# Distributional SAdS BH Spacetime-Induced Vacuum Dominance

Jaykov Foukzon, A Potapov, E Men'Kova

► To cite this version:

Jaykov Foukzon, A Potapov, E Men'Kova. Distributional SAdS BH Spacetime-Induced Vacuum Dominance. British Journal of Mathematics & Computer Science, 2016, 13, pp.1 - 54. 10.9734/bjmcs/2016/19235. hal-04572404

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

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# Distributional SAdS BH Spacetime-Induced Vacuum Dominance

Jaykov Foukzon, A Potapov, E Men'Kova

► To cite this version:

Jaykov Foukzon, A Potapov, E Men'Kova. Distributional SAdS BH Spacetime-Induced Vacuum Dominance. British Journal of Mathematics & Computer Science, 2016, 13, pp.1 - 54. 10.9734/bjmcs/2016/19235. hal-04572404

HAL Id: hal-04572404

https://hal.science/hal-04572404v1

Submitted on 17 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
Distributional SAdS BH Spacetime-Induced Vacuum
Dominance

Jaykov Foukzon<fnref n="1" />*, A. Potapov² and E. Menkova³

DOI:

Abstract

This paper dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [5]-[30] and [31]-[15]. The regularizations of singularities present in some Colombeau solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this paper essentially new class Colombeau solutions to Einstein 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 $r_+$: $\langle\Phi^2(r)\rangle^{-|r-r_+|^{-2}}$. A CHALLENGE TO THE BRIGHTNESS TEMPERATURE LIMIT OF THE QUASAR 3C273 explained successfully.

Keywords: Colombeau nonlinear generalized functions; Distributional Riemannian; Geometry; Distributional Schwarzschild Geometry; Schwarzschild singularity; Schwarzschild Horizon; smooth regularization; nonsmooth regularization; quantum fields; curved spacetime; vacuum fluctuations; vacuum dominance.

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

<footnote n="1">¹Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel.</footnote>

<footnote n="1">¹Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia.</footnote>

<footnote n="1">¹All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia.</footnote>

<footnote>*Corresponding author: E-mail: jaykovfoukzon@list.ru;</footnote>
Distributional SAdS BH Spacetime-Induced Vacuum
Dominance

Jaykov Foukzon<fnref n="1" />*, A. Potapov² and E. Menkova³

DOI:

**Abstract**

This paper dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [5]-[30] and [31]-[15]. The regularizations of singularities present in some Colombeau solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this paper essentially new class Colombeau solutions to Einstein 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 $r_+$: $\langle\Phi^2(r)\rangle^{-|r-r_+|^{-2}}$. A CHALLENGE TO THE BRIGHTNESS TEMPERATURE LIMIT OF THE QUASAR 3C273 explained successfully.

Keywords: Colombeau nonlinear generalized functions; Distributional Riemannian; Geometry; Distributional Schwarzschild Geometry; Schwarzschild singularity; Schwarzschild Horizon; smooth regularization; nonsmooth regularization; quantum fields; curved spacetime; vacuum fluctuations; vacuum dominance.

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

<footnote n="1">¹Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel.</footnote>

<footnote n="1">¹Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia.</footnote>

<footnote n="1">¹All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia.</footnote>

<footnote>*Corresponding author: E-mail: jaykovfoukzon@list.ru;</footnote>
**Physics and Astronomy Classification Scheme:** Primary: 04.20.Cv; secondary: 04.20.Jb, 04.20.Dw, 02.30.Sa, 02.40.Vh.

# 1 Introduction

## 1.1 Remarks on Linear Distributional Geometry in General Relativity

A degenerate (singular) semi-Riemannian manifold $(M; g)$ is a differentiable manifold $M$ endowed with a symmetric bilinear form $g \in T_2^0 M$ named metric. Note that the metric $g$ is not required to be non-degenerate. In particular, if the metric is non-degenerate, $(M; g)$ is a semi-Riemannian manifold. If in addition $g$ is positive definite, $(M; g)$ is a Riemannian manifold.

This paper dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions such that Schwarzschild solution etc. of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. Note that in physical literature these singular solutions many years were mistakenly considered as vacuum solutions of the Einstein field equations, see for example [3],[4].

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

**Remark 1.1.1.** Let $(R_{bcd,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^4)$ be Colombeau generalized function obtained using the standard definition of the Riemann curvature in a coordinate basis, i.e.

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

where $(\Gamma_{bc,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^n)$ 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 $(g_{ab,\epsilon})_\epsilon, (g_{ab}^{ab})_\epsilon \in \mathcal{G}(\mathbb{R}^4)$, $(\det(g_{ab,\epsilon}))_\epsilon \ne 0_{\mathbb{R}}$. It has been shown by many authors (see for example [22]) that under appropriate 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

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

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 $(R_{bcd,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^4)$ 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 $(R_{bcd,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^4)$ classical
**Physics and Astronomy Classification Scheme:** Primary: 04.20.Cv; secondary: 04.20.Jb, 04.20.Dw, 02.30.Sa, 02.40.Vh.

# 1 Introduction

## 1.1 Remarks on Linear Distributional Geometry in General Relativity

A degenerate (singular) semi-Riemannian manifold $(M; g)$ is a differentiable manifold $M$ endowed with a symmetric bilinear form $g \in T_2^0 M$ named metric. Note that the metric $g$ is not required to be non-degenerate. In particular, if the metric is non-degenerate, $(M; g)$ is a semi-Riemannian manifold. If in addition $g$ is positive definite, $(M; g)$ is a Riemannian manifold.

This paper dealing with Colombeau extension of the Einstein field equations using apparatus of the Colombeau generalized function [1]-[2] and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions such that Schwarzschild solution etc. of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. Note that in physical literature these singular solutions many years were mistakenly considered as vacuum solutions of the Einstein field equations, see for example [3],[4].

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

**Remark 1.1.1.** Let $(R_{bcd,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^4)$ be Colombeau generalized function obtained using the standard definition of the Riemann curvature in a coordinate basis, i.e.

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

where $(\Gamma_{bc,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^n)$ 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 $(g_{ab,\epsilon})_\epsilon, (g_{ab}^{ab})_\epsilon \in \mathcal{G}(\mathbb{R}^4)$, $(\det(g_{ab,\epsilon}))_\epsilon \ne 0_{\mathbb{R}}$. 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 Schwarzschild 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

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

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 $(R_{bcd,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^4)$ 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 $(R_{bcd,\epsilon}^a) \in \mathcal{G}_\delta(\mathbb{R}^4)$ 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 authors 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]:

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

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

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

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 $(h_\epsilon)_\epsilon = a/(\sqrt{\hat{r}^2 + \epsilon^2})_\epsilon$ and the function $\frac{\hat{x}^\alpha \hat{x}^\beta}{\hat{r}^2}$ as $\frac{\hat{x}^\alpha \hat{x}^\beta}{(\hat{r}^2 + \epsilon^2)_\epsilon}$ with $\epsilon \in (0, 1]$, we replace now the singular metric (1.1.3) by the Colombeau generalized metric

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

where

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

and therefore

$$ \begin{aligned} \kappa \left( \hat{T}_{0,\epsilon}^0 (\hat{x}) \right)_\epsilon &= - \left( \frac{a\epsilon^2}{(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_\epsilon, \quad \kappa \left( \hat{T}_0^\alpha (\hat{x}; \epsilon) \right)_\epsilon = 0_{\mathbb{R}}, \quad \kappa \left( \hat{T}_\alpha^0 (\hat{x}; \epsilon) \right)_\epsilon = 0_{\mathbb{R}}, \\ \kappa \left( \hat{T}_\alpha^\beta (\hat{x}; \epsilon) \right)_\epsilon &= \delta_\alpha^\beta \left( \frac{3a\epsilon^2}{2(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_\epsilon - \\ & \qquad \left( \left( \frac{\hat{x}^\alpha \hat{x}^\beta}{\hat{r}^2 + \epsilon^2} \right)_\epsilon \left( \frac{a\epsilon^2}{(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_\epsilon \left( \frac{5}{2} + \left( \frac{\epsilon^2}{\hat{r}^2 + \epsilon^2} \right)_\epsilon \right) \right)_\epsilon. \end{aligned} \qquad (1.1.7) $$

Note that from Eq. (1.1.7) one obtains

$$ \hat{T}_\mu^\nu(\hat{x}) \triangleq w \lim_{\epsilon \to 0} \hat{T}_{\mu,\epsilon}^\nu(\hat{x}) \sim -Mc^2 \delta_\mu^0 \delta_0^\nu \delta^{(3)}(\hat{x}). \qquad (1.1.8) $$
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 authors 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]:

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

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

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

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 $(h_\epsilon)_\epsilon = a/((\hat{r}^2 + \epsilon^2)_\epsilon)$ and the function $\frac{\hat{x}^\alpha \hat{x}^\beta}{\hat{r}^2}$ as $\frac{\hat{x}^\alpha \hat{x}^\beta}{(\hat{r}^2 + \epsilon^2)_\epsilon}$ with $\epsilon \in (0, 1]$, we replace now the singular metric (1.1.3) by the Colombeau generalized metric

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

where

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

and therefore

$$ \begin{aligned} \kappa \left( \hat{T}_{0,\epsilon}^0 (\hat{x}) \right)_\epsilon &= - \left( \frac{a\epsilon^2}{(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_\epsilon, \quad \kappa \left( \hat{T}_0^\alpha (\hat{x}; \epsilon) \right)_\epsilon = 0_{\mathbb{R}}, \quad \kappa \left( \hat{T}_\alpha^0 (\hat{x}; \epsilon) \right)_\epsilon = 0_{\mathbb{R}}, \\ \kappa \left( \hat{T}_\alpha^\beta (\hat{x}; \epsilon) \right)_\epsilon &= \delta_\alpha^\beta \left( \frac{3a\epsilon^2}{2(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_\epsilon - \\ & \qquad \left( \left( \frac{\hat{x}^\alpha \hat{x}^\beta}{\hat{r}^2 + \epsilon^2} \right)_\epsilon \left( \frac{a\epsilon^2}{(\hat{r}^2 + \epsilon^2)^{5/2}} \right)_\epsilon \left( \frac{5}{2} + \left( \frac{\epsilon^2}{\hat{r}^2 + \epsilon^2} \right)_\epsilon \right) \right)_\epsilon. \end{aligned} \tag{1.1.7} $$

Note that from Eq. (1.1.7) one obtains

$$ \hat{T}_\mu^\nu(\hat{x}) \triangleq w \lim_{\epsilon \to 0} \hat{T}_{\mu,\epsilon}^\nu(\hat{x}) \sim -Mc^2 \delta_\mu^0 \delta_0^\nu \delta^{(3)}(\hat{x}). \tag{1.1.8} $$
**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:

