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

HAL
open science

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

Foukzon J, Men’Kova E R, Potapov a A

► To cite this version:

Foukzon J, Men'Kova E R, Potapov a A. SINGULAR GENERAL RELATIVITY USING COLOMBEAU APPROUCH. PART I. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH REGULARIZATION VIA HORIZON. 2023. hal-04129642

HAL Id: hal-04129642

https://hal.science/hal-04129642

Preprint submitted on 4 Jul 2023

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

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

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
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,
Moscow 119361, Russian Federation

³Kotelnikov Institute of Radio engineering and Electronics, Russian Academy
of Sciences, 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:** 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

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

We studied Schwarzschild solution using Colombeau distributional geometry, thus without leaving Schwarzschild coordinates $(t, r, \theta, \varphi)$. We argue that Schwarzschild solution impossible treated classically since, the Levi-Cività connection is not available on whole Schwarzschild spacetime $(\mathbf{Sch}, g_{ij}^{Sch}(t, r, \theta, \varphi))$, where $\mathbf{Sch} = \mathbb{R} \times ((\{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbf{S}^2)$, but only by using an embedding of the Schwarzschild metric tensor $g_{ij}^{Sch}$; $i, j = 1,2,3,4$ into Colombeau type algebra $\mathcal{G}(\mathbb{R}^4, \Sigma), \Sigma = \{r = 2m\} \cup \{r = 0\}$. This meant that classical Schwarzschild spacetime
extending up to distributional semi Riemannian manifold $(\mathbf{Sch}, (g_{ij}^{Sch}(t, r, \theta, \varphi, \varepsilon))_{\varepsilon})$
endowed on the tangent bundle with Colombeau distributional metric tensor
$(g_{ij,\varepsilon}^{Sch})_{\varepsilon} \triangleq (g_{ij}^{Sch}(t, r, \theta, \varphi, \varepsilon))_{\varepsilon}, \varepsilon \in (0,\delta]$ [M. Kunzinger, R.Steinbauer, Trans. Amer.
Math. Soc. **354** (2002), 4179-4199

https://doi.org/10.1090/S0002-9947-02-03058-1]. In this paper originally the
distributional metric tensor $(g_{ij,\varepsilon}^{Sch})_{\varepsilon}$ presented as Colombeau solution of the Einstein
field equations. Using Colombeau distributional techniques we calculate the
generalized Ricci tensor of the distributional Schwarzschild geometry. It turns out to
be a well-defined Colombeau tensor-distribution concentrated on the {$r = 2m$} $\cup$
{$r = 0$} region, but not only on the {$r = 0$} - region in contrast with classical result
[H Balasin and H Nachbagauer, Classical and Quantum Gravity, Volume 10, Number
11 https://doi.org/10.1088/0264-9381/10/11/010]. We obtain the distributional Ricci
tensor are of $\delta$-type, i.e. $(R_{0,\varepsilon}^0(r))_{\varepsilon} = (R_{1,\varepsilon}^1(r))_{\varepsilon} = -\varepsilon m \delta(r - 2m)$, $\varepsilon > 0$ and
$(R_{2,\varepsilon}^2(r))_{\varepsilon} = (R_{3,\varepsilon}^3(r))_{\varepsilon} = 0$. However the distributional square scalars are
essentially non classical Colombeau type distributions: $(R_{\varepsilon}^{\mu\nu}(r)R_{\mu\nu,\varepsilon}(r))_{\varepsilon}$,
$(R_{\varepsilon}^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu,\varepsilon}(r))_{\varepsilon} \in G(\mathbb{R})\backslash D'(\mathbb{R})$. This provides a new physical interpretation for
the distributional square scalars of this geometry [J Foukzon, E R Men'kova,
A. A. Potapov and S A Podosenov, 2018 J. Phys.: Conf. Ser. 1141 012100
https://doi.org/10.1088/1742-6596/1141/1/012100]

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.SINGULAR SEMI RIEMANNIAN GEOMETRY USING COLOMBEAU APPROUCH

The classical Cartan's structural equations¹,² show in a compact way the relation
between a connection and its curvature and reveals their geometric interpretation in
terms of moving frames. In order to study the mathematical properties of
singularities, we need to study the geometry of manifolds endowed on the tangent
bundle with a symmetric bilinear form which is degenerated (singular) on semi Riemannian manifold $(M, g)$ or on submanifolds $(M', g')$ of semi Riemannian manifold $(M, g)$, $M' \subset M$. But if the fundamental tensor is degenerated there are some principal obstructions in constructing the geometric objects normally associated to the fundamental tensor. Also, local orthonormal frames and coframes no longer exist, as well as the metric connection and its curvature operator$^{3,4,5,6,7}$.

