# The point free Colombeau geometry in singular general relativity. Part II

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

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Jaykov Foukzon, Potapov Alexander A., Elena R Men'Kova. The point free Colombeau geometry in singular general relativity. Part II. 2023. hal-04129522

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# The point free Colombeau geometry in singular general relativity. Part II.

Jaykov Foukzon<fnref n="1" />, Alexander A. Potapov<fnref n="2" />,<fnref n="3" />, Elena R. Men'kova<fnref n="4" />

## Abstract

**The problem statement.** Let $(t, r, \theta, \varphi)$ be coordinates on $\mathbb{R}^4$, with $(r, \theta, \varphi)$ the usual spherical polar coordinates on $\mathbb{R}^3$. The Schwarzschild solution is well defined on the subset $\mathbb{R} \times ((\{r > 2m\} \cup \{0 < r < 2m\}) \times \mathbf{S}^2)$ of $\mathbb{R}^4$. In this region, the metric line element takes the form

$$ds^2 = -\left(1 - \frac{2m_s}{r}\right)(dt)^2 + \left(1 - \frac{2m}{r}\right)^{-1}(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta(d\varphi)^2].$$

We studied the Schwarzschild solution using Colombeau distributional geometry without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$. We left the neighbourhood of the singularity at the origin and turn to the singularity at the horizon.

**The aim.** The development of new physical interpretation for the distributional curvature scalar $(R_\varepsilon(r))_\varepsilon$ and square scalar $(R_\varepsilon^{\mu\nu}(r)R_{\mu\nu,\varepsilon}(r))_\varepsilon$, $(R_\varepsilon^{\rho\sigma\mu\nu}(r)R_{\rho\delta\mu\nu,\varepsilon}(r))_\varepsilon$ is aimed.

**Results.** The Schwarzschild solution using Colombeau distributional geometry without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$ is studied. We obtain that the distributional Ricci tensor and the curvature scalar are $\delta$-type on the horizon $(R_\varepsilon(r))_\varepsilon = -\epsilon m\delta(r - 2m)$, $\epsilon > 0$.

**The practical value.** A new physical interpretation of the distributional Schwarzschild geometry is presented [J. Foukzon, E. R. Men'kova, A. A. Potapov, and S. A. Podosenov,

<footnote n="1">¹Center for Mathematical Sciences, Israel Institute of Technology, http://www.technion.ac.il
Haifa 3200003, Israel</footnote>

<footnote n="2">² V. A. Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences. http://cplire.ru Moscow 125009, Russian Federation</footnote>

<footnote n="3">³Joint Chinese-Russian Laboratory of Informational Technologies and Signals Fractal Processing, Jinan University, Guangzhou, 510632, China</footnote>

<footnote n="4">⁴All-Russian Research Institute for Optical and Physical Measurements, http://www.vniiofi.ru
Moscow 119361, Russian Federation</footnote>

<footnote n="1">1 - jaykovfoukzon@list.ru, 2,3 - potapov@cplire.ru, 4 - e_menkova@mail.ru</footnote>
Journal of Physics: Conference Series 1141, 012100 (2018). https://doi: org/10.1088/1742-6596/1141/1/012100].

As distributional square scalars are essentially nonclassical Colombeau type distributions: $(R_{\epsilon}^{\mu\nu}(r)R_{\mu\nu,\epsilon}(r))_{\epsilon}$, $(R_{\epsilon}^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\epsilon}(r))_{\epsilon} \in G(\mathbb{R}^3)\backslash D'(\mathbb{R}^3)$, this provides a new physical interpretation for the distributional curvature scalar $(R_{\epsilon}(r))_{\epsilon}$ and square scalars $(R_{\epsilon}^{\mu\nu}(r)R_{\mu\nu,\epsilon}(r))_{\epsilon}$, $(R_{\epsilon}^{\rho\sigma\mu\nu}(r)R_{\rho\delta\mu\nu,\epsilon}(r))_{\epsilon}$.

Using nonlinear superdistributional geometry and supergeneralized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild.

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.

# 1. POINT FREE DISTRIBUTIONAL GEOMETRY IN SINGULAR GENERAL RELATIVITY

**Remark 1.1.** In the canonical Schwarzschild coordinates $(t, r, \theta, \varphi)$, where $r \neq a$, the Schwarzschild metric (1.1)

$$ds^2 = -(1 - \frac{a}{r}) (d\hat{x}^0)^2 + (1 - \frac{a}{r})^{-1} (d\hat{r})^2 + \hat{r}^2 [(d\theta)^2 + \sin^2\theta (d\varphi)^2]. \quad (1.1)$$

takes the following forms:

(1) above the horizon where $r > 2m$:

$$ds^{+2} = h^{+}(r)dt^{2} - [h^{+}(r)]^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (1.2)$$

$$h^{+}(r) = -1 + \frac{a}{r} = -\frac{r-a}{r}, \quad a = 2m;$$

and

(2) below the horizon where $r < 2m$:

$$ds^{-2} = h^{-}(r)dt^{2} - h^{-}(r)^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (1.3)$$

$$h^{-}(r) = -1 + \frac{a}{r} = \frac{a-r}{r}.$$
By examining the Schwarzschild metrics in equations (1.2) – (1.3) in the neighborhood of the horizon, we see that, whereas $h^+(r)$ and $h^-(r)$ are smooth, $[h^+(r)]^{-1}$ and $[h^-(r)]^{-1}$ are not even (note that the origin is now always excluded from our considerations, and 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 replacing the equations for above the horizon $r > 2m$ and below the horizon $r < 2m$ correspondingly with [1-3]

$$r \ge 2m : \quad r - 2m \mapsto \sqrt{(r - 2m)^2 + \epsilon^2}, \qquad (1.4)$$

$$r < 2m : \quad 2m - r \mapsto \sqrt{(2m - r)^2 + \epsilon^2}. \qquad (1.5)$$

By inserting Eq. (1.4) and Eq. (1.5) into Eq. (1.2) and Eq. (1.3), respectively, we obtain the Colombeau generalized object $\{(ds_\epsilon^{+2})_\epsilon, (ds_\epsilon^{-2})_\epsilon\}$ modeling the singular Schwarzschild metric above (below) the horizon, i.e.,

$$(ds_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ h_{\epsilon}^{+}(r) = -\frac{\sqrt{(r-2m)^{2} + \epsilon^{2}}}{r}, \qquad (1.6)$$

$$(ds_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ h_{\epsilon}^{-}(r) = \frac{\sqrt{(2m-r)^{2} + \epsilon^{2}}}{r}. \qquad (1.7)$$

The generalized Ricci tensor above the horizon $([R_\epsilon^+]_\alpha^\beta)_\epsilon$ may now be calculated componentwise using the classical formulas [1, 2]

$$([R_\epsilon^+](r))_0^0_\epsilon = ([R_\epsilon^+](r))_1^1_\epsilon = \frac{1}{2}((h_\epsilon^{++}(r))_\epsilon + \frac{2}{r}(h_\epsilon^{+\prime}(r))_\epsilon), \qquad (1.8)$$

$$([R_\epsilon^+](r))_2^2_\epsilon = ([R_\epsilon^+](r))_3^3_\epsilon = \frac{(h_\epsilon^{+\prime}(r))_\epsilon}{r} + \frac{1 + (h_\epsilon^+(r))_\epsilon}{r^2}, \qquad (1.9)$$

where

$$h_{\epsilon}^{+\prime}(r) = -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{\frac{1}{2}}} + \frac{[(r-2m)^2 + \epsilon^2]^{\frac{1}{2}}}{r^2}, \qquad (1.10)$$

$$h_{\epsilon}^{+\prime\prime}(r) = -\left(\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}}\right)' + \left(\frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2}\right)' = \\ -\frac{1}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} +$$
$$ \frac{r-2m}{r^2[(r-2m)^2 + \varepsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \varepsilon^2]^{\frac{1}{2}}}{r^3}. \quad (1.11) $$

