# Singular general relativity using the Colombeau approach. Part I. Distributional Schwarzschild geometry from nonsmooth regularization via horizon

Jaykov Foukzon, Elena R Men'Kova, Alexander A Potapov

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Jaykov Foukzon, Elena R Men'Kova, Alexander A Potapov. Singular general relativity using the Colombeau approach. Part I. Distributional Schwarzschild geometry from nonsmooth regularization via horizon. Physics Essays, 2020, 33 (2), pp.180-199. 10.4006/0836-1398-33.2.180. hal-04561999

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SINGULAR GENERAL RELATIVITY USING COLOMBEAU APPROUCH.
PART I. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH
REGULARIZATION VIA HORIZON

Jaykov Foukzon¹ᵃ), Elena R. Menʼkova², Alexander A. Potapov³,⁴

¹ᵃ)jaykovfoukzon@list.ru

Center for Mathematical Sciences, Israel Institute of Technology, Haifa
3200003, Israel

²All-Russian Research Institute for Optical and Physical Measurements,
Ozernaya str. 46, Moscow 119361, Russian Federation

³Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Mokhovaya str. 11, building 7, Moscow 125009, Russian Federation

⁴JNU-IREE Joint Laboratory of Fractal Method and Signal Processing,
Department of Electronic Engineering, College of Information Science and
Technology, Jinan University, Guangzhou, People's Republic of China

**Abstract:** We studied Schwarzschild solution using Colombeau distributional geometry [M. Kunzinger, R. Steinbauer, Trans. Amer. Math. Soc. **354** (2002), 4179-4199 https://doi.org/10.1090/S0002-9947-02-03058-1], thus without leaving Schwarzschild coordinates.

**Résumé:** Nous avons étudié la solution de Schwarzschild en utilisant la géométrie distributionnelle de Colombeau, donc sans laisser de coordonnées Schwarzschild.

Key words: Singular Semi Riemannian Geometry; Colombeau Generalized Functions; Colombeau Generalized numbers; Distributional Schwarzschild Spacetime; Point Free
SINGULAR GENERAL RELATIVITY USING COLOMBEAU APPROUCH.
PART I. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH
REGULARIZATION VIA HORIZON

Jaykov Foukzon¹ª), Elena R. Men’kova², Alexander A. Potapov³,⁴

¹ª)jaykovfoukzon@list.ru

Center for Mathematical Sciences, Israel Institute of Technology, Haifa
3200003, Israel

²All-Russian Research Institute for Optical and Physical Measurements,
Ozernaya str. 46, Moscow 119361, Russian Federation

³Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Mokhovaya str. 11, building 7, Moscow 125009, Russian Federation

⁴JNU-IREE Joint Laboratory of Fractal Method and Signal Processing,
Department of Electronic Engineering, College of Information Science and
Technology, Jinan University, Guangzhou, People's Republic of China

**Abstract:** We studied Schwarzschild solution using Colombeau distributional geometry [M. Kunzinger, R. Steinbauer, Trans. Amer. Math. Soc. **354** (2002), 4179-4199 https://doi.org/10.1090/S0002-9947-02-03058-1], thus without leaving Schwarzschild coordinates.

**Résumé:** Nous avons étudié la solution de Schwarzschild en utilisant la géométrie distributionnelle de Colombeau, donc sans laisser de coordonnées Schwarzschild.

Key words: Singular Semi Riemannian Geometry; Colombeau Generalized Functions; Colombeau Generalized numbers; Distributional Schwarzschild Spacetime; Point Free
Distributional Geometry; Generalized Einstein Field Equations; Colombeau solutions of the Einstein Field Equations.

# I. BASIC NOTIONS OF COLOMBEAU GENERALIZED FUNCTIONS AND COLOMBEAU GENERALIZED NUMBERS

In contemporary mathematics, a Colombeau algebra of Colombeau generalized functions is an algebra of a certain kind containing the space of Schwartz distributions $D'(\mathbb{R}^n)$. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras¹⁻⁸ provide a rigorous framework for this.

Remark 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. Colombeau¹.

As a mathematical tool, Colombeau algebras¹⁻⁸ 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, microlocal analysis and general relativity.

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

$$ \sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_\varepsilon(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\varepsilon^{-N}), \varepsilon \to 0, \quad (1) $$

with addition and multiplication defined by the natural way:

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

$$ (f_{\varepsilon}(x))_{\varepsilon} \times (g_{\varepsilon}(x))_{\varepsilon} = (f_{\varepsilon}(x) \times g_{\varepsilon}(x))_{\varepsilon}. \quad (2) $$

**Definition 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 $O(\varepsilon^N)$ for all $N > 0$, i.e.

$$ \sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_{\varepsilon}(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\varepsilon^N), \varepsilon \to 0. \quad (3) $$

**Definition 2.** The Colombeau Algebra $G_\delta(\mathbb{R}^n)$ is defined as the quotient algebra¹ $G_\delta(\mathbb{R}^n) = \mathcal{E}_M^\infty(\mathbb{R}^n)/\mathcal{N}_\delta(\mathbb{R}^n)$. Elements of $G_\delta(\mathbb{R}^n)$ are denoted by:

$$ u = \mathrm{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + N_\delta(\mathbb{R}^n). \quad (4) $$

Note that we often abbreviate $(u_\varepsilon)_\varepsilon$ instead $\mathrm{cl}[(u_\varepsilon)_\varepsilon]$.
Distributional Geometry; Generalized Einstein Field Equations; Colombeau solutions of the Einstein Field Equations.

# I. BASIC NOTIONS OF COLOMBEAU GENERALIZED FUNCTIONS AND COLOMBEAU GENERALIZED NUMBERS

In contemporary mathematics, a Colombeau algebra of Colombeau generalized functions is an algebra of a certain kind containing the space of Schwartz distributions $D'(\mathbb{R}^n)$. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras¹⁻⁸ provide a rigorous framework for this.

Remark 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. Colombeau¹.

As a mathematical tool, Colombeau algebras¹⁻⁸ 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, microlocal analysis and general relativity.

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

$$ \sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_\varepsilon(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\varepsilon^{-N}), \varepsilon \to 0, \quad (1) $$

with addition and multiplication defined by the natural way:

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

$$ (f_{\varepsilon}(x))_{\varepsilon} \times (g_{\varepsilon}(x))_{\varepsilon} = (f_{\varepsilon}(x) \times g_{\varepsilon}(x))_{\varepsilon}. \quad (2) $$

**Definition 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 $O(\varepsilon^N)$ for all $N > 0$, i.e.

$$ \sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_{\varepsilon}(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\varepsilon^N), \varepsilon \to 0. \quad (3) $$

**Definition 2.** The Colombeau Algebra $G_\delta(\mathbb{R}^n)$ is defined as the quotient algebra¹ $G_\delta(\mathbb{R}^n) = \mathcal{E}_M^\infty(\mathbb{R}^n)/\mathcal{N}_\delta(\mathbb{R}^n)$. Elements of $G_\delta(\mathbb{R}^n)$ are denoted by:

$$ u = \mathrm{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + N_\delta(\mathbb{R}^n). \quad (4) $$

Note that we often abbreviate $(u_\varepsilon)_\varepsilon$ instead $\mathrm{cl}[(u_\varepsilon)_\varepsilon]$.
Embedding of Schwartz distributions

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

Remark 2. Note, that the embedding $\iota : D'(\mathbb{R}^n) \to G_\delta(\mathbb{R}^n)$ is non-canonical, because it depends on the choice of the $\delta$-net.

Example 1. Delta function $\delta(x) \in D'(\mathbb{R})$ for example has the following different representatives in Colombeau algebra $G_\delta(\mathbb{R})$:

$$
\frac{1}{2} \left( \frac{1}{\sqrt{\pi \epsilon}} \exp \left( -\frac{x^2}{4\epsilon} \right) \right)_{\epsilon} \in G_{\delta}(\mathbb{R}),
$$

$$
\frac{1}{\pi} \left( \frac{1}{x} \sin \left( \frac{x}{\epsilon} \right) \right)_{\epsilon} \in G_{\delta}(\mathbb{R}),
$$

$$
\frac{1}{\pi} \left( \frac{\epsilon}{x^2 + \epsilon^2} \right)_{\epsilon} \in G_{\delta}(\mathbb{R}),
$$

$$
\frac{1}{\pi} \left( \frac{1}{x^2} \sin^2 \left( \frac{x}{\epsilon} \right) \right)_{\epsilon} \in G_{\delta}(\mathbb{R}),
$$

since in $D'(\mathbb{R})$ as $\epsilon \to 0$

$$
\frac{1}{2} \frac{1}{\sqrt{\pi \epsilon}} \exp \left( -\frac{x^2}{4\epsilon} \right) \to \delta(x),
$$

$$
\frac{1}{\pi x} \sin \left( \frac{x}{\epsilon} \right) \to \delta(x),
$$

$$
\frac{1}{\pi} \frac{\epsilon}{x^2 + \epsilon^2} \to \delta(x),
$$

$$
\frac{1}{\pi x^2} \sin^2 \left( \frac{x}{\epsilon} \right) \to \delta(x) . (5)
$$

Remark 3. However, note that the embedding $\iota : D'(\mathbb{R}^n) \hookrightarrow G_\delta(\mathbb{R}^n)$ does not mean the full equivalence of the Schwartz distributions and corresponding by this embedding Colombeau generalized functions. In contrast with the Schwartz distributions Colombeau generalized functions have well defined values at any point $x \in \mathbb{R}^n$, and these point values of the Colombeau generalized functions are the Colombeau generalized numbers$^{1,8}$.

Example 2. Delta function $\delta(x)$ is ill defined, at point $x = 0$ since $\delta(0) = \infty$.
However any Colombeau generalized functions has well defined point value at point $x = 0$. For example

$$
\frac{1}{2} \left( \frac{1}{\sqrt{\pi \epsilon}} \exp \left( -\frac{x^2}{4\epsilon} \right) \right) \Big|_{\epsilon=0} = \frac{1}{2\sqrt{\pi \epsilon}} \left( \frac{1}{\sqrt{\epsilon}} \right)_{\epsilon} \in \tilde{\mathbb{R}}_{\delta},
$$
## Embedding of Schwartz distributions

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

**Remark 2.** Note, that the embedding $\iota: D'(\mathbb{R}^n) \to G_\delta(\mathbb{R}^n)$ is non-canonical, because it depends on the choice of the $\delta$-net.

**Example 1.** Delta function $\delta(x) \in D'(\mathbb{R})$ for example has the following different representatives in Colombeau algebra $G_\delta(\mathbb{R})$:

$$ \frac{1}{2} \left( \frac{1}{\sqrt{\pi \epsilon}} \exp \left( -\frac{x^2}{4\epsilon} \right) \right)_\epsilon \in G_\delta(\mathbb{R}), $$

$$ \frac{1}{\pi} \left( \frac{1}{x} \sin \left( \frac{x}{\epsilon} \right) \right)_\epsilon \in G_\delta(\mathbb{R}), $$

$$ \frac{1}{\pi} \left( \frac{\epsilon}{x^2 + \epsilon^2} \right)_\epsilon \in G_\delta(\mathbb{R}), $$

$$ \frac{1}{\pi} \left( \frac{1}{x^2} \sin^2 \left( \frac{x}{\epsilon} \right) \right)_\epsilon \in G_\delta(\mathbb{R}), $$

since in $D'(\mathbb{R})$ as $\epsilon \to 0$

$$ \frac{1}{2} \frac{1}{\sqrt{\pi \epsilon}} \exp \left( -\frac{x^2}{4\epsilon} \right) \to \delta(x), $$

$$ \frac{1}{\pi x} \sin \left( \frac{x}{\epsilon} \right) \to \delta(x), $$

$$ \frac{1}{\pi} \frac{\epsilon}{x^2 + \epsilon^2} \to \delta(x), $$

$$ \frac{1}{\pi x^2} \sin^2 \left( \frac{x}{\epsilon} \right) \to \delta(x) . (5) $$

**Remark 3.** However, note that the embedding $\iota: D'(\mathbb{R}^n) \hookrightarrow G_\delta(\mathbb{R}^n)$ does not mean the full equivalence of the Schwartz distributions and corresponding by this embedding Colombeau generalized functions. In contrast with the Schwartz distributions Colombeau generalized functions have well defined values at any point $x \in \mathbb{R}^n$, and these point values of the Colombeau generalized functions are the Colombeau generalized numbers¹,⁸.

**Example 2.** Delta function $\delta(x)$ is ill defined, at point $x = 0$ since $\delta(0) = \infty$. However any Colombeau generalized functions has well defined point value at point $x = 0$. For example

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

Designation 1. (1) We denote by $\tilde{\mathbb{R}}_{\delta}, \delta \ll 1$ the ring of real Colombeau generalized numbers$^{1,8}$. Recall that by definition $\tilde{\mathbb{R}}_{\delta} = \varepsilon_{\delta}(\mathbb{R})/\mathcal{N}_{\delta}(\mathbb{R})$ where

$$ \varepsilon_{\delta}(\mathbb{R}) = \{(x_{\varepsilon})_{\varepsilon} \in \mathbb{R}^{(0,\delta)} | (\exists a \in R_{+}) (\exists \varepsilon_{0} \in (0,1)) (\forall \varepsilon \le \varepsilon_{0}) [|x_{\varepsilon}| \le \varepsilon^{-a}]\}, $$

$$ \mathcal{N}_{\delta}(\mathbb{R}) = \{(x_{\varepsilon})_{\varepsilon} \in \mathbb{R}^{(0,\delta)} | (\forall a \in R_{+}) (\exists \varepsilon_{0} \in (0,1)) (\forall \varepsilon \le \varepsilon_{0}) [|x_{\varepsilon}| \le \varepsilon^{a}]\}. $$

(2) Notice that the ring $\tilde{\mathbb{R}}_{\delta}$ arises naturally as the ring of constants of the Colombeau algebras $G_{\delta}(\Omega)$. Recall that there exists natural embedding $\tilde{r} : \mathbb{R} \to \tilde{\mathbb{R}}_{\delta}$ such that for all $r \in \mathbb{R}, r = (r_{\varepsilon})_{\varepsilon}$ where $r_{\varepsilon} \equiv r$ for all $\varepsilon \in (0, \delta]$.

Definition 3. We say that $r$ is a standard number and abbreviate $r \in \mathbb{R}$. The ring $\mathbb{R}$ can be endowed with the structure of a partially ordered ring: for $r, s \in \tilde{\mathbb{R}}_{\delta}$ and $\eta \in \mathbb{R}_{+}, \eta \le \delta$ we abbreviate $r \le_{\tilde{\mathbb{R}}_{\delta}, \eta} s$ or simply $r \le_{\tilde{\mathbb{R}}_{\delta}} s$ if and only if there are representatives $(r_{\varepsilon})_{\varepsilon}$ and $(s_{\varepsilon})_{\varepsilon}$ with $r_{\varepsilon} \le s_{\varepsilon}$ for all $\varepsilon \in (0, \eta]$.

Designation 2. (1) Colombeau generalized number $u \in \tilde{\mathbb{R}}_{\delta}$ with representative $(u_{\varepsilon})_{\varepsilon}$ we denote by $\text{cl}[(u_{\varepsilon})_{\varepsilon}]$. (2) Colombeau generalized numbers with representative $(u_{\varepsilon})_{\varepsilon} \in \mathcal{N}_{\delta}(\mathbb{R})$ we denote by $0_{\tilde{\mathbb{R}}_{\delta}}$.

Definition 4. (1) Let $\check{\delta} = \text{cl}[(\delta_{\varepsilon})_{\varepsilon}] \in \tilde{\mathbb{R}}_{\delta}$. We say that $\check{\delta}$ is infinite small Colombeau generalized number and abbreviate $\check{\delta} \approx_{\tilde{\mathbb{R}}_{\delta}} 0$ if there exists representative $(\delta_{\varepsilon})_{\varepsilon}$ and some $q \in \mathbb{N}$ such that $|\delta_{\varepsilon}| = O(\varepsilon^q)$ as $\varepsilon \to 0$.

(2) We usually abbreviate $x \approx_{\tilde{\mathbb{R}}} y$ if $|x - y| \approx_{\tilde{\mathbb{R}}} 0$.

(3) We often abbreviated $\check{\delta} \approx 0$ instead $\check{\delta} \approx_{\tilde{\mathbb{R}}_{\delta}} 0$.

(4) Let $\Delta \in \tilde{\mathbb{R}}_{\delta}$. We say that $\Delta$ is infinite large Colombeau generalized number and abbreviate $\Delta \approx_{\tilde{\mathbb{R}}_{\delta}} \infty$ if $\Delta^{-1} \approx_{\tilde{\mathbb{R}}_{\delta}} 0$.

(5) We often abbreviated $\Delta \approx \infty$ instead $\Delta \approx_{\tilde{\mathbb{R}}_{\delta}} \infty$.

(6) Let $\mathbb{R}_{\pm\infty}$ be $\mathbb{R} \cup \{\pm\infty\}$ We say that $\Theta \in \tilde{\mathbb{R}}_{\delta; \pm\infty}$ is infinite Colombeau generalized number and abbreviate $\Theta = \pm\infty_{\tilde{\mathbb{R}}_{\delta}}$ if there exists representative $(\Theta_{\varepsilon})_{\varepsilon}$ such that $|\Theta_{\varepsilon}| = \infty$ for all $\varepsilon \in (0, \delta]$. Here we abbreviate $\varepsilon_{\delta}(\tilde{\mathbb{R}}_{\delta \pm\infty}) = \varepsilon_{\delta}(\mathbb{R} \cup \{\pm\infty\})$, $\mathcal{N}_{\delta}(\mathbb{R}_{\pm\infty}) = \mathcal{N}_{\delta}(\mathbb{R} \cup \{\pm\infty\})$ and $\tilde{\mathbb{R}}_{\delta \pm\infty} = \varepsilon_{\delta}(\mathbb{R}_{\pm;\infty})/\mathcal{N}_{\delta}(\mathbb{R}_{\pm;\infty})$.

(7) We define the set of finite Colombeau numbers $\mathcal{F}(\tilde{\mathbb{R}}_{\delta})$ as follows: $\mathcal{F}(\tilde{\mathbb{R}}_{\delta}) = \{x \in \tilde{\mathbb{R}}_{\delta} ||x| < n \text{ for some } n \in \mathbb{N}\}$.

(8) We define the set of infinite small Colombeau numbers $\mathfrak{I}(\tilde{\mathbb{R}}_{\delta})$ as follows: $\mathfrak{I}(\tilde{\mathbb{R}}_{\delta}) = \{x \in \tilde{\mathbb{R}}_{\delta} ||x| < 1/n \text{ for all } n \in \mathbb{N}\}$.
$$ \frac{1}{\pi} \left( \frac{\varepsilon}{x^2 + \varepsilon^2} \right) \Big|_{x=0} = \frac{1}{\pi} \left( \frac{1}{\varepsilon} \right) \varepsilon \in \tilde{\mathbb{R}}_{\delta}. $$

Designation 1. (1) We denote by $\tilde{\mathbb{R}}_{\delta}, \delta \ll 1$ the ring of real Colombeau generalized numbers$^{1,8}$. Recall that by definition $\tilde{\mathbb{R}}_{\delta} = \varepsilon_{\delta}(\mathbb{R})/\mathcal{N}_{\delta}(\mathbb{R})$ where

$$ \varepsilon_{\delta}(\mathbb{R}) = \{ (x_{\varepsilon})_{\varepsilon} \in \mathbb{R}^{(0,\delta)} \mid (\exists a \in R_{+}) (\exists \varepsilon_{0} \in (0,1)) (\forall \varepsilon \leq \varepsilon_{0}) [|x_{\varepsilon}| \leq \varepsilon^{-a}] \}, $$

$$ \mathcal{N}_{\delta}(\mathbb{R}) = \{ (x_{\varepsilon})_{\varepsilon} \in \mathbb{R}^{(0,\delta)} \mid (\forall a \in R_{+}) (\exists \varepsilon_{0} \in (0,1)) (\forall \varepsilon \leq \varepsilon_{0}) [|x_{\varepsilon}| \leq \varepsilon^{a}] \}. $$

(2) Notice that the ring $\tilde{\mathbb{R}}_{\delta}$ arises naturally as the ring of constants of the Colombeau algebras $G_{\delta}(\Omega)$. Recall that there exists natural embedding $\tilde{r} : \mathbb{R} \to \tilde{\mathbb{R}}_{\delta}$ such that for all $r \in \mathbb{R}, r = (r_{\varepsilon})_{\varepsilon}$ where $r_{\varepsilon} \equiv r$ for all $\varepsilon \in (0, \delta]$.

Definition 3. We say that $r$ is a standard number and abbreviate $r \in \mathbb{R}$. The ring $\mathbb{R}$ can be endowed with the structure of a partially ordered ring: for $r, s \in \tilde{\mathbb{R}}_{\delta}$ and $\eta \in \mathbb{R}_{+}, \eta \leq \delta$ we abbreviate $r \leq_{\tilde{\mathbb{R}}_{\delta}, \eta} s$ or simply $r \leq_{\tilde{\mathbb{R}}_{\delta}} s$ if and only if there are representatives $(r_{\varepsilon})_{\varepsilon}$ and $(s_{\varepsilon})_{\varepsilon}$ with $r_{\varepsilon} \leq s_{\varepsilon}$ for all $\varepsilon \in (0, \eta]$.

Designation 2. (1) Colombeau generalized number $u \in \tilde{\mathbb{R}}_{\delta}$ with representative $(u_{\varepsilon})_{\varepsilon}$ we denote by $\text{cl}[(u_{\varepsilon})_{\varepsilon}]$. (2) Colombeau generalized numbers with representative $(u_{\varepsilon})_{\varepsilon} \in \mathcal{N}_{\delta}(\mathbb{R})$ we denote by $0_{\tilde{\mathbb{R}}_{\delta}}$.

Definition 4. (1) Let $\check{\delta} = \text{cl}[(\delta_{\varepsilon})_{\varepsilon}] \in \tilde{\mathbb{R}}_{\delta}$. We say that $\check{\delta}$ is infinite small Colombeau generalized number and abbreviate $\check{\delta} \approx_{\tilde{\mathbb{R}}_{\delta}} 0$ if there exists representative $(\delta_{\varepsilon})_{\varepsilon}$ and some $q \in \mathbb{N}$ such that $|\delta_{\varepsilon}| = O(\varepsilon^q)$ as $\varepsilon \to 0$.

(2) We usually abbreviate $x \approx_{\tilde{\mathbb{R}}} y$ if $|x - y| \approx_{\tilde{\mathbb{R}}} 0$.

(3) We often abbreviated $\check{\delta} \approx 0$ instead $\check{\delta} \approx_{\tilde{\mathbb{R}}_{\delta}} 0$.

(4) Let $\Delta \in \tilde{\mathbb{R}}_{\delta}$. We say that $\Delta$ is infinite large Colombeau generalized number and abbreviate $\Delta \approx_{\tilde{\mathbb{R}}_{\delta}} \infty$ if $\Delta^{-1} \approx_{\tilde{\mathbb{R}}_{\delta}} 0$.

(5) We often abbreviated $\Delta \approx \infty$ instead $\Delta \approx_{\tilde{\mathbb{R}}_{\delta}} \infty$.

(6) Let $\mathbb{R}_{\pm\infty}$ be $\mathbb{R} \cup \{\pm\infty\}$ We say that $\Theta \in \tilde{\mathbb{R}}_{\delta; \pm\infty}$ is infinite Colombeau generalized number and abbreviate $\Theta = \pm\infty_{\tilde{\mathbb{R}}_{\delta}}$ if there exists representative $(\Theta_{\varepsilon})_{\varepsilon}$ such that $|\Theta_{\varepsilon}| = \infty$ for all $\varepsilon \in (0, \delta]$. Here we abbreviate $\varepsilon_{\delta}(\tilde{\mathbb{R}}_{\delta \pm\infty}) = \varepsilon_{\delta}(\mathbb{R} \cup \{\pm\infty\})$, $\mathcal{N}_{\delta}(\mathbb{R}_{\pm\infty}) = \mathcal{N}_{\delta}(\mathbb{R} \cup \{\pm\infty\})$ and $\tilde{\mathbb{R}}_{\delta \pm\infty} = \varepsilon_{\delta}(\mathbb{R}_{\pm\infty})/\mathcal{N}_{\delta}(\mathbb{R}_{\pm\infty})$.