**Definition 1.** (1) Semi Riemannian manifold $(M, g)$ is a nonclassical if the fundamental tensor $g$ is allowed to be degenerate or singular, (2) semi Riemannian manifold $(M, g)$ is internally nonclassical if the fundamental tensor $g$ is not allowed to be degenerate or singular but there exists semi Riemannian submanifold $(M', g')$, $M' \subset M$, $g' = g|_{M'}$ such that the fundamental tensor $g'$ is allowed to be degenerate or singular, (iii) otherwise we shall say that $(M, g)$ is classical.

**Remark 1.** In nonclassical case the main problem arises from the degeneracy of the $\det(g_{ij}(\hat{x}))$ on some isolated points: $\det(g_{ij}(\hat{x}^0)) = 0, \hat{x}^0 \in M$ or on some submanifold $\det(g_{ij}(\hat{x})) = 0$ (1)

for all $\hat{x} \in M' \subset M$ and consequently, the corresponding Christoffel symbols become infinity. Many years ago, A. Einstein$^8$ pointed out that under condition $\det(g_{ij}(\hat{x})) = 0$ the tensors $R^i_{klm}$ and $R_{kl}$ obviously take on the form 0/0 and therefore the field equations

$$R_{kl} = R^m_{klm} = 0 \quad (2)$$

become to absurdum

$$R_{kl} = R^m_{klm} = 0/0.$$

**Remark 2.** Note that in classical handbooks$^{8,9,10,11}$ the singularities

$$\det^{-1}(g_{ij}(\hat{x})) = \infty \quad (3)$$

which required from Eq. (1) are mistakenly considered as *unphysical* pure coordinate singularities since it has been easily deleted by using appropriate singular transform.

In mathematical literature more than 80 years it was accepted that a nonclassical semi Riemannian manifold mentioned above is *impossible to treat classically*, i.e. by using canonical apparatus of the semi Riemannian geometry$^{3,4,5,6}$. An up-to date bibliography on this topic is given in Pambira$^5$. However, in the contemporary mathematical literature, manifolds with degenerate metric tensors have been studied successfully only for some special case called a Reinhart manifolds$^{4,5}$.

**Remark 3.** In this paper we studied nonclassical semi Riemannian manifolds $(M, g)$ extrinsically i.e. as degenerate standard part of Colombeau generalized semi Riemannian manifolds furnished with nondegenerate Colombeau generalized

fundamental tensor$^{12}$ $(g_{ij,\epsilon})_{\epsilon} \in G_0^2(M)$, or super generalized fundamental tensor$^{13,14}$ $(g_{ij,\epsilon})_{\epsilon} \in G_0^2(M, \Sigma)$, see sect. III-V.

**Definition 2.** We shall say that Colombeau generalized semi Riemannian manifold
$(M, (g_{i,j,\epsilon}(\hat{x}))_{\epsilon})$ is the Colombeau extension of the nonclassical semi Riemannian manifold $(M, g_{i,j,0}(\hat{x}))$ if there exists the canonical imbedding $(M, g_{i,j,0}(\hat{x})) \hookrightarrow (M, (g_{i,j,\epsilon}(\hat{x}))_{\epsilon})$, see sect. III, where such imbedding is obtained for Schwarzschild spacetime with metric line element given by Eq. (8) in canonical Schwarzschild coordinates $(t, r, \theta, \varphi)$.

**Remark 4.** (1) The canonical imbedding $(M, g_{i,j,0}(\hat{x})) \hookrightarrow (M, (g_{i,j,\epsilon}(\hat{x}))_{\epsilon})$ is easily obtained by appropriate $\epsilon$-regularization$^{12-17}$ of the tensor $g_{i,j,0}(\hat{x})$, see section III.