From Eq. (1.8), Eq. (1.9), Eq. (1.10) and Eq. (1.11), we obtain

$$ (r^2[R_{\varepsilon}^{+}(r)]_2^2)_{\varepsilon} = (r^2[R_{\varepsilon}^{+}(r)]_3^3)_{\varepsilon} = r(h_{\varepsilon}^{++}(r))_{\varepsilon} + 1 + (h_{\varepsilon}^{+}(r))_{\varepsilon} = $$

$$ \begin{aligned} & r \left\{ -\frac{r-2m}{r[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \varepsilon^2]^{1/2}}{r^2} \right\} + 1 - \frac{\sqrt{(r-2m)^2 + \varepsilon^2}}{r} \\ &= -\frac{r-2m}{[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \varepsilon^2]^{1/2}}{r} + 1 - \frac{\sqrt{(r-2m)^2 + \varepsilon^2}}{r} = \\ & \quad -\frac{r-2m}{[(r-2m)^2 + \varepsilon^2]^{1/2}} + 1 \end{aligned} \quad (1.12) $$

and

$$ (2r^2[R_{\varepsilon}^{+}(r)]_0^0)_{\varepsilon} = (2r^2[R_{\varepsilon}^{+}(r)]_1^1)_{\varepsilon} = $$

$$ \begin{aligned} & r^2(h_{\varepsilon}^{++}(r))_{\varepsilon} + 2r(h_{\varepsilon}^{+}(r))_{\varepsilon} = \\ & r^2 \left\{ -\frac{1}{r[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2 + \varepsilon^2]^{3/2}} + \right. \\ & \qquad \left. \frac{r-2m}{r^2[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \varepsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \varepsilon^2]^{1/2}}{r^3} \right\} + \\ & 2r \left\{ -\frac{r-2m}{r[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \varepsilon^2]^{1/2}}{r^2} \right\} = \\ & -\frac{r}{[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \varepsilon^2]^{3/2}} + \\ & \qquad \frac{r-2m}{[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{r-2m}{[(r-2m)^2 + \varepsilon^2]^{1/2}} - \\ & \qquad \frac{2[(r-2m)^2 + \varepsilon^2]^{1/2}}{r} - \\ & \qquad \frac{2(r-2m)}{[(r-2m)^2 + \varepsilon^2]^{1/2}} + \frac{2[(r-2m)^2 + \varepsilon^2]^{1/2}}{r} = \end{aligned} $$
$$ \frac{r}{[(r-2m)^2 + \varepsilon^2]^{\frac{1}{2}}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \varepsilon^2]^{\frac{3}{2}}} \quad (1.13) $$

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 letting $\Phi(x)$ 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 we have the following:

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

Then, for any function $\Phi(x) \in S_{2m}(\mathbb{R}^3)$, we obtain the following:

$$ \begin{aligned} \left(\left[R_{\varepsilon}^{+}(\Phi)\right]_{2}^{2}\right)_{\varepsilon} &= \left(\left[R_{\varepsilon}^{+}(\Phi)\right]_{3}^{3}\right)_{\varepsilon} = \\ & \int_{K^{+}} \left(\left[R_{\varepsilon}^{+}(r)\right]_{2}^{2}\right)_{\varepsilon} \Phi(x) d^{3}x = \int_{K^{+}} \left(\left[R_{\varepsilon}^{+}(r)\right]_{3}^{3}\right)_{\varepsilon} \Phi(x) d^{3}x = \\ & \int_{2m}^{R} (r^2 \left[R_{\varepsilon}^{+}(r)\right]_{2}^{2})_{\varepsilon} \hat{\Phi}(r) dr = \int_{2m}^{R} (r^2 \left[R_{\varepsilon}^{+}(r)\right]_{3}^{3})_{\varepsilon} \hat{\Phi}(r) dr = \\ & \int_{2m}^{R} \left(r(h_{\varepsilon}^{+\prime}(r))_{\varepsilon} + 1 + (h_{\varepsilon}^{+}(r))_{\varepsilon}\right) \hat{\Phi}(r) dr = \\ & \int_{2m}^{R} \left\{-\frac{r-2m}{[(r-2m)^2 + \varepsilon^2]^{1/2}}\right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr. \end{aligned} \quad (1.14) $$

By replacing $r - 2m = u$, from Eq. (1.14), we obtain:

$$ \int_{2m}^{R} \left\{ -\frac{r-2m}{[(r-2m)^2 + \varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr = -\int_{0}^{R-2m} \frac{u \hat{\Phi}(u+2m) du}{(u^2 + \varepsilon^2)^{\frac{1}{2}}} + \int_{0}^{R-2m} \hat{\Phi}(u+2m) du. \quad (1.15) $$

By replacing $u = \varepsilon\eta$, from Eq. (1.15), we obtain the expression:

$$ I_3^+(\varepsilon) = \int_K \left( \left[ R_{\varepsilon}^+ \right]_3^3 \right)_{\varepsilon} \Phi(x) d^3 x = I_2^+(\varepsilon) = \int_K \left( \left[ R_{\varepsilon}^+ \right]_2^2 \right)_{\varepsilon} \Phi(x) d^3 x = \\ -\varepsilon \times \left( \int_0^{\frac{R-2m}{\varepsilon}} \frac{\eta \hat{\Phi}(\varepsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{\frac{1}{2}}} - \int_0^{\frac{R-2m}{\varepsilon}} \hat{\Phi}(\varepsilon\eta + 2m) d\eta \right). \quad (1.16) $$

From Eq. (1.16) we obtain:

$$ I_3^+(\varepsilon) = I_2^+(\varepsilon) = -\varepsilon \frac{\hat{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} - 1 \right] - $$
$$ -\frac{\epsilon^2}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta = -\epsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1 - \left(\frac{R-2m}{\epsilon} - 1\right)} - \frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{\frac{1}{2}}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta, \quad (1.17) $$

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

$$ \hat{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \hat{\Phi}^{(n)}(\xi), \\ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 \quad (1.18) $$

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

Then Eq. (1.17) – Eq. (1.18) gives

$$ \begin{aligned} \lim_{\epsilon \to 0} I_3^+(\epsilon) &= \lim_{\epsilon \to 0} I_2^+(\epsilon) = \\ & \lim_{\epsilon \to 0} \left\{ -\epsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1 + \frac{R-2m}{\epsilon}} \right] + \right. \\ & \left. \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta \right\} = 0. \right\} \end{aligned} \quad (1.19) $$

Thus, for $S'_{2m}(B_R^+(2m)) \subset S'_{2m}(\mathbb{R}^3) \subset D'(\mathbb{R}^3)$, where

$$ B^+(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\} $$

from Eq. (1.19), we obtain

$$ \begin{aligned} \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(\Phi)]_{3}^{3} &= \lim_{\epsilon \to 0} I_{3}^{+}(\epsilon) = 0, \\ \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(\Phi)]_{2}^{2} &= \lim_{\epsilon \to 0} I_{2}^{+}(\epsilon) = 0. \end{aligned} \quad (1.20) $$

For $([R_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon}$, $([R_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon}$ we obtain the following:

$$ 2 \int_K ([R_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^3 x = 2 \int_K ([R_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^3 x = \\ \int_{2m}^{R} (r^2 (h_{\epsilon}^{++}(r))_{\epsilon} + 2r (h_{\epsilon}^{+'}(r))_{\epsilon}) \hat{\Phi}(r) dr = $$
$$ \int_{2m}^{R} \left\{ -\frac{r}{[(r-2m)^2 + \epsilon^2]^{\frac{1}{2}}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{\frac{3}{2}}} \right\} \hat{\Phi}(r) dr. \quad (1.21) $$

By replacing $r - 2m = u$, from Eq. (1.21), we obtain

$$ \begin{aligned} I_1^+(\epsilon) &= [R_\epsilon^+(\Phi)]_1^1 = \\ & 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x = [R_\epsilon^+(\Phi)]_0^0 = I_0^+(\epsilon) = \\ & 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x = \\ & \int_{2m}^R (r^2(h_\epsilon^{++})_\epsilon + 2r(h_\epsilon^{+\prime})_\epsilon) \hat{\Phi}(r) dr = \\ & \int_0^{R-2m} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{\frac{1}{2}}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{\frac{3}{2}}} \right\} \hat{\Phi}(u+2m) du. \end{aligned} \quad (1.22) $$