(7) We define the set of finite Colombeau numbers $\mathcal{F}(\tilde{\mathbb{R}}_{\delta})$ as follows: $\mathcal{F}(\tilde{\mathbb{R}}_{\delta}) = \{x \in \tilde{\mathbb{R}}_{\delta} \mid |x| < n \text{ for some } n \in \mathbb{N}\}$.

(8) We define the set of infinite small Colombeau numbers $\mathfrak{I}(\tilde{\mathbb{R}}_{\delta})$ as follows: $\mathfrak{I}(\tilde{\mathbb{R}}_{\delta}) = \{x \in \tilde{\mathbb{R}}_{\delta} \mid |x| < 1/n \text{ for all } n \in \mathbb{N}\}$.
(9) We define the set of infinite large Colombeau numbers $\mathcal{L}(\tilde{\mathbb{R}}_\delta)$ as follows:

$$ \mathcal{L}(\tilde{\mathbb{R}}_\delta) = \{x \in \tilde{\mathbb{R}}_\delta \mid |x| > n \text{ for all } n \in \mathbb{N}\}. $$

**Definition 5. (Standard Part Mapping).** (1) The standard part mapping st: $\mathcal{F}(\tilde{\mathbb{R}}_\delta) \to \mathbb{R}$ is defined by the formula:

$$ \mathrm{st}(x) = \sup\{r \in \mathbb{R} \mid r \le \tilde{\mathbb{R}}_\delta x\}. \quad (6) $$

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

(2) The mapping st: $\tilde{\mathbb{R}}_\delta_{\pm\infty} \to \mathbb{R} \cup \{\pm\infty\}$ is defined by (1) and by $\mathrm{st}(x) = \pm\infty$ for $x \in \mathcal{F}(\tilde{\mathbb{R}}_\delta; \pm\infty)$ and for $x \in \tilde{\mathbb{R}}_\delta; \pm\infty \setminus \mathcal{F}(\tilde{\mathbb{R}}_\delta; \pm\infty)$ respectively.

(3) The standard part mapping is a ring homomorphism from $\mathcal{F}(\tilde{\mathbb{R}}_\delta)$ onto $\mathbb{R}$, i.e. for every $x, y \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ we have: (a) $\mathrm{st}(x \pm y) = \mathrm{st}(x) \pm \mathrm{st}(y)$, (b) $\mathrm{st}(x \cdot y) = \mathrm{st}(x) \cdot \mathrm{st}(y)$, (c) $\mathrm{st}(x/y) = \mathrm{st}(x)/\mathrm{st}(y)$, whenever $\mathrm{st}(y) \ne 0$.

(4) Every finite Colombeau generalized number $x \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ has a unique expansion $x = \mathrm{st}(x) + dx$, where $dx \in \mathfrak{I}(\tilde{\mathbb{R}}_\delta)$.

**Definition 6.** Let $(f_\epsilon(x))_\epsilon \in \mathcal{G}_\delta(\mathbb{R}^n)$ and $\tilde{x} \in \mathbb{R}^n$, then $\mathrm{cl}[(f_\epsilon(x))_\epsilon] \in \tilde{\mathbb{R}}_\delta$. We say that Colombeau generalized number $\mathrm{cl}[(f_\epsilon(x))_\epsilon]$ is a point value of Colombeau generalized function $(f_\epsilon(x))_\epsilon$ at point $x \in \mathbb{R}^n$.

**Definition 7. (Principal value mapping)** The principal value mapping p. v: $\tilde{\mathbb{R}}_\delta \to \mathbb{R}$ of Colombeau generalized function $(f_\epsilon(x))_\epsilon$ at point $x \in \mathbb{R}^n$ is defined by the formula

$$ \text{p. v. } \{\mathrm{cl}[(f_\epsilon(x))_\epsilon]\} = \sup_{\epsilon \in (0,1]} |f_\epsilon(x)|. \quad (7) $$

**Designation 3.** We write for short $\mathrm{p. v.}[(f_\epsilon(x))_\epsilon]$ instead $\mathrm{p. v.}\{\mathrm{cl}[(f_\epsilon(x))_\epsilon]\}$

Let $u = (u_\epsilon(x))_\epsilon$ be Colombeau generalized function such that $u \in \mathcal{E}_M^\infty(\mathbb{R})$ and let $\mu$ be Colombeau generalized number $\mu \in \tilde{\mathbb{R}}_\delta$ with representative $(\mu_\epsilon)_\epsilon \in \mathcal{E}_\delta(\mathbb{R})$. Thus we have a mapping $\tilde{u}: \mathcal{E}_\delta(\mathbb{R}) \to \mathcal{E}_\delta(\mathbb{R})$ is defined by the formula

$$ \tilde{u}[(\mu_\epsilon)_\epsilon] = (u_\epsilon(\mu_\epsilon))_\epsilon \in \mathcal{E}_\delta(\mathbb{R}) \quad (8) $$

in natural way. Let $u_1 = (u_{1,\epsilon}(x))_\epsilon$ and $u_2 = (u_{2,\epsilon}(x))_\epsilon$ be Colombeau generalized function such that $u_1, u_2 \in \mathcal{E}_M^\infty(\mathbb{R})$. The addition $\tilde{u}_1 + \tilde{u}_2$ and multiplication $\tilde{u}_1 \cdot \tilde{u}_2$ by using Eq. (8) is defined in the natural way by formulas:

$$ (\tilde{u}_1 + \tilde{u}_2)[(\mu_\epsilon)_\epsilon] = (u_{1,\epsilon}(\mu_\epsilon))_\epsilon + (u_\epsilon(\mu_\epsilon))_\epsilon = (u_{1,\epsilon}(\mu_\epsilon) + u_{2,\epsilon}(\mu_\epsilon))_\epsilon, $$

$$ (\tilde{u}_1 \cdot \tilde{u}_2)[(\mu_\epsilon)_\epsilon] = (u_{1,\epsilon}(\mu_\epsilon))_\epsilon \cdot (u_\epsilon(\mu_\epsilon))_\epsilon = (u_{1,\epsilon}(\mu_\epsilon) \cdot u_{2,\epsilon}(\mu_\epsilon))_\epsilon. \quad (9) $$

**Definition 8.** The algebra moderate functions $\mathcal{E}_M^\infty(\tilde{\mathbb{R}})$ on $\tilde{\mathbb{R}}$ is the algebra of a
(9) We define the set of infinite large Colombeau numbers $\mathcal{L}(\tilde{\mathbb{R}}_\delta)$ as follows:

$$ \mathcal{L}(\tilde{\mathbb{R}}_\delta) = \{x \in \tilde{\mathbb{R}}_\delta \mid |x| > n \text{ for all } n \in \mathbb{N}\}. $$

**Definition 5. (Standard Part Mapping).** (1) The standard part mapping st: $\mathcal{F}(\tilde{\mathbb{R}}_\delta) \to \mathbb{R}$ is defined by the formula:

$$ \mathrm{st}(x) = \sup\{r \in \mathbb{R} \mid r \le \tilde{\mathbb{R}}_\delta x\}. \quad (6) $$

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

(2) The mapping st: $\tilde{\mathbb{R}}_\delta_{\pm\infty} \to \mathbb{R} \cup \{\pm\infty\}$ is defined by (1) and by $\mathrm{st}(x) = \pm\infty$ for $x \in \mathcal{F}(\tilde{\mathbb{R}}_\delta; \pm\infty)$ and for $x \in \tilde{\mathbb{R}}_\delta; \pm\infty \setminus \mathcal{F}(\tilde{\mathbb{R}}_\delta; \pm\infty)$ respectively.

(3) The standard part mapping is a ring homomorphism from $\mathcal{F}(\tilde{\mathbb{R}}_\delta)$ onto $\mathbb{R}$, i.e. for every $x, y \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ we have: (a) $\mathrm{st}(x \pm y) = \mathrm{st}(x) \pm \mathrm{st}(y)$, (b) $\mathrm{st}(x \cdot y) = \mathrm{st}(x) \cdot \mathrm{st}(y)$, (c) $\mathrm{st}(x/y) = \mathrm{st}(x)/\mathrm{st}(y)$, whenever $\mathrm{st}(y) \ne 0$.

(4) Every finite Colombeau generalized number $x \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ has a unique expansion $x = \mathrm{st}(x) + dx$, where $dx \in \mathfrak{I}(\tilde{\mathbb{R}}_\delta)$.

**Definition 6.** Let $(f_\varepsilon(x))_\varepsilon \in \mathcal{G}_\delta(\mathbb{R}^n)$ and $\tilde{x} \in \mathbb{R}^n$, then $\mathrm{cl}[(f_\varepsilon(x))_\varepsilon] \in \tilde{\mathbb{R}}_\delta$. We say that Colombeau generalized number $\mathrm{cl}[(f_\varepsilon(x))_\varepsilon]$ is a point value of Colombeau generalized function $(f_\varepsilon(x))_\varepsilon$ at point $x \in \mathbb{R}^n$.

**Definition 7. (Principal value mapping)** The principal value mapping p. v: $\tilde{\mathbb{R}}_\delta \to \mathbb{R}$ of Colombeau generalized function $(f_\varepsilon(x))_\varepsilon$ at point $x \in \mathbb{R}^n$ is defined by the formula

$$ \text{p. v. } \{\mathrm{cl}[(f_\varepsilon(x))_\varepsilon]\} = \sup_{\varepsilon \in (0,1]} |f_\varepsilon(x)|. \quad (7) $$

**Designation 3.** We write for short $\mathrm{p. v.}[(f_\varepsilon(x))_\varepsilon]$ instead $\mathrm{p. v.}\{\mathrm{cl}[(f_\varepsilon(x))_\varepsilon]\}$

Let $u = (u_\varepsilon(x))_\varepsilon$ be Colombeau generalized function such that $u \in \mathcal{E}_M^\infty(\mathbb{R})$ and let $\mu$ be Colombeau generalized number $\mu \in \tilde{\mathbb{R}}_\delta$ with representative $(\mu_\varepsilon)_\varepsilon \in \mathcal{E}_\delta(\mathbb{R})$. Thus we have a mapping $\tilde{u}: \mathcal{E}_\delta(\mathbb{R}) \to \mathcal{E}_\delta(\mathbb{R})$ is defined by the formula

$$ \tilde{u}[(\mu_\varepsilon)_\varepsilon] = (u_\varepsilon(\mu_\varepsilon))_\varepsilon \in \mathcal{E}_\delta(\mathbb{R}) \quad (8) $$

in natural way. Let $u_1 = (u_{1,\varepsilon}(x))_\varepsilon$ and $u_2 = (u_{2,\varepsilon}(x))_\varepsilon$ be Colombeau generalized function such that $u_1, u_2 \in \mathcal{E}_M^\infty(\mathbb{R})$. The addition $\tilde{u}_1 + \tilde{u}_2$ and multiplication $\tilde{u}_1 \cdot \tilde{u}_2$ by using Eq. (8) is defined in the natural way by formulas:

$$ (\tilde{u}_1 + \tilde{u}_2)[(\mu_\varepsilon)_\varepsilon] = (u_{1,\varepsilon}(\mu_\varepsilon))_\varepsilon + (u_\varepsilon(\mu_\varepsilon))_\varepsilon = (u_{1,\varepsilon}(\mu_\varepsilon) + u_{2,\varepsilon}(\mu_\varepsilon))_\varepsilon, $$

$$ (\tilde{u}_1 \cdot \tilde{u}_2)[(\mu_\varepsilon)_\varepsilon] = (u_{1,\varepsilon}(\mu_\varepsilon))_\varepsilon \cdot (u_\varepsilon(\mu_\varepsilon))_\varepsilon = (u_{1,\varepsilon}(\mu_\varepsilon) \cdot u_{2,\varepsilon}(\mu_\varepsilon))_\varepsilon. \quad (9) $$

**Definition 8.** The algebra moderate functions $\mathcal{E}_M^\infty(\tilde{\mathbb{R}})$ on $\tilde{\mathbb{R}}$ is the algebra of a
functions $\tilde{u}: \mathcal{E}_\delta(\mathbb{R}) \to \mathcal{E}_\delta(\mathbb{R})$ defined by the Eq. (8) and such that for all compact subsets $K$ of $\mathbb{R}$ and all $m \in \mathbb{N}$ there is an $N > 0$ and $\varepsilon_0 \in \mathbb{R}_+$ such that for all $\varepsilon \le \varepsilon_0$:

$$ \left( \sup_{\mu_{\varepsilon} \in K} \left| \frac{d^m u_{\varepsilon}(\mu_{\varepsilon})}{d\mu_{\varepsilon}^m} \right| \right)_{\varepsilon} = O((\varepsilon^{-N})_{\varepsilon}), \quad (10) $$

and with addition and multiplication defined by Eq. (9).

Definition 9. The ideal $\mathcal{N}_\delta(\tilde{\mathbb{R}})$ of negligible functions is defined in the same way but with the derivatives instead bounded by $O((\varepsilon^N)_\varepsilon)$ for all $N > 0$, i.e.,

$$ \left( \sup_{\mu_{\varepsilon} \in K} \left| \frac{d^m u_{\varepsilon}(\mu_{\varepsilon})}{d\mu_{\varepsilon}^m} \right| \right)_{\varepsilon} = O((\varepsilon^N)_{\varepsilon}). \quad (11) $$

for all $\varepsilon \le \varepsilon_0$.

Definition 10. The Colombeau Algebra $G_\delta(\tilde{\mathbb{R}})$ is defined as the quotient algebra $G_\delta(\tilde{\mathbb{R}}) = \mathcal{E}_M^\infty(\tilde{\mathbb{R}})/\mathcal{N}_\delta(\tilde{\mathbb{R}})$. Elements of $G_\delta(\tilde{\mathbb{R}})$ are denoted by: $\tilde{u} = \text{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + N_\delta(\tilde{\mathbb{R}})$.

Definition 11. The Colombeau Algebra $G_\delta(\tilde{\mathbb{R}}^n)$ is defined as the quotient algebra $G_\delta(\tilde{\mathbb{R}}) = \mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)/\mathcal{N}_\delta(\tilde{\mathbb{R}}^n)$ where $\mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)$ and $\mathcal{N}_\delta(\tilde{\mathbb{R}}^n)$ is defined similarly as $\mathcal{E}_M^\infty(\tilde{\mathbb{R}})$ and $\mathcal{N}_\delta(\tilde{\mathbb{R}})$ by obvious way. (1) Let $u = (u_\varepsilon(x))_\varepsilon$ be Colombeau generalized function such that $u \in \mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)$ and let $\mu$ be a vector $(\mu_1, \dots, \mu_i, \dots, \mu_n)$, where $\mu_i, i = 1, \dots, n$ are Colombeau generalized numbers $\mu_i \in \mathcal{R}_\delta$ with representatives $(\mu_{i,\varepsilon})_\varepsilon \in \mathcal{E}_\delta(\tilde{\mathbb{R}})$. Thus we have a mapping $\tilde{u}: \mathcal{E}_\delta(\tilde{\mathbb{R}}^n) \to \mathcal{E}_\delta(\tilde{\mathbb{R}})$ is defined by the formula

$$ \tilde{u}[(\mu_{\varepsilon})_{\varepsilon}] = (u_{\varepsilon}(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon}))_{\varepsilon} \in \mathcal{E}_{\delta}(\tilde{\mathbb{R}}). \quad (12) $$

(2) The algebra moderate functions $\mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)$ on $\tilde{\mathbb{R}}^n$ is the algebra of a functions $\tilde{u}: \mathcal{E}_\delta(\tilde{\mathbb{R}}^n) \to \mathcal{E}_\delta(\tilde{\mathbb{R}})$ defined by the Eq. (32) and such that for all compact subsets $K$ of $\tilde{\mathbb{R}}^n$ and all multi indices $\alpha$ there is an $N > 0$ and $\varepsilon_0 \in \mathbb{R}_+$ such that for all $\varepsilon \le \varepsilon_0$

$$ \left( \sup_{x \in K} \left| \frac{\partial |\alpha| u_{\varepsilon}(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon})}{(\partial \mu_{1,\varepsilon})^{\alpha_1} \dots (\partial \mu_{n,\varepsilon})^{\alpha_n}} \right| \right)_{\varepsilon} = O((\varepsilon^{-N})_{\varepsilon}), \quad (13) $$

and with addition and multiplication defined by natural way by formulas

$$ \begin{aligned} (\tilde{u}_1 + \tilde{u}_2)[(\mu_\varepsilon)_\varepsilon] &= (u_{1,\varepsilon}(\mu_\varepsilon))_\varepsilon + (u_\varepsilon(\mu_\varepsilon))_\varepsilon = (u_{1,\varepsilon}(\mu_\varepsilon) + u_{2,\varepsilon}(\mu_\varepsilon))_\varepsilon \\ (\tilde{u}_1 \cdot \tilde{u}_2)[(\mu_\varepsilon)_\varepsilon] &= (u_{1,\varepsilon}(\mu_\varepsilon))_\varepsilon \cdot (u_\varepsilon(\mu_\varepsilon))_\varepsilon = (u_{1,\varepsilon}(\mu_\varepsilon) \cdot u_{2,\varepsilon}(\mu_\varepsilon))_\varepsilon. \end{aligned} \quad (14) $$

(3) The ideal $\mathcal{N}_\delta(\tilde{\mathbb{R}})$ of negligible functions on $\tilde{\mathbb{R}}$ is defined in the same way but with the derivatives instead bounded by $O((\varepsilon^N)_\varepsilon)$ for all $N > 0$, i.e.,

$$ \left( \sup_{x \in K} \left| \frac{\partial |\alpha| u_{\varepsilon}(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon})}{(\partial \mu_{1,\varepsilon})^{\alpha_1} \dots (\partial \mu_{n,\varepsilon})^{\alpha_n}} \right| \right)_{\varepsilon} = O((\varepsilon^N)_{\varepsilon}), \quad (15) $$

for all $\varepsilon \le \varepsilon_0$.

(4) Elements of $G_\delta(\tilde{\mathbb{R}}^n)$ are denoted by: $\tilde{u} = \text{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + N_\delta(\tilde{\mathbb{R}})$.

## II. POINT FREE DISTRIBUTIONAL GEOMETRY IN SINGULAR 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. If in addition $g$ is positive definite, $(M; g)$ is a degenerate (singular) Riemannian manifold. This paper dealing with Colombeau extension of the Einstein field equations using
functions $\tilde{u}: \mathcal{E}_\delta(\mathbb{R}) \to \mathcal{E}_\delta(\mathbb{R})$ defined by the Eq. (8) and such that for all compact subsets $K$ of $\mathbb{R}$ and all $m \in \mathbb{N}$ there is an $N > 0$ and $\varepsilon_0 \in \mathbb{R}_+$ such that for all $\varepsilon \le \varepsilon_0$:

$$ \left( \sup_{\mu_{\varepsilon} \in K} \left| \frac{d^m u_{\varepsilon}(\mu_{\varepsilon})}{d\mu_{\varepsilon}^m} \right| \right)_{\varepsilon} = O((\varepsilon^{-N})_{\varepsilon}), \quad (10) $$

and with addition and multiplication defined by Eq. (9).

Definition 9. The ideal $\mathcal{N}_\delta(\tilde{\mathbb{R}})$ of negligible functions is defined in the same way but with the derivatives instead bounded by $O((\varepsilon^N)_\varepsilon)$ for all $N > 0$, i.e.,

$$ \left( \sup_{\mu_{\varepsilon} \in K} \left| \frac{d^m u_{\varepsilon}(\mu_{\varepsilon})}{d\mu_{\varepsilon}^m} \right| \right)_{\varepsilon} = O((\varepsilon^N)_\varepsilon). \quad (11) $$

for all $\varepsilon \le \varepsilon_0$.

Definition 10. The Colombeau Algebra $G_\delta(\tilde{\mathbb{R}})$ is defined as the quotient algebra $G_\delta(\tilde{\mathbb{R}}) = \mathcal{E}_M^\infty(\tilde{\mathbb{R}})/\mathcal{N}_\delta(\tilde{\mathbb{R}})$. Elements of $G_\delta(\tilde{\mathbb{R}})$ are denoted by: $\tilde{u} = \text{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + N_\delta(\tilde{\mathbb{R}})$.

Definition 11. The Colombeau Algebra $G_\delta(\tilde{\mathbb{R}}^n)$ is defined as the quotient algebra $G_\delta(\tilde{\mathbb{R}}) = \mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)/\mathcal{N}_\delta(\tilde{\mathbb{R}}^n)$ where $\mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)$ and $\mathcal{N}_\delta(\tilde{\mathbb{R}}^n)$ is defined similarly as $\mathcal{E}_M^\infty(\tilde{\mathbb{R}})$ and $\mathcal{N}_\delta(\tilde{\mathbb{R}})$ by obvious way. (1) Let $u = (u_\varepsilon(x))_\varepsilon$ be Colombeau generalized function such that $u \in \mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)$ and let $\mu$ be a vector $(\mu_1, \dots, \mu_i, \dots, \mu_n)$, where $\mu_i, i = 1, \dots, n$ are Colombeau generalized numbers $\mu_i \in \mathcal{R}_\delta$ with representatives $(\mu_{i,\varepsilon})_\varepsilon \in \mathcal{E}_\delta(\tilde{\mathbb{R}})$. Thus we have a mapping $\tilde{u}: \mathcal{E}_\delta(\tilde{\mathbb{R}}^n) \to \mathcal{E}_\delta(\tilde{\mathbb{R}})$ is defined by the formula

$$ \tilde{u}[(\mu_\varepsilon)_\varepsilon] = (u_\varepsilon(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon}))_\varepsilon \in \mathcal{E}_\delta(\tilde{\mathbb{R}}). \quad (12) $$

(2) The algebra moderate functions $\mathcal{E}_M^\infty(\tilde{\mathbb{R}}^n)$ on $\tilde{\mathbb{R}}^n$ is the algebra of a functions $\tilde{u}: \mathcal{E}_\delta(\tilde{\mathbb{R}}^n) \to \mathcal{E}_\delta(\tilde{\mathbb{R}})$ defined by the Eq. (32) and such that for all compact subsets $K$ of $\tilde{\mathbb{R}}^n$ and all multi indices $\alpha$ there is an $N > 0$ and $\varepsilon_0 \in \mathbb{R}_+$ such that for all $\varepsilon \le \varepsilon_0$

$$ \left( \sup_{x \in K} \left| \frac{\partial |\alpha| u_\varepsilon(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon})}{(\partial \mu_{1,\varepsilon})^{\alpha_1} \dots (\partial \mu_{n,\varepsilon})^{\alpha_n}} \right| \right)_\varepsilon = O((\varepsilon^{-N})_\varepsilon), \quad (13) $$

and with addition and multiplication defined by natural way by formulas

$$ \begin{aligned} (\tilde{u}_1 + \tilde{u}_2)[(\mu_\varepsilon)_\varepsilon] &= (u_{1,\varepsilon}(\mu_\varepsilon))_\varepsilon + (u_\varepsilon(\mu_\varepsilon))_\varepsilon = (u_{1,\varepsilon}(\mu_\varepsilon) + u_{2,\varepsilon}(\mu_\varepsilon))_\varepsilon \\ (\tilde{u}_1 \cdot \tilde{u}_2)[(\mu_\varepsilon)_\varepsilon] &= (u_{1,\varepsilon}(\mu_\varepsilon))_\varepsilon \cdot (u_\varepsilon(\mu_\varepsilon))_\varepsilon = (u_{1,\varepsilon}(\mu_\varepsilon) \cdot u_{2,\varepsilon}(\mu_\varepsilon))_\varepsilon. \end{aligned} \quad (14) $$

(3) The ideal $\mathcal{N}_\delta(\tilde{\mathbb{R}})$ of negligible functions on $\tilde{\mathbb{R}}$ is defined in the same way but with the derivatives instead bounded by $O((\varepsilon^N)_\varepsilon)$ for all $N > 0$, i.e.,

$$ \left( \sup_{x \in K} \left| \frac{\partial |\alpha| u_\varepsilon(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon})}{(\partial \mu_{1,\varepsilon})^{\alpha_1} \dots (\partial \mu_{n,\varepsilon})^{\alpha_n}} \right| \right)_\varepsilon = O((\varepsilon^N)_\varepsilon), \quad (15) $$

for all $\varepsilon \le \varepsilon_0$.

(4) Elements of $G_\delta(\tilde{\mathbb{R}}^n)$ are denoted by: $\tilde{u} = \text{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + N_\delta(\tilde{\mathbb{R}})$.