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

where

$$ \begin{aligned} \left(\hat{R}_{\varepsilon}(\hat{r})\right)_{\varepsilon} = \left(\hat{R}_{\mu, \varepsilon}^{\mu}(\hat{r})\right)_{\varepsilon} = & -\left(\frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} + \left(\frac{2a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} \end{aligned} \qquad (1.1.10) $$

**Remark 1.1.7.** Notethattheregularizedscalarcurvature$_\varepsilon$ has the well-defined weak limit $\hat{R}_w$ in $D'(\mathbb{R}^n)$

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

**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

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

(ii) for any $(\hat{r}_\varepsilon)_\varepsilon \in \tilde{\mathbb{R}}$ such that $(\hat{r}_\varepsilon)_\varepsilon \approx \tilde{0}$ (see Definition 1.2.4) from Eq.(1.1.10) it follows that

$$ ((\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon}) \approx \tilde{0}, \qquad (1.1.13) $$

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

$$ ((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^O))_{\varepsilon}) = \tilde{0} - \left(\frac{a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right)_{\varepsilon} = \tilde{0} - \frac{a}{(\varepsilon^3)_{\varepsilon}}, \qquad (1.1.14) $$

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]:

$$ \begin{aligned} \hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}) \hat{R}_{\mu\nu,\varepsilon}(\hat{r}) &= \frac{1}{2} \left[ \frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right]^2 + 2 \left[ \frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right]^2 = \frac{13}{2} \left[ \frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right]^2 \\ \hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}) \hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}) &= \frac{4a^2}{\hat{r}^2 + \varepsilon^2} \left[ \frac{3}{(\hat{r}^2 + \varepsilon^2)^2} \right] - \frac{12a^2\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^4} + \\ & \qquad + \frac{9a^2\varepsilon^4}{(\hat{r}^2 + \varepsilon^2)^5}. \end{aligned} \qquad (1.1.15) $$

**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
**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:

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

where

$$ \begin{aligned} (\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon} = (\hat{R}_{\mu, \varepsilon}^{\mu}(\hat{r}))_{\varepsilon} = & - \left( \frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right)_{\varepsilon} + \left( \frac{2a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right)_{\varepsilon} = - \left( \frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right)_{\varepsilon} \end{aligned} \qquad (1.1.10) $$

**Remark 1.1.7.** Notethattheregularizedscalarcurvature$_\varepsilon$ has the well-defined weak limit $\hat{R}_w$ in $D'(\mathbb{R}^n)$

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

**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

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

(ii) for any $(\hat{r}_\varepsilon)_\varepsilon \in \tilde{\mathbb{R}}$ such that $(\hat{r}_\varepsilon)_\varepsilon \approx \tilde{0}$ (see Definition 1.2.4) from Eq.(1.1.10) it follows that

$$ ((\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon}) \approx \tilde{0}, \qquad (1.1.13) $$

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

$$ ((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^O))_{\varepsilon}) = \tilde{0} - \left(\frac{a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right)_{\varepsilon} = \tilde{0} - \frac{a}{(\varepsilon^3)_{\varepsilon}}, \qquad (1.1.14) $$

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]:

$$ \begin{aligned} \hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}) \hat{R}_{\mu\nu,\varepsilon}(\hat{r}) &= \frac{1}{2} \left[ \frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right]^2 + 2 \left[ \frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right]^2 = \frac{13}{2} \left[ \frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right]^2 \\ \hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}) \hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}) &= \frac{4a^2}{\hat{r}^2 + \varepsilon^2} \left[ \frac{3}{(\hat{r}^2 + \varepsilon^2)^2} \right] - \frac{12a^2\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^4} + \frac{9a^2\varepsilon^4}{(\hat{r}^2 + \varepsilon^2)^5}. \end{aligned} \qquad (1.1.15) $$

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

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

$$
\begin{align}
\langle \hat{R}_{\epsilon}^{\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon} &= \frac{1}{2} \left[ \left( \frac{3a\epsilon^2}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon} \right]^2 + 2 \left[ \left( \frac{a\epsilon^2}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon} \right]^2, \nonumber \\
\langle \hat{R}_{\epsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon} &= \frac{12a^2}{((\hat{r}_{\epsilon}^2 + \epsilon^2)^3)_{\epsilon}} - \left( \frac{12a^2\epsilon^2}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^4} \right)_{\epsilon} + \nonumber \\
&\quad + \left( \frac{9a^2\epsilon^4}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^5} \right)_{\epsilon}. \tag{1.1.17}
\end{align}
$$

**Remark 1.1.12.** Note that Colombeau quadratic scalars $\langle \hat{R}_{\epsilon}^{\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon}$ and $\langle \hat{R}_{\epsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon}$ 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 $\langle \hat{r}_{\epsilon}^{O} \rangle_{\epsilon} = 0_{\tilde{\mathbb{R}}}$ we get

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

**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

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

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

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

Inserting (1.1.16) into (1.1.2) we obtain Colombeau generalized object modeling the singular Schwarzschild metric above (below) horizon, i.e.,

$$
\begin{align}
 ds_{\epsilon}^{+2})_{\epsilon} &= (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \nonumber \\
 ds_{\epsilon}^{-2})_{\epsilon} &= (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2} \tag{1.1.21}
\end{align}
$$

The generalized Ricci tensor above horizon $\langle [\mathbf{R}_{\epsilon}^{+}]_{\alpha}^{\beta} \rangle_{\epsilon}$ may now be calculated componentwise using
$$
\begin{align}
\hat{R}^{\mu\nu}(\hat{x})\hat{R}_{\mu\nu}(\hat{x}) &\triangleq \lim_{\epsilon\to 0} \hat{R}^{\mu\nu}(\hat{x};\epsilon)\hat{R}_{\mu\nu}(\hat{x};\epsilon) \sim 40\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2, \nonumber \\
\hat{R}^{\rho\sigma\mu\nu}(\hat{x})\hat{R}_{\rho\sigma\mu\nu}(\hat{x}) &\triangleq \lim_{\epsilon\to 0} \hat{R}^{\rho\sigma\mu\nu}(\hat{x};\epsilon)\hat{R}_{\rho\sigma\mu\nu}(\hat{x};\epsilon) \tag{1.1.16} \\
&\sim \frac{12a^2}{\hat{r}^6} + \frac{16\pi a^2}{3} \frac{1}{\hat{r}^3} \delta^{(3)}(\hat{x}) + 16\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2. \nonumber
\end{align}
$$

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

$$
\begin{align}
\langle \hat{R}_{\epsilon}^{\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon} &= \frac{1}{2} \left[ \left( \frac{3a\epsilon^2}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon} \right]^2 + 2 \left[ \left( \frac{a\epsilon^2}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^{5/2}} \right)_{\epsilon} \right]^2, \nonumber \\
\langle \hat{R}_{\epsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon} &= \frac{12a^2}{((\hat{r}_{\epsilon}^2 + \epsilon^2)^3)_{\epsilon}} - \left( \frac{12a^2\epsilon^2}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^4} \right)_{\epsilon} + \nonumber \\
&\quad + \left( \frac{9a^2\epsilon^4}{(\hat{r}_{\epsilon}^2 + \epsilon^2)^5} \right)_{\epsilon}. \tag{1.1.17}
\end{align}
$$

**Remark 1.1.12.** Note that Colombeau quadratic scalars $\langle \hat{R}_{\epsilon}^{\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon}$ and $\langle \hat{R}_{\epsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\epsilon}) \hat{R}_{\rho\sigma\mu\nu,\epsilon}(\hat{r}_{\epsilon}) \rangle_{\epsilon}$ 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 $\langle \hat{r}_{\epsilon}^{O} \rangle_{\epsilon} = 0_{\tilde{\mathbb{R}}}$ we get

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

**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

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

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

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

Inserting (1.1.16) into (1.1.2) we obtain Colombeau generalized object modeling the singular Schwarzschild metric above (below) horizon, i.e.,

$$
\begin{align}
 ds_{\epsilon}^{+2})_{\epsilon} &= (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \nonumber \\
 ds_{\epsilon}^{-2})_{\epsilon} &= (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2} \tag{1.1.21}
\end{align}
$$

The generalized Ricci tensor above horizon $\langle [\mathbf{R}_{\epsilon}^{+}]_{\alpha}^{\beta} \rangle_{\epsilon}$ may now be calculated componentwise using
the classical formulae

$$ 
\begin{aligned}
\left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{0}^{0} \right)_{\varepsilon} &= \left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{1}^{1} \right)_{\varepsilon} = \frac{1}{2} \left( (h_{\varepsilon}^{+''})_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+'})_{\varepsilon} \right) \\
\left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{2}^{2} \right)_{\varepsilon} &= \left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{3}^{3} \right)_{\varepsilon} = \frac{(h_{\varepsilon}^{+'})_{\varepsilon}}{r} + \frac{1 + (h_{\varepsilon}^{+})_{\varepsilon}}{r^2},
\end{aligned}
\quad (1.1.22) $$

where

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

From Eq. (1.1.18) - Eq. (1.1.23) we obtain (see sect.3)

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

**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,
the classical formulae

$$ 
\begin{aligned}
\left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{0}^{0} \right)_{\varepsilon} &= \left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{1}^{1} \right)_{\varepsilon} = \frac{1}{2} \left( (h_{\varepsilon}^{+''})_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+'})_{\varepsilon} \right) \\
\left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{2}^{2} \right)_{\varepsilon} &= \left( \left[ \mathbf{R}_{\varepsilon}^{+}(r) \right]_{3}^{3} \right)_{\varepsilon} = \frac{(h_{\varepsilon}^{+'})_{\varepsilon}}{r} + \frac{1 + (h_{\varepsilon}^{+})_{\varepsilon}}{r^2},
\end{aligned}
\quad (1.1.22) $$

where

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

From Eq. (1.1.18) - Eq. (1.1.23) we obtain (see sect.3)

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

**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 $(f_\varepsilon(x))_\varepsilon \triangleq (f_\varepsilon(x))_\varepsilon, x \in \mathbb{R}^n, \varepsilon \in (0, \delta], \delta \le 1$ (smooth $\varepsilon$-regularisations, where $\varepsilon$ is the regularization parameter), such that: (i) for all compact subsets $K$ of $\mathbb{R}^n$ and all multiindices $3b1$, there is an $N > 0$ such that

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

with addition and multiplication defined by natural way:

$$ (f_\varepsilon(x))_\varepsilon + (g_\varepsilon(x))_\varepsilon = (f_\varepsilon(x) + g_\varepsilon(x))_\varepsilon \qquad (1.2.2) \text{ and} $$

$$ (f_\varepsilon(x))_\varepsilon \times (g_\varepsilon(x))_\varepsilon = (f_\varepsilon(x) \times g_\varepsilon(x))_\varepsilon. \qquad (1.2.3) $$

**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 $O(3b5^N)$ for all $N > 0$, i.e.

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

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

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

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

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

## Embedding of distributions

The space of Schwartz distributions $\mathcal{D}'(\mathbb{R}^n)$ can be **embedded** into the Colombeau algebra $\mathcal{G}(\mathbb{R}^n)$ by (component-wise) convolution with any element $(3c6_{3b5})_\varepsilon$ of the algebra $\mathcal{G}_\delta(\mathbb{R}^n)$ having as representative a $3b4$-net, i.e. a family of smooth functions $3c6_{3b5}$ such that $3c6_{3b5} \to 3b4$ in $\mathcal{D}'(\mathbb{R}^n)$ as $3b5 \to 0$.