By replacing $u = \epsilon\eta$, from Eq. (1.22) we obtain

$$ \begin{aligned} 2 \int_K ([R_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3x &= 2 \int_K ([R_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3x = \\ & \int_{2m}^R (r^2(h_\epsilon^{++})_\epsilon + 2r(h_\epsilon^{+\prime})_\epsilon) \hat{\Phi}(r) dr = \\ & \epsilon \left[ \int_0^{\frac{R-2m}{\epsilon}} d\eta \left\{ -\frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \hat{\Phi}(\epsilon\eta + 2m) \right. \\ & \left. - \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^2\eta\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \right. \\ & \left. \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^4\eta^3\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} + 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^3\eta^2\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = \right. \\ & \left. \epsilon \left[ - \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta^3\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{3/2}} \right] + \right. \\ & \left. 2m \left[ - \int_0^{\frac{R-2m}{\epsilon}} \frac{\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta^2\hat{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{3/2}} \right] \right]. \end{aligned} \quad (1.23) $$

From Eq. (1.23) we obtain

$$ I_0^+(\epsilon) = I_1^+(\epsilon) = 2m \frac{\hat{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] + $$
$$
\begin{equation}
\begin{aligned}
& \frac{\epsilon}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta + \\
& \frac{\epsilon \hat{\Phi}(2m)}{0!} \int_{\frac{-2m}{\epsilon}}^{\frac{R-2m}{\epsilon}} d\eta \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] + \\
& \frac{\epsilon^2}{1!} \int_0^{\frac{R-2m}{\epsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{\frac{1}{2}}} + \frac{\eta^2}{(\eta^2+1)^{\frac{3}{2}}} \right] \eta,
\end{aligned}
\tag{1.24}
\end{equation}
$$

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

$$
\begin{gather*}
\hat{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \hat{\Phi}^{\alpha\beta(n)}(\xi), \\
\xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1
\end{gather*}
$$

with $\hat{\Phi}^{(l)}(\xi) \triangleq d^l \hat{\Phi}(\xi)/d\xi^l$. Then, Eq. (1.24) gives

$$
\begin{equation}
\begin{split}
w - \lim_{\epsilon \to 0} I_0^+(\epsilon) &= w - \lim_{\epsilon \to 0} I_1^+(\epsilon) = \\
& 2m \hat{\Phi}(2m) \lim_{\epsilon \to 0} \left\{ \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta \right\} = \\
& 2m \hat{\Phi}(2m) \lim_{s \to \infty} \left[ \int_0^s \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^s \frac{d\eta}{(\eta^2+1)^{1/2}} \right] = -2m \hat{\Phi}(2m).
\end{split}
\tag{1.25}
\end{equation}
$$

The following 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 for $S'_{2m}(B^+(2m, R)) \subset S'_{2m}(\mathbb{R}^3)$, we obtain

$$
\lim_{\epsilon \to 0} [R_{\epsilon}^{+}(\Phi)]_{1}^{1} = \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(\Phi)]_{0}^{0} = -m\hat{\Phi}(2m). \quad (1.26)
$$

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

$$
\begin{align}
([R_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} = ([R_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} &= \frac{1}{2} \left( (h_{\epsilon}^{-\prime\prime}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{-\prime}(r))_{\epsilon} \right), \nonumber \\
([R_{\epsilon}^{-}(r)]_{2}^{2})_{\epsilon} = ([R_{\epsilon}^{-}(r)]_{3}^{3})_{\epsilon} &= \frac{(h_{\epsilon}^{-\prime}(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{-}(r))_{\epsilon}}{r^2}. \tag{1.27}
\end{align}
$$

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

$$ (1) \operatorname{supp} (\Phi(x)) \subset K^{-} = \{x|0 \le \|x\| < 2m\}, \text{ and} $$

$$ (2) \hat{\Phi}(r) \in C^{\infty}(\mathbb{R}). $$

Then, similarly as above, for any function $\Phi(x) \in S_{2m}^{-}(\mathbb{R}^3)$ from Eq. (56) we obtain

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^{3}x = \\ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^{3}x = m \hat{\Phi}(2m), \qquad (1.28) $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{2}^{2})_{\epsilon} \Phi(x) d^{3}x = \\ \lim_{\epsilon \to 0} \int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{3}^{3})_{\epsilon} \Phi(x) d^{3}x = 0. \qquad (1.29) $$

## II. GENERALIZED COLOMBEAU CALCULUS

We use [4-12] 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 $\mathbb{R}^n$. Standard 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 the classical Colombeau theory of nonlinear generalized functions [4-8] is regularization using sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter $\epsilon$. Use $(u_\epsilon)_{\epsilon \in (0, \delta]}$ 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 2.1.** The classical Colombeau algebra of the generalized functions on $M$ is defined as the quotient:
$$G_{\delta}(M) = \mathcal{E}_{M}(M)/\mathcal{N}(M) \qquad (2.1)$$

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

$$\mathcal{E}_M(M) \triangleq \{(u_\varepsilon)_\varepsilon | \forall K(K \subset \subset M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N})$$

$$\forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in X(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\varepsilon(p)| = O(\varepsilon^{-N}) \text{ as } \varepsilon \to 0 \right] \right\}, (2.2)$$

$$\mathcal{N}(M) \triangleq \{(u_\varepsilon)_\varepsilon | \forall K(K \subset \subset M), \forall k(k \in N_0) \forall q(q \in \mathbb{N})$$

$$\forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in X(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\varepsilon(p)| = O(\varepsilon^q) \text{ as } \varepsilon \to 0 \right] \right\}. (2.3)$$

**Remark 2.1.** In the definition, the Landau symbol $a_\varepsilon = O(\psi(\varepsilon))$ appears, and it has the following meaning:

$$\exists C(C > 0) \exists \varepsilon_0 (\varepsilon_0 \in (0,1]) \forall \varepsilon (\varepsilon < \varepsilon_0) [a_\varepsilon \le C\psi(\varepsilon)].$$

**Definition 2.2.** The elements of $G_\delta(M)$ are denoted by the following:

$$u = \mathrm{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + \mathcal{N}(M). \qquad (2.4)$$

**Remark 2.2.** With component-wise operations $(\cdot, \pm)$ $G_\delta(M)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by

$$L_{\xi} u \triangleq \mathrm{cl}[(L_{\xi} u_{\varepsilon})_{\varepsilon}].$$

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_\delta(M)$ 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, the 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 the Lie derivatives are replaced by
partial derivatives.

**Remark 2.3.** Smooth functions $f \in C^{\infty}(M)$ are simply embedded into $G(M)$ by the constant embedding $\sigma$, i.e., $\sigma(f) = \text{cl}[(f)_{\varepsilon}]$; 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

$$
G_s^r(M) = G_{\delta}(M) \otimes T_s^r(M), \qquad (2.5)
$$

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, such as 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 using generalized vector and covector fields in
generalized functions, i.e., for $G_\delta(M)$-modules, we have

$$
G_s^r(M) \cong L_{(M)}(G_1^0(M)^r, G_0^1(M)^s; G_\delta(M)). \quad (2.6)
$$

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 $\epsilon$) 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

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

for all small enough $\epsilon$. This notion captures the intuitive idea that a generalized metric is 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 a 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 $G_0^1(M)$ to $G_1^0(M)$ 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 $((g_{\epsilon})_{ab})_{\epsilon}$). Additionally, the generalized Levi-Cività connection and the generalized Riemann, Ricci, and Einstein tensors of a generalized metric are defined simply by the usual coordinate formulas based on the level of representatives [13].

### III.DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH REGULARIZATION VIA THE HORIZON OF A NONSPINNING GRAVITATIONAL SINGULARITY

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

Note that the Colombeau generalized object $\{(ds_{\epsilon}^{+2})_{\epsilon}, (ds_{\epsilon}^{-2})_{\epsilon}\}$, given by Eq.(1.6) – Eq.(1.7) is a natural way that 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 is

$$ ([R_{\epsilon}(r)]_0^0)_{\epsilon} = ([R_{\epsilon}(r)]_1^1)_{\epsilon} = \frac{1}{2} \left( (\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} + \frac{2}{r} (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} \right), \quad (3.1) $$

$$ ([R_{\epsilon}(r)]_2^2)_{\epsilon} = ([R_{\epsilon}(r)]_3^3)_{\epsilon} = \frac{(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}(r))_{\epsilon}}{r^2}, \quad (3.2) $$

where $(h_{\epsilon})_{\epsilon} = \{(h_{\epsilon}^{+})_{\epsilon}, (h_{\epsilon}^{-})_{\epsilon}\}$.