## II. POINT FREE DISTRIBUTIONAL GEOMETRY IN SINGULAR 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. If in addition $g$ is positive definite, $(M; g)$ is a degenerate (singular) Riemannian manifold. This paper dealing with Colombeau extension of the Einstein field equations using
apparatus of the Colombeau generalized functions<sup>1-7</sup> and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry<sup>7,9-18</sup> . The *regularizations of singularities* presented in some solutions of the Einstein equations is an important part of this approach. Any singularities presented in some solutions such that Schwarzschild solution etc. of the Einstein field equations recognized in the sense of Colombeau generalized functions are not classical. Note that in physical literature these singular solutions many years were mistakenly considered as vacuum solutions of the Einstein field equations<sup>7,14</sup>.

Remark 2. Let $(R^a_{bcd,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ be Colombeau generalized function obtained using the standard definition of the Riemann curvature in a coordinate basis, i.e.

$$ (R^a_{bcd,\epsilon})_\epsilon = (\Gamma^a_{db,c,\epsilon})_\epsilon - (\Gamma^a_{cb,d,\epsilon})_\epsilon + (\Gamma^a_{cf,\epsilon}\Gamma^f_{ab,\epsilon})_\epsilon - (\Gamma^a_{df,\epsilon}\Gamma^f_{cb,\epsilon})_\epsilon, \quad (16) $$

connection coefficients in terms of the regularized metric $(g_{ab,\epsilon})_\epsilon$, where $\epsilon \in (0, \delta]$ such that $(g_{ab,\epsilon})_\epsilon, (g^a_\epsilon)_{\epsilon} \in G_\delta(\mathbb{R}^4), (\det(g_{ab,\epsilon}))_\epsilon \neq 0_{\mathbb{R}}$. It has been shown by many authors that under appropriate regularization using the Eq. (16) one can defines the curvature scalar as a classical Schwartz distribution in $D'(\mathbb{R}^4)$.

Remark 3. This is the case even for the well-known Schwarzschild spacetime, which is given in the Schwarzschild coordinates $(\hat{x}^0, \hat{r}, \theta, \varphi)$, by the metric

$$ ds^2 = -(1 - \frac{a}{\hat{r}})(d\hat{x}^0)^2 + (1 - \frac{a}{\hat{r}})^{-1}(d\hat{r})^2 + \hat{r}^2[(d\theta)^2 + \sin^2\theta (d\varphi)^2]. \quad (17) $$

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 spacetime, respectively. Obviously, the fundamental tensor corresponding to $ds^2$ has the components which are degenerate (singular): (i) at $\hat{r} = 0$ and (ii) at $\hat{r} = a$.

Remark 4. Note that: (1) in canonical papers<sup>19-30</sup> the Colombeau distributional metric tensor $(g_{ab,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ related to $(R^a_{bcd,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ by Eq. (16) is not never considered as the Colombeau solution<sup>31-38</sup> of the Einstein field equations.

(2) Colombeau nonlinear distributional geometry 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 $(R^a_{bcd,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ classical Schwartz distributions in $D'(\mathbb{R}^4)$, (3) there are no any important physical applications of the classical linear distributional geometry were obtained.

Remark 5. Fundamental physical applications of the Colombeau nonlinear distributional geometry have been originally obtained in authors paper<sup>11,14,15</sup>.

By using the Cartesian coordinates $(\hat{x}^0, \hat{x}^1, \hat{x}^2, \hat{x}^3)$, which are related to $(\hat{x}^0, \hat{r}, \theta, \varphi)$ through the canonical relations: $\hat{x}^1 = \hat{r} \cos \varphi \sin \theta$, $\hat{x}^2 = \hat{r} \sin \varphi \sin \theta$, $\hat{x}^3 = \hat{r} \cos \theta$, the metric (17) reads

$$ ds^2 = \hat{g}_{\mu\nu} d\hat{x}^\mu d\hat{x}^\nu, \text{ where at points } \hat{r} \neq 0, \hat{r} \neq a \text{ the metric } \hat{g}_{\mu\nu} \text{ is given by} $$

$$ \hat{g}_{00} = -(1-h), \quad \hat{g}_{0\alpha} = 0, $$

$$ \hat{g}_{\alpha\beta} = \delta^{\alpha\beta} + h(1-h)^{-1} \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2}, \quad \alpha, \beta = 1, 2, 3 \quad (18) $$

with $h \triangleq a/\hat{r}$. It is known that at points $\hat{r} \neq 0, \hat{r} \neq a$.
apparatus of the Colombeau generalized functions<sup>1-7</sup> and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry<sup>7,9-18</sup> . The *regularizations of singularities* presented in some solutions of the Einstein equations is an important part of this approach. Any singularities presented in some solutions such that Schwarzschild solution etc. of the Einstein field equations recognized in the sense of Colombeau generalized functions are not classical. Note that in physical literature these singular solutions many years were mistakenly considered as vacuum solutions of the Einstein field equations<sup>7,14</sup>.

Remark 2. Let $(R^a_{bcd,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ be Colombeau generalized function obtained using the standard definition of the Riemann curvature in a coordinate basis, i.e.

$$ (R^a_{bcd,\epsilon})_\epsilon = (\Gamma^a_{db,c,\epsilon})_\epsilon - (\Gamma^a_{cb,d,\epsilon})_\epsilon + (\Gamma^a_{cf,\epsilon}\Gamma^f_{ab,\epsilon})_\epsilon - (\Gamma^a_{df,\epsilon}\Gamma^f_{cb,\epsilon})_\epsilon, \quad (16) $$

connection coefficients in terms of the regularized metric $(g_{ab,\epsilon})_\epsilon$, where $\epsilon \in (0, \delta]$ such that $(g_{ab,\epsilon})_\epsilon, (g^a_\epsilon)_{\epsilon} \in G_\delta(\mathbb{R}^4)$, $(\det(g_{ab,\epsilon}))_\epsilon \neq 0_{\mathbb{R}}$. It has been shown by many authors that under appropriate regularization using the Eq. (16) one can defines the curvature scalar as a classical Schwartz distribution in $D'(\mathbb{R}^4)$.

Remark 3. This is the case even for the well-known Schwarzschild spacetime, which is given in the Schwarzschild coordinates $(\hat{x}^0, \hat{r}, \theta, \varphi)$, by the metric

$$ ds^2 = -(1 - \frac{a}{\hat{r}})(d\hat{x}^0)^2 + (1 - \frac{a}{\hat{r}})^{-1}(d\hat{r})^2 + \hat{r}^2[(d\theta)^2 + \sin^2\theta(d\varphi)^2]. \quad (17) $$

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 spacetime, respectively. Obviously, the fundamental tensor corresponding to $ds^2$ has the components which are degenerate (singular): (i) at $\hat{r} = 0$ and (ii) at $\hat{r} = a$.

Remark 4. Note that: (1) in canonical papers<sup>19-30</sup> the Colombeau distributional metric tensor $(g_{ab,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ related to $(R^a_{bcd,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ by Eq. (16) is not never considered as the Colombeau solution<sup>31-38</sup> of the Einstein field equations.

(2) Colombeau nonlinear distributional geometry 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 $(R^a_{bcd,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ classical Schwartz distributions in $D'(\mathbb{R}^4)$, (3) there are no any important physical applications of the classical linear distributional geometry were obtained.

Remark 5. Fundamental physical applications of the Colombeau nonlinear distributional geometry have been originally obtained in authors paper<sup>11,14,15</sup>.

By using the Cartesian coordinates $(\hat{x}^0, \hat{x}^1, \hat{x}^2, \hat{x}^3)$, which are related to $(\hat{x}^0, \hat{r}, \theta, \varphi)$ through the canonical relations: $\hat{x}^1 = \hat{r} \cos \varphi \sin \theta$, $\hat{x}^2 = \hat{r} \sin \varphi \sin \theta$, $\hat{x}^3 = \hat{r} \cos \theta$, the metric (17) 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

$$ \hat{g}_{00} = -(1-h), \quad \hat{g}_{0\alpha} = 0, $$

$$ \hat{g}_{\alpha\beta} = \delta^{\alpha\beta} + h(1-h)^{-1} \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2}, \quad \alpha, \beta = 1, 2, 3 \quad (18) $$

with $h \triangleq a/\hat{r}$. It is known that at points $\hat{r} \neq 0, \hat{r} \neq a$
$$\kappa \hat{T}_0^0 = -\frac{h'}{\hat{r}} - \frac{h}{\hat{r}^2},$$

$$\kappa \hat{T}_0^\alpha = 0,$$

$$\kappa \hat{T}_{\alpha}^{0} = 0, \kappa \hat{T}_{\alpha}^{\beta} = \delta_{\alpha}^{\beta} \left(-\frac{h''}{2} - \frac{h'}{\hat{r}}\right) + \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2} \left(\frac{h''}{2} - \frac{h}{\hat{r}^2}\right), \quad (19)$$

where the hatted symbols $\hat{T}_{\mu}^{\nu}$ represent the quantity $T_{\mu}^{\nu}$ 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 5. We extend now the quantity (18)-(19) in point $\hat{r} = 0$ as Colombeau generalized functions from Colombeau algebra $G_{\delta}(\mathbb{R}^4)$. Regularizing now the function $h = a/\hat{r}$ as $(h_{\varepsilon})_{\varepsilon} = a/(\sqrt{\hat{r}^2 + \varepsilon^2})_{\varepsilon}$

and the function $\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2}$ as $\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{(\hat{r}^2 + \varepsilon^2)_{\varepsilon}}$

with $\varepsilon \in (0, \delta]$, we replace now the ill-defined singular metric (18) by the well defined Colombeau generalized metric

$$ds^2 = (\hat{g}_{\mu\nu,\varepsilon} d\hat{x}^{\mu} d\hat{x}^{\nu})_{\varepsilon}, \quad (20)$$

where

$$(\hat{g}_{00,\varepsilon})_{\varepsilon} = -(1-h_{\varepsilon}), \quad (\hat{g}_{0\alpha,\varepsilon})_{\varepsilon} = 0_{R},$$

$$(\hat{g}_{\alpha\beta,\varepsilon})_{\varepsilon} = \delta^{\alpha\beta} + ((h_{\varepsilon}(1-h_{\varepsilon})^{-1})_{\varepsilon}) \left(\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2 + \varepsilon^2}\right)_{\varepsilon}, \quad \alpha, \beta = 1,2,3 \quad (21)$$

and therefore

$$\kappa(\hat{T}_{0,\varepsilon}^{0}(\hat{x}))_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon}, \quad \kappa(\hat{T}_{0}^{\alpha}(\hat{x};\varepsilon))_{\varepsilon} = 0_{R}, \quad \kappa(\hat{T}_{\alpha}^{0}(\hat{x};\varepsilon))_{\varepsilon} = 0_{R},$$

$$\kappa(\hat{T}_{\alpha}^{\beta}(\hat{x}; \varepsilon))_{\varepsilon} = \delta_{\alpha}^{\beta} \left(\frac{3a\varepsilon^2}{2(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} - \left(\left(\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{(\hat{r}^2+\varepsilon^2)_{\varepsilon}}\right)_{\varepsilon}\left(\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon}\left(\frac{5}{2} + \left(\frac{\varepsilon^2}{\hat{r}^2+\varepsilon^2}\right)_{\varepsilon}\right)\right). \quad (22)$$

Note that from Eq. (41) one obtains¹⁷

$$\hat{T}_{\mu}^{\nu}(\hat{x}) \triangleq w \lim_{\varepsilon \to 0} \hat{T}_{\mu,\varepsilon}^{\nu}(\hat{x}) = -Mc^2 \delta_{\mu}^{0} \delta_{0}^{\nu} \delta^{(3)}(\hat{x}). \quad (23)$$

Remark 6. Note that $(\hat{T}_0^0(\hat{x}; \varepsilon))_{\varepsilon}, (\hat{T}_{\alpha}^{\beta}(\hat{x}; \varepsilon))_{\varepsilon} \in G_{\delta}(\mathbb{R}^4)$. Thus the generalized Einstein field equations related to Eq. (20)-Eq. (23) in Colombeau notations read⁷

$$(\hat{g}_{\mu,\varepsilon})^{\nu}_{\varepsilon} = (\hat{R}_{\mu,\varepsilon})^{\nu}_{\varepsilon} - \frac{1}{2} \delta_{\mu}^{\nu} (\hat{R}_{\varepsilon})^{\nu}_{\varepsilon} = \kappa (\hat{T}_{\mu,\varepsilon})^{\nu}_{\varepsilon}, \quad (24)$$

where

$$(\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon} = (\hat{R}^{\mu}_{\mu,\varepsilon}(\hat{r}))_{\varepsilon} = -\left(\frac{3a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} + \left(\frac{2a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} \quad (25)$$

Remark 6. Note that the regularized scalar curvature $\hat{R}_{\varepsilon}$ has the well-defined weak limit $\hat{R}_{w}$ in $D'(\mathbb{R}^4)$

$$\hat{R}_{w}(\hat{x}) \triangleq w \lim_{\varepsilon \to 0} \hat{R}_{\varepsilon}(\hat{x}) = -\frac{4}{3} \pi a \delta^{(3)}(\hat{x}). \quad (26)$$

Remark 7. Note that: (1) for any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ such that $\text{st}((\hat{r}_{\varepsilon})_{\varepsilon}) = \hat{r}_{\text{fin}} \neq 0$, (see Definition 5) where $\hat{r}_{\text{fin}} \in \mathbb{R}$ from Eq. (44) it follows

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

(2) for any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ such that $(\hat{r}_{\varepsilon})_{\varepsilon} < \tilde{\mathbb{R}}_{\delta}(\varepsilon)_{\varepsilon}$ (see Definition 4) from Eq. (25) it follows
$$\kappa \hat{T}_0^0 = -\frac{h'}{\hat{r}} - \frac{h}{\hat{r}^2},$$

$$\kappa \hat{T}_0^\alpha = 0,$$

$$\kappa \hat{T}_{\alpha}^{0} = 0, \kappa \hat{T}_{\alpha}^{\beta} = \delta_{\alpha}^{\beta} \left(-\frac{h''}{2} - \frac{h'}{\hat{r}}\right) + \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2} \left(\frac{h''}{2} - \frac{h}{\hat{r}^2}\right), \quad (19)$$

where the hatted symbols $\hat{T}_{\mu}^{\nu}$ represent the quantity $T_{\mu}^{\nu}$ 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 5. We extend now the quantity (18)-(19) in point $\hat{r} = 0$ as Colombeau generalized functions from Colombeau algebra $G_{\delta}(\mathbb{R}^4)$. Regularizing now the function $h = a/\hat{r}$ as $(h_{\varepsilon})_{\varepsilon} = a/(\sqrt{\hat{r}^2 + \varepsilon^2})_{\varepsilon}$

and the function $\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2}$ as $\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{(\hat{r}^2 + \varepsilon^2)_{\varepsilon}}$

with $\varepsilon \in (0, \delta]$, we replace now the ill-defined singular metric (18) by the well defined Colombeau generalized metric

$$ds^2 = (\hat{g}_{\mu\nu,\varepsilon} d\hat{x}^{\mu} d\hat{x}^{\nu})_{\varepsilon}, \quad (20)$$

where

$$(\hat{g}_{00,\varepsilon})_{\varepsilon} = -(1 - h_{\varepsilon}), \quad (\hat{g}_{0\alpha,\varepsilon})_{\varepsilon} = 0_{R},$$

$$(\hat{g}_{\alpha\beta,\varepsilon})_{\varepsilon} = \delta^{\alpha\beta} + ((h_{\varepsilon}(1 - h_{\varepsilon})^{-1})_{\varepsilon}) \left(\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2 + \varepsilon^2}\right)_{\varepsilon}, \quad \alpha, \beta = 1,2,3 \quad (21)$$

and therefore

$$\kappa(\hat{T}_{0,\varepsilon}^{0}(\hat{x}))_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon}, \quad \kappa(\hat{T}_{0}^{\alpha}(\hat{x}; \varepsilon))_{\varepsilon} = 0_{R}, \quad \kappa(\hat{T}_{\alpha}^{0}(\hat{x}; \varepsilon))_{\varepsilon} = 0_{R},$$

$$\kappa(\hat{T}_{\alpha}^{\beta}(\hat{x}; \varepsilon))_{\varepsilon} = \delta_{\alpha}^{\beta} \left(\frac{3a\varepsilon^2}{2(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} - \left(\left(\frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{(\hat{r}^2+\varepsilon^2)_{\varepsilon}}\right)_{\varepsilon}\left(\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon}\left(\frac{5}{2} + \left(\frac{\varepsilon^2}{\hat{r}^2+\varepsilon^2}\right)_{\varepsilon}\right)\right). \quad (22)$$

Note that from Eq. (41) one obtains¹⁷

$$\hat{T}_{\mu}^{\nu}(\hat{x}) \triangleq w \lim_{\varepsilon \to 0} \hat{T}_{\mu,\varepsilon}^{\nu}(\hat{x}) = -Mc^2 \delta_{\mu}^{0} \delta_{0}^{\nu} \delta^{(3)}(\hat{x}). \quad (23)$$

Remark 6. Note that $(\hat{T}_0^0(\hat{x}; \varepsilon))_{\varepsilon}, (\hat{T}_{\alpha}^{\beta}(\hat{x}; \varepsilon))_{\varepsilon} \in G_{\delta}(\mathbb{R}^4)$. Thus the generalized Einstein field equations related to Eq. (20)-Eq. (23) in Colombeau notations read⁷

$$(\hat{g}_{\mu,\varepsilon})^{\nu}_{\varepsilon} = (\hat{R}_{\mu,\varepsilon})^{\nu}_{\varepsilon} - \frac{1}{2} \delta_{\mu}^{\nu} (\hat{R}_{\varepsilon})^{\nu}_{\varepsilon} = \kappa (\hat{T}_{\mu,\varepsilon})^{\nu}_{\varepsilon}, \quad (24)$$

where

$$(\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon} = (\hat{R}^{\mu}_{\mu,\varepsilon}(\hat{r}))_{\varepsilon} = -\left(\frac{3a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} + \left(\frac{2a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon} \quad (25)$$

Remark 6. Note that the regularized scalar curvature $\hat{R}_{\varepsilon}$ has the well-defined weak limit $\hat{R}_{w}$ in $D'(\mathbb{R}^4)$

$$\hat{R}_{w}(\hat{x}) \triangleq w \lim_{\varepsilon \to 0} \hat{R}_{\varepsilon}(\hat{x}) = -\frac{4}{3} \pi a \delta^{(3)}(\hat{x}). \quad (26)$$

Remark 7. Note that: (1) for any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ such that $\mathrm{st}((\hat{r}_{\varepsilon})_{\varepsilon}) = \hat{r}_{\mathrm{fin}} \neq 0$, (see Definition 5) where $\hat{r}_{\mathrm{fin}} \in \mathbb{R}$ from Eq. (44) it follows

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

(2) for any $(\hat{r}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}_{\delta}$ such that $(\hat{r}_{\varepsilon})_{\varepsilon} < \tilde{\mathbb{R}}_{\delta}(\varepsilon)_{\varepsilon}$ (see Definition 4) from Eq. (25) it follows
$$((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}) \approx -\infty,$$

(3) at origin $(\hat{r}_{\varepsilon}^{0})_{\varepsilon} = 0_{\mathbb{R}}$ (see Definition 4) one obtains

$$((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon}) = -\left(\frac{a\varepsilon^2}{(\varepsilon^2)^{\frac{5}{2}}}\right)_{\varepsilon} = -\frac{a}{(\varepsilon^3)_{\varepsilon}} \approx -\infty,$$

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

For the regularized quadratic scalars, one obtains²⁷

$$\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}) = \frac{1}{2}\left[\frac{3a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right]^2 + 2\left[\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right]^2 = \frac{13}{2}\left[\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right]^2, \quad (45)$$

$$\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}) = \frac{4a^2}{\hat{r}^2+\varepsilon^2}\left[\frac{3}{(\hat{r}^2+\varepsilon^2)^2}\right] - \frac{12a^2\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^4} + \frac{9a^2\varepsilon^4}{(\hat{r}^2+\varepsilon^2)^5}. \quad (27)$$

**Remark 8.** 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

$$\hat{R}^{\mu\nu}(\hat{x}) \hat{R}_{\mu\nu}(\hat{x}) \triangleq \lim_{\varepsilon \to 0} \hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}) \hat{R}_{\mu\nu,\varepsilon}(\hat{r}) \sim 40\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2,$$

$$\hat{R}^{\rho\sigma\mu\nu}(\hat{x}) \hat{R}_{\rho\sigma\mu\nu}(\hat{x}) \triangleq \lim_{\varepsilon \to 0} \left( \hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}, \varepsilon) \hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}, \varepsilon) \right) \sim \dots$$

$$\frac{12a^2}{\hat{r}^6} + \frac{16\pi a^2}{3} \frac{1}{\hat{r}^3} \delta^{(3)}(\hat{x}) + 16\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2. \quad (28)$$

**Remark 9.** However Colombeau quadratic scalars $(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ and $(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ are well defined as Colombeau generalized functions in $G_{\delta}(\mathbb{R}^3)$, since

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

$$(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}))_{\varepsilon} = \left\{ \frac{12a^2}{((\hat{r}_{\varepsilon}^2+\varepsilon^2)^3)_{\varepsilon}} - \left(\frac{12a^2\varepsilon^2}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^4}\right)_{\varepsilon} + \left(\frac{9a^2\varepsilon^4}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^5}\right)_{\varepsilon} \right\} \in G_{\delta}(\mathbb{R}^3). \quad (29)$$

**Remark 10.** Note that Colombeau quadratic scalars $(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ and $(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ can be treated only nonclassical as Colombeau generalized

functions extended on $\tilde{\mathbb{R}}_{\delta}^3 = \tilde{\mathbb{R}}_{\delta} \times \tilde{\mathbb{R}}_{\delta} \times \tilde{\mathbb{R}}_{\delta}$ and therefore at origin $(\hat{r}_{\varepsilon}^{0})_{\varepsilon} = 0_{\tilde{\mathbb{R}}_{\delta}}$ we get

$$(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon}^{0})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon} = \frac{1}{2}\left\{\left[\left(\frac{3a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right)_{\varepsilon}\right]^2 + 2\left[\left(\frac{a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right)_{\varepsilon}\right]^2\right\} \in \tilde{\mathbb{R}}_{\delta},$$

$$(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon}^{0})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon} = \left\{\frac{12a^2}{(\varepsilon^6)_{\varepsilon}} - \left(\frac{12a^2\varepsilon^2}{(\varepsilon^8)_{\varepsilon}}\right)_{\varepsilon} + \left(\frac{9a^2\varepsilon^4}{(\varepsilon^{10})_{\varepsilon}}\right)_{\varepsilon}\right\} \in \tilde{\mathbb{R}}_{\delta}. \quad (30)$$

**Remark 11.** In the canonical Schwarzschild coordinates $(t, r, \theta, \varphi)$, $r \neq a$ the Schwarzschild metric (29) takes the form (1) above the horizon: $r > 2m$:

$$ds^{+2} = h^{+}(r)dt^{2} - [h^{+}(r)]^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (31)$$

$$h^{+}(r) = -1 + \frac{a}{r} = -\frac{r-a}{r}, a = 2m$$
$$((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}) \approx -\infty,$$

(3) at origin $(\hat{r}_{\varepsilon}^{0})_{\varepsilon} = 0_{\mathbb{R}}$ (see Definition 4) one obtains

$$((\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon}) = -\left(\frac{a\varepsilon^2}{(\varepsilon^2)^{\frac{5}{2}}}\right)_{\varepsilon} = -\frac{a}{(\varepsilon^3)_{\varepsilon}} \approx -\infty,$$

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

For the regularized quadratic scalars, one obtains²⁷

$$\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}) = \frac{1}{2}\left[\frac{3a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right]^2 + 2\left[\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right]^2 = \\ \frac{13}{2}\left[\frac{a\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^{5/2}}\right]^2, \quad (45)$$

$$\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}) = \frac{4a^2}{\hat{r}^2+\varepsilon^2}\left[\frac{3}{(\hat{r}^2+\varepsilon^2)^2}\right] - \frac{12a^2\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^4} + \frac{9a^2\varepsilon^4}{(\hat{r}^2+\varepsilon^2)^5}. \quad (27)$$

**Remark 8.** 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

$$\hat{R}^{\mu\nu}(\hat{x}) \hat{R}_{\mu\nu}(\hat{x}) \triangleq \lim_{\varepsilon \to 0} \hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}) \hat{R}_{\mu\nu,\varepsilon}(\hat{r}) \sim 40\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2,$$

$$\hat{R}^{\rho\sigma\mu\nu}(\hat{x}) \hat{R}_{\rho\sigma\mu\nu}(\hat{x}) \triangleq \lim_{\varepsilon \to 0} \left( \hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}, \varepsilon) \hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}, \varepsilon) \right) \sim \dots$$

$$\frac{12a^2}{\hat{r}^6} + \frac{16\pi a^2}{3} \frac{1}{\hat{r}^3} \delta^{(3)}(\hat{x}) + 16\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2. \quad (28)$$

**Remark 9.** However Colombeau quadratic scalars $(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ and $(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ are well defined as Colombeau generalized functions in $G_{\delta}(\mathbb{R}^3)$, since

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

$$(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}))_{\varepsilon} = \left\{ \frac{12a^2}{((\hat{r}_{\varepsilon}^2+\varepsilon^2)^3)_{\varepsilon}} - \left( \frac{12a^2\varepsilon^2}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^4} \right)_{\varepsilon} + \left( \frac{9a^2\varepsilon^4}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^5} \right)_{\varepsilon} \right\} \in G_{\delta}(\mathbb{R}^3). \quad (29)$$

**Remark 10.** Note that Colombeau quadratic scalars $(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ and

$(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ can be treated only nonclassical as Colombeau generalized functions extended on $\tilde{\mathbb{R}}_{\delta}^3 = \tilde{\mathbb{R}}_{\delta} \times \tilde{\mathbb{R}}_{\delta} \times \tilde{\mathbb{R}}_{\delta}$ and therefore at origin $(\hat{r}_{\varepsilon}^{0})_{\varepsilon} = 0_{\tilde{\mathbb{R}}_{\delta}}$ we get

$$(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon}^{0})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon} = \frac{1}{2}\left\{\left[\frac{3a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right]_{\varepsilon}^2 + 2\left[\frac{a\varepsilon^2}{(\varepsilon^2)^{5/2}}\right]_{\varepsilon}^2\right\} \in \tilde{\mathbb{R}}_{\delta},$$

$$(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon}^{0})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon} = \left\{ \frac{12a^2}{(\varepsilon^6)_{\varepsilon}} - \left( \frac{12a^2\varepsilon^2}{(\varepsilon^8)_{\varepsilon}} \right)_{\varepsilon} + \left( \frac{9a^2\varepsilon^4}{(\varepsilon^{10})_{\varepsilon}} \right)_{\varepsilon} \right\} \in \tilde{\mathbb{R}}_{\delta}. \quad (30)$$

**Remark 11.** In the canonical Schwarzschild coordinates $(t, r, \theta, \varphi)$, $r \neq a$ the Schwarzschild metric (29) takes the form (1) above the horizon: $r > 2m$:

$$ds^{+2} = h^{+}(r)dt^{2} - [h^{+}(r)]^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (31)$$

$$h^{+}(r) = -1 + \frac{a}{r} = -\frac{r-a}{r}, a = 2m$$
and (2) below the horizon $r < 2m$ :

$$ds^{-2} = h^{-}(r)dt^{2} - h^{-}(r)^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (32)$$

$$h^{-}(r) = -1 + \frac{a}{r} = \frac{a-r}{r}$$

Examining the Schwarzschild metric (31)-(32) in a neighborhood of the horizon, we see that, whereas $h^{+}(r)$ and $h^{-}(r)$ is smooth, $[h^{+}(r)]^{-1}$ and $[h^{-}(r)]^{-1}$ is not even in (note that the origin is now always excluded from our considerations; the space we are working on is $\mathbb{R}^{3}\setminus\{0\}$).