**Remark 1.2.2.** Note that the embedding $\iota: \mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{G}_\delta(\mathbb{R}^n)$ is non-canonical, because it depends on the choice of the $3b4$-net.

**Example 1.2.1.** Delta function $3b4(x) \in \mathcal{D}'(\mathbb{R})$ for example has the following different representatives in Colombeau algebra $\mathcal{G}_\delta(\mathbb{R})$:

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

since

$$ \frac{1}{2} \frac{1}{\sqrt{\pi \varepsilon}} \exp \left( -\frac{x^2}{4\varepsilon} \right) \to 3b4(x), \quad \frac{1}{\pi x} \sin \left( \frac{x}{\varepsilon} \right) \to 3b4(x), \\ \frac{1}{\pi x^2 + \varepsilon^2} \to 3b4(x), \quad \frac{1}{\pi x^2} \sin^2 \left( \frac{x}{\varepsilon} \right) \to 3b4(x) \qquad (1.2.8) $$

in $\mathcal{D}'$ as $3b5 \to 0$.
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 $(f_\varepsilon(x))_\varepsilon \triangleq (f_\varepsilon(x))_\varepsilon, x \in \mathbb{R}^n, \varepsilon \in (0, \delta], \delta \le 1$ (smooth $\varepsilon$-regularisations, where $\varepsilon$ is the regularization parameter), such that: (i) for all compact subsets $K$ of $\mathbb{R}^n$ and all multiindices $3b1$, there is an $N > 0$ such that

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

with addition and multiplication defined by natural way:

$$ (f_\varepsilon(x))_\varepsilon + (g_\varepsilon(x))_\varepsilon = (f_\varepsilon(x) + g_\varepsilon(x))_\varepsilon \qquad (1.2.2) \text{ and} $$

$$ (f_\varepsilon(x))_\varepsilon \times (g_\varepsilon(x))_\varepsilon = (f_\varepsilon(x) \times g_\varepsilon(x))_\varepsilon. \qquad (1.2.3) $$

**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 $O(3b5^N)$ for all $N > 0$, i.e.

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

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

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

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

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

## Embedding of distributions

The space of Schwartz distributions $\mathcal{D}'(\mathbb{R}^n)$ can be **embedded** into the Colombeau algebra $\mathcal{G}(\mathbb{R}^n)$ by (component-wise) convolution with any element $(3c6_{3b5})_\varepsilon$ of the algebra $\mathcal{G}_\delta(\mathbb{R}^n)$ having as representative a $3b4$-net, i.e. a family of smooth functions $3c6_{3b5}$ such that $3c6_{3b5} \to 3b4$ in $\mathcal{D}'(\mathbb{R}^n)$ as $3b5 \to 0$.

**Remark 1.2.2.** Note that the embedding $\iota: \mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{G}_\delta(\mathbb{R}^n)$ is non-canonical, because it depends on the choice of the $3b4$-net.

**Example 1.2.1.** Delta function $3b4(x) \in \mathcal{D}'(\mathbb{R})$ for example has the following different representatives in Colombeau algebra $\mathcal{G}_\delta(\mathbb{R})$:

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

since

$$ \frac{1}{2} \frac{1}{\sqrt{\pi \varepsilon}} \exp \left( -\frac{x^2}{4\varepsilon} \right) \to 3b4(x), \quad \frac{1}{\pi x} \sin \left( \frac{x}{\varepsilon} \right) \to 3b4(x), \\ \frac{1}{\pi x^2 + \varepsilon^2} \to 3b4(x), \quad \frac{1}{\pi x^2} \sin^2 \left( \frac{x}{\varepsilon} \right) \to 3b4(x) \qquad (1.2.8) $$

in $\mathcal{D}'$ as $3b5 \to 0$.
**Remark 1.2.3.** However note that embedding $\mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{G}(\mathbb{R}^n)$ 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 $3b4(x)$ ill defined at point $x = 0$ since $3b4(0) = \infty$. However any Colombeau generalized function defined by Eq.(1.2.7) has well defined point value at point $x = 0$. For example

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

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 $\tilde{\mathbb{R}}_\delta = \mathbb{E}_{\tilde{\mathbb{R}},\delta}(\mathbb{R}) / \mathbb{N}_\delta(\mathbb{R})$ where [34],[36],[37]:

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

(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 $\text{cl}[(r_\epsilon)_\epsilon]$.

**Definition 1.2.4.** (i) Let $\check{\delta} = \text{cl}[(\delta_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}$. We say that $\check{\delta}$ is infinite small Colombeau generalized number and abbreviate $\check{\delta} \approx_{\tilde{\mathbb{R}}} 0$ 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 $\Delta =_{\tilde{\mathbb{R}}} \infty$ if $\Delta_{\tilde{\mathbb{R}}}^{-1} \approx_{\tilde{\mathbb{R}}} 0$. (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 $\Theta =_{\tilde{\mathbb{R}}} \pm\infty_{\tilde{\mathbb{R}}}$ if there exists representative $(\Theta_\epsilon)_\epsilon$ where $|\Theta_\epsilon| = \infty$ for all $\epsilon \in (0, 1]$. Here we abbreviate $\mathcal{E}_M(\mathbb{R}_{\pm\infty}) = \mathcal{E}_M(\mathbb{R} \cup \{\pm\infty\})$, $\mathcal{N}(\mathbb{R}_{\pm\infty}) = \mathcal{N}(\mathbb{R} \cup \{\pm\infty\})$ 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 $x \in \mathbb{R}$.
**Remark 1.2.3.** However note that embedding $\mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{G}(\mathbb{R}^n)$ 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 $3b4(x)$ ill defined at point $x = 0$ since $3b4(0) = \infty$. However any Colombeau generalized function defined by Eq.(1.2.7) has well defined point value at point $x = 0$. For example

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

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 $\tilde{\mathbb{R}}_\delta = \mathbb{E}_{\tilde{\mathbb{R}},\delta}(\mathbb{R}) / \mathbb{N}_\delta(\mathbb{R})$ where [34],[36],[37]:

$$ \mathbb{E}_{\mathbb{R},\delta}(\mathbb{R}) = \left\{ (x_\epsilon)_\epsilon \in \mathbb{R}^{(0,\delta)} \middle| \begin{array}{l} (\exists a \in \mathbb{R}_+) (\exists \epsilon_0 \in (0,1)) (\forall \epsilon \le \epsilon_0) [|x_\epsilon| \le \epsilon^{-a}] \\ (\forall a \in \mathbb{R}_+) (\exists \epsilon_0 \in (0,1)) (\forall \epsilon \le \epsilon_0) [|x_\epsilon| \le \epsilon^{a}] \end{array} \right\}, \qquad (1.2.10) $$

(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 $\check{\delta} = \mathbb{cl}[(\delta_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}$. We say that $\check{\delta}$ is infinite small Colombeau generalized number and abbreviate $\check{\delta} \approx_{\tilde{\mathbb{R}}} 0$ 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 $\Delta =_{\tilde{\mathbb{R}}} \infty$ if $\Delta_{\tilde{\mathbb{R}}}^{-1} \approx_{\tilde{\mathbb{R}}} 0$. (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 $\Theta =_{\tilde{\mathbb{R}}} \pm\infty_{\tilde{\mathbb{R}}}$ if there exists representative $(\Theta_\epsilon)_\epsilon$ where $|\Theta_\epsilon| = \infty$ for all $\epsilon \in (0, 1]$. Here we abbreviate $\mathcal{E}_M(\mathbb{R}_{\pm\infty}) = \mathcal{E}_M(\mathbb{R} \cup \{\pm\infty\})$, $\mathcal{N}(\mathbb{R}_{\pm\infty}) = \mathcal{N}(\mathbb{R} \cup \{\pm\infty\})$ 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 $\mathbb{st}: \tilde{\mathbb{R}} \to \mathbb{R}$ is defined by the formula:

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

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

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

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

**Definition 1.2.7.** (Principal value mapping) The principal value mapping $\mathbf{p.v.}: \tilde{\mathbb{R}} \to \mathbb{R}$

of Colombeau generalized function $(f_\epsilon(x))_\epsilon$ at point $\tilde{x} \in \mathbb{R}$ is defined by the formula:

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

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

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

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

itbpF2.6446in1.8455in0inFigure

Pic.3.Plot of the function $R(a, \tilde{r}, \epsilon) = \frac{a\epsilon^2}{(\tilde{r}^2 + \epsilon^2)^{5/2}}$, $a = 1, \tilde{r} = 10^{-3}, z = \epsilon \in (0, 0.01]$, $R(10^{-3}, 7 \times 10^{-4}) = 1.808 \times 10^8$.

$$ \mathbf{p.v.} \left[ \left( \hat{R}_{\epsilon}(\tilde{r}, a) \right)_{\epsilon} \right] \approx 1.808 \times 10^8. $$

## 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 $(f_\epsilon(x))_\epsilon \in \mathcal{G}_\delta(\mathbb{R})$ and $\mathbf{cl}[(\tilde{x}_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}_\delta$. Assume that $\mathbf{cl}[(f_\epsilon(\tilde{x}_\epsilon))_\epsilon] \in \tilde{\mathbb{R}}_\delta$.

We will say that Colombeau generalized number $\mathbf{cl}[(f_\epsilon(\tilde{x}_\epsilon))_\epsilon]$ is a point values of Colombeau generalized function $(f_\epsilon(x))_\epsilon$ at point $(\tilde{x}_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_\delta$.
$\tilde{\mathbb{R}}_{\pm\infty}$, respectively.

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

**Definition 1.2.7.** (Principal value mapping) The principal value mapping **p.v.** : $\tilde{\mathbb{R}} \to \mathbb{R}$

of Colombeau generalized function $(f_\epsilon(x))_\epsilon$ at point $\tilde{x} \in \mathbb{R}$ is defined by the formula:

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

We will be write for short $\mathrm{p.v.} \left[ (f_\epsilon(\tilde{x}))_\epsilon \right]$.