From Eq. (1.6) – Eq. (1.7) we obtain

$$ \begin{aligned} & ([h_{\epsilon}]_{2m})_{\epsilon} = (h_{\epsilon}^{+}(2m + 0))_{\epsilon} - (h_{\epsilon}^{-}(2m - 0))_{\epsilon} = \\ & (h_{\epsilon}^{+}(2m) - h_{\epsilon}^{-}(2m))_{\epsilon} = -\left(\frac{\epsilon}{2m}\right)_{\epsilon} - \left(\frac{\epsilon}{2m}\right)_{\epsilon} = -\frac{(\epsilon)_{\epsilon}}{m}. \end{aligned} \quad (3.3) $$

From Eq. (3.3) we obtain

$$ (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r < 2m, \end{cases} \quad (3.4) $$

where $([h]_{2m})_{\epsilon} = -(\epsilon)_{\epsilon}/m$ and therefore
$$ (\bar{h}_{\varepsilon}^{\prime}(r))_{\varepsilon} = \begin{cases} (h_{\varepsilon}^{+\prime}(r))_{\varepsilon} - \frac{(\varepsilon)_{\varepsilon}}{m} \delta(r-2m) & r \ge 2m \\ (h_{\varepsilon}^{-\prime}(r))_{\varepsilon} - \frac{(\varepsilon)_{\varepsilon}}{m} \delta(r-2m) & r < 2m. \end{cases} \quad (3.5) $$

From Eq. (1.10) we obtain

$$ \begin{aligned} ([h_{\varepsilon}']_{2m})_{\varepsilon} &= (h_{\varepsilon}^{+\prime}(2m+0))_{\varepsilon} - (h_{\varepsilon}^{-\prime}(2m-0))_{\varepsilon} = \\ & \qquad (h_{\varepsilon}^{+\prime}(2m))_{\varepsilon} - (h_{\varepsilon}^{-\prime}(2m))_{\varepsilon} = \\ & \qquad \frac{(\varepsilon)_{\varepsilon}}{4m^2} - \left(-\frac{(\varepsilon)_{\varepsilon}}{4m^2}\right) = \frac{(\varepsilon)_{\varepsilon}}{2m^2}. \end{aligned} \quad (3.6) $$

We obtain

$$ (\bar{h}_{\varepsilon}''(r))_{\varepsilon} = \begin{cases} h_{\varepsilon}^{+''}(r) + [h_{\varepsilon}]_{2m} \delta'(r-2m) + [h_{\varepsilon}']_{2m} \delta(r-2m) & r \ge 2m \\ h_{\varepsilon}^{-''}(r) + [h_{\varepsilon}]_{2m} \delta(r-2m) + [h_{\varepsilon}']_{x_0} \delta(r-2m) & r < 2m, \end{cases} \quad (3.7) $$

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

$$ \bar{h}_{\varepsilon}''(r) = \begin{cases} (h_{\varepsilon}^{+''}(r))_{\varepsilon} - \frac{(\varepsilon)_{\varepsilon}}{m} \delta'(r-2m) + \frac{(\varepsilon)_{\varepsilon}}{2m^2} \delta(r-2m) & r \ge 2m \\ h_{\varepsilon}^{-''}(r) - \frac{(\varepsilon)_{\varepsilon}}{m} \delta(r-2m) + \frac{(\varepsilon)_{\varepsilon}}{2m^2} \delta(r-2m) & r < 2m. \end{cases} \quad (3.8) $$

Inserting Eq. (3.5) and Eq. (3.8) into Eq. (3.1) and into Eq. (3.2) finally we obtain for $r \ge 2m$:

$$ \begin{aligned} (\bar{[R}_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = (\bar{[R}_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2} \left( (\bar{h}_{\varepsilon}^{+''}(r))_{\varepsilon} + \frac{2}{r} (\bar{h}_{\varepsilon}^{+'}(r))_{\varepsilon} \right) = \\ \frac{1}{2} \left( (h_{\varepsilon}^{+''}(r))_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+'}(r))_{\varepsilon} \right) - \\ - \frac{(\varepsilon)_{\varepsilon}}{2m} \delta'(r-2m) + \frac{(\varepsilon)_{\varepsilon}}{2m^2} \delta(r-2m) - \frac{2(\varepsilon)_{\varepsilon}}{rm} \delta(r-2m). \end{aligned} \quad (3.9) $$

$$ \begin{aligned} (\bar{[R}_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = (\bar{[R}_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{(\bar{h}_{\varepsilon}')_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}(r))_{\varepsilon}}{r^2} = \\ \frac{(h_{\varepsilon}^{\prime\prime}(r))_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{\prime}(r))_{\varepsilon}}{r^2} - \frac{(\varepsilon)_{\varepsilon}}{rm} \delta(r-2m). \end{aligned} \quad (3.10) $$

Similarly, we obtain the following for $r < 2m$:

$$ \begin{aligned} (\bar{[R}_{\varepsilon}^{-}(r)]_{0}^{0})_{\varepsilon} = (\bar{[R}_{\varepsilon}^{-}(r)]_{1}^{1})_{\varepsilon} = \\ \frac{1}{2} \left( (\bar{h}_{\varepsilon}''(r))_{\varepsilon} + \frac{2}{r} (\bar{h}_{\varepsilon}'(r))_{\varepsilon} \right) = \end{aligned} $$
$$ \frac{1}{2}((h_{\varepsilon}''(r))_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}'(r))_{\varepsilon}) - \frac{(\varepsilon)_{\varepsilon}}{2m}\delta'(r-2m) + \frac{(\varepsilon)_{\varepsilon}}{2m^2}\delta(r-2m) - \frac{2(\varepsilon)_{\varepsilon}}{rm}\delta(r-2m). \quad (3.11) $$

$$ \begin{aligned} \left(\left[\bar{R}_{\varepsilon}^{-}(r)\right]_{2}^{2}\right)_{\varepsilon} &= \left(\left[\bar{R}_{\varepsilon}^{-}(r)\right]_{3}^{3}\right)_{\varepsilon} = \frac{\left(\bar{h}_{\varepsilon}'(r)\right)_{\varepsilon}}{r} + \frac{1 + (h_{\varepsilon}(r))_{\varepsilon}}{r^2} = \\ & \frac{(h_{\varepsilon}'(r))_{\varepsilon}}{r} + \frac{1 + (h_{\varepsilon}^{-}(r))_{\varepsilon}}{r^2} - \frac{(\varepsilon)_{\varepsilon}}{rm}\delta(r-2m). \end{aligned} \quad (3.12) $$

**Remark 3.1.** Note that in $D'([2m, \infty))$

$$ w - \lim_{\varepsilon \to 0} \frac{2\pi^{-1}\varepsilon}{(r - 2m)^2 + \varepsilon^2} = \delta(r - 2m). \quad (3.13) $$

and therefore in $D'([2m, \infty))$

$$ \begin{aligned} w - \lim_{\varepsilon \to 0} \frac{d}{dr} \frac{2\pi^{-1}\varepsilon}{(r - 2m)^2 + \varepsilon^2} &= \\ w - \lim_{\varepsilon \to 0} \frac{4\pi^{-1}\varepsilon(r - 2m)}{[(r - 2m)^2 + \varepsilon^2]^2} &= \delta'(r - 2m). \end{aligned} \quad (3.14) $$

By using Eq. (3.13) and Eq. (3.14), one obtains the natural embeddings

$$ \delta(r - 2m) \mapsto G_{\delta}(\mathbb{R}) : $$

$$ \delta(r - 2m) \mapsto (\delta(r - 2m; \varepsilon))_{\varepsilon} = \left( \frac{2\pi^{-1}\varepsilon}{(r - 2m)^2 + \varepsilon^2} \right)_{\varepsilon} \in G_{\delta}(\mathbb{R}) \quad (3.15) $$

and

$$ \delta'(r - 2m) \mapsto G_{\delta}(\mathbb{R}) : $$

$$ \delta'(r - 2m) \mapsto (\delta'(r - 2m; \varepsilon))_{\varepsilon} = \left( \frac{4\pi^{-1}\varepsilon(r - 2m)}{[(r - 2m)^2 + \varepsilon^2]^2} \right)_{\varepsilon} \in G_{\delta}(\mathbb{R}). \quad (3.16) $$

**Remark 3.2.** Note, that: in $D'([2m, \infty))$

$$ \begin{aligned} w - \lim_{\varepsilon \to 0} \frac{4\pi^{-1}\varepsilon^2(r - 2m)}{[(r - 2m)^2 + \varepsilon^2]^2} &= c_1\delta(r - 2m), \quad (3.17) \\ c_1 &= 2\pi^{-1}. \end{aligned} $$