Following the above discussion we embed the metric coefficients $h^{+}(r)$, $[h^{+}(r)]^{-1}$ and $h^{-}(r)$, $[h^{-}(r)]^{-1}$ into Colombeau algebra $G_{\delta}(\mathbb{R}^{3})$ by replacements above the horizon $r > 2m$ and below the horizon $r < 2m$ correspondingly$^{10,11,15}$

$$r \ge 2m : \quad r - 2m \mapsto \sqrt{(r - 2m)^{2} + \varepsilon^{2}}, \quad (33)$$

$$r < 2m : \quad 2m - r \mapsto \sqrt{(2m - r)^{2} + \varepsilon^{2}}. \quad (34)$$

Inserting Eq. (33) and Eq. (34) into Eq. (31) and Eq. (32) correspondingly, we obtain Colombeau generalized object $\{(ds_{\varepsilon}^{+})_{\varepsilon}, (ds_{\varepsilon}^{-})_{\varepsilon}\}$ modeling the singular Schwarzschild metric above (below) the horizon, i.e.,

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

$$h_{\varepsilon}^{+}(r) = -\frac{\sqrt{(r-2m)^{2}+\varepsilon^{2}}}{r}, \quad (35)$$

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

$$h_{\varepsilon}^{-}(r) = \frac{\sqrt{(2m-r)^{2}+\varepsilon^{2}}}{r}. \quad (36)$$

The generalized Ricci tensor above the horizon $([R_{\varepsilon}^{+}]_{\alpha}^{\beta})_{\varepsilon}$ may now be calculated componentwise using the classical formulas$^{10,11}$

$$([R_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2}((h_{\varepsilon}^{+''})_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}^{+'})_{\varepsilon}), \quad (37)$$

$$([R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{(h_{\varepsilon}^{+'})_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+})_{\varepsilon}}{r^{2}}, \quad (38)$$

where

$$h_{\varepsilon}^{+'}(r) = -\frac{r-2m}{r[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} + \frac{[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}{r^{2}}, \quad (39)$$

$$\begin{aligned} h_{\varepsilon}^{+''}(r) = & -\left(\frac{r-2m}{r[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}\right)' + \left(\frac{[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}{r^{2}}\right)' \\ = & -\frac{1}{r[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} + \frac{(r-2m)^{2}}{r[(r-2m)^{2}+\varepsilon^{2}]^{3/2}} + \frac{r-2m}{r^{2}[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} \\ & + \frac{r-2m}{r^{2}[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} - \frac{2[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}{r^{3}}. \end{aligned} \quad (40)$$

From Eq. (37), Eq. (38), Eq. (39) and Eq. (40) we obtain

$$(r^2 [R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = (r^2 [R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} =$$

$$r(h_{\varepsilon}^{+'}(r))_{\varepsilon} + 1 + (h_{\varepsilon}^{+}(r))_{\varepsilon} =$$
and (2) below the horizon $r < 2m$ :

$$ds^{-2} = h^{-}(r)dt^{2} - h^{-}(r)^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (32)$$

$$h^{-}(r) = -1 + \frac{a}{r} = \frac{a-r}{r}$$

Examining the Schwarzschild metric (31)-(32) in a neighborhood of the horizon, we see that, whereas $h^{+}(r)$ and $h^{-}(r)$ is smooth, $[h^{+}(r)]^{-1}$ and $[h^{-}(r)]^{-1}$ is not even in (note that the origin is now always excluded from our considerations; the space we are working on is $\mathbb{R}^{3}\setminus\{0\}$).

Following the above discussion we embed the metric coefficients $h^{+}(r), [h^{+}(r)]^{-1}$ and $h^{-}(r), [h^{-}(r)]^{-1}$ into Colombeau algebra $G_{\delta}(\mathbb{R}^{3})$ by replacements above the horizon $r > 2m$ and below the horizon $r < 2m$ correspondingly$^{10,11,15}$

$$r \ge 2m : \quad r - 2m \mapsto \sqrt{(r - 2m)^{2} + \varepsilon^{2}}, \quad (33)$$

$$r < 2m : \quad 2m - r \mapsto \sqrt{(2m - r)^{2} + \varepsilon^{2}}. \quad (34)$$

Inserting Eq. (33) and Eq. (34) into Eq. (31) and Eq. (32) correspondingly, we obtain Colombeau generalized object $\{(ds_{\varepsilon}^{+})_{\varepsilon}, (ds_{\varepsilon}^{-})_{\varepsilon}\}$ modeling the singular Schwarzschild metric above (below) the horizon, i.e.,

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

$$h_{\varepsilon}^{+}(r) = -\frac{\sqrt{(r-2m)^{2}+\varepsilon^{2}}}{r}, \quad (35)$$

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

$$h_{\varepsilon}^{-}(r) = \frac{\sqrt{(2m-r)^{2}+\varepsilon^{2}}}{r}. \quad (36)$$

The generalized Ricci tensor above the horizon $([R_{\varepsilon}^{+}]_{\alpha}^{\beta})_{\varepsilon}$ may now be calculated componentwise using the classical formulas$^{10,11}$

$$([R_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2}((h_{\varepsilon}^{+''})_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}^{+'})_{\varepsilon}), \quad (37)$$

$$([R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{(h_{\varepsilon}^{+'})_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+})_{\varepsilon}}{r^2}, \quad (38)$$

where

$$h_{\varepsilon}^{+'}(r) = -\frac{r-2m}{r[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} + \frac{[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}{r^{2}}, \quad (39)$$

$$\begin{aligned} h_{\varepsilon}^{+''}(r) = & -\left(\frac{r-2m}{r[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}\right)' + \left(\frac{[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}{r^{2}}\right)' \\ = & -\frac{1}{r[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} + \frac{(r-2m)^{2}}{r[(r-2m)^{2}+\varepsilon^{2}]^{3/2}} + \frac{r-2m}{r^{2}[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} \\ & + \frac{r-2m}{r^{2}[(r-2m)^{2}+\varepsilon^{2}]^{1/2}} - \frac{2[(r-2m)^{2}+\varepsilon^{2}]^{1/2}}{r^{3}}. \end{aligned} \quad (40)$$

From Eq. (37), Eq. (38), Eq. (39) and Eq. (40) we obtain

$$(r^2 [R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = (r^2 [R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} =$$

$$r(h_{\varepsilon}^{+'}(r))_{\varepsilon} + 1 + (h_{\varepsilon}^{+}(r))_{\varepsilon} =$$
$$
\begin{aligned}
& r \left\{ -\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2} \right\} + 1 - \frac{\sqrt{r-2m}^2+\varepsilon^2}{r} = \\
& -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r} + 1 - \frac{\sqrt{r-2m}^2+\varepsilon^2}{r} = \\
& -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} + 1 \quad (41)
\end{aligned}
$$

and

$$
\begin{aligned}
& (2r^2[R_{\varepsilon}^{+}(r)]_0^0)_{\varepsilon} = (2r^2[R_{\varepsilon}^{+}(r)]_1^1)_{\varepsilon} = \\
& r^2(h_{\varepsilon}^{++}(r))_{\varepsilon} + 2r(h_{\varepsilon}^{++}(r))_{\varepsilon} = \\
& r^2 \left\{ -\frac{1}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2+\varepsilon^2]^{3/2}} + \right. \\
& \qquad \left. \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} - \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r^3} \right\} + \\
& \qquad + 2r \left\{ -\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2} \right\} = \\
& \qquad -\frac{r}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r^2[(r-2m)^2+\varepsilon^2]^{3/2}} + \\
& \qquad \left. \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} - \right. \\
& \qquad \left. \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r} - \right. \\
& \qquad \left. - \frac{2(r-2m)}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r} = \right. \\
& \qquad \left. - \frac{r}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2+\varepsilon^2]^{3/2}}. \quad (42)
\end{aligned}
$$

Investigating the weak limit of the angular components of the Ricci tensor by using
the abbreviation $\hat{\Phi}(r) = \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\phi \, \Phi(x)$

and let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$, where by $S_{2m}^{+}(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with compact support such that:

(1) $\mathrm{supp}(\Phi(x)) \subset K^+ = \{x \mid \|x\| \ge 2m\}$, (2) $\hat{\Phi}(r) \in C^\infty(\mathbb{R})$.

Then for any function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$ we get:

$$
\int_{K^+} ([R_{\varepsilon}^{+}(r)]_2^2)_{\varepsilon} \Phi(x) d^3 x = \int_{K^+} ([R_{\varepsilon}^{+}(r)]_3^3)_{\varepsilon} \Phi(x) d^3 x = \\
\int_{2m}^{R} (r^2 [R_{\varepsilon}^{+}(r)]_2^2)_{\varepsilon} \hat{\Phi}(r) dr = \int_{2m}^{R} (r^2 [R_{\varepsilon}^{+}(r)]_3^3)_{\varepsilon} \hat{\Phi}(r) dr =
$$

$$
\int_{2m}^{R} \left( r \left( h_{\varepsilon}^{++}(r) \right)_{\varepsilon} + 1 + (h_{\varepsilon}^{++}(r))_{\varepsilon} \right) \hat{\Phi}(r) dr = \\
\int_{2m}^{R} \left\{ - \frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr. \quad (43)
$$

By replacement $r - 2m = u$, from Eq. (43) we obtain

$$
\int_{2m}^{R} \left\{ - \frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr = \\
- \int_{0}^{R-2m} \frac{u \hat{\Phi}(u+2m) du}{(u^2+\varepsilon^2)^{1/2}} + \int_{0}^{R-2m} \hat{\Phi}(u+2m) du. \quad (44)
$$

By replacement $u = \varepsilon\eta$, from Eq. (44) we obtain the expression

$$
I_3^+(\varepsilon) = \int_K ([R_\varepsilon^+(r)]_3^3)_\varepsilon \Phi(x) d^3x = I_2^+(\varepsilon) = \int_K ([R_\varepsilon^+(r)]_2^2)_\varepsilon \Phi(x) d^3x = \\
-\varepsilon \times \left( \int_0^{\frac{R-2m}{\varepsilon}} \frac{\eta \hat{\Phi}(\varepsilon\eta+2m)d\eta}{(\eta^2+1)^{1/2}} - \int_0^{\frac{R-2m}{\varepsilon}} \hat{\Phi}(\varepsilon\eta+2m)d\eta \right). \quad (45)
$$
$$
\begin{aligned}
& r \left\{ -\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2} \right\} + 1 - \frac{\sqrt{r-2m}^2+\varepsilon^2}{r} = \\
& -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r} + 1 - \frac{\sqrt{r-2m}^2+\varepsilon^2}{r} = \\
& -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} + 1 \quad (41)
\end{aligned}
$$

and

$$
\begin{aligned}
& (2r^2[R_{\varepsilon}^{+}(r)]_0^0)_{\varepsilon} = (2r^2[R_{\varepsilon}^{+}(r)]_1^1)_{\varepsilon} = \\
& r^2(h_{\varepsilon}^{++}(r))_{\varepsilon} + 2r(h_{\varepsilon}^{++}(r))_{\varepsilon} = \\
& r^2 \left\{ -\frac{1}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2+\varepsilon^2]^{3/2}} + \right. \\
& \qquad \left. \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} - \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r^3} \right\} + \\
& \qquad + 2r \left\{ -\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2} \right\} = \\
& \qquad -\frac{r}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r-2m[(r-2m)^2+\varepsilon^2]^{3/2}} + \\
& \qquad \left. \frac{r-2m}{r-2m[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r-2m}{r-2m[(r-2m)^2+\varepsilon^2]^{1/2}} - \right. \\
& \qquad \left. \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r} - \right. \\
& \qquad \left. - \frac{2(r-2m)}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r} = \right. \\
& \qquad \left. - \frac{r}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2+\varepsilon^2]^{3/2}}. \quad (42)
\end{aligned}
$$

Investigating the weak limit of the angular components of the Ricci tensor by using
the abbreviation $\hat{\Phi}(r) = \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\phi \, \Phi(x)$

and let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$, where by $S_{2m}^{+}(\mathbb{R}^3)$ we denote the
class of all functions $\Phi(x)$ with compact support such that:

(1) $\mathrm{supp}(\Phi(x)) \subset K^+ = \{x \mid \|x\| \ge 2m\}$, (2) $\hat{\Phi}(r) \in C^\infty(\mathbb{R})$.

Then for any function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$ we get:

$$
\int_{K^+} ([R_{\varepsilon}^{+}(r)]_2^2)_{\varepsilon} \Phi(x) d^3 x = \int_{K^+} ([R_{\varepsilon}^{+}(r)]_3^3)_{\varepsilon} \Phi(x) d^3 x = \\
\int_{2m}^{R} (r^2 [R_{\varepsilon}^{+}(r)]_2^2)_{\varepsilon} \hat{\Phi}(r) dr = \int_{2m}^{R} (r^2 [R_{\varepsilon}^{+}(r)]_3^3)_{\varepsilon} \hat{\Phi}(r) dr =
$$

$$
\int_{2m}^{R} \left( r \left( h_{\varepsilon}^{++}(r) \right)_{\varepsilon} + 1 + (h_{\varepsilon}^{++}(r))_{\varepsilon} \right) \hat{\Phi}(r) dr = \\
\int_{2m}^{R} \left\{ - \frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr. (43)
$$

By replacement $r - 2m = u$, from Eq. (43) we obtain

$$
\int_{2m}^{R} \left\{ - \frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr = \\
- \int_{0}^{R-2m} \frac{u \hat{\Phi}(u+2m) du}{(u^2+\varepsilon^2)^{1/2}} + \int_{0}^{R-2m} \hat{\Phi}(u+2m) du. (44)
$$

By replacement $u = \varepsilon\eta$, from Eq. (44) we obtain the expression

$$
I_3^+(\varepsilon) = \int_K ([R_\varepsilon^+(r)]_3^3)_\varepsilon \Phi(x) d^3x = I_2^+(\varepsilon) = \int_K ([R_\varepsilon^+(r)]_2^2)_\varepsilon \Phi(x) d^3x = \\
-\varepsilon \times \left( \int_0^{\frac{R-2m}{\varepsilon}} \frac{\eta \hat{\Phi}(\varepsilon\eta+2m)d\eta}{(\eta^2+1)^{1/2}} - \int_0^{\frac{R-2m}{\varepsilon}} \hat{\Phi}(\varepsilon\eta+2m)d\eta \right). \quad (45)
$$
From Eq. (45) we obtain

$$
\begin{align*}
I_3^+(\varepsilon) = I_2^+(\varepsilon) = -\varepsilon \frac{\hat{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] & - \\
-\frac{\varepsilon^2}{1!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta & = \\
-\varepsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\varepsilon}\right)^2 + 1} - \left(\frac{R-2m}{\varepsilon} - 1\right) \right] & - \\
-\frac{\varepsilon^2}{1} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta, & (46)
\end{align*}
$$

where we have expressed the function $\hat{\Phi}(\varepsilon\eta + 2m)$ as

$$
\hat{\Phi}(\varepsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\varepsilon\eta)^l + \frac{1}{n!} (\varepsilon\eta)^n \hat{\Phi}^{(n)}(\xi), \\
\xi \triangleq \theta\varepsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 \quad (47)
$$

with $\hat{\Phi}^{(l)}(\xi) \triangleq d^l \hat{\Phi}/d\xi^l$.

The Eq. (46) - (Eq. (47) gives

$$
\begin{align*}
\lim_{\epsilon \to 0} I_3^+(\epsilon) &= \lim_{\epsilon \to 0} I_2^+(\epsilon) = \\
&\quad \lim_{\epsilon \to 0} \left\{ -\epsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1 + 1 - \frac{R-2m}{\epsilon}} \right] + \right. \\
&\qquad \left. + \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta \right\} = 0. \quad (48)
\end{align*}
$$

Thus in $S'_{2m}(B_R^+(2m)) \subset S'_{2m}(\mathbb{R}^3) \subset D'(\mathbb{R}^3)$,
where $B^+(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}$
from the Eq. (66) we obtain

$$
\begin{align*}
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(r)]_{3}^{3} &= \lim_{\epsilon \to 0} I_{3}^{+}(\epsilon) = 0, \\
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(r)]_{2}^{2} &= \lim_{\epsilon \to 0} I_{2}^{+}(\epsilon) = 0. \quad (49)
\end{align*}
$$

For $([R_\epsilon^+(r)]_1^1)_\epsilon, ([R_\epsilon^+(r)]_0^0)_\epsilon$ we get:

$$
\begin{align*}
& 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x = \\
& \int_{2m}^R \left( r^2 (h_\epsilon^+''(r)) + 2r (h_\epsilon^+'(r)) \right)_\epsilon \hat{\Phi}(r) dr = \\
& = \int_{2m}^R \left\{ -\frac{r}{[(r-2m)^2+\epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2+\epsilon^2]^{3/2}} \right\} \hat{\Phi}(r) dr. \quad (50)
\end{align*}
$$

By replacement $r - 2m = u$, from Eq. (50) we obtain

$$
\begin{align*}
I_1^+(\epsilon) &= 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = I_2^+(\epsilon) = 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x \\
&= \int_{2m}^R (r^2 (h_\epsilon^+)^2)_\epsilon + 2r (h_\epsilon^+)_\epsilon \hat{\Phi}(r) dr \\
&= \int_0^{R-2m} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{1/2}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \hat{\Phi}(u+2m) du. \quad (51)
\end{align*}
$$

By replacement $u = \epsilon\eta$, from Eq. (51) we obtain

$$
\begin{align*}
& 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x = \\
&= \int_{2m}^R (r^2 (h_\epsilon^+)^2)_\epsilon + 2r (h_\epsilon^+)_\epsilon \hat{\Phi}(r) dr = \\
&= \epsilon \int_0^{\frac{R-2m}{\epsilon}} d\eta \left\{ -\frac{\epsilon\eta+2m}{(\epsilon^2\eta^2+\epsilon^2)^{1/2}} + \frac{\epsilon^2\eta^2(\epsilon\eta+2m)}{(\epsilon^2\eta^2+\epsilon^2)^{3/2}} \right\} \hat{\Phi}(\epsilon\eta+2m) = \\
& - \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^2\eta\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{1/2}} - 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{1/2}} + \\
& \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^4\eta^3\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{3/2}} + 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^3\eta^2\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{3/2}} = \\
& \epsilon \left[ - \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\eta^2+1)^{1/2}} + \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta^3\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\eta^2+1)^{3/2}} \right] +
\end{align*}
$$
From Eq. (45) we obtain

$$
\begin{align*}
I_3^+(\varepsilon) = I_2^+(\varepsilon) = -\varepsilon \frac{\hat{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] & - \\
-\frac{\varepsilon^2}{1!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta & = \\
-\varepsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\varepsilon}\right)^2 + 1} - \left(\frac{R-2m}{\varepsilon} - 1\right) \right] & - \\
-\frac{\varepsilon^2}{1} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta, & (46)
\end{align*}
$$

where we have expressed the function $\hat{\Phi}(\varepsilon\eta + 2m)$ as

$$
\hat{\Phi}(\varepsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\varepsilon\eta)^l + \frac{1}{n!} (\varepsilon\eta)^n \hat{\Phi}^{(n)}(\xi),
$$
$$
\xi \triangleq \theta\varepsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 \quad (47)
$$

with $\hat{\Phi}^{(l)}(\xi) \triangleq d^l \hat{\Phi}/d\xi^l$.