**Remark 5.** Note that the notion of the *classical* Riemannian curvature comes from the study of parallel transport on the *classical* Riemannian manifold. For instance, if a vector $A_i(\hat{x})$ is moved around a closed contour (or a loop) on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This failure to return to the initial vector is measured by the holonomy of the connection. The classical holonomy presented below by the Eq.(4) for the change $\Delta A_k$ in a smooth vector $A_i(\hat{x})$ after parallel displacement around the infinitesimal closed contour $\Gamma$. The classical formula for the change in a smooth vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$ or integral measure corresponding to *holonomy* of the connection on the surface $\Sigma_{\Gamma}$ spanning by $\Gamma$ reads$^{1,2,4,9}$

$$ \Delta A_k(\Gamma) = \oint_{\Gamma} \delta A_k = \oint_{\Gamma} \Gamma_{kl}^{i}(\hat{x}) A_k dx^l. \quad (4) $$

Now applying classical Stokes' theorem to the integral in Eq. (4) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f^{im})$, one obtains$^{1,2,4,9}$

$$ \Delta A_k = \frac{1}{2} R_{klm}^i(x) A_i(x) \Delta f^{im}, \quad (5) $$

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

$$ R_{klm}^i(x) = \frac{\partial(\Gamma_{km}^i(x))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(x))}{\partial x^m} + \Gamma_{nl}^i(x)\Gamma_{km}^n(x) - \Gamma_{nm}^i(x)\Gamma_{kl}^n(x). \quad (6) $$

The tensor field $R_{klm}^i(x)$ is called the classical curvature tensor or the classical Riemann tensor. The classical Riemann tensor that is infinitesimal tensorial measure of holonomy.

(2) Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy. (3) Note that in classical case the change $\Delta A_k(\Gamma)$ is always finite, i.e. $\Delta A_k(\Gamma) < \infty$.

(4) If semi Riemannian manifold $(M, g)$ is nonclassical the condition $\Delta A_k(\Gamma) < \infty$ is not always holds and we shall say that the contour $\Gamma = \Gamma_{\text{reg}}$ is regular if $\Delta A_k(\Gamma) < \infty$, see Fig.1 (v) if $\Delta A_k(\Gamma) = \infty$ we shall say that the contour $\Gamma = \Gamma_{\text{sing}}$ is a singular contour or a singular loop.
<figure><img src="image_3.png" /><figcaption>Fig.1. Regular contour $\Gamma_{\text{reg}}$ beyond a black hole horizon, $\Delta A_k(\Gamma_{\text{reg}}) < \infty$.</figcaption></figure>

**Definition 3.** We shall say that a point $\hat{x}_0 \in \Sigma_\Gamma$ is a singular point of the surface $\Sigma_\Gamma \subset M$ if the Levi-Civitá connection is not available at point $\hat{x}_0$, i.e. some of the Christoffel symbols become infinite at point $\hat{x}_0$.

**Remark 6.** Note that any singular contour $\Gamma_{\text{sing}}$ contains at least one singular point

$\hat{x}_0 \in \Gamma_{\text{sing}}$, see Fig.2 and we shall abbreviate $\Gamma_{\text{sing}}^{\hat{x}_0}$ if $\hat{x}_0 \in \Gamma_{\text{sing}}^{\hat{x}_0}$.

**Remark 7.** (1) Note that the classical formula (4) holds only for regular loops but obviously, the Eq. (3) breaks down on singular loops $\Gamma_{\text{sing}}$, since $\Delta A_k(\Gamma_{\text{sing}}) = \infty$.

(2) Note that Eq. (5) and Eq. (6) no longer hold for singular loops $\Gamma_{\text{sing}}$. And therefore, the classical Ricci scalar and the classical Kretschmar scalar is not holds.
at BH horizon with $\Delta A_k(\Gamma_{sing}^{\hat{x}_0}) = \infty$. However, in classical literature$^{8,9,10,11}$ it was just pulled in general to the BH horizon by the ears.