Inserting Eq. (3.15) – Eq. (3.16) and Eq. (3.17) into Eq. (3.9) and into Eq. (3.10), we obtain the following for $r \ge 2m$:

$$ \left(\left[\bar{R}_{\varepsilon}^{+}(r)\right]_{0}^{0}\right)_{\varepsilon} = \left(\left[\bar{R}_{\varepsilon}^{+}(r)\right]_{1}^{1}\right)_{\varepsilon} = \frac{1}{2}\left(\left(\bar{h}_{\varepsilon}''(r)\right)_{\varepsilon} + \frac{2}{r}\left(\bar{h}_{\varepsilon}'(r)\right)_{\varepsilon}\right) = $$
$$
\begin{aligned}
& \frac{1}{2} \left( (h_{\varepsilon}^{++}(r))_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+}(r))_{\varepsilon} \right) - \\
& \qquad \frac{(\varepsilon)_{\varepsilon}}{2m} (\delta'(r - 2m; \varepsilon))_{\varepsilon} + \left[ \frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{rm} \right] (\delta(r - 2m; \varepsilon))_{\varepsilon} = \\
& \qquad \frac{1}{2} \left( (h_{\varepsilon}^{++}(r))_{\varepsilon} + \frac{2}{r} (h_{\varepsilon}^{+}(r))_{\varepsilon} \right) - \\
& \qquad \left[ \frac{c_1}{2m} - (\varepsilon)_{\varepsilon} \left( \frac{1}{2m^2} - \frac{2}{rm} \right) \right] (\delta(r - 2m; \varepsilon))_{\varepsilon} \in G_{\delta}(\mathbb{R}^3),
\end{aligned}
\quad (3.18)
$$

$$
\begin{aligned}
& ([\bar{R}_{\varepsilon}^{+}(r)]_{2})_{\varepsilon} = ([\bar{R}_{\varepsilon}^{+}(r)]_{3})_{\varepsilon} = \frac{(\bar{h}_{\varepsilon}')(r)}{r} + \frac{1 + (h_{\varepsilon}(r))_{\varepsilon}}{r^2} = \\
& \qquad \frac{(h_{\varepsilon}^{+}(r))_{\varepsilon}}{r} + \frac{1 + (h_{\varepsilon}^{+}(r))_{\varepsilon}}{r^2} - \frac{(\varepsilon)_{\varepsilon}}{rm} (\delta(r - 2m; \varepsilon))_{\varepsilon} \in G_{\delta}(\mathbb{R}^3).
\end{aligned}
\quad (3.19)
$$

By 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 letting $\Phi(x)$ 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 the following hold:

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

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

$$
\begin{align}
\lim_{\epsilon \to 0} \int_K ( [\bar{R}_{\epsilon}^{+}(r)]_1^1 )_{\epsilon} \Phi(x) d^3 x &= \nonumber \\
&\quad \lim_{\epsilon \to 0} \int_K ( [\bar{R}_{\epsilon}^{+}(r)]_0^0 )_{\epsilon} \Phi(x) d^3 x = -\epsilon_1 m \hat{\Phi}(2m), \tag{3.20} \\
&\qquad \epsilon_1 > 0. \nonumber
\end{align}
$$

$$
\begin{align}
& \lim_{\epsilon \to 0} \int_K ([\bar{R}_\epsilon^+(r)]_1^1)_\epsilon \Phi(x) d^3 x = \nonumber \\
& \lim_{\epsilon \to 0} \int_K ([\bar{R}_\epsilon^+(r)]_0^0)_\epsilon \Phi(x) d^3 x = 0. \tag{3.21}
\end{align}
$$

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

(1) $\text{supp}(\Phi(x)) \subset K^- = \{x \mid 0 \le \|x\| < 2m\}$, and (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 to the above, we obtain

$$
\lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^{3}x =
$$
$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{1})_{\epsilon} \Phi(x) d^{3}x = \epsilon_{2} m \hat{\Phi}(2m), \quad (3.22) \\ \epsilon_{2} < 0. $$

$$ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{2})_{\epsilon} \Phi(x) d^{3}x = \\ \lim_{\epsilon \to 0} \int_{K^{-}} ([\bar{R}_{\epsilon}^{-}(r)]_{3})_{\epsilon} \Phi(x) d^{3}x = 0. \quad (3.23) $$

Let $(\tilde{t}, \tilde{r}, \tilde{\theta}, \tilde{\phi})$ be the generalized Colombeau coordinates on $\mathbb{R}_{\delta}^{4}$, with $(\tilde{r}, \tilde{\theta}, \tilde{\phi})$ being the usual spherical polar coordinates on $\mathbb{R}_{\delta}^{3}$. Here, we abbreviate for short $\tilde{t} = (t_{\epsilon})_{\epsilon}$, $\tilde{r} = (r_{\epsilon})_{\epsilon}$, $\tilde{\theta} = (\theta_{\epsilon})_{\epsilon}$, and $\tilde{\phi} = (\varphi_{\epsilon})_{\epsilon}$. This meant that the classical Schwarzschild spacetime was extended up to distributional semi Riemannian manifold $(\widetilde{\mathbf{Sch}}, (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon})$, where

$$ \widetilde{\mathbf{Sch}} = \mathbb{R}_{\delta} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbf{S}_{\delta}^{2}), $$

and endowed on the tangent bundle with the Colombeau distributional metric tensor

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

such that

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

**Remark 3.3.** Note that the distributional semi Riemannian manifold

$(\widetilde{\mathbf{Sch}}, (g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon})$ is endowed with point free geometry [14] in a natural way, since $(g_{ij}^{Sch}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in \mathcal{G}_{\delta}(\mathbb{R}_{\delta}^{4})$, where $i, j = 0,1,2,3$.

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

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

where

$$ h_{\epsilon}^{+}(r_{\epsilon}) = - \frac{\sqrt{(r_{\epsilon} - 2m)^{2} + \epsilon^{2}}}{r_{\epsilon}}, \quad (3.24) $$

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

and
$$h_{\epsilon}^{-}(r_{\epsilon}) = \frac{\sqrt{(2m - r_{\epsilon})^{2} + \epsilon^{2}}}{r_{\epsilon}}. \qquad (3.25)$$

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 the Colombeau generalized object $(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}$ above the horizon $([R_{\epsilon}^{+}(r_{\epsilon})]_{\alpha}^{\beta})_{\epsilon}$ is

$$([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{0}^{0})_{\epsilon} = ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{1}^{1})_{\epsilon} = \\ \frac{1}{2}((h_{\epsilon}^{++}(r_{\epsilon}))_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{+\prime}(r_{\epsilon}))_{\epsilon}) - \\ -\frac{(\epsilon)_{\epsilon}}{2m}(\delta'(r_{\epsilon} - 2m; \epsilon))_{\epsilon} + \left[\frac{(\epsilon)_{\epsilon}}{2m^{2}} - \frac{2(\epsilon)_{\epsilon}}{rm}\right](\delta(r_{\epsilon} - 2m; \epsilon))_{\epsilon} \qquad (3.26)$$

and

$$([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{2}^{2})_{\epsilon} = ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{3}^{3})_{\epsilon} = \\ \frac{(h_{\epsilon}^{+\prime}(r_{\epsilon}))_{\epsilon}}{(r_{\epsilon})_{\epsilon}} + \frac{1+(h_{\epsilon}^{+}(r_{\epsilon}))_{\epsilon}}{(r_{\epsilon})_{\epsilon}^{2}} - \frac{(\epsilon)_{\epsilon}}{(r_{\epsilon})_{\epsilon}m} (\delta(r_{\epsilon} - 2m; \epsilon))_{\epsilon}. \qquad (3.27)$$