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

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

itbpF2.6446in1.8455in0inFigure

Pic.3.Plot of the function $R(a, \tilde{r}, \epsilon) = \frac{a\epsilon^2}{(\tilde{r}^2 + \epsilon^2)^{5/2}}$, $a = 1, \tilde{r} = 10^{-3}, z = \epsilon \in (0, 0.01]$, $R(10^{-3}, 7 \times 10^{-4}) = 1.808 \times 10^8$.

$$ \mathrm{p.v.} \left[ \left( \hat{R}_{\epsilon}(\tilde{r}, a) \right)_{\epsilon} \right] \approx 1.808 \times 10^8. $$

## 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 $(f_\epsilon(x))_\epsilon \in \mathcal{G}_\delta(\mathbb{R})$ and $\mathrm{cl}[(\tilde{x}_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}_\delta$. Assume that $\mathrm{cl}[(f_\epsilon(\tilde{x}_\epsilon))_\epsilon] \in \tilde{\mathbb{R}}_\delta$.

We will say that Colombeau generalized number $\mathrm{cl}[(f_\epsilon(\tilde{x}_\epsilon))_\epsilon]$ is a point values of Colombeau generalized function $(f_\epsilon(x))_\epsilon$ at point $(\tilde{x}_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_\delta$.
**Example 1.3.1.** For any $(\hat{r}_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_\delta, \epsilon \in (0, \delta]$ the point values $(\hat{R}_\epsilon(\hat{r}_\epsilon))_\epsilon$ of Colombeau generalized function $(\hat{R}_\epsilon(\hat{r}))_\epsilon$ (see Eq.(1.1.10) reads

$$ (\hat{R}_\epsilon(\hat{r}_\epsilon))_\epsilon = (\hat{R}_{\mu,\epsilon}^\mu(\hat{r}_\epsilon))_\epsilon = -\left(\frac{a\epsilon^2}{(\hat{r}_\epsilon^2 + \epsilon^2)^{5/2}}\right)_\epsilon = -a\frac{(\epsilon^2)_\epsilon}{[(\hat{r}_\epsilon^2)_\epsilon + (\epsilon^2)_\epsilon]^{5/2}} = -a\frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[(\hat{r}_\epsilon^2)_\epsilon + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}}, \quad (1.3.1) $$

where $\epsilon_1 \in (0, 1]$.

**Remark 1.3.2.** We choose now $(\hat{r}_\epsilon)_\epsilon = (\hat{r}_\epsilon^*)_\epsilon = \eta(\epsilon)_\epsilon = \eta\delta(\epsilon_1)_{\epsilon_1}$ and from Eq.(1.3.1) we get

$$ (\hat{R}_\epsilon(\hat{r}_\epsilon^*))_\epsilon = -a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[(\hat{r}_\epsilon^{*2})_\epsilon + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}} = -a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[\eta\delta(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}} = -a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[\eta^2\delta^2(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}} = -a \frac{\delta^3(\epsilon_1^3)_{\epsilon_1}}{[\eta^2 + 1]^{5/2}}. \quad (1.3.2) $$

Thus in "point limit" $\delta \asymp 0$ the curvature scalar $(\hat{R}_\epsilon(\hat{r}_\epsilon^*))_\epsilon$ diverges as $\delta^{-3}(\epsilon_1^{-3})_{\epsilon_1}$.

**Remark 1.3.2.** In order prevent the divergence mentioned above, we assume now that there exist fundamental generalized length $l^* = \text{cl}[(l_{\epsilon_1}^*)_{\epsilon_1}] = \eta \text{cl}[(\epsilon_1)_{\epsilon_1}]$, $\epsilon_1 \in (0, 1]$, $\eta \in \mathbb{R}$, $\delta \ll \eta \ll 1$, such that: $(\hat{r}_{\epsilon_1})_{\epsilon_1} \ge l^*$, see [36] sec.2. Thus from Eq.(1.3.1) we get now instead Eq.(1.3.2)

$$ \begin{aligned} \left| (\hat{R}_\epsilon(\hat{r}_\epsilon))_\epsilon \right| &\le \left( \frac{a\epsilon^2}{(l_{\epsilon_1}^{*2} + \epsilon^2)^{5/2}} \right)_\epsilon = a \frac{a(\epsilon^2)_\epsilon}{[(l_{\epsilon_1}^{*2})_\epsilon + (\epsilon^2)_\epsilon]^{5/2}} = \\ &a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[\eta^2(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{1/2}} \frac{1}{[\eta^2(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{1/2}} \frac{1}{[(\hat{r}_{\epsilon_1})_{\epsilon_1} + (\epsilon^2)_\epsilon]^{3/2}} = \\ &\frac{a\delta^2}{[\eta^2 + \delta^2]} \frac{1}{[(\hat{r}_{\epsilon_1}^2)_{\epsilon_1} + (\epsilon^2)_\epsilon]^{3/2}} = \frac{a\delta^2}{[\eta^2 + \delta^2]} \frac{1}{[(\hat{r}_\epsilon^2)_{\epsilon} + (\epsilon^2)_\epsilon]^{3/2}}. \end{aligned} \quad (1.3.3) $$

## 1.4 The Point-Free Loop Quantum Gravity.

We remind that canonical quantization of GRT can be expressed as an SU(2) gauge theory on the 3 dimensional manifold $\Sigma$ 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:

$$ \{A_a^i(x), E_j^b(x')\} = 8\pi\gamma\delta_j^i\delta_a^b\delta(x - x'), \quad (1.4.1) $$

with G the gravitational constant and $\gamma$ the Immirzi parameter. The conjugate pair are constraint to satisfy the system
**Example 1.3.1.** For any $(\hat{r}_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_\delta, \epsilon \in (0, \delta]$ the point values $(\hat{R}_\epsilon(\hat{r}_\epsilon))_\epsilon$ of Colombeau generalized function $(\hat{R}_\epsilon(\hat{r}))_\epsilon$ (see Eq.(1.1.10) reads

$$ (\hat{R}_\epsilon(\hat{r}_\epsilon))_\epsilon = (\hat{R}_{\mu,\epsilon}^\mu(\hat{r}_\epsilon))_\epsilon = -\left(\frac{a\epsilon^2}{(\hat{r}_\epsilon^2 + \epsilon^2)^{5/2}}\right)_\epsilon = -a\frac{(\epsilon^2)_\epsilon}{[(\hat{r}_\epsilon^2)_\epsilon + (\epsilon^2)_\epsilon]^{5/2}} = -a\frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[(\hat{r}_\epsilon^2)_\epsilon + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}}, \quad (1.3.1) $$

where $\epsilon_1 \in (0, 1]$.

**Remark 1.3.2.** We choose now $(\hat{r}_\epsilon)_\epsilon = (\hat{r}_\epsilon^*)_\epsilon = \eta(\epsilon)_\epsilon = \eta\delta(\epsilon_1)_{\epsilon_1}$ and from Eq.(1.3.1) we get

$$ (\hat{R}_\epsilon(\hat{r}_\epsilon^*))_\epsilon = -a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[(\hat{r}_\epsilon^{*2})_\epsilon + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}} = -a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[\eta\delta(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}} = -a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[\eta^2\delta^2(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{5/2}} = -a \frac{\delta^3(\epsilon_1^3)_{\epsilon_1}}{[\eta^2 + 1]^{5/2}}. \quad (1.3.2) $$

Thus in "point limit" $\delta \asymp 0$ the curvature scalar $(\hat{R}_\epsilon(\hat{r}_\epsilon^*))_\epsilon$ diverges as $\delta^{-3}(\epsilon_1^{-3})_{\epsilon_1}$.

**Remark 1.3.2.** In order prevent the divergence mentioned above, we assume now that there exist fundamental generalized length $l^* = \text{cl}[(l_{\epsilon_1}^*)_{\epsilon_1}] = \eta \text{cl}[(\epsilon_1)_{\epsilon_1}]$, $\epsilon_1 \in (0, 1]$, $\eta \in \mathbb{R}$, $\delta \ll \eta \ll 1$, such that: $(\hat{r}_{\epsilon_1})_{\epsilon_1} \ge l^*$, see [36] sec.2. Thus from Eq.(1.3.1) we get now instead Eq.(1.3.2)

$$ \begin{aligned} \left| (\hat{R}_\epsilon(\hat{r}_\epsilon))_\epsilon \right| &\le \left( \frac{a\epsilon^2}{(l_{\epsilon_1}^{*2} + \epsilon^2)^{5/2}} \right)_\epsilon = a \frac{a(\epsilon^2)_\epsilon}{[(l_{\epsilon_1}^{*2})_\epsilon + (\epsilon^2)_\epsilon]^{5/2}} = \\ &a \frac{\delta^2(\epsilon_1^2)_{\epsilon_1}}{[\eta^2(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{1/2}} \frac{1}{[\eta^2(\epsilon_1^2)_{\epsilon_1} + \delta^2(\epsilon_1^2)_{\epsilon_1}]^{1/2}} \frac{1}{[(\hat{r}_{\epsilon_1})_{\epsilon_1} + (\epsilon^2)_\epsilon]^{3/2}} = \\ &\frac{a\delta^2}{[\eta^2 + \delta^2]} \frac{1}{[(\hat{r}_{\epsilon_1}^2)_{\epsilon_1} + (\epsilon^2)_\epsilon]^{3/2}} = \frac{a\delta^2}{[\eta^2 + \delta^2]} \frac{1}{[(\hat{r}_\epsilon^2)_{\epsilon_1} \in (0,1)] + (\epsilon^2)_\epsilon]^{3/2}}. \end{aligned} \quad (1.3.3) $$

## 1.4 The Point-Free Loop Quantum Gravity.

We remind that canonical quantization of GRT can be expressed as an SU(2) gauge theory on the 3 dimensional manifold $\Sigma$ 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:

$$ \{A_a^i(x), E_j^b(x')\} = 8\pi\gamma\delta_j^i\delta_a^b\delta(x - x'), \quad (1.4.1) $$

with G the gravitational constant and $\gamma$ the Immirzi parameter. The conjugate pair are constraint to satisfy the system
$$ 
\begin{aligned}
G_i &= D_a E_i^a = 0, H_b = E_i^a F_{ab}^i = 0, \\
H &= \epsilon_{ijk} F_{ab}^i E_j^a E_k^b - 2(1 + \gamma^2) K_{[a}^i K_{b]}^j E_i^a E_j^b = 0,
\end{aligned} 
\qquad (1.4.2) 
$$

which are called Gauss, spatial diffeomorphism and Hamiltonian constraints respectively. In fact, the task of finding a metric satisfying the Einstein's equations, describing configuration of a gravitating system, is now replaced by finding a conjugate pair (A, E) satisfying the constraint system (1.4.1). On quantization, one smears the basic fields (A, E) to holonomies of $A_i^a$ 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.