<figure><img src="image_4.png" /><figcaption>Fig.2. Singular contour $\Gamma_{sing}^{\hat{x}_0}$ with a singular point</figcaption></figure>

$\hat{x}_0$ on a black hole horizon, $\Delta A_k(\Gamma_{sing}^{\hat{x}_0}) = \infty$.

Remark 8. Note that for truncated singular contour $\Gamma_{sing}^{\#} = \Gamma_{sing}^{\hat{x}_0}\{\hat{x}_0\}$, see Fig.3, the Levi-Civiltà connection is available at the whole contour $\Gamma_{sing}^{\#}$ and classical formula (3) reads

$$ \Delta A_k(\Gamma_{sing}^{\#}) = \oint_{\Gamma_{sing}^{\hat{x}_0}\{\hat{x}_0\}} \delta A_k = \oint_{\Gamma_{sing}^{\hat{x}_0}\{\hat{x}_0\}} \Gamma_{kl}^{i}(\hat{x}) A_k dx^l. \quad (7) $$

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

$\Delta A_k(\Gamma_{\hat{x}_0}\{\hat{x}_0\}) = \infty$ and therefore the Eq. (5) and Eq. (6) again no longer hold for truncated singular contour $\Gamma_{sing}^{\#}$.
<figure><img src="image_5.png" /><figcaption>Fig.3. Truncated singular contour $\Gamma^{\#} = \Gamma_{\text{sing}}^{\#} = \Gamma_{\text{sing}}^{\hat{x}_0} \setminus \{\hat{x}_0\}$ with deleted singular point $\hat{x}_0$ on a black hole horizon, $(\Delta A_k(\Gamma_{\text{sing}}^{\#}) = \infty)$.</figcaption></figure>

**Definition 4.** We shall say that a semi Riemannian manifold $(M, g)$ is a semi classical if the Levi-Cività connection is available at whole manifold $M$ and there exists at least one truncated singular contour $\Gamma_{\text{sing}}^{\#} = \Gamma_{\text{sing}}^{\hat{x}_0} \setminus \{\hat{x}_0\}$ such that $\Gamma_{\text{sing}}^{\#} \subset M$ and therefore $\Delta A_k(\Gamma_{\text{sing}}^{\#}) = \infty$.

**Remark 9.** Note that any semi Riemannian semiclassical manifolds $(M, g)$ are essentially different in comparison with the classical semi Riemannian manifold $(M^*, g^*)$, since there is no exists any truncated singular contour $\Gamma_{\text{sing}}^{\#}$ such that $\Gamma_{\text{sing}}^{\#} \subset M^*$.

**Definition 5.** (1) A semi Riemannian manifold $(M', g')$ is said to be an extension¹⁵ of a semi Riemannian manifold $(M, g)$ if there exists an isometric embedding $i: M \to M'$. A semi Riemannian manifold $(M, g)$. ) (2) A semi Riemannian manifold $(M, g)$ is extendible¹⁵ if there exists an extension $(M', g')$ of $(M, g)$.

**Example 1.** (1) 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 exterior Schwarzschild solution is defined on the subset $r > 2m$ of $\mathbb{R}^4$, which is topologically $\mathbb{R}^2 \times \mathbb{R}^2$. In this region, the metric takes the form

$$ds^2 = -\left(1 - \frac{r_s}{r}\right) (dt)^2 + \left(1 - \frac{r_s}{r}\right)^{-1} (dr)^2 + r^2 \left[(d\theta)^2 + \sin^2 \theta (d\varphi)^2\right]. \quad (8)$$

The Schwarzschild metric (8) is degenerated and singular as $r \to 2m$. We may rewrite the metric (8) in the following form
$$ds^2 = -(1 - \frac{r_s}{r}) \left[ dt - \left(1 - \frac{r_s}{r}\right)^{-1} dr \right] \left[ dt + \left(1 - \frac{r_s}{r}\right)^{-1} dr \right] + r^2 d\Omega^2$$

and define a new coordinate, $v \in \mathbb{R}$ by

$$dv = dt + \left(1 - \frac{r_s}{r}\right)^{-1} dr.$$

Rewriting the metric in terms of the coordinates $(v, r, \theta, \varphi)$, it then takes the form