From Eq. (3.26) and Eq. (3.15) – Eq. (3.16), we obtain

$$([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{0}^{0})_{\epsilon} = ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{1}^{1})_{\epsilon} = \\ -\frac{1}{2(r_{\epsilon})_{\epsilon}}\left(\frac{1}{[(r_{\epsilon}-2m)^{2} + \epsilon^{2}]^{1/2}}\right)_{\epsilon} + \\ \frac{1}{2(r_{\epsilon})_{\epsilon}}\left(\frac{(r_{\epsilon}-2m)^{2}}{[(r_{\epsilon}-2m)^{2} + \epsilon^{2}]^{3/2}}\right)_{\epsilon} - \\ \frac{(\epsilon)_{\epsilon}}{2m}(\delta'(r_{\epsilon}-2m; \epsilon))_{\epsilon} + \left[\frac{(\epsilon)_{\epsilon}}{2m^{2}} - \frac{2(\epsilon)_{\epsilon}}{(r_{\epsilon})_{\epsilon}m}\right](\delta(r_{\epsilon}-2m; \epsilon))_{\epsilon} = \\ -\frac{1}{2(r_{\epsilon})_{\epsilon}}\left(\frac{1}{[(r_{\epsilon}-2m)^{2} + \epsilon^{2}]^{1/2}}\right)_{\epsilon} + \\ \frac{1}{2(r_{\epsilon})_{\epsilon}}\left(\frac{(r_{\epsilon}-2m)^{2}}{[(r_{\epsilon}-2m)^{2} + \epsilon^{2}]^{3/2}}\right)_{\epsilon} - \\ \frac{(\epsilon)_{\epsilon}}{2m}\left(\frac{4\pi^{-1}\epsilon(r_{\epsilon}-2m)}{[(r_{\epsilon}-2m)^{2} + \epsilon^{2}]^{2}}\right)_{\epsilon} + \left[\frac{(\epsilon)_{\epsilon}}{2m^{2}} - \frac{2(\epsilon)_{\epsilon}}{(r_{\epsilon})_{\epsilon}m}\right]\left(\frac{2\pi^{-1}\epsilon}{(r_{\epsilon}-2m)^{2} + \epsilon^{2}}\right)_{\epsilon}. \qquad (3.28)$$

For any $(r_{\epsilon})_{\epsilon}$ such that $(r_{\epsilon})_{\epsilon} - 2m \neq 0$, from Eq. (3.28), one obtains

$$([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{0}^{0})_{\epsilon} = ([\tilde{R}_{\epsilon}^{+}(r_{\epsilon})]_{1}^{1})_{\epsilon} =$$
$$ \frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^2)_{\varepsilon}]^{1/2}} + \frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^2)_{\varepsilon}]^{3/2}} $$

$$ \frac{(\varepsilon)_{\varepsilon}}{2m((r_{\varepsilon}-2m)^3)_{\varepsilon}} \left( \frac{4\pi^{-1}\varepsilon}{[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}\varepsilon^2]^2} \right) + \left[ \frac{(\varepsilon)_{\varepsilon}}{2m^2} - \frac{2(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \right] \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^2+\varepsilon^2} \right)_{\varepsilon} . \quad (3.29) $$

For any $(r_\varepsilon)_\varepsilon$ such that $(r_\varepsilon)_\varepsilon - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ and $\delta[(r_\varepsilon)_\varepsilon - 2m]^{-2} < 1$, from Eq. (3.29) one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} \approx (\varepsilon^2)_{\varepsilon} \quad (3.30) $$

and therefore

$$ \mathrm{st}(( [\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon}) = \mathrm{st}(( [\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon}) = 0, \quad (3.31) $$

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

It follows from Eq. (3.30) for any $(r_\varepsilon)_\varepsilon - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ such that
$\delta^2[(r_\varepsilon)_\varepsilon - 2m]^{-2} < 1$ that

$$ \mathrm{st}(( [R_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon}) = \mathrm{st}(( [R_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon}) = 0. \quad (3.32) $$

Thus, for any $(r_\varepsilon)_\varepsilon - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ 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 $((r_\varepsilon)_\varepsilon - 2m) = 0_{\tilde{\mathbb{R}}}$, from Eq. (3.28), one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(2m)]_{0}^{0})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(2m)]_{1}^{1})_{\varepsilon} = -\frac{(\varepsilon^{-1})_{\varepsilon}}{2m} \approx -\infty. \quad (3.33) $$

From Eq. (3.27) and Eq. (3.15), we obtain

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = -\frac{(r_{\varepsilon})_{\varepsilon}-2m}{(r_{\varepsilon}^{2})_{\varepsilon}[( (r_{\varepsilon})_{\varepsilon}-2m)^{2}+(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon}. \quad (3.34) $$

For any $(r_\varepsilon)_\varepsilon$ such that $(r_\varepsilon)_\varepsilon - 2m \neq 0_{\tilde{\mathbb{R}}}$, from Eq. (3.34), one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = -\frac{(r_{\varepsilon})_{\varepsilon}-2m}{(r_{\varepsilon}^{2})_{\varepsilon}((r_{\varepsilon})_{\varepsilon}-2m)[1+((r_{\varepsilon})_{\varepsilon}-2m)^{-2}(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon} = -\frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}[1+((r_{\varepsilon})_{\varepsilon}-2m)^{-2}(\varepsilon^{2})_{\varepsilon}]} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} - \frac{(\varepsilon)_{\varepsilon}}{(r_{\varepsilon})_{\varepsilon}m} \left( \frac{2\pi^{-1}\varepsilon}{(r_{\varepsilon}-2m)^{2}+\varepsilon^{2}} \right)_{\varepsilon}. \quad (3.35) $$

For any $(r_\varepsilon)_\varepsilon$ such that $(r_\varepsilon)_\varepsilon - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_\delta)$ and $\delta^2[(r_\varepsilon)_\varepsilon - 2m]^{-2} < 1$, from Eq. (3.35),
one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} \approx ((r_{\varepsilon})_{\varepsilon} - 2m)^{-2}(\varepsilon^{2})_{\varepsilon} \approx 0. \quad (3.36) $$

For any $(r_\varepsilon)_\varepsilon$ such that $((r_\varepsilon)_\varepsilon - 2m) = 0_{\mathbb{R}}$, from Eq. (108), one obtains

$$ ([\tilde{R}_{\varepsilon}^{+}(2m)]_{2}^{2})_{\varepsilon} = ([\tilde{R}_{\varepsilon}^{+}(2m)]_{3}^{3})_{\varepsilon} = O(m^{-2}). \quad (3.37) $$

The distributional square scalar $(R^{+\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}^{+}(r_\varepsilon, \varepsilon))_\varepsilon$ above the horizon is [3]

$$ (R^{\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon = 2\left(\left[\frac{1}{r_\varepsilon}\tilde{h}_\varepsilon^{+'} + \frac{-h_\varepsilon^+ + 1}{r_\varepsilon^2}\right]^2\right)_\varepsilon + 2\left(\left[\frac{1}{2}\tilde{h}_\varepsilon^{+''} + \frac{1}{r_\varepsilon}\tilde{h}_\varepsilon^{+'}\right]^2\right)_\varepsilon, \quad (3.38) $$

where $\tilde{h}_\varepsilon^{+'}$ and $\tilde{h}_\varepsilon^{+''}$ are given by Eq. (3.4) and Eq. (3.8) correspondingly.

From Eq. (3.38) one obtains

$$ \begin{aligned} & (R^{\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon \\ & = 2\left(\left(\left[\left(\frac{1}{r_\varepsilon^3}\sqrt{\varepsilon^2 + (r_\varepsilon - 2m)^2} + \frac{1}{r_\varepsilon^2}\right)_\varepsilon - 2\frac{m}{r_\varepsilon^3}\frac{r_\varepsilon - 2m}{\sqrt{\varepsilon^2 + (r_\varepsilon - 2m)^2}}\right]^2\right)_\varepsilon \right. \\ & \qquad \left. + 2\left(\left[\frac{4m(r_\varepsilon - 2m)^3 + (r_\varepsilon - 4m)\varepsilon^2}{r_\varepsilon^3[(r_\varepsilon - 2m)^2 + \varepsilon^2]^{3/2}} - 2\frac{m}{r_\varepsilon^3}\frac{r_\varepsilon - 2m}{\sqrt{\varepsilon^2 + (r_\varepsilon - 2m)^2}}\right]^2\right)_\varepsilon \right) + \dots, \end{aligned} \quad (3.39) $$

From Eq. (3.38) at the horizon $(r_\varepsilon)_\varepsilon = 2m$ it follows that

$$ (R^{+\mu\nu}(2m, \varepsilon)R_{\mu\nu}^{+}(2m, \varepsilon))_{\varepsilon} = K(2m) + \left(\frac{1}{4m^4\varepsilon^2}\right)_{\varepsilon} \approx \infty, \quad (3.40) $$

where $K(r)$ is the Kretschman scalar.