The Eq. (46) - (Eq. (47) gives

$$
\begin{align*}
\lim_{\epsilon \to 0} I_3^+(\epsilon) &= \lim_{\epsilon \to 0} I_2^+(\epsilon) = \\
&\quad \lim_{\epsilon \to 0} \left\{ -\epsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1 + 1 - \frac{R-2m}{\epsilon}} \right] + \right. \\
&\qquad \left. + \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta \right\} = 0. \right\} \tag{48}
\end{align*}
$$

Thus in $S'_{2m}(B_R^+(2m)) \subset S'_{2m}(\mathbb{R}^3) \subset D'(\mathbb{R}^3)$,
where $B^+(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}$
from the Eq. (66) we obtain

$$
\begin{align*}
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(r)]_{3}^{3} &= \lim_{\epsilon \to 0} I_{3}^{+}(\epsilon) = 0, \\
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(r)]_{2}^{2} &= \lim_{\epsilon \to 0} I_{2}^{+}(\epsilon) = 0. \quad (49)
\end{align*}
$$

For $([R_\epsilon^+(r)]_1^1)_\epsilon, ([R_\epsilon^+(r)]_0^0)_\epsilon$ we get:

$$
\begin{align*}
& 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x = \\
& \int_{2m}^R (r^2 (h_\epsilon^+''(r))_\epsilon + 2r (h_\epsilon^+'(r))_\epsilon) \hat{\Phi}(r) dr = \\
& = \int_{2m}^R \left\{ -\frac{r}{[(r-2m)^2+\epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2+\epsilon^2]^{3/2}} \right\} \hat{\Phi}(r) dr. \quad (50)
\end{align*}
$$

By replacement $r - 2m = u$, from Eq. (50) we obtain

$$
\begin{align*}
I_1^+(\epsilon) &= 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = I_2^+(\epsilon) = 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x \\
&= \int_{2m}^R (r^2 (h_\epsilon^+)^2)_\epsilon + 2r (h_\epsilon^+)_\epsilon) \hat{\Phi}(r) dr \\
&= \int_0^{R-2m} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{1/2}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \hat{\Phi}(u+2m) du. \quad (51)
\end{align*}
$$

By replacement $u = \epsilon\eta$, from Eq. (51) we obtain

$$
\begin{align*}
& 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x = \\
&= \int_{2m}^R (r^2 (h_\epsilon^+)^2)_\epsilon + 2r (h_\epsilon^+)_\epsilon) \hat{\Phi}(r) dr = \\
&= \epsilon \int_0^{\frac{R-2m}{\epsilon}} d\eta \left\{ -\frac{\epsilon\eta+2m}{(\epsilon^2\eta^2+\epsilon^2)^{1/2}} + \frac{\epsilon^2\eta^2(\epsilon\eta+2m)}{(\epsilon^2\eta^2+\epsilon^2)^{3/2}} \right\} \hat{\Phi}(\epsilon\eta+2m) = \\
&\quad - \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^2\eta\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{1/2}} - 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{1/2}} + \\
&\quad \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^4\eta^3\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{3/2}} + 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^3\eta^2\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+\epsilon^2)^{3/2}} = \\
&\quad \epsilon \left[ - \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+1)^{1/2}} + \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta^3\hat{\Phi}(\epsilon\eta+2m)d\eta}{(\epsilon^2\eta^2+1)^{3/2}} \right] +
\end{align*}
$$
$$2m \left[ - \int_{0}^{\frac{R-2m}{\epsilon}} \frac{\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^{2}+1)^{1/2}} + \int_{0}^{\frac{R-2m}{\epsilon}} \frac{\eta^2 \hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^{2}+1)^{3/2}} \right]. (52)$$

From Eq. (52) we obtain

$$ 
\begin{aligned}
I_0^+(\epsilon) = I_1^+(\epsilon) = 2m \frac{\hat{\Phi}(2m)}{0!} & \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] + \\
& + \frac{\epsilon}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] \eta + \\
& + \frac{\epsilon \hat{\Phi}(2m)}{0!} \int_{-\frac{R-2m}{\epsilon}}^{\frac{R-2m}{\epsilon}} d\eta \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] + \\
& + \frac{\epsilon^2}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] \eta,
\end{aligned} 
\quad (53) $$

where we have expressed the function $\hat{\Phi}(\epsilon\eta + 2m)$ as

$$ \hat{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \hat{\Phi}^{\alpha\beta(n)}(\xi), $$

$$ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 $$

with $\hat{\Phi}^{(l)}(\xi) \triangleq d^l \hat{\Phi}(\xi)/d\xi^l$. The Eq. (53) gives

$$ w - \lim_{\epsilon \to 0} I_0^+(\epsilon) = w - \lim_{\epsilon \to 0} I_1^+(\epsilon) = $$

$$ 2m\hat{\Phi}(2m)\lim_{\epsilon \to 0} \left\{ \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] d\eta \right\} = $$

$$ 2m\hat{\Phi}(2m)\lim_{\epsilon \to 0} \left\{ \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] d\eta \right\} = $$

$$ 2m\hat{\Phi}(2m)\lim_{S \to \infty} \left[ \int_0^S \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^S \frac{d\eta}{(\eta^2+1)^{1/2}} \right] = -2m\hat{\Phi}(2m). \quad (54) $$

where the relation is used

$$ \lim_{S \to \infty} \left[ \int_0^S \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^S \frac{d\eta}{(u^2+1)^{1/2}} \right] = -1 $$

Thus in $S'_{2m}(B^+(2m, R)) \subset S'_{2m}(R^3)$ we obtain

$$ w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}]_{1}^{1} = w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}]_{0}^{0} = -2m\hat{\Phi}(2m). \quad (55) $$

The Colombeau generalized Ricci tensor below the horizon $[R_{\epsilon}^{-}]_{\alpha}^{\beta} = [R_{\epsilon}^{-}]_{\alpha}^{\beta}$ may now be calculated componentwise using the classical formulae

$$ ([R_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} = ([R_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{-''}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{-'}(r))_{\epsilon} \right), $$

$$ ([R_{\epsilon}^{-}(r)]_{2}^{2})_{\epsilon} = ([R_{\epsilon}^{-}(r)]_{3}^{3})_{\epsilon} = \frac{(h_{\epsilon}^{-'}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}^{-}(r))_{\epsilon}}{r^2}. \quad (56) $$

Let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$, where by $S_{2m}^{-}(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with compact support such that:

(1) $\text{supp}(\Phi(x)) \subset K^{-} = \{x|0 \le \|x\| < 2m\}$, (2) $\hat{\Phi}(r) \in C^{\infty}(\mathbb{R})$.

Then for any function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$ from Eq. (56) similarly as above we obtain
$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^3 x = $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^3 x = m\hat{\Phi}(2m). \quad (57) $$
$$2m \left[ - \int_{0}^{\frac{R-2m}{\epsilon}} \frac{\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^{2}+1)^{1/2}} + \int_{0}^{\frac{R-2m}{\epsilon}} \frac{\eta^2 \hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^{2}+1)^{3/2}} \right]. (52)$$

From Eq. (52) we obtain

$$ 
\begin{aligned}
I_0^+(\epsilon) = I_1^+(\epsilon) = 2m \frac{\hat{\Phi}(2m)}{0!} & \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] + \\
& + \frac{\epsilon}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] \eta + \\
& + \frac{\epsilon \hat{\Phi}(2m)}{0!} \int_{\frac{R-2m}{\epsilon}}^{\frac{R-2m}{\epsilon}} d\eta \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] + \\
& + \frac{\epsilon^2}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] \eta,
\end{aligned} 
\quad (53) $$

where we have expressed the function $\hat{\Phi}(\epsilon\eta + 2m)$ as

$$ \hat{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \hat{\Phi}^{\alpha\beta(n)}(\xi), $$

$$ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 $$

with $\hat{\Phi}^{(l)}(\xi) \triangleq d^l \hat{\Phi}(\xi)/d\xi^l$. The Eq. (53) gives

$$ w - \lim_{\epsilon \to 0} I_0^+(\epsilon) = w - \lim_{\epsilon \to 0} I_1^+(\epsilon) = $$

$$ 2m\hat{\Phi}(2m)\lim_{\epsilon \to 0} \left\{ \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] d\eta \right\} = $$

$$ 2m\hat{\Phi}(2m)\lim_{\epsilon \to 0} \left\{ \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^{2}+1)^{1/2}} + \frac{\eta^2}{(\eta^{2}+1)^{3/2}} \right] d\eta \right\} = $$

$$ 2m\hat{\Phi}(2m)\lim_{S \to \infty} \left[ \int_0^S \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^S \frac{d\eta}{(\eta^2+1)^{1/2}} \right] = -2m\hat{\Phi}(2m). \quad (54) $$

where the relation is used

$$ \lim_{S \to \infty} \left[ \int_0^S \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^S \frac{d\eta}{(u^2+1)^{1/2}} \right] = -1 $$

Thus in $S'_{2m}(B^+(2m, R)) \subset S'_{2m}(R^3)$ we obtain

$$ w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}]_{1}^{1} = w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}]_{0}^{0} = -2m\hat{\Phi}(2m). \quad (55) $$

The Colombeau generalized Ricci tensor below the horizon $[R_{\epsilon}^{-}]_{\alpha}^{\beta} = [R_{\epsilon}^{-}]_{\alpha}^{\beta}$ may now be calculated componentwise using the classical formulae

$$ ([R_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} = ([R_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{-''}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{-'}(r))_{\epsilon} \right), $$

$$ ([R_{\epsilon}^{-}(r)]_{2}^{2})_{\epsilon} = ([R_{\epsilon}^{-}(r)]_{3}^{3})_{\epsilon} = \frac{(h_{\epsilon}^{-'}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}^{-}(r))_{\epsilon}}{r^2}. \quad (56) $$

Let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$, where by $S_{2m}^{-}(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with compact support such that:

(1) $\text{supp}(\Phi(x)) \subset K^{-} = \{x|0 \le \|x\| < 2m\},$ (2) $\hat{\Phi}(r) \in C^{\infty}(\mathbb{R})$.

Then for any function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$ from Eq. (56) similarly as above we obtain
$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^3 x = $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^3 x = m\hat{\Phi}(2m). \quad (57) $$
$$
\lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{2})_{\epsilon} \Phi(x) d^{3}x = \\
\lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{3})_{\epsilon} \Phi(x) d^{3}x = 0 \quad (58)
$$

III. GENERALIZED COLOMBEAU CALCULUS

We use 1-9 as standard references for the foundations and various applications of the
standard Colombeau theory. We briefly recall the basic Colombeau construction.
Throughout the paper Ω will denote an open subset of ℝⁿ. Stanford Colombeau
generalized functions on Ω are defined as equivalence classes u = [(u_ε)ε] of nets
of smooth functions u_ε ∈ C∞(Ω) (regularizations) subjected to asymptotic norm
conditions with respect to ε ∈ (0, δ] for their derivatives on compact sets.

The basic idea of classical Colombeau theory of nonlinear generalized functions¹⁻⁵ is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter ε. Let (u_ε)_{ε∈(0,δ]} with (u_ε)_{ε∈C∞(M)} for all ε ∈ ℝ+, where M is a separable, smooth orientable Hausdorff manifold of dimension n.

Definition 11. The classical Colombeau algebra of generalized functions on M is
defined as the quotient:

$$
\mathcal{G}_{\delta}(M) = \mathcal{E}_{M}(M)/\mathcal{N}(M) \quad (59)
$$

of the space $\mathcal{E}_M(M)$ of sequences of moderate growth modulo the space $\mathcal{N}(M)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates, where $\mathcal{X}(M)$ denoting the space of smooth vector fields on $M$ :

$$
\mathcal{E}_M(M) \triangleq \{(u_\epsilon)_\epsilon \mid \forall K(K \subset\subset M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\
\forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}
$$

$$
\mathcal{N}(M) \triangleq \{(u_\epsilon)_\epsilon \mid \forall K(K \subset\subset M), \forall k(k \in N_0) \forall q(q \in \mathbb{N}) \\
\forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \} \quad (61)
$$

Remark 12. In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0(\epsilon_0 \in (0,1]) \forall \epsilon(\epsilon < \epsilon_0)[a_\epsilon \le C\psi(\epsilon)]$. Definition 12. Elements of $G_\delta(M)$ are denoted by:

$$
u = \mathrm{cl}[(u_{\epsilon})_{\epsilon}] \triangleq (u_{\epsilon})_{\epsilon} + \mathcal{N}(M). \quad (62)
$$

Remark 13. With component wise operations (·, ±) $G_\delta(M)$ is a fine sheaf of
differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \mathrm{cl}[(L_\xi u_\epsilon)_\epsilon]$.
The spaces of moderate resp. negligible sequences and hence the algebra itself
may be characterized locally, i.e., $u \in G_\delta(M)$ iff $u \circ \psi_\alpha \in G_\delta(\psi_\alpha(V_\alpha))$ for all charts
$(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie
derivatives are replaced by partial derivatives.
$$ 
\begin{aligned}
\lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{2})_{\epsilon} \Phi(x) d^{3}x &= \\
\lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{3})_{\epsilon} \Phi(x) d^{3}x = 0 \quad (58)
\end{aligned}
$$

### III. GENERALIZED COLOMBEAU CALCULUS

We use 1-9 as standard references for the foundations and various applications of the standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper $\Omega$ will denote an open subset of $\mathbb{R}^n$. Stanford Colombeau generalized functions on $\Omega$ are defined as equivalence classes $u = [(u_\epsilon)_\epsilon]$ of nets of smooth functions $u_\epsilon \in C^\infty(\Omega)$ (regularizations) subjected to asymptotic norm conditions with respect to $\epsilon \in (0, \delta]$ for their derivatives on compact sets.

The basic idea of classical Colombeau theory of nonlinear generalized functions 1-5 is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter $\epsilon$. Let $(u_\epsilon)_{\epsilon \in (0, \delta]}$ with $(u_\epsilon)_\epsilon \in C^\infty(M)$ for all $\epsilon \in \mathbb{R}_+$, where $M$ is a separable, smooth orientable Hausdorff manifold of dimension $n$.

Definition 11. The classical Colombeau algebra of generalized functions on $M$ is defined as the quotient:

$$ G_{\delta}(M) = \mathcal{E}_{M}(M)/\mathcal{N}(M) \quad (59) $$

of the space $\mathcal{E}_M(M)$ of sequences of moderate growth modulo the space $\mathcal{N}(M)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates, where $\mathcal{X}(M)$ denoting the space of smooth vector fields on $M$:

$$ 
\begin{aligned}
\mathcal{E}_M(M) &\triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subset\subset M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\
&\qquad \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0\} \quad (60)
\end{aligned}
$$

$$ 
\begin{aligned}
\mathcal{N}(M) &\triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subset\subset M), \forall k(k \in N_0) \forall q(q \in \mathbb{N}) \\
&\qquad \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0\} \quad (61)
\end{aligned}
$$

Remark 12. In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0(\epsilon_0 \in (0,1]) \forall \epsilon(\epsilon < \epsilon_0)[a_\epsilon \le C\psi(\epsilon)]$. Definition 12. Elements of $G_\delta(M)$ are denoted by:

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}(M). \quad (62) $$

Remark 13. With component wise operations $(\cdot, \pm)$ $G_\delta(M)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \mathrm{cl}[(L_\xi u_\epsilon)_\epsilon]$. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_\delta(M)$ iff $u \circ \psi_\alpha \in G_\delta(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.
The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_\delta(M)$ iff $u \circ \psi_\alpha \in G_\delta(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

Remark 14. Smooth functions $f \in C^\infty(M)$ are embedded into $G_\delta(M)$ simply by the constant embedding $\sigma$, i.e., $\sigma(f) = \text{cl}[(f)_\epsilon]$, hence $C^\infty(M)$ is a faithful subalgebra of $G_\delta(M)$.

The $G_\delta(M)$-module of generalized sections in vector bundles-especially the space of generalized tensor fields $T_s^r(M)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

$$G_s^r(M) = G_\delta(M) \otimes T_s^r(M), \quad (63)$$

where $T_s^r(M)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $C^\infty(M)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting for example the Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $G_\delta(M)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $G_\delta(M)$-modules we have

$$G_s^r(M) \cong L_{(M)}(G_1^0(M)^r, G_0^1(M)^s; G_\delta(M)). \quad (64)$$

In particular a generalized metric is defined to be a symmetric, generalized $(0,2)$-tensor field $g_{ab} = [((g_\epsilon)_{ab})_\epsilon]$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in Colombeau algebra $G_\delta(M)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have

$$\forall K \subset \subset M \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^m] \quad (65)$$

for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric iff (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $G_\delta(M)$-linear isomorphism from $G_1^0(M)$ to $G_0^0(M)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $G_0^2(M)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also, the generalized Levi-Cività connection as well as the generalized Riemann, Ricci and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives¹².

## Super generalized Colombeau calculus

We briefly recall now the basic super generalized Colombeau construction¹⁰,¹¹. Colombeau super generalized functions on $\Omega \subset \mathbb{R}^n$, where $\dim(\Omega) = n$ are defined
The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_\delta(M)$ iff $u \circ \psi_\alpha \in G_\delta(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

Remark 14. Smooth functions $f \in C^\infty(M)$ are embedded into $G_\delta(M)$ simply by the constant embedding $\sigma$, i.e., $\sigma(f) = \text{cl}[(f)_\epsilon]$, hence $C^\infty(M)$ is a faithful subalgebra of $G_\delta(M)$.

The $G_\delta(M)$-module of generalized sections in vector bundles-especially the space of generalized tensor fields $T_s^r(M)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

$$G_s^r(M) = G_\delta(M) \otimes T_s^r(M), \quad (63)$$

where $T_s^r(M)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $C^\infty(M)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting for example the Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $G_\delta(M)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $G_\delta(M)$-modules we have

$$G_s^r(M) \cong L_{(M)}(G_1^0(M)^r, G_0^1(M)^s; G_\delta(M)). \quad (64)$$

In particular a generalized metric is defined to be a symmetric, generalized $(0,2)$-tensor field $g_{ab} = [((g_\epsilon)_{ab})_\epsilon]$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in Colombeau algebra $G_\delta(M)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have

$$\forall K \subset \subset M \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^m] \quad (65)$$

for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric iff (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $G_\delta(M)$-linear isomorphism from $G_1^0(M)$ to $G_0^0(M)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $G_0^2(M)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also, the generalized Levi-Cività connection as well as the generalized Riemann, Ricci and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives¹².

## Super generalized Colombeau calculus

We briefly recall now the basic super generalized Colombeau construction¹⁰,¹¹. Colombeau super generalized functions on $\Omega \subset \mathbb{R}^n$, where $\dim(\Omega) = n$ are defined
as equivalence classes $u = [(u_\epsilon)_\epsilon]$ of nets of functions $u_\epsilon \in C^\infty(\Omega\setminus\Sigma)$, $\epsilon \in (0, \delta]$ such that any $u_\epsilon$ is a net of functions smooth on $\Omega\setminus\Sigma$ and has a discontinuity on a subset $\Sigma \subset \Omega$, where $\dim(\Sigma) < n$. We assume that for any $\epsilon \in (0, \delta]$ the derivative

$\frac{\partial^m u_\epsilon}{\partial x_1^{k_1} \dots \partial x_n^{k_n}}$ exists in the sense of the theory of generalized functions and $\frac{\partial^m u_\epsilon}{\partial x_1^{k_1} \dots \partial x_n^{k_n}} \in \mathcal{D}'(\Omega)$. The basic idea of generalized Colombeau theory of nonlinear super generalized functions$^{10-11}$ is regularization by sequences (nets) of non-smooth functions with derivatives in $\mathcal{D}'(\Omega)$ and the use of asymptotic estimates in terms of a regularization parameter $\epsilon \in (0, \delta]$. Let $(u_\epsilon)_{\epsilon \in (0,1]}$ with $u_\epsilon$ such that:

(1) $u_\epsilon \in C^\infty(M\setminus\Sigma)$ and
(i2) $L_{\xi_1}...L_{\xi_k}u_\epsilon \in \mathcal{D}'(M)$, for all $\epsilon \in (0, \delta]$, where $M$ a separable, smooth orientable Hausdorff manifold of dimension $n$.

Definition 13. The super generalized Colombeau algebra $G_\delta(M, \Sigma)$ of super generalized functions on $M$, where $\Sigma \not\subseteq M$, $\dim(M) = n$, $\dim(\Sigma) < n$, is defined as the quotient:

$$G_\delta(M, \Sigma) = \mathcal{E}_M(M, \Sigma) / \mathcal{N}(M, \Sigma)$$

of the space $\mathcal{E}_M(M, \Sigma)$ of sequences of moderate growth modulo the space $\mathcal{N}(M, \Sigma)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where $X(M\setminus\Sigma)$ denoting the space of smooth vector fields on $M\setminus\Sigma$):

$$\mathcal{E}_M(M, \Sigma) \triangleq \{(u_\epsilon)_\epsilon \mid \forall K(K \subset\subset M\setminus\Sigma)\forall k(k \in \mathbb{N})\exists N(N \in \mathbb{N}) \sup_{p \in K} |L_{\xi_1}...L_{\xi_k}u_\epsilon(p)| = O(\epsilon^{-N}), \epsilon \to 0\} \&$$

$$\forall K(K \subset\subset M)\forall k(k \in \mathbb{N})\exists N(N \in \mathbb{N})\forall f(f \in C^\infty(M))\forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in X(M\setminus\Sigma)) \\ \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon\| = \left( \sup_{f \in C^\infty(M\setminus\Sigma)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^{-N}), \epsilon \to 0 \right], (65)$$

$$\mathcal{N}(M, \Sigma) \triangleq \{(u_\epsilon)_\epsilon \mid \forall K(K \subset\subset M\setminus\Sigma), \forall k(k \in N_0)\forall q(q \in \mathbb{N})$$

$$\forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q), \epsilon \to 0 \right] \& \\ \forall K(K \subset\subset M)\forall k(k \in N_0)\exists N(N \in \mathbb{N})\forall f(f \in C^\infty(M))\forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma))$$

$$\left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon\| = \left( \sup_{f \in C^\infty(M\setminus\Sigma)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^q), \epsilon \to 0 \right], (66)$$

where $L_{\xi_k}^w$ denotes the weak Lie derivative in L.Schwartz sense. In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0(\epsilon_0 \in (0,1]) \forall \epsilon(\epsilon < \epsilon_0)[a_\epsilon \le C\psi(\epsilon)]$. Definition 14. Elements of $G_\delta(M, \Sigma)$ are denoted by
as equivalence classes $u = [(u_\epsilon)_\epsilon]$ of nets of functions $u_\epsilon \in C^\infty(\Omega\setminus\Sigma)$, $\epsilon \in (0, \delta]$ such that any $u_\epsilon$ is a net of functions smooth on $\Omega\setminus\Sigma$ and has a discontinuity on a subset $\Sigma \subset \Omega$, where $\dim(\Sigma) < n$. We assume that for any $\epsilon \in (0, \delta]$ the derivative

$\frac{\partial^m u_\epsilon}{\partial x_1^{k_1} \dots \partial x_n^{k_n}}$ exists in the sense of the theory of generalized functions and $\frac{\partial^m u_\epsilon}{\partial x_1^{k_1} \dots \partial x_n^{k_n}} \in \mathcal{D}'(\Omega)$. The basic idea of generalized Colombeau theory of nonlinear super generalized functions$^{10-11}$ is regularization by sequences (nets) of non-smooth functions with derivatives in $\mathcal{D}'(\Omega)$ and the use of asymptotic estimates in terms of a regularization parameter $\epsilon \in (0, \delta]$. Let $(u_\epsilon)_{\epsilon \in (0,1]}$ with $u_\epsilon$ such that:

(1) $u_\epsilon \in C^\infty(M\setminus\Sigma)$ and
(i2) $L_{\xi_1}...L_{\xi_k}u_\epsilon \in \mathcal{D}'(M)$, for all $\epsilon \in (0, \delta]$, where $M$ a separable, smooth orientable Hausdorff manifold of dimension $n$.

Definition 13. The super generalized Colombeau algebra $G_\delta(M, \Sigma)$ of super generalized functions on $M$, where $\Sigma \not\subseteq M$, $\dim(M) = n$, $\dim(\Sigma) < n$, is defined as the quotient:

$$G_\delta(M, \Sigma) = \mathcal{E}_M(M, \Sigma) / \mathcal{N}(M, \Sigma)$$

of the space $\mathcal{E}_M(M, \Sigma)$ of sequences of moderate growth modulo the space $\mathcal{N}(M, \Sigma)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where $X(M\setminus\Sigma)$ denoting the space of smooth vector fields on $M\setminus\Sigma$):

$$\mathcal{E}_M(M, \Sigma) \triangleq \{(u_\epsilon)_\epsilon \mid \forall K(K \subset\subset M\setminus\Sigma) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \sup_{p \in K} |L_{\xi_1}...L_{\xi_k}u_\epsilon(p)| = O(\epsilon^{-N}), \epsilon \to 0\} \&$$

$$\forall K(K \subset\subset M)\forall k(k \in \mathbb{N})\exists N(N \in \mathbb{N})\forall f(f \in C^\infty(M))\forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma)) \\ \left[ \|L^w_{\xi_1} \dots L^w_{\xi_k} u_\epsilon\| = \left( \sup_{f \in C^\infty(M\setminus\Sigma)} |L^w_{\xi_1} \dots L^w_{\xi_k} u_\epsilon(f)| \right) = O(\epsilon^{-N}), \epsilon \to 0 \right] \}, (65)$$

$$\mathcal{N}(M, \Sigma) \triangleq \{(u_\epsilon)_\epsilon \mid \forall K(K \subset\subset M\setminus\Sigma), \forall k(k \in N_0)\forall q(q \in \mathbb{N})$$

$$\forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q), \epsilon \to 0 \right] \& \\ \forall K(K \subset\subset M)\forall k(k \in N_0)\exists N(N \in \mathbb{N})\forall f(f \in C^\infty(M))\forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma)) \\ \left[ \|L^w_{\xi_1} \dots L^w_{\xi_k} u_\epsilon\| = \left( \sup_{f \in C^\infty(M\setminus\Sigma)} |L^w_{\xi_1} \dots L^w_{\xi_k} u_\epsilon(f)| \right) = O(\epsilon^q), \epsilon \to 0 \right] \}, (66)$$

where $L^w_{\xi_k}$ denotes the weak Lie derivative in L.Schwartz sense. In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0(\epsilon_0 \in (0,1]) \forall \epsilon(\epsilon < \epsilon_0)[a_\epsilon \le C\psi(\epsilon)]$. Definition 14. Elements of $G_\delta(M, \Sigma)$ are denoted by
$$ : u = \mathrm{cl}[(u_{\varepsilon})_{\varepsilon}] \triangleq (u_{\varepsilon})_{\varepsilon} + \mathcal{N}(M, \Sigma). \quad (67) $$

Remark 15. With component wise operations $(\cdot, \pm)$ $G_{\delta}(M, \Sigma)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_{\xi}u \triangleq \mathrm{cl}[(L_{\xi}u_{\varepsilon})_{\varepsilon}]$. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_{\delta}(M, \Sigma)$ if $u \circ \psi_{\alpha} \in G_{\delta}(\psi_{\alpha}(V_{\alpha}))$ for all charts $(V_{\alpha}, \psi_{\alpha})$, where on the open set $\psi_{\alpha}(V_{\alpha}) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

Remark 16. Smooth functions $f \in C^{\infty}(M\setminus\Sigma)$ are embedded into $G_{\delta}(M, \Sigma)$ simply by the constant embedding $\sigma$, i.e., $\sigma(f) = \mathrm{cl}[(f)_{\varepsilon}]$, hence $C^{\infty}(M\setminus\Sigma)$ is a faithful subalgebra of $G_{\delta}(M, \Sigma)$.