$$ds^2 = -(1 - \frac{r_s}{r}) \left[ dv - 2 \left(1 - \frac{r_s}{r}\right)^{-1} dr \right] dv + r^2 d\Omega^2 =$$

$$ds^2 = -(1 - \frac{r_s}{r}) dv^2 + 2drdv + r^2 d\Omega^2. \quad (9)$$

We may use the Eq. (9) to define a new metric tensor $g'$, given by the above coordinate expression on the manifold, $M'$, defined by $v \in \mathbb{R}, r \in (0, \infty), (\theta, \varphi) \in S^2$. On restriction to the original manifold $M$, where $r > 2m$, the metric $g'|_M$ is isometric to the original metric $g$.

(2) Note that the original manifold $M$, where $r > 2m$ is a semi classical Lorentzian Manifold, since there obviously exist truncated singular loops $\Gamma_{\text{sing}}^{\#} \subset M$, see Fig.3.

(3) Note that in contrast with the original manifold $M$, where $r > 2m$ the manifold, $M'$ is the classical Lorentzian manifold, since $i(\Gamma_{\text{sing}}^{\#})=\Gamma_{\text{reg}}^{\#} = i(\Gamma_{\text{sing}}^{\hat{x}_0})\{i(\hat{x})\}$ and $\Delta A_k(\Gamma_{\text{reg}}^{\#}) < \infty$.

**Remark 10.** The semi Riemannian manifold $(M, g)$ which contain only truncated singular contours $\Gamma_{\text{sing}}^{\hat{x}_0}$ such that $\Gamma_{\text{sing}}^{\hat{x}_0} \subset M, \hat{x}_0 \notin M$, seems as classical semi Riemannian manifold, since (1) the Levi-Cività connection is available on whole space-time and (2) there exists an isometric embedding $i: M \to M'$ such that the metric $g'|_M$ is isometric to the original metric $g$ and the manifold $M'$ is allowed to be the classical semi Riemannian manifold. However, by example 1, the manifold $(M, g)$ is not the classical semi Riemannian manifold but only the semi classical semi Riemannian manifold.

**Remark 11.** Note that in order to obtain the full geometrical properties of BH horizon one needs the nonclassical definition of the integral representation of the holonomy corresponding to the surface $\Sigma_{\Gamma_{\text{sing}}}$ spanning by singular loop $\Gamma_{\text{sing}}$ such that $\Delta A_k(\Gamma_{\text{sing}})$ is well defined Colombeau generalized number$^{18}$, see sect. II.

**Definition 6.** By using contemporary Colombeau approach$^{16-25}$ one obtains appropriate definition of the nonclassical integral measure$^{25}$ $\Delta A_k(\Gamma_{\text{sing}})$ corresponding to the holonomy as direct generalization in natural way of the classical Eq. (4)

$$\Delta A_k(\Gamma_{\text{sing}}) \triangleq (\Delta A_{k,\varepsilon})_{\varepsilon} = (\oint_{\Gamma_{\text{sing}}^{\hat{x}_0}} \delta A_{k,\varepsilon})_{\varepsilon} = (\oint_{\Gamma_{\text{sing}}^{\hat{x}_0}} \Gamma_{kl,\varepsilon}^{i}(\hat{x}) A_{k,\varepsilon} dx^l)_{\varepsilon} \quad (10)$$

(see APPENDIX E), where point $\hat{x}^0$ belongs to BH horizon and $\hat{x}^0 \in \Gamma_{\text{sing}}^{\hat{x}^0}$ and $\Delta A_k(\Gamma_{\text{sing}}) \in \mathbb{R}$, where $\mathbb{R}$ is a ring of Colombeau generalized numbers$^{18,26}$, see sect. II for the full explanation.

**Definition 7.** Let $(M, g)$ be the nonclassical semi Riemannian manifold We shall say that a surface $\Sigma_{\Gamma_{\text{sing}}} \subset M$ is a singular surface if $\Sigma_{\Gamma_{\text{sing}}}$ it has at least one singular
point $\hat{x}_0 \in \Sigma_{\Gamma}^{\text{sing}}$, see Fig.4.