**Remark 3.4.** Note that in contrast with abnormal canonical calculation [15] the square scalar $(R^{+\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}^{+}(r_\varepsilon, \varepsilon))_\varepsilon$ is singular at the horizon $(r_\varepsilon)_\varepsilon = 2m$.
The distributional square scalar $(R^{+\rho\sigma\mu\nu}(r_\epsilon, \epsilon)R_{\rho\sigma\mu\nu}^+(r_\epsilon, \epsilon))_\epsilon$ above the horizon is

$$ \begin{aligned} (R^{+\rho\sigma\mu\nu}(r_\epsilon, \epsilon)R_{\rho\sigma\mu\nu}^+(r_\epsilon, \epsilon))_\epsilon &= \left( (\bar{h}_\epsilon^{+''})^2 + 2\left(\frac{\bar{h}_\epsilon^{+'}}{r_\epsilon}\right)^2 \right)_\epsilon + \\ & \quad 4\left(\left[\frac{1}{r_\epsilon^2}(1-h_\epsilon^+)^2\right]_\epsilon + 2\left(\left[\frac{\bar{h}_\epsilon^{+'}}{r_\epsilon}\right]^2\right)_\epsilon \right) \\ &= \left(\bar{h}_\epsilon^{+''2} + 4\frac{\bar{h}_\epsilon^{+''2}}{r_\epsilon^2}\right)_\epsilon + 4\left(\frac{1}{r_\epsilon^4}(1-h_\epsilon^+)^2\right)_\epsilon. \end{aligned} \quad (3.41) $$

From Eq. (3.41) one obtains

$$ \begin{aligned} (R^{+\rho\sigma\mu\nu}(r_\epsilon, \epsilon)R_{\rho\sigma\mu\nu}^+(r_\epsilon, \epsilon))_\epsilon = & \\ & \left( \left[ \frac{4m(r_\epsilon - 2m)^3 + (r_\epsilon - 4m)\epsilon^2}{r_\epsilon^3[(r_\epsilon - 2m)^2 + \epsilon^2]^{3/2}} \right]^2 \right)_\epsilon - \frac{8m^2(r_\epsilon - 2m)^2}{r_\epsilon^6[(r_\epsilon - 2m)^2 + \epsilon^2]} + \\ & \frac{4}{r_\epsilon^4}(1 + r_\epsilon^{-1}\sqrt{(r_\epsilon - 2m)^2 + \epsilon^2})^2 + \dots \end{aligned} \quad (3.42). $$

From Eq. (3.42) at the horizon $(r_\epsilon)_\epsilon = 2m$ it follows that:

$$ \begin{aligned} (R^{\rho\sigma\mu\nu}(2m, \epsilon)R_{\rho\sigma\mu\nu}(2m, \epsilon))_{\epsilon} = & \\ & K(2m) + \left( \frac{\epsilon^4}{4m^4[\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_{\epsilon} \Big|_{(r_\epsilon)_\epsilon = 2m} = \\ & K(2m) + (\epsilon^{-2})_{\epsilon}. \end{aligned} \quad (3.43) $$

## IV. CONCLUSION

We argue that the canonical interpretation of the Schwarzschild spacetime in contemporary general relativity is wrong and the revision is needed. We studied the Schwarzschild solution using Colombeau distributional geometry, thus without leaving Schwarzschild coordinates
(t, r, θ, φ). We argue that the Schwarzschild solution is impossible to treat classically since the Levi-Cività connection is not available for the whole Schwarzschild spacetime ($\mathbf{Sch}, g_{ij}^{Sch}(t, r, θ, φ)$), where $\mathbf{Sch} = \mathbb{R} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbf{S}^2)$, and it can only be treated by using an embedding of the classical Schwarzschild metric tensor $g_{ij}^{Sch}$; $i, j = 1,2,3,4$ into Colombeau algebra

$G_\delta(\mathbb{R}^4, \Sigma), \Sigma = \{r = 2m\} \cup \{r = 0\}$ supergeneralized functions. This meant that the classical Schwarzschild spacetime could be extended up to the distributional semi Riemannian manifold $(\mathbf{Sch}, (g_{ij}^{Sch}(t, r, θ, φ, \varepsilon))_\varepsilon)$ endowed on the tangent bundle with the Colombeau distributional metric tensor $(g_{ij,\varepsilon}^{Sch})_\varepsilon \triangleq (g_{ij}^{Sch}(t, r, θ, φ, \varepsilon))_{\varepsilon} \in (0, \delta], (g_{ij,\varepsilon}^{Sch})_\varepsilon \in G_\delta(\mathbb{R}^4, \Sigma)$.

We obtain that the distributional Ricci tensor and the curvature scalar are $\delta$-type, i.e., $(R_\varepsilon(r))_\varepsilon = -\varepsilon m \delta(r - 2m), \varepsilon > 0$. However, the distributional square scalars are essentially nonclassical Colombeau type distributions: $(R_\varepsilon^{\mu\nu}(r)R_{\mu\nu,\varepsilon}(r))_\varepsilon$, $(R_\varepsilon^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\varepsilon}(r))_\varepsilon \in G(\mathbb{R}^3)\setminus\mathcal{D}'(\mathbb{R}^3)$. This provides a new physical interpretation for the distributional curvature scalar $(R_\varepsilon(r))_\varepsilon$ and square scalars $(R_\varepsilon^{\mu\nu}(r)R_{\mu\nu,\varepsilon}(r))_\varepsilon$, $(R_\varepsilon^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\varepsilon}(r))_\varepsilon$ of this geometry [3, 16].

## References

1. Foukzon J. *British Journal of Mathematics & Computer Science*. SCIENCEDOMAIN International, 2015, **11**(1): 1. Article no. BJMCS. 16961.

DOI: https://doi.org/10.9734/BJMCS/2015/16961

2. Foukzon J., Men'kova E. R., Potapov A. A. *British Journal of Mathematics & Computer Science*, 2016, **13**(6): 1. Article no. BJMCS.19235. DOI: [10.9734/BJMCS/2016/19235](https://doi.org/10.9734/BJMCS/2016/19235)
3. Foukzon J., Men'kova E. R., Potapov A. A. *Journal of High Energy Physics, Gravitation and Cosmology*, 2018, **4**, No. 2. Paper ID 84304, 80 p. DOI: https://doi.org/10.4236/jhepgc.2018.42023

4. Colombeau J. F. New Generalized Functions and Multiplication of Distributions. 1984. North Holland, Amsterdam.

5. Nedeljkov M., Pilipovic S., Scarpalezos D. Linear Theory of Colombeau Generalized Functions. London, United Kingdom, 01 Nov. 1998.

6. Grosser M., Kunzinger M., Oberguggenberger M., Steinbauer R. Geometric Theory of Generalized Functions with Applications to General Relativity. Netherlands, April 2013.

7. Biagioni H. A. A Nonlinear Theory of Generalized Functions. Springer, 14 Nov. 2006. Mathematics. 218 p.

8. Grosser M., Farkas E., Kunzinger M., Steinbauer R. On the Foundations of Nonlinear Generalized Functions I and II, Issue 729. American Mathematical Soc., 2001.

9. Rosinger E. E. Non-Linear Partial Differential Equations. Elsevier, 22 Nov. 1990.

10. Foukzon J., Men'kova E. R., Potapov A. A. Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced vacuum dominance. Book Publisher International 1. Published Nov. 14, 2019. DOI: https://doi.org/10.9734/bpi/mono/978-93-89562-22-4

11. Vernaeve H. *Comm. Alg.*, 2010, **38** (6): 2199.

DOI: https://doi.org/10.1080/00927870903055222, arXiv:0707.0698 [math.RA]

12. Kunzinger M., Steinbauer R. *Trans. Amer. Math. Soc.*, 2002, **354**: 4179. DOI: https://doi.org/10.1090/S0002-9947-02-03058-1

13. Grant J. D. E., Kunzinger M., Sämann C. *Ann Glob Anal Geom* (Dordr), 2019, **55**(1):

133. DOI: 10.1007/s10455-018-9637-x

14. Tozzi A. and Peters J. F. *Br. J. Neurosci. Res.* 2016, 94: 351.

15. Müller T., Grave F. arXiv: 0904.4184 [gr-qc] https://arxiv.org/abs/0904.4184v2

16. Foukzon J., Men'kova E. R., Potapov A. A., Podosenov S. A. *J. Phys.: Conf. Ser.*, 2018, **1141**: 012100. DOI: https://doi.org/10.1088/1742-6596/1141/1/012100
**Information about the authors**

Foukzon Jaykov - Doctor of phys. and math., Prof.,
Israel Institute of Technology, Haifa, 3200003, Israel.

