map a space of