Definition 15. The $G_{\delta}(M, \Sigma)$-module of supergeneralized sections in vector bundles- especially the space of super generalized tensor fields $T_s^r(M\setminus\Sigma)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields $G_s^r(M, \Sigma) = G_{\delta}(M, \Sigma) \otimes T_s^r(M\setminus\Sigma)$, where $T_s^r(M\setminus\Sigma)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $C^{\infty}(M\setminus\Sigma)$. Hence super generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting, the Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $G_{\delta}(M, \Sigma)$-multilinear maps taking generalized vector and co vector fields to generalized functions, i.e., as $G_{\delta}(M, \Sigma)$-modules we have

$$ G_s^r(M, \Sigma) \cong L_{(M)}(G_1^0(M, \Sigma)^r, G_0^1(M, \Sigma)^s; G_{\delta}(M, \Sigma)) \quad (68) $$

In particular a super generalized metric is defined to be a symmetric, super generalized (0,2)-tensor field $g_{ab} = [((g_{\varepsilon})_{ab})_{\varepsilon}]$ (with its index independent of $\varepsilon$ and) whose determinant $\det(g_{ab})$ is invertible in $G_{\delta}(M, \Sigma)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_{\varepsilon})_{ab})_{\varepsilon}$ of $\det(g_{ab})$ we have $\forall K \subset M\setminus\Sigma \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\varepsilon))| \ge \varepsilon^q]$ for all $\varepsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric if (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_{\varepsilon})_{ab})_{\varepsilon}$ of $g_{ab}$ such that for fixed $\varepsilon$ (small enough) $(g_{\varepsilon})_{ab} = g_{ab}(\varepsilon)$ (resp. $(g_{\varepsilon})_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $G(M, \Sigma)$-linear isomorphism from $G_0^1(M, \Sigma)$ to $G_1^0(M, \Sigma)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\varepsilon))_{\varepsilon}]$ is a well defined element of $G_0^2(M, \Sigma)$ (i.e., independent of the representative $((g_{\varepsilon})_{ab})_{\varepsilon}$). Also, the super generalized Levi-Cività connection as well as the super generalized Riemann, Ricci and Einstein tensor of a super generalized metric are defined simply by the usual coordinate formulae on the level of representatives.
Let $u = \text{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}(M, \Sigma). (67)$

**Remark 15.** With component wise operations $(\cdot, \pm)$, $G_\delta(M, \Sigma)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \text{cl}[(L_\xi u_\epsilon)_\epsilon]$. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_\delta(M, \Sigma)$ if $u \circ \psi_\alpha \in G_\delta(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 16.** Smooth functions $f \in C^\infty(M\setminus\Sigma)$ are embedded into $G_\delta(M, \Sigma)$ simply by the constant embedding $\sigma$, i.e., $\sigma(f) = \text{cl}[(f)_\epsilon]$, hence $C^\infty(M\setminus\Sigma)$ is a faithful subalgebra of $G_\delta(M, \Sigma)$.

**Definition 15.** The $G_\delta(M, \Sigma)$-module of supergeneralized sections in vector bundles- especially the space of super generalized tensor fields $T_s^r(M\setminus\Sigma)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields $G_s^r(M, \Sigma) = G_\delta(M, \Sigma) \otimes T_s^r(M\setminus\Sigma)$, where $T_s^r(M\setminus\Sigma)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $C^\infty(M\setminus\Sigma)$. Hence super generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting, the Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $G_\delta(M, \Sigma)$-multilinear maps taking generalized vector and co vector fields to generalized functions, i.e., as $G_\delta(M, \Sigma)$-modules we have

$$G_s^r(M, \Sigma) \cong L_{(M)}(G_1^0(M, \Sigma)^r, G_0^1(M, \Sigma)^s; G_\delta(M, \Sigma)) \quad (68)$$

In particular a super generalized metric is defined to be a symmetric, super generalized (0,2)-tensor field $g_{ab} = [((g_\epsilon)_{ab})_\epsilon]$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in $G_\delta(M, \Sigma)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have $\forall K \subset M\setminus\Sigma \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^q]$ for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric if (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $G(M, \Sigma)$-linear isomorphism from $G_0^1(M, \Sigma)$ to $G_1^0(M, \Sigma)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $G_0^2(M, \Sigma)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also, the super generalized Levi-Cività connection as well as the super generalized Riemann, Ricci and Einstein tensor of a super generalized metric are defined simply by the usual coordinate formulae on the level of representatives.
Let $f(x)$ be a function of a single variable $x \in \mathbb{R}$ that has a jump discontinuity at $x = x_0$ of magnitude $[f]_{x_0} = f(x_0 + 0) - f(x_0 - 0)$ but has a continuous derivative everywhere else except $x = x_0$:

$$f(x) = \begin{cases} f_1(x) & x < x_0 \\ f_2(x) & x \ge x_0 \end{cases}$$

Let the derivative in the interval $x \le x_0$ and $x > x_0$ be denoted $\{f'(x)\}$, where $\{f'(x)\} = f'(x)$ iff $x \ne x_0$. This derivative $\{f'(x)\}$ is undefined at $x = x_0$. With the help of generalized functions, however, the distributional derivative $\bar{f}'(x)$ is obtained by setting:

$$g(x) = f(x) - [f]_{x_0} H(x - x_0), (69)$$

where $H(x - x_0)$ is the Heaviside function. The function $g(x)$ is continuous at $x = x_0$. Its derivative coincides with that of $f(x)$ on both sides of $x_0$. Accordingly, we differentiate both sides of Eq. (69) and therefore one obtains

$$\{f'(x)\} = \bar{f}'(x) - [f]_{x_0} \delta(x - x_0).$$

Thus, finally we get

$$\bar{f}'(x) = \{f'(x)\} + [f]_{x_0} \delta(x - x_0). \quad (70)$$

or

$$\bar{f}'(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta(x - x_0) & x \le x_0 \\ f_2'(x) + [f]_{x_0} \delta(x - x_0) & x > x_0 \end{cases} \quad (71)$$

Eq. (71) is easily generalized to a function $f(x)$ that has jumps of magnitude $[f]_{x_0}$, $[f]_{x_1}$, ..., $[f]_{x_k}$ at $x_0, x_1, \dots, x_k$. The result is

$$\bar{f}''(x) = \{f''(x)\} + \sum_{i=0}^{k} [f]_{x_i} \delta(x - x_i). \quad (72)$$

Let us now consider a function $f(x)$ that admits derivatives up to the second order on both sides of the point $x_0$, that has a jump discontinuity of strength $[f]_{x_0}$, and whose derivative has a jump discontinuity of strength $[f']_{x_0}$ at this point. To obtain $\bar{f}''(x)$, one substitutes $f'(x)$ for $f(x)$ in Eq. (70) and immediately one obtains

$$\bar{f}''(x) = \{f''(x)\} + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0),$$

where $\{f''(x)\} = f''(x)$ iff $x \ne x_0$,

or

$$\bar{f}''(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0) & x < x_0 \\ f_2'(x) + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0) & x \ge x_0 \end{cases} \quad (73)$$

This process can be continued for higher derivatives and for singularities at several points. Thus, a function $f(x)$ that admits continuous derivatives up to the $m$-th order in each of the intervals $(x_{j-1}, x_j)$, $j = 1, 2, \dots, l$ has $m$-th order distributional derivative $\bar{f}^{(m)}(x)$:

$$\bar{f}^{(m)}(x) = \{f^{(m)}(x)\} + \sum_{j=1}^{l} a_j [\delta^{(m-1)}(x - x_j) + b_j \delta^{(m-2)}(x - x_j) + \dots + f_j \delta(x - x_j)], (74)$$

where $a_j = [f(x)]_{x_j}, b_j = [f'(x)]_{x_j}, \dots, f_j = [f^{(m-1)}(x)]_{x_j}$ and $[\cdot]_{x_j}$ stands for the jump in the quantity across the point $x_j$.

# IV. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH REGULARIZATION VIA HORIZON. NON-SPINNING GRAVITATIONAL
Let $f(x)$ be a function of a single variable $x \in \mathbb{R}$ that has a jump discontinuity at $x = x_0$ of magnitude $[f]_{x_0} = f(x_0 + 0) - f(x_0 - 0)$ but has a continuous derivative everywhere else except $x = x_0$:

$$f(x) = \begin{cases} f_1(x) & x < x_0 \\ f_2(x) & x \ge x_0 \end{cases}$$

Let the derivative in the interval $x \le x_0$ and $x > x_0$ be denoted $\{f'(x)\}$, where $\{f'(x)\} = f'(x)$ iff $x \ne x_0$. This derivative $\{f'(x)\}$ is undefined at $x = x_0$. With the help of generalized functions, however, the distributional derivative $\bar{f}'(x)$ is obtained by setting:

$$g(x) = f(x) - [f]_{x_0} H(x - x_0), (69)$$

where $H(x - x_0)$ is the Heaviside function. The function $g(x)$ is continuous at $x = x_0$. Its derivative coincides with that of $f(x)$ on both sides of $x_0$. Accordingly, we differentiate both sides of Eq. (69) and therefore one obtains

$$\{f'(x)\} = \bar{f}'(x) - [f]_{x_0} \delta(x - x_0).$$

Thus, finally we get

$$\bar{f}'(x) = \{f'(x)\} + [f]_{x_0} \delta(x - x_0). (70)$$

or

$$\bar{f}'(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta(x - x_0) & x \le x_0 \\ f_2'(x) + [f]_{x_0} \delta(x - x_0) & x > x_0 \end{cases} \quad (71)$$

Eq. (71) is easily generalized to a function $f(x)$ that has jumps of magnitude $[f]_{x_0}$, $[f]_{x_1}$, ..., $[f]_{x_k}$ at $x_0, x_1, ..., x_k$. The result is

$$\bar{f}''(x) = \{f''(x)\} + \sum_{i=0}^{k} [f]_{x_i} \delta(x - x_i). (72)$$

Let us now consider a function $f(x)$ that admits derivatives up to the second order on both sides of the point $x_0$, that has a jump discontinuity of strength $[f]_{x_0}$, and whose derivative has a jump discontinuity of strength $[f']_{x_0}$ at this point. To obtain $\bar{f}''(x)$, one substitutes $f'(x)$ for $f(x)$ in Eq. (70) and immediately one obtains

$$\bar{f}''(x) = \{f''(x)\} + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0),$$

where $\{f''(x)\} = f''(x)$ iff $x \ne x_0$,

or

$$\bar{f}''(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0) & x < x_0 \\ f_2'(x) + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0) & x \ge x_0 \end{cases} \quad (73)$$

This process can be continued for higher derivatives and for singularities at several points. Thus, a function $f(x)$ that admits continuous derivatives up to the $m$-th order in each of the intervals $(x_{j-1}, x_j)$, $j = 1, 2, ..., l$ has $m$-th order distributional derivative $\bar{f}^{(m)}(x)$:

$$\bar{f}^{(m)}(x) = \{f^{(m)}(x)\} + \sum_{j=1}^{l} a_j [\delta^{(m-1)}(x - x_j) + b_j \delta^{(m-2)}(x - x_j) + \dots + f_j \delta(x - x_j)], (74)$$

where $a_j = [f(x)]_{x_j}, b_j = [f'(x)]_{x_j}, \dots, f_j = [f^{(m-1)}(x)]_{x_j}$ and $[\cdot]_{x_j}$ stands for the jump in the quantity across the point $x_j$.

# IV. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH REGULARIZATION VIA HORIZON. NON-SPINNING GRAVITATIONAL
SINGULARITY

In this section we studied Schwarzschild solution using Colombeau distributional geometry, thus without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$.

Note that the Colombeau generalized object $\{(ds_{\epsilon}^{+2})_{\epsilon}, (ds_{\epsilon}^{-2})_{\epsilon}\}$, given by Eq. (35)-Eq. (36) that is a natural way is a super generalized (0,2) -tensor field $g_{ab} = [(g_{\epsilon})_{ab}]_{\epsilon}$ whose determinant $\det(g_{ab})$ is invertible in $G_{\delta}(\mathbb{R}^4, \{r=2m\})$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_{\epsilon}]$ is a well defined element of $G_{0,\delta}^2(\mathbb{R}^4, \{r=2m\})$.

Thus the full super generalized Ricci tensor above the horizon reads

$$ ([\bar{R}_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} = ([\bar{R}_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2}((\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} + \frac{2}{r}(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}), \quad (75) $$

$$ ([\bar{R}_{\epsilon}^{+}(r)]_{2}^{2})_{\epsilon} = ([\bar{R}_{\epsilon}^{+}(r)]_{3}^{3})_{\epsilon} = \frac{(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}(r))_{\epsilon}}{r^{2}}, \quad (76) $$

where $(h_{\epsilon})_{\epsilon} = \{(h_{\epsilon}^{+})_{\epsilon}, (h_{\epsilon}^{-})_{\epsilon}\}$ and the distributional derivatives $\bar{h}_{\epsilon}^{\prime}(r)$ and $\bar{h}_{\epsilon}^{\prime\prime}(r)$ is defined by Eq. (71) and by Eq. (73) correspondingly.

From Eq. (35)-Eq. (36) we obtain

$$ ([h_{\epsilon}]_{2m})_{\epsilon} = (h_{\epsilon}^{+}(2m + 0))_{\epsilon} - (h_{\epsilon}^{-}(2m - 0))_{\epsilon} = (h_{\epsilon}^{+}(2m) - h_{\epsilon}^{-}(2m))_{\epsilon} = $$

$$ -\left(\frac{\epsilon}{2m}\right)_{\epsilon} - \left(\frac{\epsilon}{2m}\right)_{\epsilon} = -(\epsilon)_{\epsilon}/m \quad (77) $$

From Eq. (77) and Eq. (70) we obtain

$$ (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r < 2m \end{cases} \quad (78) $$

where $([h]_{2m})_{\epsilon} = -(\epsilon)_{\epsilon}/m$ and therefore

$$ (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m}\delta(r - 2m) & r < 2m \end{cases} \quad (79) $$

From Eq. (39) we obtain

$$ ([h_{\epsilon}']_{2m})_{\epsilon} = (h_{\epsilon}^{+\prime}(2m + 0))_{\epsilon} - (h_{\epsilon}^{-\prime}(2m - 0))_{\epsilon} = $$

$$ (h_{\epsilon}^{+\prime}(2m))_{\epsilon} - (h_{\epsilon}^{-\prime}(2m))_{\epsilon} = $$

$$ \frac{(\epsilon)_{\epsilon}}{4m^2} - \left(-\frac{(\epsilon)_{\epsilon}}{4m^2}\right) = \frac{(\epsilon)_{\epsilon}}{2m^2}. \quad (80) $$

From Eq. (73) we obtain

$$ (\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta'(r - 2m) + ([h_{\epsilon}']_{2m})_{\epsilon}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) + ([h_{\epsilon}']_{x_0})_{\epsilon}\delta(r - 2m) & r < 2m \end{cases} \quad (81) $$

where $([h_{\epsilon}']_{2m})_{\epsilon} = (\epsilon)_{\epsilon}/2m^2$ and therefore
SINGULARITY

In this section we studied Schwarzschild solution using Colombeau distributional geometry, thus without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$.

Note that the Colombeau generalized object $\{(ds_{\epsilon}^{+2})_{\epsilon}, (ds_{\epsilon}^{-2})_{\epsilon}\}$, given by Eq. (35)-Eq. (36) that is a natural way is a super generalized (0,2) -tensor field $g_{ab} = [(g_{\epsilon})_{ab}]_{\epsilon}$ whose determinant $\det(g_{ab})$ is invertible in $G_{\delta}(\mathbb{R}^4, \{r=2m\})$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_{\epsilon}]$ is a well defined element of $G_{0,\delta}^2(\mathbb{R}^4, \{r=2m\})$.

Thus the full super generalized Ricci tensor above the horizon reads

$$ ([\bar{R}_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} = ([\bar{R}_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2}((\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} + \frac{2}{r}(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}), \quad (75) $$

$$ ([\bar{R}_{\epsilon}^{+}(r)]_{2}^{2})_{\epsilon} = ([\bar{R}_{\epsilon}^{+}(r)]_{3}^{3})_{\epsilon} = \frac{(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}(r))_{\epsilon}}{r^2}, \quad (76) $$

where $(h_{\epsilon})_{\epsilon} = \{(h_{\epsilon}^{+})_{\epsilon}, (h_{\epsilon}^{-})_{\epsilon}\}$ and the distributional derivatives $\bar{h}_{\epsilon}^{\prime}(r)$ and $\bar{h}_{\epsilon}^{\prime\prime}(r)$ is defined by Eq. (71) and by Eq. (73) correspondingly.

From Eq. (35)-Eq. (36) we obtain

$$ ([h_{\epsilon}]_{2m})_{\epsilon} = (h_{\epsilon}^{+}(2m + 0))_{\epsilon} - (h_{\epsilon}^{-}(2m - 0))_{\epsilon} = (h_{\epsilon}^{+}(2m) - h_{\epsilon}^{-}(2m))_{\epsilon} = -\left(\frac{\epsilon}{2m}\right)_{\epsilon} - \left(\frac{\epsilon}{2m}\right)_{\epsilon} = -(\epsilon)_{\epsilon}/m \quad (77) $$

From Eq. (77) and Eq. (70) we obtain

$$ (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r < 2m \end{cases} \quad (78) $$

where $([h]_{2m})_{\epsilon} = -(\epsilon)_{\epsilon}/m$ and therefore

$$ (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m}\delta(r - 2m) & r < 2m \end{cases} \quad (79) $$

From Eq. (39) we obtain

$$ ([h_{\epsilon}']_{2m})_{\epsilon} = (h_{\epsilon}^{+\prime}(2m + 0))_{\epsilon} - (h_{\epsilon}^{-\prime}(2m - 0))_{\epsilon} = (h_{\epsilon}^{+\prime}(2m))_{\epsilon} - (h_{\epsilon}^{-\prime}(2m))_{\epsilon} = -\left(\frac{\epsilon}{4m^2}\right)_{\epsilon} - \left(\frac{\epsilon}{4m^2}\right)_{\epsilon} = -(\epsilon)_{\epsilon}/4m^2 \quad (80) $$

From Eq. (73) we obtain

$$ (\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta'(r - 2m) + ([h_{\epsilon}']_{2m})_{\epsilon}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) + ([h_{\epsilon}']_{x_0})_{\epsilon}\delta(r - 2m) & r < 2m \end{cases} \quad (81) $$

where $([h_{\epsilon}']_{2m})_{\epsilon} = (\epsilon)_{\epsilon}/2m^2$ and therefore
$$ \bar{h}_{\varepsilon}''(r) = \begin{cases} (h_{\varepsilon}^{+''}(r))_{\varepsilon} - \frac{\varepsilon}{m} \delta'(r-2m) + \frac{\varepsilon}{2m^2} \delta(r-2m) & r \ge 2m \\ h_{\varepsilon}^{-''}(r) - \frac{\varepsilon}{m} \delta'(r-2m) + \frac{\varepsilon}{2m^2} \delta(r-2m) & r < 2m \end{cases} \quad (82) $$

Inserting Eq. (79) and Eq. (82) into Eq. (75) and into Eq. (76) finally we get for

$$ \begin{aligned} r \ge 2m : \quad (&[\bar{R}_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2}((\bar{h}_{\varepsilon}^{+''}(r))_{\varepsilon} + \frac{2}{r}(\bar{h}_{\varepsilon}^{+'}(r))_{\varepsilon}) = \\ & \frac{1}{2}((h_{\varepsilon}^{+''}(r))_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}^{+'}(r))_{\varepsilon}) - \\ & - \frac{\varepsilon}{2m}\delta'(r-2m) + \frac{\varepsilon}{2m^2}\delta(r-2m) - \frac{2\varepsilon}{rm}\delta(r-2m). \end{aligned} \quad (83) $$

$$ ([\bar{R}_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{(\bar{h}_{\varepsilon}^{'}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}(r))_{\varepsilon}}{r^2} = $$

$$ \frac{(h_{\varepsilon}^{+'}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+}(r))_{\varepsilon}}{r^2} - \frac{(\varepsilon)_{\varepsilon}}{rm} \delta(r-2m). \quad (84) $$

And similarly, we get for $r < 2m$ :

$$ \begin{aligned} ([\bar{R}_{\varepsilon}^{-}(r)]_{0}^{0})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{-}(r)]_{1}^{1})_{\varepsilon} = & \\ \frac{1}{2}((\bar{h}_{\varepsilon}^{''}(r))_{\varepsilon} + \frac{2}{r}(\bar{h}_{\varepsilon}^{'}(r))_{\varepsilon}) = & \\ \frac{1}{2}((h_{\varepsilon}^{-''}(r))_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}^{-'}(r))_{\varepsilon}) - & \\ - \frac{\varepsilon}{2m}\delta'(r-2m) + \frac{\varepsilon}{2m^2}\delta(r-2m) - \frac{2\varepsilon}{rm}\delta(r-2m) & . \end{aligned} \quad (85) $$

$$ ([\bar{R}_{\varepsilon}^{-}(r)]_{2}^{2})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{-}(r)]_{3}^{3})_{\varepsilon} = \frac{(\bar{h}_{\varepsilon}^{'}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}(r))_{\varepsilon}}{r^2} = $$

$$ \frac{(h_{\varepsilon}^{-'}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{-}(r))_{\varepsilon}}{r^2} - \frac{(\varepsilon)_{\varepsilon}}{rm} \delta(r-2m) . \quad (86) $$

Remark 17. Note that in $\mathcal{D}'([2m, \infty))$ :

$$ w - \lim_{\varepsilon \to 0} \frac{2\pi^{-1}\varepsilon}{(r-2m)^2 + \varepsilon^2} = \delta(r-2m), \quad (87) $$

and therefore in $\mathcal{D}'([2m, \infty))$ :

$$ \begin{aligned} w - \lim_{\varepsilon \to 0} \frac{d}{dr} \frac{2\pi^{-1}\varepsilon}{(r-2m)^2 + \varepsilon^2} &= \\ w - \lim_{\varepsilon \to 0} \frac{4\pi^{-1}\varepsilon(r-2m)}{[(r-2m)^2 + \varepsilon^2]^2} &= \delta'(r-2m) . \end{aligned} \quad (88) $$

By using Eq. (87) and Eq. (88) one obtains the natural imbedding

$$ \delta(r-2m) \mapsto \mathcal{G}_{\delta}(\mathbb{R}) : $$
$$
\bar{h}_{\epsilon}''(r) = \begin{cases} (h_{\epsilon}^{+''}(r))_{\epsilon} - \frac{\langle \epsilon \rangle_{\epsilon}}{m} \delta'(r-2m) + \frac{\langle \epsilon \rangle_{\epsilon}}{2m^2} \delta(r-2m) & r \ge 2m \\ h_{\epsilon}^{-''}(r) - \frac{\langle \epsilon \rangle_{\epsilon}}{m} \delta'(r-2m) + \frac{\langle \epsilon \rangle_{\epsilon}}{2m^2} \delta(r-2m) & r < 2m \end{cases} \quad (82)
$$