**Remark 1.4.2.** Canonical quantization of nonlinear distributional GRT can be expressed as an $\overline{SU}(2)$ 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 $\overline{SU}(2)$ Colombeau distributional connection and $(E_{i,\epsilon}^a(x_\epsilon))_\epsilon$ is the distributional densitised triad vector field, conjugate to $(A_\epsilon)_\epsilon$:

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

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

$$ 
\begin{aligned}
(G_{i,\epsilon})_{\tilde{\mathbb{R}}} &= (D_{a,\epsilon} E_{i,\epsilon}^a)_{\tilde{\mathbb{R}}} = 0, (H_{b,\epsilon})_{\tilde{\mathbb{R}}} = (E_{i,\epsilon}^a F_{ab,\epsilon})_{\tilde{\mathbb{R}}} = \tilde{\mathbb{R}}_0, \\
(H_{\epsilon})_{\tilde{\mathbb{R}}} &= \epsilon_{ijk} (F_{ab,\epsilon}^i E_{j,\epsilon}^a E_{k,\epsilon}^b)_{\tilde{\mathbb{R}}} - 2(1 + \gamma^2) (K_{[a,\epsilon}^i K_{b,\epsilon]}^j E_{i,\epsilon}^a E_{j,\epsilon}^b)_{\tilde{\mathbb{R}}} = \tilde{\mathbb{R}}_0.
\end{aligned} 
\qquad (1.4.4) 
$$

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

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 $\overline{SU}(2)$ representations $j_l$.
$$ 
\begin{aligned}
G_i &= D_a E_i^a = 0, H_b = E_i^a F_{ab}^i = 0, \\
H &= \epsilon_{ijk} F_{ab}^i E_j^a E_k^b - 2(1 + \gamma^2) K_{[a}^i K_{b]}^j E_i^a E_j^b = 0,
\end{aligned} 
\qquad (1.4.2) 
$$

which are called Gauss, spatial diffeomorphism and Hamiltonian constraints respectively. In fact, the task of finding a metric satisfying the Einstein's equations, describing configuration of a gravitating system, is now replaced by finding a conjugate pair $(A, E)$ satisfying the constraint system (1.4.1). On quantization, one smears the basic fields $(A, E)$ to holonomies of $A_i^a$ 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.

**Remark 1.4.2.** 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 $(A_\epsilon)_\epsilon$:

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

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

$$ 
\begin{aligned}
(G_{i,\epsilon})_{\tilde{\mathbb{R}}} &= (D_{a,\epsilon} E_{i,\epsilon}^a)_{\tilde{\mathbb{R}}} = 0, (H_{b,\epsilon})_{\tilde{\mathbb{R}}} = (E_{i,\epsilon}^a F_{ab,\epsilon})_{\tilde{\mathbb{R}}} = \tilde{\mathbb{R}}_0, \\
(H_{\epsilon})_{\tilde{\mathbb{R}}} &= \epsilon_{ijk} (F_{ab,\epsilon}^i E_{j,\epsilon}^a E_{k,\epsilon}^b)_{\tilde{\mathbb{R}}} - 2(1 + \gamma^2) (K_{[a,\epsilon}^i K_{b,\epsilon]}^j E_{i,\epsilon}^a E_{j,\epsilon}^b)_{\tilde{\mathbb{R}}} = \tilde{\mathbb{R}}_0.
\end{aligned} 
\qquad (1.4.4) 
$$

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

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 $\tilde{\mathbb{T}}$, consisting of all Colombeau complex numbers with absolute value 1 under multiplication, i.e. $\tilde{\mathbb{T}} = \{z \in \tilde{\mathbb{C}} | |z| = 1\}$.

(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 $((x_\varepsilon)_\varepsilon, \theta, \phi)$, 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:

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

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 $((\eta_\varepsilon(x_\varepsilon))_\varepsilon, (P_\varepsilon^{\eta_\varepsilon}(x_\varepsilon))_\varepsilon)$:

$$ (\Omega_\varepsilon)_\varepsilon = \frac{1}{2\gamma G} \int dx(dA_x \wedge dE^x + d(\gamma K_\phi) \wedge dE^\phi + 2d\eta \wedge dP^\eta), \quad (1.4.8) $$

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:
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 $\tilde{\mathbb{T}}$, consisting of all Colombeau complex numbers with absolute value 1 under multiplication, i.e. $\tilde{\mathbb{T}} = \{z \in \tilde{\mathbb{C}} | |z| = 1\}$.

(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 $((x_\varepsilon)_\varepsilon, \theta, \phi)$, 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:

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

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 $((\eta_\varepsilon(x_\varepsilon))_\varepsilon, (P_\varepsilon^{\eta_\varepsilon}(x_\varepsilon))_\varepsilon)$:

$$ (\Omega_\varepsilon)_\varepsilon = \frac{1}{2\gamma G} \int dx(dA_x \wedge dE^x + d(\gamma K_\phi) \wedge dE^\phi + 2d\eta \wedge dP^\eta), \quad (1.4.8) $$

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:
$$ (G_{\varepsilon}[\lambda_{\varepsilon}])_{\varepsilon} = \left(dx_{\varepsilon}\lambda_{\varepsilon}(E_{\varepsilon}^{x'_{\varepsilon}} + P_{\varepsilon}^{\eta_{\varepsilon}})\right)_{\varepsilon} = \mathbb{R} 0_{\mathbb{R}}, \qquad (1.4.9) $$

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

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

## 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 $\mathbb{R}$ are defined as $(h_{\varepsilon}^{(\gamma_{\varepsilon})})_{\varepsilon} \equiv \exp\left[\frac{i}{2}(\gamma_{\varepsilon} A_{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon}\right]$ 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 $(\eta_{\varepsilon})_{\varepsilon} \in \tilde{S}^{1}$, 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 $(T_{g,k,\mu,\varepsilon})_{\varepsilon}$:

$$ \prod_{e \in g} \exp \left( \frac{i}{2} k_{\varepsilon} (e dx_{\varepsilon} A_{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon} \right) \left( \prod_{\nu \in V(g)} \exp (i \mu_{\nu, \varepsilon} \gamma K_{\phi_{\varepsilon}, \varepsilon}(\nu)) \exp (i k_{\nu} \eta_{\varepsilon}(\nu)) \right)_{\varepsilon}. \qquad (1.4.11) $$

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 $\tilde{\mathbb{R}}_{\text{Bohr}}$ representations $(\mu_{\nu,\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ and irreducible $\tilde{S}^1$ representation $k_{\nu} \in \tilde{\mathbb{Z}}$, 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

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

$$ dx_{\varepsilon} \hat{E}_{\varepsilon}^{\phi_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon} = \gamma \ell_{p}^{2} \sum_{\nu} \mu_{\nu,\varepsilon} T_{g,k,\mu,\varepsilon}, \qquad (1.4.13) $$

$$ dx_{\varepsilon} \hat{P}_{\varepsilon}^{\eta_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon} = 2\gamma \ell_{p}^{2} \sum_{\nu} k_{\nu,\varepsilon} T_{g,k,\mu,\varepsilon}. \qquad (1.4.14) $$

The generalized volume operator can be expressed as

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

which is diagonal in spin network representation

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

where
$$ (G_{\varepsilon}[\lambda_{\varepsilon}])_{\varepsilon} = \left(dx_{\varepsilon}\lambda_{\varepsilon}(E_{\varepsilon}^{x'_{\varepsilon}} + P_{\varepsilon}^{\eta_{\varepsilon}})\right)_{\varepsilon} = \mathbb{R} 0_{\mathbb{R}}, \qquad (1.4.9) $$

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

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

## 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 $\mathbb{R}$ are defined as $(h_{\varepsilon}^{(\gamma_{\varepsilon})})_{\varepsilon} \equiv \exp\left[\frac{i}{2}(\gamma_{\varepsilon} A_{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon}\right]$ 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 $(\eta_{\varepsilon})_{\varepsilon} \in \tilde{S}^{1}$, 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 $(T_{g,k,\mu,\varepsilon})_{\varepsilon}$:

$$ \prod_{e \in g} \exp \left( \frac{i}{2} k_{\varepsilon} ({}_e dx_{\varepsilon} A_{x_{\varepsilon}}(x_{\varepsilon}))_{\varepsilon} \right) \left( \prod_{\nu \in V(g)} \exp (i \mu_{\nu, \varepsilon} \gamma K_{\phi_{\varepsilon}, \varepsilon}(\nu)) \exp (i k_{\nu} \eta_{\varepsilon}(\nu)) \right)_{\varepsilon} . \qquad (1.4.11) $$

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 $(\mu_{\nu,\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ and irreducible $\tilde{S}^{1}$ representation $k_{\nu} \in \tilde{\mathbb{Z}}$, 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

$$ (\hat{E}_{\varepsilon}^{x_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon})_{\varepsilon} = \gamma \frac{\ell_{p}^{2}}{2} ((k_{e+}(x_{\varepsilon}) + k_{e-}(x_{\varepsilon})) T_{g,k,\mu,\varepsilon})_{\varepsilon}, \qquad (1.4.12) $$

$$ dx_{\varepsilon} \hat{E}_{\varepsilon}^{\phi_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon} = \gamma \ell_{p}^{2} \sum_{\nu} \mu_{\nu,\varepsilon} T_{g,k,\mu,\varepsilon}, \qquad (1.4.13) $$

$$ dx_{\varepsilon} \hat{P}_{\varepsilon}^{\eta_{\varepsilon}}(x_{\varepsilon}) T_{g,k,\mu,\varepsilon} = 2\gamma \ell_{p}^{2} \sum_{\nu} k_{\nu,\varepsilon} T_{g,k,\mu,\varepsilon}. \qquad (1.4.14) $$

The generalized volume operator can be expressed as

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

which is diagonal in spin network representation

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

where
$$ (V_{k,m,\epsilon})_{\epsilon} = 4\pi\gamma^{3/2}\ell_p^3 \left( \sum_{\nu} |\mu_{\nu}| \sqrt{\frac{1}{2} |k_{e+(x_{\epsilon})} + k_{e-(x_{\epsilon})}|} \right)_{\epsilon} \quad (1.4.17) $$

Implementing the Gauss constraint as an operator on spin networks to select the gauge invariant states, leads to a restriction on labels

$$ (\hat{G}_{\epsilon}[\lambda_{\epsilon}]T_{g,k,\mu,\epsilon})_{\epsilon} = \gamma \ell_p^2 (\sum_{\nu} \lambda_{\epsilon}(\nu)(k_{e+(x_{\epsilon})} - k_{e-(x_{\epsilon})} + 2k_{\nu})T_{g,k,\mu,\epsilon})_{\epsilon} \quad (1.4.18) $$

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

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

$$ (T_{g,k,\mu,\epsilon})_{\epsilon} = \prod_{e \in g} \exp \left[ \frac{i}{2} k_e (e \, dx_e (A_{x_e}(x_e) + \eta'_e))_{\epsilon} \right] \left( \prod_{\nu \in V(g)} \exp (i \mu_{\nu} \gamma K_{\phi_{\nu}, \epsilon}(\nu)) \right)_{\epsilon} \quad (1.4.20) $$

## 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, θ, φ) 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 of partitions space-time into 2 regions: I(xᵢx⁺), 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.