<figure><img src="image_6.png" /><figcaption>Fig.4. Singular surface $\Sigma_{\Gamma}^{\text{sing}}$ with singular point $r = 0$, spanning by regular contour $\Gamma$.</figcaption></figure>

**Remark 12.** Note that for the case of the singular surface $\Sigma_{\Gamma}^{\text{sing}}$ by impossibility classical Stokes' theorem, again problem arises even if the surface $\Sigma_{\Gamma}^{\text{sing}}$ is spanned by the regular contour $\Gamma$, see Fig.4. In this case the nonclassical correct integral measure $\Delta A_k(\Gamma, \Sigma_{\Gamma}^{\text{sing}})$ corresponding to the holonomy reads:

$$ \Delta A_k(\Gamma, \Sigma_{\Gamma}^{\text{sing}}) \triangleq (\Delta A_{k,\varepsilon})_{\varepsilon} = (\oint_{\Gamma} \delta A_{k,\varepsilon})_{\varepsilon} = (\oint_{\Gamma} \Gamma_{kl,\varepsilon}^{i}(\hat{x}, \hat{x}_0) A_{k,\varepsilon} dx^l)_{\varepsilon}. \quad (11) $$

**Definition 8.** Let $M$ be a differentiable manifold equipped with it canonical topology $\mathfrak{F}$. Let $M^{\#}$ be a submanifold $M^{\#} \subset M$. The closure $M_{\mathfrak{F}}^{\#}$ of a submanifold $M^{\#} \subset M$ of points in a topological space $(M, \mathfrak{F})$ consists of all points in $M^{\#}$ together with all limit points of $M^{\#}$. The closure $M_{\mathfrak{F}}^{\#}$ may equivalently be defined as the union of $M^{\#}$ and its boundary $\partial M^{\#}$, and also as the intersection of all closed sets containing $M^{\#}$. Intuitively, the closure can be thought of as all the points that are either in $M^{\#}$ or "near" $M^{\#}$. A point which is in the closure of $M^{\#}$ is a point of closure of $M^{\#}$.

**Definition 9.** Let $(M, g)$ and $(M^{\#}, g^{\#})$ are semi Riemannian manifolds such that $(1) \ M^{\#} \subset M$, (ii) $M_{\mathfrak{F}}^{\#} = M$ and (iii) $g^{\#} = g|_{M^{\#}}$. Assume that (a) the Levi-Cività connection $\Gamma_{kj}^{\#l} = \frac{1}{2}[g^{lm}][(g_{mk,j}^{\#} + g_{mj,k}^{\#} - g_{kj,m}^{\#})]$ corresponding to metric tensor $g^{\#}$ is available at whole semi Riemannian manifold $(M^{\#}, g^{\#})$ and (b) the Levi-Civita connection $\Gamma_{kj}^{l} = \frac{1}{2}[g^{lm}][(g_{mk,j} + g_{mj,k} - g_{kj,m}^{\#})]$ corresponding to metric tensor $g$ is not available at its boundary $\partial M^{\#}$, i.e. $(M, g)$ is a nonclassical semi Riemannian manifold. Then, we shall say that $(M^{\#}, g^{\#})$ is a *semiclassical* semi Riemannian manifold.

**Remark 13.** It follows from consideration above that semiclassical semi Riemannian Manifolds are obviously *impossible to treat classically* as nonclassical semi Riemannian manifolds mentioned above.

**Example 2.** Obviously, the Levi-Cività connection is available at whole Schwarzschild
spacetime $(\mathbf{Sch}, g_{ij}^{\text{sc}\square}(t, r, \theta, \varphi))$, where $\mathbf{Sch} = (S^2 \times \{r > 2m\} \cup \{0 < r < 2m\}) \times \mathbb{R}$, but spacetime $\mathbf{Sch}$ contains truncated singular loops $\Gamma_{\text{sing}}^{\#} = \Gamma_{\hat{x}_0} \setminus \{\hat{x}_0\} \subset \mathbf{Sch}$, see Fig.5, and in particular the Levi-Cività connection is not available at whole it topological closure $(\overline{\mathbf{Sch}}, g_{ij}^{\text{sch}}(t, r, \theta, \varphi))$, where $\overline{\mathbf{Sch}} = (S^2 \times (\{r \ge 2m\} \cup \{0 \le r \le 2m\})) \times \mathbb{R}$ and $\hat{x}_0 \in \overline{\mathbf{Sch}}$.