Inserting Eq. (79) and Eq. (82) into Eq. (75) and into Eq. (76) finally we get for

$$
\begin{align*}
r \ge 2m :
& ( [\bar{R}_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} = ( [\bar{R}_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2} \left( (\bar{h}_{\epsilon}^{+''}(r))_{\epsilon} + \frac{2}{r} (\bar{h}_{\epsilon}^{+'}(r))_{\epsilon} \right) = \\
& \frac{1}{2} \left( (h_{\epsilon}^{+''}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{+'}(r))_{\epsilon} \right) - \\
& \qquad - \frac{\langle \epsilon \rangle_{\epsilon}}{2m} \delta'(r-2m) + \frac{\langle \epsilon \rangle_{\epsilon}}{2m^2} \delta(r-2m) - \frac{2\langle \epsilon \rangle_{\epsilon}}{rm} \delta(r-2m). \tag{83}
\end{align*}
$$

$$
\begin{align*}
([\bar{R}_{\epsilon}^{+}(r)]_{2})_{\epsilon} &= ([\bar{R}_{\epsilon}^{+}(r)]_{3})_{\epsilon} = \frac{(\bar{h}_{\epsilon}')_{\epsilon}}{r} + \frac{1+(h_{\epsilon}(r))_{\epsilon}}{r^2} = \\
\frac{(h_{\epsilon}^{+'}(r))_{\epsilon}}{r} &+ \frac{1+(h_{\epsilon}^{+}(r))_{\epsilon}}{r^2} - \frac{(\epsilon)_{\epsilon}}{rm} \delta(r-2m). \quad (84)
\end{align*}
$$

And similarly, we get for $r < 2m$ :

$$
\begin{align*}
([\bar{R}_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} &= ([\bar{R}_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} = \\
\frac{1}{2} \left( \left( \bar{h}_{\epsilon}^{''}(r) \right)_{\epsilon} + \frac{2}{r} \left( \bar{h}_{\epsilon}'(r) \right)_{\epsilon} \right) &= \\
\frac{1}{2} \left( \left( h_{\epsilon}^{-''}(r) \right)_{\epsilon} + \frac{2}{r} \left( h_{\epsilon}'(r) \right)_{\epsilon} \right) - \\
&\quad - \frac{(\epsilon)_{\epsilon}}{2m} \delta'(r-2m) + \frac{(\epsilon)_{\epsilon}}{2m^2} \delta(r-2m) - \frac{2(\epsilon)_{\epsilon}}{rm} \delta(r-2m) . \tag{85}
\end{align*}
$$

$$
\begin{align*}
([\bar{R}_{\epsilon}^{-}(r)]_{2})_{\epsilon} &= ([\bar{R}_{\epsilon}^{-}(r)]_{3})_{\epsilon} = \frac{(\bar{h}_{\epsilon}')_{\epsilon}}{r} + \frac{1+(h_{\epsilon}(r))_{\epsilon}}{r^2} = \\
\frac{(h_{\epsilon}^{-'}(r))_{\epsilon}}{r} &+ \frac{1+(h_{\epsilon}^{-}(r))_{\epsilon}}{r^2} - \frac{(\epsilon)_{\epsilon}}{rm} \delta(r-2m) . \quad (86)
\end{align*}
$$

Remark 17. Note that in $\mathcal{D}'([2m, \infty))$ :

$$
w - \lim_{\epsilon \to 0} \frac{2\pi^{-1}\epsilon}{(r-2m)^2 + \epsilon^2} = \delta(r-2m), \quad (87)
$$

and therefore in $\mathcal{D}'([2m, \infty))$ :

$$
\begin{align*}
w - \lim_{\epsilon \to 0} \frac{d}{dr} \frac{2\pi^{-1}\epsilon}{(r-2m)^2 + \epsilon^2} &= \\
w - \lim_{\epsilon \to 0} \frac{4\pi^{-1}\epsilon(r-2m)}{[(r-2m)^2 + \epsilon^2]^2} &= \delta'(r-2m) . \quad (88)
\end{align*}
$$

By using Eq. (87) and Eq. (88) one obtains the natural imbedding

$$
\delta(r-2m) \mapsto G_{\delta}(\mathbb{R}) :
$$
$$
\delta(r - 2m) \hookrightarrow (\delta(r - 2m; \varepsilon))_{\varepsilon} = \left( \frac{2\pi^{-1}\varepsilon}{(r-2m)^2+\varepsilon^2} \right)_{\varepsilon} \in G_{\delta}(\mathbb{R}) \quad (89)
$$

and

$$
\begin{align}
\delta'(r - 2m) \hookrightarrow \mathcal{G}_{\delta}(\mathbb{R}) : \nonumber \\
\delta'(r - 2m) \hookrightarrow (\delta'(r - 2m; \varepsilon))_{\varepsilon} = \left( \frac{4\pi^{-1}\varepsilon(r-2m)}{[(r-2m)^2+\varepsilon^2]^2} \right)_{\varepsilon} \in \mathcal{G}_{\delta}(\mathbb{R}) \tag{90}
\end{align}
$$

Remark 18. Note, that, in $\mathcal{D}'([2m, \infty))$ :

$$
\begin{align}
w - \lim_{\epsilon \to 0} \frac{4\pi^{-1}\epsilon^2(r-2m)}{[(r-2m)^2+\epsilon^2]^2} &= c_1 \delta(r-2m), \quad (91) \\
c_1 &= 2\pi^{-1}.
\end{align}
$$

Inserting Eq. (89)-Eq. (90) and Eq. (91) into Eq. (83) and into Eq. (84) we get for

$$
\begin{align*}
& r \ge 2m : \\
& ([\bar{R}_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2} \left( (\bar{h}_{\varepsilon}^{''}(r))_{\varepsilon} + \frac{2}{r} (\bar{h}_{\varepsilon}^{'}(r))_{\varepsilon} \right) = \\
& \frac{1}{2} \left( (h_{\varepsilon}^{++}(r))_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+'}(r))_{\varepsilon} \right) - \\
& - \frac{(\varepsilon)_{\varepsilon}}{2m} (\delta'(r-2m;\varepsilon))_{\varepsilon} + \left[ \frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{rm} \right] (\delta(r-2m;\varepsilon))_{\varepsilon} = \\
& \frac{1}{2} \left( (h_{\varepsilon}^{++}(r))_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+'}(r))_{\varepsilon} \right) - \\
& - \left[ \frac{c_1}{2m} - (\varepsilon)_{\varepsilon} \left( \frac{1}{2m^2} - \frac{2}{rm} \right) \right] (\delta(r-2m;\varepsilon))_{\varepsilon} \in \mathcal{G}_{\delta}(\mathbb{R}^3). \tag{92}
\end{align*}
$$

$$
\begin{align}
([\bar{R}_{\varepsilon}^{+}(r)]_{2})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{+}(r)]_{3})_{\varepsilon} &= \frac{(\bar{h}'_{\varepsilon}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}(r))_{\varepsilon}}{r^2} = \nonumber \\
\frac{(h_{\varepsilon}^{+'}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+}(r))_{\varepsilon}}{r^2} - \frac{(\varepsilon)_{\varepsilon}}{rm} (\delta(r-2m;\varepsilon))_{\varepsilon} \in \mathcal{G}_{\delta}(\mathbb{R}^3) \tag{93}
\end{align}
$$

Investigating the weak limit of the angular components of the Ricci tensor by using
the abbreviation $\hat{\Phi}(r) = \int_0^\pi \sin\theta \, d\theta \int_0^{2\pi} d\varphi \, \Phi(x)$ and let $\Phi(x)$ be the function
$\Phi(x) \in S_{2m}^+(\mathbb{R}^3)$, where by $S_{2m}^+(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with
compact support such that:

(1) $\text{supp}(\Phi(x)) \subset K^+ = \{x \mid \|x\| \ge 2m\}$, (2) $\hat{\Phi}(r) \in C^\infty(\mathbb{R})$.

Then for any function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$ we get:

$$
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{1})_{\epsilon} \Phi(x) d^3 x = \\
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{0})_{\epsilon} \Phi(x) d^3 x = - \epsilon_1 m \hat{\Phi}(2m), (94)
$$

$\epsilon_1 > 0.$

$$
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{1})_{\epsilon} \Phi(x) d^3 x = \\
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{0})_{\epsilon} \Phi(x) d^3 x = 0. \quad (95)
$$

Let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$, where by $S_{2m}^{-}(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with compact support such that:

(1) $\text{supp}(\Phi(x)) \subset K^{-} = \{x|0 \le \|x\| < 2m\}$, (2) $\hat{\Phi}(r) \in C^{\infty}(\mathbb{R})$.

Then for any function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$ from Eq. (85)-Eq. (86) similarly as above we
$$
\delta(r - 2m) \hookrightarrow (\delta(r - 2m; \varepsilon))_{\varepsilon} = \left( \frac{2\pi^{-1}\varepsilon}{(r-2m)^2+\varepsilon^2} \right)_{\varepsilon} \in G_{\delta}(\mathbb{R}) \quad (89)
$$

and

$$
\delta'(r - 2m) \hookrightarrow \mathcal{G}_{\delta}(\mathbb{R}) :
$$

$$
\delta'(r - 2m) \hookrightarrow (\delta'(r - 2m; \varepsilon))_{\varepsilon} = \left( \frac{4\pi^{-1}\varepsilon(r-2m)}{[(r-2m)^2+\varepsilon^2]^2} \right)_{\varepsilon} \in G_{\delta}(\mathbb{R}) \quad (90)
$$

Remark 18. Note, that, in $\mathcal{D}'([2m, \infty))$ :

$$
w - \lim_{\epsilon \to 0} \frac{4\pi^{-1}\epsilon^2(r-2m)}{[(r-2m)^2+\epsilon^2]^2} = c_1\delta(r-2m), \quad (91)
$$

$c_1 = 2\pi^{-1}.$

Inserting Eq. (89)-Eq. (90) and Eq. (91) into Eq. (83) and into Eq. (84) we get for

$r \ge 2m:$

$$
([\bar{R}_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} = ([\bar{R}_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2}((\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} + \frac{2}{r}(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}) = \\
\frac{1}{2}((h_{\epsilon}^{+\prime\prime}(r))_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{+\prime}(r))_{\epsilon}) -
$$

$$
- \frac{(\epsilon)_{\epsilon}}{2m} (\delta'(r - 2m; \epsilon))_{\epsilon} + \left[ \frac{(\epsilon)_{\epsilon}}{2m^2} - \frac{2(\epsilon)_{\epsilon}}{rm} \right] (\delta(r - 2m; \epsilon))_{\epsilon} =
$$

$$
\frac{1}{2} \left( \left( h_{\epsilon}^{+}''(r) \right)_{\epsilon} + \frac{2}{r} \left( h_{\epsilon}^{+'}(r) \right)_{\epsilon} \right) -
$$

$$
- \left[ \frac{c_1}{2m} - (\epsilon)_\epsilon \left( \frac{1}{2m^2} - \frac{2}{rm} \right) \right] (\delta(r - 2m; \epsilon))_\epsilon \in G_\delta(\mathbb{R}^3). \quad (92)
$$

$$
([\bar{R}_{\epsilon}^{+}(r)]_{2})_{\epsilon} = ([\bar{R}_{\epsilon}^{+}(r)]_{3})_{\epsilon} = \frac{(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}(r))_{\epsilon}}{r^2} =
$$

$$
\frac{(h_{\epsilon}^{+'}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}^{+}(r))_{\epsilon}}{r^2} - \frac{(\epsilon)_{\epsilon}}{rm} (\delta(r - 2m; \epsilon))_{\epsilon} \in G_{\delta}(\mathbb{R}^3) \quad (93)
$$

Investigating the weak limit of the angular components of the Ricci tensor by using
the abbreviation $\hat{\Phi}(r) = \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\varphi \, \Phi(x)$ and let $\Phi(x)$ be the function
$\Phi(x) \in S_{2m}^+(\mathbb{R}^3)$, where by $S_{2m}^+(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with
compact support such that:

$$
(1) \operatorname{supp} (\Phi(x)) \subset K^+ = \{x \mid \|x\| \ge 2m\}, \quad (2) \ \hat{\Phi}(r) \in C^\infty(\mathbb{R}).
$$

Then for any function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$ we get:

$$
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{1})_{\epsilon} \Phi(x) d^3 x =
$$

$$
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{0})_{\epsilon} \Phi(x) d^3 x = -\epsilon_1 m \hat{\Phi}(2m), (94)
$$

$\epsilon_1 > 0.$

$$
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{1})_{\epsilon} \Phi(x) d^3 x =
$$

$$
\lim_{\epsilon \to 0} \int_K ([\bar{R}_{\epsilon}^{+}(r)]_{0})_{\epsilon} \Phi(x) d^3 x = 0. \quad (95)
$$

Let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$, where by $S_{2m}^{-}(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with compact support such that:

$$
(1) \supp (\Phi(x)) \subset K^{-} = \{x | 0 \le \|x\| < 2m\}, (2) \hat{\Phi}(r) \in C^{\infty}(\mathbb{R}).
$$

Then for any function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$ from Eq. (85)-Eq. (86) similarly as above we
obtain

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^{3}x = $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^{3}x = \epsilon_{2} m \hat{\Phi}(2m), \quad (96) \\ \epsilon_{2} > 0. $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{2}^{2})_{\epsilon} \Phi(x) d^{3}x = $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{3}^{3})_{\epsilon} \Phi(x) d^{3}x = 0. \quad (97) $$

Let $(\tilde{t}, \tilde{r}, \tilde{\theta}, \tilde{\phi})$ be the generalized Colombeau coordinates on $\tilde{\mathbb{R}}_{\delta}^{4}$, with $(\tilde{r}, \tilde{\theta}, \tilde{\phi})$ the usual spherical polar coordinates on $\tilde{\mathbb{R}}_{\delta}^{3}$. Here we abbreviate for short $\tilde{t} = (t_{\epsilon})_{\epsilon}$, $\tilde{r} = (r_{\epsilon})_{\epsilon}$, $\tilde{\theta} = (\theta_{\epsilon})_{\epsilon}$, $\tilde{\phi} = (\varphi_{\epsilon})_{\epsilon}$. This meant that classical Schwarzschild spacetime extending up to distributional semi Riemannian manifold $(\tilde{\mathcal{S}}\tilde{h}, (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon})$, where $\tilde{\mathcal{S}}\tilde{h} = \mathbb{R}_{\delta} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbf{S}_{\delta}^{2})$, endowed on the tangent bundle with Colombeau distributional metric tensor $(g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \triangleq (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in (0, \delta]$, $\delta < 1$, such that $(g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in G_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4}), i, j = 0,1,2,3$.

**Remark 19.** Note that the distributional semi Riemannian manifold $(\tilde{\mathcal{S}}\tilde{h}, (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon})$ is endowed point free geometry$^{41,42,43}$ in natural way, since $(g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in G_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4}), i, j = 0,1,2,3$.

Thus, Colombeau super generalized object $\{(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}, (d\tilde{s}_{\epsilon}^{-2})_{\epsilon}\}$ modeling the singular Schwarzschild metric above (below) the horizon reads

$$ (d\tilde{s}_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r_{\epsilon})dt_{\epsilon}^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r_{\epsilon})]^{-1}dr_{\epsilon}^{2})_{\epsilon} + (r_{\epsilon}^{2}d\Omega_{\epsilon}^{2})_{\epsilon} $$

where

$$ h_{\epsilon}^{+}(r_{\epsilon}) = - \frac{\sqrt{(r_{\epsilon}-2m)^{2}+\epsilon^{2}}}{r_{\epsilon}}, \quad (98) $$

$$ (d\tilde{s}_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r_{\epsilon})dt_{\epsilon}^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r_{\epsilon})]^{-1}dr_{\epsilon}^{2})_{\epsilon} + (r_{\epsilon}^{2}d\Omega_{\epsilon}^{2})_{\epsilon}, $$

where

$$ h_{\epsilon}^{-}(r_{\epsilon}) = \frac{\sqrt{(2m-r_{\epsilon})^{2}+\epsilon^{2}}}{r_{\epsilon}}. \quad (99) $$

Note that $(h_{\epsilon}^{+}(r_{\epsilon}))_{\epsilon}, (h_{\epsilon}^{-}(r_{\epsilon}))_{\epsilon} \in G_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4})$.

The Colombeau generalized Ricci tensor corresponding to Colombeau generalized object $(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}$ above the horizon $([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{\alpha}^{\beta})_{\epsilon}$ reads

$$ ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{0}^{0})_{\epsilon} = ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{1}^{1})_{\epsilon} = $$

$$ \frac{1}{2}((h_{\epsilon}^{++}(r_{\epsilon}))_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{++'}(r_{\epsilon}))_{\epsilon}) - $$
obtain

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^{3}x = $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^{3}x = \epsilon_{2} m \hat{\Phi}(2m), \quad (96) $$

$\epsilon_2 > 0.$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{2}^{2})_{\epsilon} \Phi(x) d^{3}x = $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{3}^{3})_{\epsilon} \Phi(x) d^{3}x = 0. \quad (97) $$

Let $(\tilde{t}, \tilde{r}, \tilde{\theta}, \tilde{\phi})$ be the generalized Colombeau coordinates on $\tilde{\mathbb{R}}_{\delta}^{4}$, with $(\tilde{r}, \tilde{\theta}, \tilde{\phi})$ the usual spherical polar coordinates on $\tilde{\mathbb{R}}_{\delta}^{3}$. Here we abbreviate for short $\tilde{t} = (t_{\epsilon})_{\epsilon}$, $\tilde{r} = (r_{\epsilon})_{\epsilon}$, $\tilde{\theta} = (\theta_{\epsilon})_{\epsilon}$, $\tilde{\phi} = (\varphi_{\epsilon})_{\epsilon}$. This meant that classical Schwarzschild spacetime extending up to distributional semi Riemannian manifold $(\tilde{\mathcal{S}}\tilde{h},$

$$ (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \), \text{ where } \tilde{\mathcal{S}}\tilde{h} = \mathbb{R}_{\delta} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbf{S}_{\delta}^{2}), $$

endowed on the tangent bundle with Colombeau distributional metric tensor

$$ (g_{ij,\epsilon}^{Sch})_{\epsilon} \triangleq (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon}, \epsilon \in (0,\delta], \delta < 1, \text{ such that} $$

$$ (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in G_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4}), i, j = 0,1,2,3. $$

Remark 19. Note that the distributional semi Riemannian manifold

$$ (\tilde{\mathcal{S}}\tilde{h}, (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon}) \text{ is endowed point free geometry}^{41,42,43} \text{ in natural way, since } (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in G_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4}), i, j = 0,1,2,3. $$

Thus, Colombeau super generalized object $\{(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}, (d\tilde{s}_{\epsilon}^{-2})_{\epsilon}\}$ modeling the singular Schwarzschild metric above (below) the horizon reads

$$ (d\tilde{s}_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r_{\epsilon})dt_{\epsilon}^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r_{\epsilon})]^{-1}dr_{\epsilon}^{2})_{\epsilon} + (r_{\epsilon}^{2}d\Omega_{\epsilon}^{2})_{\epsilon} $$

where

$$ h_{\epsilon}^{+}(r_{\epsilon}) = - \frac{\sqrt{(r_{\epsilon}-2m)^{2}+\epsilon^{2}}}{r_{\epsilon}}, \quad (98) $$

$$ (d\tilde{s}_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r_{\epsilon})dt_{\epsilon}^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r_{\epsilon})]^{-1}dr_{\epsilon}^{2})_{\epsilon} + (r_{\epsilon}^{2}d\Omega_{\epsilon}^{2})_{\epsilon}, $$

where

$$ h_{\epsilon}^{-}(r_{\epsilon}) = \frac{\sqrt{(2m-r_{\epsilon})^{2}+\epsilon^{2}}}{r_{\epsilon}}. \quad (99) $$

$$ \text{Note that } (h_{\epsilon}^{+}(r_{\epsilon}))_{\epsilon}, (h_{\epsilon}^{-}(r_{\epsilon}))_{\epsilon} \in G_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4}). $$

The Colombeau generalized Ricci tensor corresponding to Colombeau generalized object $(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}$ above the horizon $([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{\alpha}^{\beta})_{\epsilon}$ reads

$$ ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{0}^{0})_{\epsilon} = ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{1}^{1})_{\epsilon} = $$

$$ \frac{1}{2}((h_{\epsilon}^{++}(r_{\epsilon}))_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{++'}(r_{\epsilon}))_{\epsilon}) - $$
$$ -\frac{(\varepsilon)_{\varepsilon}}{2m} (\delta'(r_{\varepsilon} - 2m; \varepsilon))_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{rm}\right] (\delta(r_{\varepsilon} - 2m; \varepsilon))_{\varepsilon} \quad (100) $$

And

$$ \begin{aligned} \left(\left[\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})\right]_{2}^{2}\right)_{\varepsilon} &= \left(\left[\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})\right]_{3}^{3}\right)_{\varepsilon} = \\ & \frac{(h_{\varepsilon}^{+}(r_{\varepsilon}))_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}} + \frac{1+(h_{\varepsilon}^{+}(r_{\varepsilon}))_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}^{2}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} (\delta(r_{\varepsilon} - 2m; \varepsilon))_{\varepsilon} \quad (101) \end{aligned} $$

From Eq. (100) and Eq. (89)-Eq. (90) we obtain

$$ \begin{aligned} \left(\left[\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})\right]_{0}^{0}\right)_{\varepsilon} &= \left(\left[\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})\right]_{1}^{1}\right)_{\varepsilon} = \\ & -\frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{1}{\left[(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}\right]^{1/2}}\right)_{\varepsilon} + \\ & \frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{(r_{\varepsilon}-2m)^{2}}{\left[(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}\right]^{3/2}}\right)_{\varepsilon} - \\ & -\frac{(\varepsilon)_{\varepsilon}}{2m}(\delta'(r_{\varepsilon}-2m; \varepsilon))_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^{2}} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m}\right](\delta(r_{\varepsilon}-2m; \varepsilon))_{\varepsilon} = \\ & -\frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{1}{\left[(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}\right]^{1/2}}\right)_{\varepsilon} + \\ & \frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{(r_{\varepsilon}-2m)^{2}}{\left[(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}\right]^{3/2}}\right)_{\varepsilon} - \\ & -\frac{(\varepsilon)_{\varepsilon}}{2m}\left(\frac{4\pi^{-1}\varepsilon(r_{\varepsilon}-2m)}{\left[(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}\right]^{2}}\right)_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^{2}} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m}\right]\left(\frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}}\right)_{\varepsilon}. \quad (102) \end{aligned} $$

For any $(r_\epsilon)_\epsilon$ such that $(r_\epsilon)_\epsilon - 2m \neq 0_{\mathbb{R}}$ from Eq. (102) one obtains

$$ \begin{aligned} \left(\left[\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})\right]_{0}^{0}\right)_{\varepsilon} &= \left(\left[\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})\right]_{1}^{1}\right)_{\varepsilon} = \\ & -\frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}\left[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^{2})_{\varepsilon}\right]^{1/2}} + \\ & \frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}\left[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^{2})_{\varepsilon}\right]^{3/2}} - \\ & -\frac{(\varepsilon)_{\varepsilon}}{2m((r_{\varepsilon}-2m)^{3})_{\varepsilon}}\left(\frac{4\pi^{-1}\varepsilon}{\left[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}\varepsilon^{2}\right]^{2}}\right)_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^{2}} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m}\right]\left(\frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}}\right)_{\varepsilon}. \quad (103) \end{aligned} $$

For any $(r_\epsilon)_\epsilon$ such that $(r_\epsilon)_\epsilon - 2m \in \mathcal{F}(\mathbb{R}_\delta)$ and $\delta[(r_\epsilon)_\epsilon - 2m]^{-2} < 1$ from Eq. (103)
$$ -\frac{(\varepsilon)_{\varepsilon}}{2m} (\delta'(r_{\varepsilon} - 2m; \varepsilon))_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{rm}\right] (\delta(r_{\varepsilon} - 2m; \varepsilon))_{\varepsilon} \quad (100) $$

And

$$ \begin{aligned} ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} &= ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = \\ & \frac{(h_{\varepsilon}^{+}(r_{\varepsilon}))_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}} + \frac{1+(h_{\varepsilon}^{+}(r_{\varepsilon}))_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}^{2}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} (\delta(r_{\varepsilon} - 2m; \varepsilon))_{\varepsilon} \quad (101) \end{aligned} $$

From Eq. (100) and Eq. (89)-Eq. (90) we obtain

$$ \begin{aligned} ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} &= ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} = \\ & -\frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{1}{[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{1/2}}\right)_{\varepsilon} + \\ & \frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{(r_{\varepsilon}-2m)^2}{[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{3/2}}\right)_{\varepsilon} - \\ & -\frac{(\varepsilon)_{\varepsilon}}{2m}(\delta'(r_{\varepsilon}-2m;\varepsilon))_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m}\right](\delta(r_{\varepsilon}-2m;\varepsilon))_{\varepsilon} = \\ & -\frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{1}{[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{1/2}}\right)_{\varepsilon} + \\ & \frac{1}{2(r_{\varepsilon})_{\varepsilon}}\left(\frac{(r_{\varepsilon}-2m)^2}{[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{3/2}}\right)_{\varepsilon} - \\ & -\frac{(\varepsilon)_{\varepsilon}}{2m}\left(\frac{4\pi^{-1}\varepsilon(r_{\varepsilon}-2m)}{[(r_{\varepsilon}-2m)^2+\varepsilon^2]^2}\right)_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m}\right]\left(\frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^2+\varepsilon^2}\right)_{\varepsilon}. \quad (102) \end{aligned} $$

For any $(r_\epsilon)_\epsilon$ such that $(r_\epsilon)_\epsilon - 2m \neq 0_{\mathbb{R}}$ from Eq. (102) one obtains

$$ \begin{aligned} ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} &= ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} = \\ & -\frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^2)_{\varepsilon}]^{1/2}} + \\ & \frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^2)_{\varepsilon}]^{3/2}} - \\ & -\frac{(\varepsilon)_{\varepsilon}}{2m((r_{\varepsilon}-2m)^3)_{\varepsilon}}\left(\frac{4\pi^{-1}\varepsilon}{[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}\varepsilon^2]^{2}}\right)_{\varepsilon} + \left[\frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m}\right]\left(\frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^2+\varepsilon^2}\right)_{\varepsilon}. \quad (103) \end{aligned} $$

For any $(r_\epsilon)_\epsilon$ such that $(r_\epsilon)_\epsilon - 2m \in \mathcal{F}(\mathbb{R}_\delta)$ and $\delta[(r_\epsilon)_\epsilon - 2m]^{-2} < 1$ from Eq. (103)
one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} \approx (\varepsilon^{2})_{\varepsilon} \quad (104) $$

and therefore

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} \approx 0, \quad (105) $$

since $(\varepsilon^2)_{\varepsilon} \approx 0$.