### 1.5.1 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

$$ e^0 = \pm \left(\frac{2m}{t} - 1\right)^{-1/2} dt; \quad e^1 = \pm t \sin \theta d\phi; \quad e^2 = \pm t d\theta; \quad e^3 = \pm \left(\frac{2m}{t} - 1\right)^{1/2} dx, \quad (1.5.4) $$
$$ (V_{k,m,\epsilon})_{\epsilon} = 4\pi\gamma^{3/2}\ell_p^3 \left( \sum_{\nu} |\mu_{\nu}| \sqrt{\frac{1}{2} |k_{e+(x_{\epsilon})} + k_{e-(x_{\epsilon})}|} \right)_{\epsilon}. \quad (1.4.17) $$

Implementing the Gauss constraint as an operator on spin networks to select the gauge invariant states, leads to a restriction on labels

$$ (\hat{G}_{\epsilon}[\lambda_{\epsilon}] T_{g,k,\mu,\epsilon})_{\epsilon} = \gamma \ell_p^2 (\sum_{\nu} \lambda_{\epsilon}(\nu) (k_{e+(x_{\epsilon})} - k_{e-(x_{\epsilon})} + 2k_{\nu}) T_{g,k,\mu,\epsilon})_{\epsilon} \quad (1.4.18) $$

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

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

$$ (T_{g,k,\mu,\epsilon})_{\epsilon} = \prod_{e \in g} \exp \left[ \frac{i}{2} k_e (e \, dx_e (A_{x_e}(x_e) + \eta'_e))_{\epsilon} \right] \left( \prod_{\nu \in V(g)} \exp (i \mu_{\nu} \gamma K_{\phi_{\nu}, \epsilon}(\nu)) \right)_{\epsilon}. \quad (1.4.20) $$

## 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, θ, φ) 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 of partitions space-time into 2 regions: I($x_\zeta x_+$), 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.

### 1.5.1 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

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

The A field can be constructed using spin connections:

$$ 
\begin{aligned}
A^3 &= \pm \gamma \left(-\frac{m}{t^2}\right) dx, \quad A^2 = \pm \gamma \left(\frac{2m}{t} - 1\right)^{1/2} d\theta, \\
A^1 &= \pm \gamma \left(\frac{2m}{t} - 1\right)^{1/2} \sin\theta d\phi, \quad A^3 = \pm \cos\theta d\phi.
\end{aligned} 
\qquad (1.5.6) 
$$

To construct the E field on $\Sigma_{in}$ 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 $\Sigma = \mathbb{R} \times S^2$. The 3 triad fields become:

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

with determinant

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

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

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

The 3 triad fields (1.5.7) define their compatible spin connection, $\Gamma^{ij} \wedge e^j + de^i = 0$:

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

and

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

Extrinsic curvature is related to A via $\gamma K = A - \Gamma$ reads

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

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

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 $\{A_a^i(x), E_j^b(x')\} = \delta_j^i \delta_a^b(x - x')$.
which gives the compatible spin connection components

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

The A field can be constructed using spin connections:

$$ 
\begin{aligned}
A^3 &= \pm \gamma \left(-\frac{m}{t^2}\right) dx, \quad A^2 = \pm \gamma \left(\frac{2m}{t} - 1\right)^{1/2} d\theta, \\
A^1 &= \pm \gamma \left(\frac{2m}{t} - 1\right)^{1/2} \sin\theta d\phi, \quad A^3 = \pm \cos\theta d\phi.
\end{aligned} 
\qquad (1.5.6) 
$$

To construct the E field on $\Sigma_{in}$ 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 $\Sigma = \mathbb{R} \times S^2$. The 3 triad fields become:

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

with determinant

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

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

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

The 3 triad fields (1.5.7) define their compatible spin connection, $\Gamma^{ij} \wedge e^j + de^i = 0$:

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

and

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

Extrinsic curvature is related to A via $\gamma K = A - \Gamma$ reads

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

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

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 $\{A_a^i(x), E_j^b(x')\} = \delta_j^i \delta_a^b(x - x')$.
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

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

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

and Hamiltonian constraint gives:

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

For the above constraints to be zero we must have

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

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

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

and

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

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

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

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

where,
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

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

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

and Hamiltonian constraint gives:

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

For the above constraints to be zero we must have

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

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

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

and

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

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

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

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

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

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

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

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 $\sim 3b4(\hat{x}) \in D'(\mathbb{R})$, see Eq. (1.1.7) and Eq. (1.1.10). This result as well established and accepted 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]:

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

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

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

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

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

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

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

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

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 $\sim 3b4(\hat{x}) \in D'(\mathbb{R})$, see Eq. (1.1.7) and Eq. (1.1.10). This result as well established and accepted 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]:

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

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

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

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

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

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

and rewrite $\mathcal{R}$ as

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

Now, quantization would be straightforward:

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

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:

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

Such operator $\hat{\mathcal{R}}|\tau, \mu\rangle$ 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 $LQG^\epsilon$ has a maximum value of

$$\left| \hat{R}_{\epsilon}(\hat{r}) \right|_{\max} \sim \frac{M \epsilon^2}{l_{Pl}^5} \left|_{\max} \right. . \qquad (1.6.9)$$

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

## 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

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

where $cl[(t_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}$. According to definition of Colombeau tetrad (frame) fields $(g_{\mu\nu,\epsilon})_\epsilon = \eta_{IJ} (e_{\mu,\epsilon}^I e_{\nu,\epsilon}^J)_\epsilon$, 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 $(g_{\mu\nu,\epsilon})_\epsilon = \eta_{IJ} (e_{\mu,\epsilon}^I e_{\nu,\epsilon}^J)_\epsilon$ 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 $((A_\epsilon)_\epsilon, (E_\epsilon)_\epsilon)$, a particular labeling must be chosen which will be clear below. The suitable choice for labeling 4 orthogonal Colombeau generalized frame fields reads
$$h_x^{(\tau)} = 1 + \epsilon \int_0^\tau dx \tau_3 + \mathcal{O}(\epsilon^2), \qquad (1.6.4)$$

and rewrite $\mathcal{R}$ as

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

Now, quantization would be straightforward:

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

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:

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

Such operator $\hat{\mathcal{R}}|\tau, \mu\rangle$ 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 $LQG^\epsilon$ has a maximum value of

$$\left| \hat{R}_{\epsilon}(\hat{r}) \right|_{\max} \sim \frac{M \epsilon^2}{l_{Pl}^5} \left|_{\max} \right. . \qquad (1.6.9)$$

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

## 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

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

where $cl[(t_\epsilon)_\epsilon] \in \tilde{\mathbb{R}}$. According to definition of Colombeau tetrad (frame) fields $(g_{\mu\nu,\epsilon})_\epsilon = \eta_{IJ} (e^I_{\mu,\epsilon} e^J_{\nu,\epsilon})_\epsilon$, 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 $(g_{\mu\nu,\epsilon})_\epsilon = \eta_{IJ} (e^I_{\mu,\epsilon} e^J_{\nu,\epsilon})_\epsilon$ 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 $((A_\epsilon)_\epsilon, (E_\epsilon)_\epsilon)$, a particular labeling must be chosen which will be clear below. The suitable choice for labeling 4 orthogonal Colombeau generalized frame fields reads
$$ (e_{\epsilon}^{0})_{\epsilon} = \pm \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{-1/2} (dt_{\epsilon})_{\epsilon} ; (e_{\epsilon}^{1})_{\epsilon} = \pm \left[ (t_{\epsilon})_{\epsilon} \right] \left[ (\sin \theta_{\epsilon})_{\epsilon} \right] \left[ (d\phi_{\epsilon})_{\epsilon} \right] ; \\ (e_{\epsilon}^{2})_{\epsilon} = \pm \left[ (t_{\epsilon})_{\epsilon} \right] \left[ (d\theta_{\epsilon})_{\epsilon} \right] ; (e_{\epsilon}^{3})_{\epsilon} = \pm \left[ \frac{2m}{(t_{\epsilon})_{\epsilon}} - 1 \right]^{1/2} \left[ (dx_{\epsilon})_{\epsilon} \right], \qquad (1.6.11) $$

which gives the compatible Colombeau generalized spin connection components

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

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

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

To construct Colombeau generalized field $(E_{\epsilon})_{\epsilon}$ 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:

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

with determinant

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

and inverse triad

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

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, $(\Gamma_{\epsilon}^{ij} \wedge e_{\epsilon}^{j}) + (de_{\epsilon}^{i})_{\epsilon} = 0_{\tilde{\mathbb{R}}}$:
$$ (e_{\varepsilon}^0)_{\varepsilon} = \pm \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{-1/2} (dt_{\varepsilon})_{\varepsilon} ; (e_{\varepsilon}^1)_{\varepsilon} = \pm \left[ (t_{\varepsilon})_{\varepsilon} \right] \left[ (\sin \theta_{\varepsilon})_{\varepsilon} \right] \left[ (d\phi_{\varepsilon})_{\varepsilon} \right]; \\ (e_{\varepsilon}^2)_{\varepsilon} = \pm \left[ (t_{\varepsilon})_{\varepsilon} \right] \left[ (d\theta_{\varepsilon})_{\varepsilon} \right] ; (e_{\varepsilon}^3)_{\varepsilon} = \pm \left[ \frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1 \right]^{1/2} \left[ (dx_{\varepsilon})_{\varepsilon} \right], \qquad (1.6.11) $$

which gives the compatible Colombeau generalized spin connection components

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

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

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

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 $\tilde{\Sigma} = \mathbb{R} \times \tilde{S}^2$. The Colombeau generalized 3 triad fields become:

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

with determinant

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

and inverse triad

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

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

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

The Colombeau generalized 3 triad fields (1.6.14) define their generalized compatible spin connection, $(\Gamma_{\varepsilon}^{ij} \wedge e_{\varepsilon}^j) + (de_{\varepsilon}^i)_{\varepsilon} = 0_{\mathbb{R}}$:
$$ (\Gamma_{\varepsilon}^{12})_{\varepsilon} = -(\Gamma_{\varepsilon}^{21})_{\varepsilon} = [(\cos \theta_{\varepsilon})_{\varepsilon}] [(d\phi_{\varepsilon})_{\varepsilon}], \qquad (1.6.18) $$

and

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

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

$$ \begin{aligned} (K_{r\varepsilon,\varepsilon}^3)_{\varepsilon} &= \frac{1}{\gamma} (A_{r\varepsilon,\varepsilon}^3)_{\varepsilon} = \pm \left(-\frac{m}{(t_{\varepsilon}^2)_{\varepsilon}}\right) [(dx_{\varepsilon})_{\varepsilon}], & (1.6.20) \\ (K_{\theta\varepsilon,\varepsilon}^2)_{\varepsilon} &= \frac{1}{\gamma} (A_{\theta\varepsilon,\varepsilon}^2)_{\varepsilon} = \pm \left[\frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1\right]^{1/2}, & \end{aligned} $$

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

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 $\{A_{a,\varepsilon}^i(x_{\varepsilon}), E_j^b(x'_{\varepsilon})\}_{\varepsilon} = \delta_j^i (\delta_a^b \delta(x_{\varepsilon} - x'_{\varepsilon}))_{\varepsilon}$.