Potapov Alexander A. – Chief researcher of V. A. Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences.
11, Mokhovaya str., Moscow, 125009, Russian Federation.
Professor, Doctor of phys. and math.

President of Joint Chinese-Russian Laboratory of Informational Technologies and Signals
Fractal Processing, Jinan University, Guangzhou, 510632, China.

Men’kova Elena R. – Ph.D., senior researcher

FGUP «All-Russian research institute for optical and physical measurements»,
46, Ozernaya str., 119361 Moscow, RF
Точечная свободная геометрия Коломбо в сингулярной общей теории
относительности. Часть 2.

Джэйков Фоукзон<fnref n="1" />, Александр Алексеевич Потапов<fnref n="2" />,³, Елена Романовна Менькова<fnref n="4" />

Аннотация

Исследовано решение Шварцшильда с применением распределительной геометрии
Коломбо в координатах Шварцшильда (t, r, θ, φ).

Пусть (t, r, θ, φ) – координаты в ℝ<fnref n="4" />, а (r, θ, φ) – сферические полярные координаты в ℝ³. Решение Шварцшильда определяется на подмножестве ℝ × (({r > 2m} ∪ {0 < r < 2m}) × S<fnref n="2" />) множества ℝ<fnref n="4" />. В этой области линейный элемент метрики имеет вид

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

В этой работе мы рассматриваем сингулярность на горизонте, отходя от сингулярности
в начале координат. Применение нелинейной сверхраспределительной геометрии и
сверхгенерализованных функций позволило показать, что сингулярность на горизонте
не определяется только сингулярностью координат Шварцшильда.

Установлено, что распределительный тензор Риччи и скаляр кривизны представляют собой величины $\delta$ -типа, $(R_{\varepsilon}(r))_{\varepsilon} = -\varepsilon m \delta(r - 2m)$, $\varepsilon > 0$. Однако распределительные скалярные квадраты фактически представляют собой неклассические распределения Коломбо $(R_{\varepsilon}^{\mu\nu}(r) R_{\mu\nu,\varepsilon}(r))_{\varepsilon} \in \mathcal{G}(\mathbb{R}^3) \setminus \mathcal{D}'(\mathbb{R}^3)$, $(R_{\varepsilon}^{\rho\sigma\mu\nu}(r) R_{\rho\sigma\mu\nu,\varepsilon}(r))_{\varepsilon} \in \mathcal{G}(\mathbb{R}^3) \setminus \mathcal{D}'(\mathbb{R}^3)$.
Представлена новая физическая интерпретация распределительной геометрии

<footnote n="1">¹Израильский технологический институт (Хайфа). 3200003 Израиль, г. Хайфа,
E-mail: jaykovfoukzon@list.ru</footnote>

<footnote n="2">²,³ potapov@cplire.ru</footnote>

<footnote n="4">⁴ e_menkova@mail.ru</footnote>
Шварцшильда [J. Foukzon, E. R. Men'kova, A. A. Potapov, and S. A. Podosenov, *Journal of Physics: Conference Series* **1141**, 012100 (2018). https://doi.org/10.1088/1742-6596/1141/1/012100].

**Ключевые слова:** Сингулярная полуримановая геометрия, обобщенные функции Коломбо, обобщенные числа Коломбо, распределенное пространство-время Шварцшильда, точечная свободная распределенная геометрия, обобщенные уравнения поля Эйнштейна, решения Коломбо уравнений поля Эйнштейна.

**Список источников**

1. Foukzon J. *British Journal of Mathematics & Computer Science*. SCIENCEDOMAIN International, 2015, **11**(1): 1. Article no. BJMCS. 16961.

DOI: https://doi.org/10.9734/BJMCS/2015/16961

2. Foukzon J., Men'kova E. R., Potapov A. A. *British Journal of Mathematics & Computer Science*, 2016, **13**(6): 1. Article no. BJMCS.19235. DOI: 10.9734/BJMCS/2016/19235

3. Foukzon J., Men'kova E. R., Potapov A. A. *Journal of High Energy Physics, Gravitation and Cosmology*, 2018, **4**, No. 2. Paper ID 84304, 80 p. DOI: 10.4236/jhepgc.2018.42023

4. Colombeau J. F. *New Generalized Functions and Multiplication of Distributions*. 1984. North Holland, Amsterdam.

5. Nedeljkov M., Pilipovic S., Scarpalezos D. *Linear Theory of Colombeau Generalized Functions*. London, United Kingdom, 01 Nov. 1998.

6. Grosser M., Kunzinger M., Oberguggenberger M., Steinbauer R. *Geometric Theory of Generalized Functions with Applications to General Relativity*. Netherlands, April 2013.

7. Biagioni H. A. *A Nonlinear Theory of Generalized Functions*. Springer, 14 Nov. 2006. Mathematics. 218 p.

8. Grosser M., Farkas E., Kunzinger M., Steinbauer R. *On the Foundations of Nonlinear Generalized Functions I and II*, Issue 729. American Mathematical Soc., 2001.

9. Rosinger E. E. *Non-Linear Partial Differential Equations*. Elsevier, 22 Nov. 1990.

10. Foukzon J., Men'kova E. R., Potapov A. A. *Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH*
Spacetime-Induced vacuum dominance. Book Publisher International 1. Published Nov. 14, 2019. DOI: https://doi.org/10.9734/bpi/mono/978-93-89562-22-4

11. Vernaeve H. Comm. Alg., 2010, **38** (6): 2199.

DOI: https://doi.org/10.1080/00927870903055222, arXiv:0707.0698 [math.RA]

12. Kunzinger M., Steinbauer R. Trans. Amer. Math. Soc., 2002, **354**: 4179. DOI: https://doi.org/10.1090/S0002-9947-02-03058-1

13. Grant J. D. E., Kunzinger M., Sämann C. Ann Glob Anal Geom (Dordr), 2019, **55**(1):

133. DOI: 10.1007/s10455-018-9637-x

14. Tozzi A. and Peters J. F. Br. J. Neurosci. Res. 2016, 94: 351.

15. Müller T., Grave F. arXiv: 0904.4184 [gr-qc] https://arxiv.org/abs/0904.4184v2

16. Foukzon J., Men'kova E. R., Potapov A. A., Podosenov S. A. J. Phys.: Conf. Ser., 2018, **1141**: 012100. DOI: https://doi:10.1088/1742-6596/1141/1/012100
Сведения об авторах

Фоукзон Джэйков (Jaykov Foukzon)

¹Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel.

Израильский технологический институт, г. Хайфа. 3200003 Израиль, г. Хайфа.

Профессор. Доктор физико-математических наук.

Адрес:

Rambam street 7-a, (flat) 2. ZIP code (Zone Improvement Plan) code (postal code) 6560119

Tel Aviv, Israel

Потапов Александр Алексеевич

Главный научный сотрудник ИРЭ им. В.А. Котельникова РАН. Профессор. Доктор
физико-математических наук. Президент совместной китайско-российской
лаборатории информационных технологий и фрактальной обработки сигналов,
Университет Гуанчжоу, Китай.

Chief researcher of V. A. Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences. 11, Mokhovaya str., Moscow, 125009, Russian Federation

Professor, Doctor of phys. and math. President of Joint Chinese-Russian Laboratory of Informational Technologies and Signals Fractal Processing, Jinan University, Guangzhou, 510632, China.

Менькова Елена Романовна

ФГУП «Всероссийский научно-исследовательский институт оптико-физических измерений» Росстандарта, город Москва. 119361 РФ, Москва, ул. Озерная, дом 46.

Старший научный сотрудник лаборатории генерирования и измерения параметров
электромагнитных импульсов. Кандидат технических наук.

Адрес

121354 Москва, ул. Кутузова, д. 8, кв. 72.

Контактные сведения: рабочий телефон 8-(495)437-28-47

домашний телефон 8-(499) 737-83-41

мобильный телефон 8-926-386-34-63