It follows from Eq. (104) for any $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\mathbb{R}_{\delta})$

such that $\delta^2[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$:

$$ \mathrm{st}(( [\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} ) = \mathrm{st}(( [\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} ) = 0. \quad (106) $$

Thus for any $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\mathbb{R}_{\delta})$ such that $\delta[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$ the nonlinear distributional Schwarzschild geometry returns the classical result.

For any $(r_{\varepsilon})_{\varepsilon}$ such that $((r_{\varepsilon})_{\varepsilon} - 2m) = 0_{\mathbb{R}}$ from Eq. (102) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(2m)]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(2m)]_{1}^{1})_{\varepsilon} = -\frac{(\varepsilon^{-1})_{\varepsilon}}{2m} \approx -\infty. \quad (107) $$

From Eq. (101) and Eq. (89) we obtain

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = -\frac{(r_{\varepsilon})_{\varepsilon}-2m}{(r_{\varepsilon}^{2})_{\varepsilon}[( (r_{\varepsilon})_{\varepsilon}-2m)^{2}+(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon}. \quad (108) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $(r_{\varepsilon})_{\varepsilon} - 2m \neq 0_{\mathbb{R}}$

from Eq. (108) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = -\frac{(r_{\varepsilon})_{\varepsilon}-2m}{(r_{\varepsilon}^{2})_{\varepsilon}((r_{\varepsilon})_{\varepsilon}-2m)[1+((r_{\varepsilon})_{\varepsilon}-2m)^{-2}(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon} = -\frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}[1+((r_{\varepsilon})_{\varepsilon}-2m)^{-2}(\varepsilon^{2})_{\varepsilon}]} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon} \quad (109) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\mathbb{R}_{\delta})$ and

$\delta^2[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$ from Eq. (109) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} \approx ((r_{\varepsilon})_{\varepsilon} - 2m)^{-2}(\varepsilon^{2})_{\varepsilon} \approx 0. \quad (110) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $((r_{\varepsilon})_{\varepsilon} - 2m) = 0_{\mathbb{R}}$

from Eq. (108) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(2m)]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(2m)]_{3}^{3})_{\varepsilon} = O(m^{-2}). \quad (111) $$

The distributional square scalar $(R^{+\mu\nu}(r_{\varepsilon}, \varepsilon)R_{\mu\nu}^{+}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ above the horizon reads

$$ (R^{\mu\nu}(r_{\varepsilon}, \varepsilon)R_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon} = 2\left(\left[\frac{1}{r_{\varepsilon}}\bar{h}_{\varepsilon}^{+'} + \frac{-h_{\varepsilon}^{+}+1}{r_{\varepsilon}^{2}}\right]_{\varepsilon}^{2} + 2\left(\left[\frac{1}{2}\bar{h}_{\varepsilon}^{+''} + \frac{1}{r_{\varepsilon}}\bar{h}_{\varepsilon}^{+'}\right]_{\varepsilon}^{2}\right)\right) = $$
one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} \approx (\varepsilon^{2})_{\varepsilon} \quad (104) $$

and therefore

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} \approx 0, \quad (105) $$

since $(\varepsilon^2)_{\varepsilon} \approx 0$.

It follows from Eq. (104) for any $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\mathbb{R}_{\delta})$

such that $\delta^2[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$:

$$ \mathrm{st}(( [\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} ) = \mathrm{st}(( [\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} ) = 0. \quad (106) $$

Thus for any $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\mathbb{R}_{\delta})$ such that $\delta[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$ the nonlinear distributional Schwarzschild geometry returns the classical result.

For any $(r_{\varepsilon})_{\varepsilon}$ such that $((r_{\varepsilon})_{\varepsilon} - 2m) = 0_{\mathbb{R}}$ from Eq. (102) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(2m)]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(2m)]_{1}^{1})_{\varepsilon} = -\frac{(\varepsilon^{-1})_{\varepsilon}}{2m} \approx -\infty. \quad (107) $$

From Eq. (101) and Eq. (89) we obtain

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = -\frac{(r_{\varepsilon})_{\varepsilon}-2m}{(r_{\varepsilon}^{2})_{\varepsilon}[( (r_{\varepsilon})_{\varepsilon}-2m)^{2}+(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon}. \quad (108) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $(r_{\varepsilon})_{\varepsilon} - 2m \neq 0_{\mathbb{R}}$

from Eq. (108) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = -\frac{(r_{\varepsilon})_{\varepsilon}-2m}{(r_{\varepsilon}^{2})_{\varepsilon}((r_{\varepsilon})_{\varepsilon}-2m)[1+((r_{\varepsilon})_{\varepsilon}-2m)^{-2}(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon} = -\frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}[1+((r_{\varepsilon})_{\varepsilon}-2m)^{-2}(\varepsilon^{2})_{\varepsilon}]} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon} \quad (109) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\mathbb{R}_{\delta})$ and

$\delta^2[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$ from Eq. (109) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} \approx ((r_{\varepsilon})_{\varepsilon} - 2m)^{-2}(\varepsilon^{2})_{\varepsilon} \approx 0. \quad (110) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $((r_{\varepsilon})_{\varepsilon} - 2m) = 0_{\mathbb{R}}$

from Eq. (108) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(2m)]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(2m)]_{3}^{3})_{\varepsilon} = O(m^{-2}). \quad (111) $$

The distributional square scalar $(R^{+\mu\nu}(r_{\varepsilon}, \varepsilon)R_{\mu\nu}^{+}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ above the horizon reads

$$ (R^{\mu\nu}(r_{\varepsilon}, \varepsilon)R_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon} = 2\left(\left[\frac{1}{r_{\varepsilon}}\bar{h}_{\varepsilon}^{+'} + \frac{-h_{\varepsilon}^{+}+1}{r_{\varepsilon}^{2}}\right]_{\varepsilon}^{2} + 2\left(\left[\frac{1}{2}\bar{h}_{\varepsilon}^{+''} + \frac{1}{r_{\varepsilon}}\bar{h}_{\varepsilon}^{+'}\right]_{\varepsilon}^{2}\right)\right) = $$
$$2 \left( \left[ \left( \frac{1}{r_{\varepsilon}^3} \sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2} + \frac{1}{r_{\varepsilon}^2} \right)_\varepsilon - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_\varepsilon + \\ 2 \left( \left[ \frac{4m(r_{\varepsilon} - 2m)^3 + (r_{\varepsilon} - 4m)\varepsilon^2}{r_{\varepsilon}^3[(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_\varepsilon + \dots \quad (112)$$

From Eq. (112) at horizon $(r_\varepsilon)_\varepsilon = 2m$ follows that

$$(R^{+\mu\nu}(2m, \varepsilon) R_{\mu\nu}^{+}(2m, \varepsilon))_{\varepsilon} = K(2m) + \left(\frac{1}{4m^4\varepsilon^2}\right)_{\varepsilon} \approx \infty, \quad (113)$$

where $K(r)$ is the Kretschman scalar. The distributional square scalar $(R^{+\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon) R_{\rho\sigma\mu\nu}^{+}(r_\varepsilon, \varepsilon))_\varepsilon$ above the horizon reads

$$(R^{+\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon) R_{\rho\sigma\mu\nu}^{+}(r_\varepsilon, \varepsilon))_\varepsilon = \left( (\bar{h}_\varepsilon^{+''})^2 + 2 \left( \frac{\bar{h}_\varepsilon^{+'}}{r_\varepsilon} \right)^2 \right)_\varepsilon + 4 \left( \left[ \frac{1}{r_\varepsilon^2} (1 - h_\varepsilon^+) \right]^2 \right)_\varepsilon + \\ 2 \left( \left[ \frac{\bar{h}_\varepsilon^{+'}}{r_\varepsilon} \right]^2 \right)_\varepsilon = \\ (\bar{h}_\varepsilon^{+''} + 4 \frac{\bar{h}_\varepsilon^{+2}}{r_\varepsilon^2})_\varepsilon + 4 \left( \frac{1}{r_\varepsilon^4} (1 - h_\varepsilon^+)^2 \right)_\varepsilon = \\ \left( \left[ \frac{4m(r_\varepsilon - 2m)^3 + (r_\varepsilon - 4m)\varepsilon^2}{r_\varepsilon^3[(r_\varepsilon - 2m)^2 + \varepsilon^2]^{3/2}} \right]^2 \right)_\varepsilon - \frac{8m^2(r_\varepsilon - 2m)^2}{r_\varepsilon^6[(r_\varepsilon - 2m)^2 + \varepsilon^2]} + \\ \frac{4}{r_\varepsilon^4} \left( 1 + r_\varepsilon^{-1} \sqrt{(r_\varepsilon - 2m)^2 + \varepsilon^2} \right)^2 + \dots \quad (114)$$

From Eq. (114) at horizon $(r_\varepsilon)_\varepsilon = 2m$ follows that:

$$(R^{\rho\sigma\mu\nu}(2m, \varepsilon) R_{\rho\sigma\mu\nu}(2m, \varepsilon))_{\varepsilon} = K(r_{\varepsilon}) + \left. \left( \frac{\varepsilon^4}{4m^4[\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} \right|_{(r_{\varepsilon})_{\varepsilon}=2m} = K(2m) + (\varepsilon^{-2})_{\varepsilon}. \quad (115)$$

**The truncated distributional Schwarzschild geometry.**

Colombeau super generalized object $\{(d\tilde{s}_\varepsilon^{+2})_\varepsilon, (d\tilde{s}_\varepsilon^{-2})_\varepsilon\}$ modeling the singular Schwarzschild metric truncated below the horizon reads

$$(d\tilde{s}_\varepsilon^{+2})_\varepsilon = (h_\varepsilon^+(r_\varepsilon) dt_\varepsilon^2)_\varepsilon - ([h_\varepsilon^+(r_\varepsilon)]^{-1} dr_\varepsilon^2)_\varepsilon + (r_\varepsilon^2 d\Omega_\varepsilon^2)_\varepsilon \quad (116)$$

where

$$h_{\varepsilon}^{+}(r_{\varepsilon}) = - \frac{\sqrt{(r_{\varepsilon}-2m)^2 + \varepsilon^2}}{r_{\varepsilon}}, \quad (117)$$

$$(d\tilde{s}_\varepsilon^{-2})_\varepsilon = (h_\varepsilon^-(r_\varepsilon) dt_\varepsilon^2)_\varepsilon - ([h_\varepsilon^-(r_\varepsilon)]^{-1} dr_\varepsilon^2)_\varepsilon + (r_\varepsilon^2 d\Omega_\varepsilon^2)_\varepsilon, \\ \text{and where} \\ h_\varepsilon^-(r_\varepsilon) = \varepsilon. \quad (118)$$
$$2 \left( \left[ \left( \frac{1}{r_{\varepsilon}^3} \sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2} + \frac{1}{r_{\varepsilon}^2} \right)_\varepsilon - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_\varepsilon + 2 \left( \left[ \frac{4m(r_{\varepsilon} - 2m)^3 + (r_{\varepsilon} - 4m)\varepsilon^2}{r_{\varepsilon}^3[(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_\varepsilon + \dots \quad (112)$$

From Eq. (112) at horizon $(r_\varepsilon)_\varepsilon = 2m$ follows that

$$(R^{+\mu\nu}(2m, \varepsilon) R_{\mu\nu}^{+}(2m, \varepsilon))_{\varepsilon} = K(2m) + \left(\frac{1}{4m^4\varepsilon^2}\right)_{\varepsilon} \approx \infty, \quad (113)$$

where $K(r)$ is the Kretschman scalar. The distributional square scalar $(R^{+\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon) R_{\rho\sigma\mu\nu}^{+}(r_\varepsilon, \varepsilon))_\varepsilon$ above the horizon reads

$$\begin{aligned} (R^{+\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon) R_{\rho\sigma\mu\nu}^{+}(r_\varepsilon, \varepsilon))_\varepsilon = & \left( (\bar{h}_\varepsilon^{+''})^2 + 2 \left(\frac{\bar{h}_\varepsilon^{+'}}{r_\varepsilon}\right)^2 \right)_\varepsilon + 4 \left( \left[ \frac{1}{r_\varepsilon^2} (1 - h_\varepsilon^+) \right]^2 \right)_\varepsilon + \\ & 2 \left( \left[ \frac{\bar{h}_\varepsilon^{+'}}{r_\varepsilon} \right]^2 \right)_\varepsilon = \\ & (\bar{h}_\varepsilon^{+''} + 4 \frac{\bar{h}_\varepsilon^{+''}}{r_\varepsilon^2})_\varepsilon + 4 \left( \frac{1}{r_\varepsilon^4} (1 - h_\varepsilon^+)^2 \right)_\varepsilon = \\ & \left( \left[ \frac{4m(r_\varepsilon - 2m)^3 + (r_\varepsilon - 4m)\varepsilon^2}{r_\varepsilon^3[(r_\varepsilon - 2m)^2 + \varepsilon^2]^{3/2}} \right]^2 \right)_\varepsilon - \frac{8m^2(r_\varepsilon - 2m)^2}{r_\varepsilon^6[(r_\varepsilon - 2m)^2 + \varepsilon^2]} + \\ & \frac{4}{r_\varepsilon^4} \left( 1 + r_\varepsilon^{-1} \sqrt{(r_\varepsilon - 2m)^2 + \varepsilon^2} \right)^2 + \dots . \end{aligned} \quad (114)$$

From Eq. (114) at horizon $(r_\varepsilon)_\varepsilon = 2m$ follows that:

$$\begin{aligned} (R^{\rho\sigma\mu\nu}(2m, \varepsilon) R_{\rho\sigma\mu\nu}(2m, \varepsilon))_{\varepsilon} = & K(r_{\varepsilon}) + \left( \frac{\varepsilon^4}{4m^4[\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} \Big|_{(r_{\varepsilon})_{\varepsilon}=2m} = \\ & K(2m) + (\varepsilon^{-2})_{\varepsilon}. \end{aligned} \quad (115)$$

**The truncated distributional Schwarzschild geometry.**

Colombeau super generalized object $\{(d\tilde{s}_\varepsilon^{+2})_\varepsilon, (d\tilde{s}_\varepsilon^{-2})_\varepsilon\}$ modeling the singular Schwarzschild metric truncated below the horizon reads

$$(d\tilde{s}_\varepsilon^{+2})_\varepsilon = (h_\varepsilon^+(r_\varepsilon)dt_\varepsilon^2)_\varepsilon - ([h_\varepsilon^+(r_\varepsilon)]^{-1}dr_\varepsilon^2)_\varepsilon + (r_\varepsilon^2d\Omega_\varepsilon^2)_\varepsilon \quad (116)$$

where

$$h_{\varepsilon}^{+}(r_{\varepsilon}) = - \frac{\sqrt{(r_{\varepsilon} - 2m)^2 + \varepsilon^2}}{r_{\varepsilon}}, \quad (117)$$

$$(d\tilde{s}_\varepsilon^{-2})_\varepsilon = (h_\varepsilon^-(r_\varepsilon)dt_\varepsilon^2)_\varepsilon - ([h_\varepsilon^-(r_\varepsilon)]^{-1}dr_\varepsilon^2)_\varepsilon + (r_\varepsilon^2d\Omega_\varepsilon^2)_\varepsilon, \\ \text{and where} \\ h_\varepsilon^-(r_\varepsilon) = \varepsilon. \quad (118)$$
By super generalized object $\{(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}, (d\tilde{s}_{\epsilon}^{-2})_{\epsilon}\}$ non-spinning gravitational singularity presented, see Fig.1. Similarly, as above one obtains that the distributional Ricci tensor and the curvature scalar at horizon are of $\delta$-type, i.e.

$$ (R_{0,\epsilon}^{0}(r))_{\epsilon} = (R_{1,\epsilon}^{1}(r))_{\epsilon} = -\mu m \delta(r - 2m), \mu > 0 \\ (R_{2,\epsilon}^{2}(r))_{\epsilon} = (R_{3,\epsilon}^{3}(r))_{\epsilon} = 0. \quad (119) $$

$$ (R_{\epsilon}(r))_{\epsilon} \approx -2\mu m \delta(r - 2m). \quad (120) $$

Remark 20. It follows from Eq. (120) and from the generalized Einstein fields equations that truncated distributional Schwarzschild spacetime has a distributional source with support on horizon $\{r = 2m\}$.

<figure><img src="image_1.png" /><figcaption>Fig.1. Non-spinning gravitational singularity.</figcaption></figure>

## V. CONCLUSIONS

In this we argue that canonical interpretation of the Schwarzschild spacetime in contemporary general relativity wrong and revision is needed. We studied Schwarzschild solution using Colombeau distributional geometry, thus without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$. We argue that Schwarzschild solution is impossible to treat classically since, the Levi-Civita connection is not available on whole Schwarzschild spacetime $(\mathbf{Sch}, g_{ij}^{Sch}(t, r, \theta, \varphi))$, where $\mathbf{Sch} = \mathbb{R} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbb{S}^2)$, but only by using an embedding of the classical Schwarzschild metric tensor $g_{ij}^{Sch}$; $i, j = 1,2,3,4$ into Colombeau algebra $G_{\delta}(\mathbb{R}^4, \Sigma)$, $\Sigma = \{r = 2m\} \cup \{r = 0\}$ super generalized functions. This meant that classical Schwarzschild spacetime extending up to distributional semi Riemannian manifold $(\mathbf{Sch}, (g_{ij}^{Sch}(t, r, \theta, \varphi, \epsilon))_{\epsilon})$ endowed on the tangent bundle with Colombeau distributional metric tensor $(g_{ij,\epsilon}^{Sch})_{\epsilon} \triangleq (g_{ij}^{Sch}(t, r, \theta, \varphi, \epsilon))_{\epsilon}, \epsilon \in (0,\delta]$, $(g_{ij,\epsilon}^{Sch})_{\epsilon} \in G_{\delta}(\mathbb{R}^4, \Sigma)$. We obtain the distributional Ricci tensor and the curvature scalar are of $\delta$-type, i.e., $(R_{\epsilon}(r))_{\epsilon} = -\epsilon m \delta(r - 2m), \epsilon > 0$. However the distributional square scalars are essentially non classical Colombeau type distributions: $(R_{\epsilon}^{\mu\nu}(r)R_{\mu\nu,\epsilon}(r))_{\epsilon}$, $(R_{\epsilon}^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\epsilon}(r))_{\epsilon} \in G(\mathbb{R}^3)\backslash\mathcal{D}'(\mathbb{R}^3)$. This provides a new physical interpretation for
By super generalized object $\{(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}, (d\tilde{s}_{\epsilon}^{-2})_{\epsilon}\}$ non-spinning gravitational singularity presented, see Fig.1. Similarly, as above one obtains that the distributional Ricci tensor and the curvature scalar at horizon are of $\delta$-type, i.e.

$$ (R_{0,\epsilon}^{0}(r))_{\epsilon} = (R_{1,\epsilon}^{1}(r))_{\epsilon} = -\mu m \delta(r - 2m), \mu > 0 \\ (R_{2,\epsilon}^{2}(r))_{\epsilon} = (R_{3,\epsilon}^{3}(r))_{\epsilon} = 0. \quad (119) $$

$$ (R_{\epsilon}(r))_{\epsilon} \approx -2\mu m \delta(r - 2m). \quad (120) $$

Remark 20. It follows from Eq. (120) and from the generalized Einstein fields equations that truncated distributional Schwarzschild spacetime has a distributional source with support on horizon $\{r = 2m\}$.

<figure><img src="image_2.png" /><figcaption>Fig.1. Non-spinning gravitational singularity.</figcaption></figure>

## V. CONCLUSIONS

In this we argue that canonical interpretation of the Schwarzschild spacetime in contemporary general relativity wrong and revision is needed. We studied Schwarzschild solution using Colombeau distributional geometry, thus without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$. We argue that Schwarzschild solution is impossible to treat classically since, the Levi-Civita connection is not available on whole Schwarzschild spacetime $(\mathbf{Sch}, g_{ij}^{Sch}(t, r, \theta, \varphi))$, where $\mathbf{Sch} = \mathbb{R} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbb{S}^2)$, but only by using an embedding of the classical Schwarzschild metric tensor $g_{ij}^{Sch}$; $i, j = 1,2,3,4$ into Colombeau algebra $G_{\delta}(\mathbb{R}^4, \Sigma)$, $\Sigma = \{r = 2m\} \cup \{r = 0\}$ super generalized functions. This meant that classical Schwarzschild spacetime extending up to distributional semi Riemannian manifold $(\mathbf{Sch}, (g_{ij}^{Sch}(t, r, \theta, \varphi, \epsilon))_{\epsilon})$ endowed on the tangent bundle with Colombeau distributional metric tensor $(g_{ij,\epsilon}^{Sch})_{\epsilon} \triangleq (g_{ij}^{Sch}(t, r, \theta, \varphi, \epsilon))_{\epsilon}, \epsilon \in (0,\delta]$, $(g_{ij,\epsilon}^{Sch})_{\epsilon} \in G_{\delta}(\mathbb{R}^4, \Sigma)$. We obtain the distributional Ricci tensor and the curvature scalar are of $\delta$-type, i.e., $(R_{\epsilon}(r))_{\epsilon} = -\epsilon m \delta(r - 2m), \epsilon > 0$. However the distributional square scalars are essentially non classical Colombeau type distributions: $(R_{\epsilon}^{\mu\nu}(r)R_{\mu\nu,\epsilon}(r))_{\epsilon}$, $(R_{\epsilon}^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\epsilon}(r))_{\epsilon} \in G(\mathbb{R}^3)\backslash\mathcal{D}'(\mathbb{R}^3)$. This provides a new physical interpretation for
the distributional curvature scalar $(R_\epsilon(r)))_\epsilon$ and square scalars $(R_\epsilon^{\mu\nu}(r)R_{\mu\nu,\epsilon}(r))_\epsilon$, $((R_\epsilon^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\epsilon}(r)))_\epsilon$ of this geometry$^{14,15}$.
the distributional curvature scalar $(R_\epsilon(r)))_\epsilon$ and square scalars $(R_\epsilon^{\mu\nu}(r)R_{\mu\nu,\epsilon}(r))_\epsilon$, $((R_\epsilon^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\epsilon}(r)))_\epsilon$ of this geometry$^{14,15}$.