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

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

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

and Hamiltonian constraint gives:

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

For the above constraints to be zero we must have

$$ \begin{aligned} \mathbf{sign} [(E_{2,\varepsilon}^{\theta\varepsilon})_{\varepsilon}] &= -\mathbf{sign} [(E_{1,\varepsilon}^{\phi\varepsilon})_{\varepsilon}], & (1.6.25) \\ \mathbf{sign} [(A_{\phi\varepsilon,\varepsilon}^3)_{\varepsilon}] &= +1, \mathbf{sign} [(A_{\phi\varepsilon,\varepsilon}^1)_{\varepsilon}] = -\mathbf{sign} [(A_{\theta\varepsilon,\varepsilon}^2)_{\varepsilon}]. & \end{aligned} $$

This leaves us with two alternatives corresponding to the residual gauge freedom

$$ \begin{aligned} & ((b_{\varepsilon})_{\varepsilon}, (p_{b\varepsilon,\varepsilon})_{\varepsilon}) \rightarrow -(b_{\varepsilon})_{\varepsilon}, -(p_{b\varepsilon,\varepsilon})_{\varepsilon}). \\ & (A_{a,\varepsilon}^i)_{\varepsilon} = [(c_{\varepsilon})_{\varepsilon}] \tau_3 [(dr_{\varepsilon})_{\varepsilon}] + [(b_{\varepsilon})_{\varepsilon}] \tau_2 [(d\theta_{\varepsilon})_{\varepsilon}] + \\ & \qquad [(\cos \theta_{\varepsilon})_{\varepsilon}] \tau_3 - [(b_{\varepsilon})_{\varepsilon}] [(\sin \theta_{\varepsilon})_{\varepsilon}] \tau_1 [(d\phi_{\varepsilon})_{\varepsilon}] \\ & \qquad (E_{i,\varepsilon}^a)_{\varepsilon} = [(p_{c_{\varepsilon},\varepsilon})_{\varepsilon}] \tau_3 [(\sin \theta_{\varepsilon})_{\varepsilon}] [(\partial_{r_{\varepsilon}})_{\varepsilon}] + \\ & \qquad [(p_{b_{\varepsilon},\varepsilon})_{\varepsilon}] \tau_2 [(\sin \theta_{\varepsilon})_{\varepsilon}] [(\partial_{\theta_{\varepsilon}})_{\varepsilon}] - [(p_{b_{\varepsilon},\varepsilon})_{\varepsilon}] \tau_1 [(\partial_{\phi_{\varepsilon}})_{\varepsilon}], \end{aligned} \qquad (1.6.26) $$

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

and

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

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

$$ \begin{aligned} (K_{r\varepsilon,\varepsilon}^3)_{\varepsilon} &= \frac{1}{\gamma} (A_{r\varepsilon,\varepsilon}^3)_{\varepsilon} = \pm \left(-\frac{m}{(t_{\varepsilon}^2)_{\varepsilon}}\right) [(dx_{\varepsilon})_{\varepsilon}], \\ (K_{\theta\varepsilon,\varepsilon}^2)_{\varepsilon} &= \frac{1}{\gamma} (A_{\theta\varepsilon,\varepsilon}^2)_{\varepsilon} = \pm \left[\frac{2m}{(t_{\varepsilon})_{\varepsilon}} - 1\right]^{1/2}, \end{aligned} \qquad (1.6.20) $$

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

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 $\{A_{a,\varepsilon}^i(x_{\varepsilon}), E_j^b(x'_{\varepsilon})\}_{\varepsilon} = \delta_j^i (\delta_a^b \delta(x_{\varepsilon} - x'_{\varepsilon}))_{\varepsilon}$.

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

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

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

and Hamiltonian constraint gives:

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

For the above constraints to be zero we must have

$$ \begin{aligned} \mathbf{sign} [(E_{2,\varepsilon}^{\theta_{\varepsilon}})_{\varepsilon}] &= -\mathbf{sign} [(E_{1,\varepsilon}^{\phi_{\varepsilon}})_{\varepsilon}], \\ \mathbf{sign} [(A_{\phi_{\varepsilon},\varepsilon}^3)_{\varepsilon}] &= +1, \mathbf{sign} [(A_{\phi_{\varepsilon},\varepsilon}^1)_{\varepsilon}] = -\mathbf{sign} [(A_{\theta_{\varepsilon},\varepsilon}^2)_{\varepsilon}]. \end{aligned} \qquad (1.6.25) $$

This leaves us with two alternatives corresponding to the residual gauge freedom

$$ \begin{aligned} & ((b_{\varepsilon})_{\varepsilon}, (p_{b_{\varepsilon},\varepsilon})_{\varepsilon}) \rightarrow -(b_{\varepsilon})_{\varepsilon}, -(p_{b_{\varepsilon},\varepsilon})_{\varepsilon}, \\ & (A_{a,\varepsilon}^i)_{\varepsilon} = [(c_{\varepsilon})_{\varepsilon}] \tau_3 [(dr_{\varepsilon})_{\varepsilon}] + [(b_{\varepsilon})_{\varepsilon}] \tau_2 [(d\theta_{\varepsilon})_{\varepsilon}] + \\ & \qquad [(\cos \theta_{\varepsilon})_{\varepsilon}] \tau_3 - [(b_{\varepsilon})_{\varepsilon}] [(\sin \theta_{\varepsilon})_{\varepsilon}] \tau_1 [(d\phi_{\varepsilon})_{\varepsilon}] \\ & \qquad (E_{i,\varepsilon}^a)_{\varepsilon} = [(p_{c_{\varepsilon},\varepsilon})_{\varepsilon}] \tau_3 [(\sin \theta_{\varepsilon})_{\varepsilon}] [(\partial_{r_{\varepsilon}})_{\varepsilon}] + \\ & \qquad [(p_{b_{\varepsilon},\varepsilon})_{\varepsilon}] \tau_2 [(\sin \theta_{\varepsilon})_{\varepsilon}] [(\partial_{\theta_{\varepsilon}})_{\varepsilon}] - [(p_{b_{\varepsilon},\varepsilon})_{\varepsilon}] \tau_1 [(\partial_{\phi_{\varepsilon}})_{\varepsilon}], \end{aligned} \qquad (1.6.26) $$

and
$$ (A_{a,\epsilon}^i)_\epsilon = [(c_\epsilon)_\epsilon] \tau_3 [(dr_\epsilon)_\epsilon] - [(b_\epsilon)_\epsilon] \tau_2 [(d\theta_\epsilon)_\epsilon] + \\ \qquad [(\cos\theta_\epsilon)_\epsilon] \tau_3 + [(b_\epsilon)_\epsilon] [(\sin\theta_\epsilon)_\epsilon] \tau_1 d\phi \\ (E_{i,\epsilon}^a)_\epsilon = [(p_{c_\epsilon,\epsilon})_\epsilon] \tau_3 [(\sin\theta_\epsilon)_\epsilon] [(\partial r_\epsilon)_\epsilon] - [(p_{c_\epsilon,\epsilon})_\epsilon] \tau_2 [(\sin\theta_\epsilon)_\epsilon] [(\partial\theta_\epsilon)_\epsilon] + \\ \qquad + [(p_{b_\epsilon,\epsilon})_\epsilon] \tau_1 [(\partial\phi_\epsilon)_\epsilon], \qquad (1.6.27) $$

where,

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

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

The momentum $(p_{c_\epsilon, \epsilon})_\epsilon = \pm (t_\epsilon^2)_\epsilon$ is a monotonic generalized function on $\tilde{\mathbb{R}}$ and can be interpreted as an internal generalized time parameter.

## Region I.

The analogous calculations for region I with Colombeau generalized line element (1.6.10) leads to the following phase space coordinates

$$ \begin{aligned} (\tilde{A}_{a,\epsilon}^i)_\epsilon = & [(\tilde{c}_\epsilon)_\epsilon] \tau_3 [(dr_\epsilon)_\epsilon] + [(\tilde{b}_\epsilon)_\epsilon] \tau_2 [(d\theta_\epsilon)_\epsilon] + \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [(\cos\theta_\epsilon)_\epsilon] \tau_3 - [(\tilde{b}_\epsilon)_\epsilon] [(\sin\theta_\epsilon)_\epsilon] \tau_1 [(d\phi_\epsilon)_\epsilon], \\ (\tilde{E}_{i,\epsilon}^a)_\epsilon = & [(\tilde{p}_{c_\epsilon,\epsilon})_\epsilon] \tau_3 [(\sin\theta_\epsilon)_\epsilon] [(\partial r_\epsilon)_\epsilon] + \\ & [(\tilde{p}_{b_\epsilon,\epsilon})_\epsilon] \tau_2 [(\sin\theta_\epsilon)_\epsilon] [(\partial\theta_\epsilon)_\epsilon] - [(\tilde{p}_{b_\epsilon,\epsilon})_\epsilon] \tau_1 [(\partial\phi_\epsilon)_\epsilon], \end{aligned} \qquad (1.6.30) $$

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

where,

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

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

$$ \begin{aligned} (A_{x_\epsilon, \epsilon})_\epsilon = (\tilde{c}_\epsilon)_\epsilon, (E_\epsilon^{x_\epsilon})_\epsilon = (\tilde{p}_{c_\epsilon, \epsilon})_\epsilon; \gamma (K_{\phi_\epsilon, \epsilon})_\epsilon = (\tilde{b}_\epsilon)_\epsilon, \\ (E_\epsilon^{\phi_\epsilon})_\epsilon = (\tilde{p}_{b_\epsilon, \epsilon})_\epsilon; \eta = (2n+1)\pi, (P_\epsilon^{\eta_\epsilon})_\epsilon = 0 \end{aligned} \qquad (1.6.33) $$

which constitute a 4 dimensional phase space.

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

$$ (\mathcal{R}_\epsilon)_\epsilon \equiv \frac{1}{2\pi\gamma G} \left\{ [(c_\epsilon)_\epsilon], \left[ \sqrt{[(|p_{c_\epsilon,\epsilon}|)_\epsilon]} \right] \right\} = \frac{\mathbf{sign}[(p_{c_\epsilon,\epsilon})_\epsilon]}{\sqrt{[(|p_{c_\epsilon,\epsilon}|)_\epsilon}}} = \frac{1}{[(t_\epsilon)_\epsilon]} \qquad (1.6.33) $$
Following the canonical methods presented in [44], we expand now the holonomy along $x$ direction of $\tilde{\Sigma} = \mathbb{R} \times \tilde{S}^2$ with oriented generalized length $(\tau_{\varepsilon})_{\varepsilon}$ as

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

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

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

Now, quantization would be straightforward:

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

Its action on $|(\tau_{\varepsilon})_{\varepsilon}, (\mu_{\varepsilon})_{\varepsilon}\rangle$ which are the simplified version of the spin network states in this reduced model (with $(\mu_{\varepsilon})_{\varepsilon}$ being the oriented length along the equator of $\tilde{S}^2$), then becomes:

$$ (\hat{\mathcal{R}}_{\varepsilon} |\tau_{\varepsilon}, \mu_{\varepsilon})_{\varepsilon} = \frac{1}{2\pi\sqrt{\gamma\ell_{Pl}}} (\sqrt{|(\tau_{\varepsilon})_{\varepsilon} + 1|} - \sqrt{|(\tau_{\varepsilon})_{\varepsilon} - 1|}) |(\tau_{\varepsilon})_{\varepsilon}, (\mu_{\varepsilon})_{\varepsilon}\rangle. \qquad (1.6.37) $$

Such operator $(\hat{\mathcal{R}}_{\varepsilon} |\tau_{\varepsilon}, \mu_{\varepsilon})_{\varepsilon}$ has a bounded spectrum with maximum value of $(\sqrt{2\pi\sqrt{\gamma\ell_{Pl}}})^{-1}$.

**Remark 1.6.3.** Thus the Colombeau generalized Kretschmann scalar curvature $(\hat{\mathcal{R}}_{\varepsilon}^{\rho\sigma\mu\nu}(t_{\varepsilon}) \hat{\mathcal{R}}_{\rho\sigma\mu\nu,\varepsilon}(t_{\varepsilon}))_{\varepsilon}$, which is classically has infinite large point value $\sim \mathrm{cl}[(\varepsilon^{-6})_{\varepsilon}] \in \tilde{\mathbb{R}}$ (see Eq.(1.1.18)), at quantum level has a maximum value of:

$$ \left. \mathrm{cl} \left[ \left( \hat{\mathcal{R}}_{\varepsilon}^{\rho\sigma\mu\nu}(t_{\varepsilon}) \hat{\mathcal{R}}_{\rho\sigma\mu\nu,\varepsilon}(t_{\varepsilon}) \right)_{\varepsilon} \right] \right|_{\max} \sim \frac{M^2}{\mathrm{cl}[(t_{\varepsilon}^6)_{\varepsilon}]} \leq_{\max} \mathrm{st} \left( \frac{M^2}{\mathrm{cl}[(t_{\varepsilon}^6)_{\varepsilon}]} \right) \leq_{\max} \frac{M^2}{\gamma^3 \pi^6 l_{Pl}^6}. \qquad (1.6.38) $$

**Remark 1.6.4.** Note that the Colombeau generalized curvature scalar $(\hat{\mathcal{R}}_{\varepsilon}(t_{\varepsilon}))_{\varepsilon}$

obtained at quantum level by point-free LQG by using similarly calculation as it has been applied above, has nonzero maximum value

$$ (|\hat{\mathcal{R}}_{\varepsilon}(t_{\varepsilon})|)_{\varepsilon} \Big|_{\max} \sim \frac{M}{l_{Pl}^3}. \qquad (1.6.39) $$

**Remark 1.6.5.** We emphasize that in contrast with trivial (zero valued) result obtained at quantum level for Colombeau generalized curvature scalar $(\hat{\mathcal{R}}_{\varepsilon}(t_{\varepsilon}))_{\varepsilon}$ by using canonical LQG, see Remark 1.6.2, Colombeau generalized curvature scalar $(\hat{\mathcal{R}}_{\varepsilon}(t_{\varepsilon}))_{\varepsilon}$ obtained at quantum level by point-free LQG has nonzero maximum value given by Eq.(1.6.39).
Following the canonical methods presented in [44], we expand now the holonomy along $x$ direction of $\tilde{\Sigma} = \mathbb{R} \times \tilde{S}^2$ with oriented generalized length $(\tau_{\varepsilon})_{\varepsilon}$ as

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

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

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

Now, quantization would be straightforward:

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

Its action on $|(\tau_{\varepsilon})_{\varepsilon}, (\mu_{\varepsilon})_{\varepsilon}\rangle$ which are the simplified version of the spin network states in this reduced model (with $(\mu_{\varepsilon})_{\varepsilon}$ being the oriented length along the equator of $\tilde{S}^2$), then becomes:

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

Such operator $(\hat{\mathcal{R}}_{\varepsilon} |\tau_{\varepsilon}, \mu_{\varepsilon}\rangle)_{\varepsilon}$ has a bounded spectrum with maximum value of $(\sqrt{2\pi}\sqrt{\gamma\ell_{Pl}})^{-1}$.

**Remark 1.6.3.** Thus the Colombeau generalized Kretschmann scalar curvature $(\hat{\mathcal{R}}_{\varepsilon}^{\rho\sigma\mu\nu}(t_{\varepsilon}) \hat{\mathcal{R}}_{\rho\sigma\mu\nu,\varepsilon}(t_{\varepsilon}))_{\varepsilon}$, which is classically has infinite large point value $\sim \mathrm{cl}[(\varepsilon^{-6})_{\varepsilon}] \in \tilde{\mathbb{R}}$ (see Eq.(1.1.18)), at quantum level has a maximum value of:

$$ \left. \mathrm{cl} \left[ \left( \hat{\mathcal{R}}_{\varepsilon}^{\rho\sigma\mu\nu} (t_{\varepsilon}) \hat{\mathcal{R}}_{\rho\sigma\mu\nu,\varepsilon} (t_{\varepsilon}) \right)_{\varepsilon} \right] \right|_{\max} \sim \frac{M^2}{\mathrm{cl}[(t_{\varepsilon}^6)_{\varepsilon}]} \leq_{\max} \mathrm{st} \left( \left. \frac{M^2}{\mathrm{cl}[(t_{\varepsilon}^6)_{\varepsilon}]} \right|_{\max} \right) \leq \frac{M^2}{\gamma^3 \pi^6 l_{Pl}^6}. \qquad (1.6.38) $$

**Remark 1.6.4.** Note that the Colombeau generalized curvature scalar $(\hat{\mathcal{R}}_{\varepsilon}(t_{\varepsilon}))_{\varepsilon}$

obtained at quantum level by point-free LQG by using similarly calculation as it has been applied above, has nonzero maximum value

$$ \left( |\hat{\mathcal{R}}_{\varepsilon}(t_{\varepsilon})| \right)_{\varepsilon} \Big|_{\max} \sim \frac{M}{l_{Pl}^3} \Big|_{\max}. \qquad (1.6.39) $$

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

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

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

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

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

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

**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:

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

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

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

**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

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

**Definition 1.7.3.** We call the Colombeau generalized contravariant tensor of the second rank, $(A_{i}^{ik})_{\epsilon}$, 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

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

and a mixed Colombeau generalized tensor transforms as follows
# 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

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

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

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

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

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

**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:

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

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

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

**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

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

**Definition 1.7.3.** We call the Colombeau generalized contravariant tensor of the second rank, $(A_{i}^{ik})_{\epsilon}$, 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

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

and a mixed Colombeau generalized tensor transforms as follows
$$ (A_{k,\epsilon}^i)_\epsilon = \left( \frac{\partial x_\epsilon^i}{\partial x_\epsilon^{il}} \frac{\partial x_\epsilon^{lm}}{\partial x_\epsilon^k} A_{m,\epsilon}^{il} \right)_\epsilon = \left( \frac{\partial x_\epsilon^i}{\partial x_\epsilon^{il}} \frac{\partial x_\epsilon^{lm}}{\partial x_\epsilon^k} \right)_\epsilon (A_{m,\epsilon}^{il})_\epsilon. \qquad (1.7.8) $$

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

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

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

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

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 $dx^i, i = 0, 1, 2, 3$:

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

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:

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

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
$$ (A_{k,\epsilon}^i)_\epsilon = \left( \frac{\partial x_\epsilon^i}{\partial x_\epsilon^{il}} \frac{\partial x_\epsilon^{lm}}{\partial x_\epsilon^k} A_{m,\epsilon}^{il} \right)_\epsilon = \left( \frac{\partial x_\epsilon^i}{\partial x_\epsilon^{il}} \frac{\partial x_\epsilon^{lm}}{\partial x_\epsilon^k} \right)_\epsilon (A_{m,\epsilon}^{il})_\epsilon. \qquad (1.7.8) $$

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

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

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

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

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 $dx^i, i = 0, 1, 2, 3$:

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

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:

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

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

$$(E_{\epsilon}^{iklm})_{\epsilon} = \left[ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x'^{p}} \frac{\partial x_{\epsilon}^{k}}{\partial x'^{r}} \frac{\partial x_{\epsilon}^{l}}{\partial x'^{s}} \frac{\partial x_{\epsilon}^{m}}{\partial x'^{t}} \right)_{\epsilon} \right] e^{prst}, \qquad (1.7.17)$$

or

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

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

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

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

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

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 $|g^{(0)im}| = -1$. 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

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

$$ (E_{\epsilon}^{iklm})_{\epsilon} = \left[ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x'^{p}} \frac{\partial x_{\epsilon}^{k}}{\partial x'^{r}} \frac{\partial x_{\epsilon}^{l}}{\partial x'^{s}} \frac{\partial x_{\epsilon}^{m}}{\partial x'^{t}} \right)_{\epsilon} \right] e^{prst}, \qquad (1.7.17) $$

or

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

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

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

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

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

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 $|g^{(0)im}| = -1$. 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

$$ \sqrt{-(g_{\epsilon})_{\epsilon}} (df_{ik,\epsilon}^{*})_{\epsilon} = \frac{1}{2} \sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm} (df_{\epsilon}^{lm})_{\epsilon}. \qquad (1.7.27) $$

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 $\sqrt{-(g_{\epsilon})_{\epsilon}}$).

## 1.8 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

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

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

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

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