Thus, Schwarzschild spacetime $\mathbf{Sch}$ is not classical but exactly is a semiclassical semi Riemannian manifold. Similarly, the Levi-Cività connection is available on the open semi Riemannian manifold $(\mathbf{Sch}^{+}, g_{ij}^{\text{sc}\square^{+}}(t, r, \theta, \varphi))$ above Schwarzschild horizon: $\mathbf{Sch}^{+} = (S^2 \times \{r > 2m\}) \times \mathbb{R}$ but it is not available at whole its topological closure $(\overline{\mathbf{Sch}}^{+}, g_{ij}^{\text{sc}\square^{+}}(t, r, \theta, \varphi)), \overline{\mathbf{Sch}}^{+} = (S^2 \times \{r \ge 2m\}) \times \mathbb{R}$

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

**Remark 14.** Note that in physical literature the spacetime $\mathbf{Sch}^+ = (S^2 \times \{r > 2m\}) \times \mathbb{R}$ is mistakenly considered as classical semi Riemannian manifold. Note that only spacetimes $(\mathbf{Sch}_{\delta}^{+}, g_{ij}^{\overline{\text{Sch}}_{\delta}^{+}})$, $\mathbf{Sch}_{\delta}^{+} = (S^2 \times \{r > 2m + \delta\}) \times \mathbb{R}, \delta > 0$ are classical semi Riemannian manifolds, see Fig.5.

<figure><img src="image_7.png" /></figure>
Fig.5. Region (I) contains a classical semi Riemannian manifolds $(\mathbf{Sch}_s^+, g_{ij}^{\overline{\mathbf{Sch}}_s^+})$. with $\delta > 0$

Remark 15. There exist canonical examples of the nonclassical semi Riemannian manifolds. However, in canonical physical literature$^{1,2,8,9,27}$ these nonclassical semi Riemannian manifolds mistakenly were considered as classical semi Riemannian manifolds.

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

$$ds^2 = -\left(1 - \frac{r_s}{r}\right)(dt)^2 + \left(1 - \frac{r_s}{r}\right)^{-1}(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta (d\varphi)^2]. \quad (12)$$

The Christoffel symbols in Schwarzschild coordinates $(t, r, \theta, \varphi)$ read$^{8,9,27}$

$$\Gamma_{tt}^r = \frac{c^2 r_s (r-r_s)}{2r^3}, \quad \Gamma_{tr}^t = \frac{r_s}{2r(r-r_s)}, \quad \Gamma_{rr}^r = -\frac{r_s}{2r(r-r_s)}, \quad \Gamma_{r\theta}^\varphi = \frac{1}{r}, \quad \Gamma_{r\varphi}^\varphi = \frac{1}{r}, \\ \Gamma_{\theta\theta}^r = -(r-r_s), \quad \Gamma_{\theta\varphi}^\varphi = \cot\theta, \quad \Gamma_{\varphi\varphi}^r = -(r-r_s)[\sin\theta]^2, \quad \Gamma_{\varphi\varphi}^\theta = -\sin\theta \cos\theta.$$

Remark 16. Note that: (1) the Schwarzschild metric (12) is _degenerated_ and _singular_ on Schwarzschild horizon {$r = r_s$}, (2) the Schwarzschild metric (12) is _degenerated_ and _singular_ on Schwarzschild singularity {$r = 0$}, (2) the Christoffel symbols $\Gamma_{tr}^t$ and $\Gamma_{rr}^r$ are infinite on Schwarzschild horizon {$r = r_s$}. Thus, the full Schwarzschild space-time is the nonclassical semi Riemannian manifold. (3) L.D. Landau$^8$ mistakenly argued that there is no physical singularity at Schwarzschild horizon {$r = r_s$}, since the determinant $\det(g_{ij}^{Sch}) = -r^4\sin\theta$ has no singularity at $r = r_s$. However L.D. Landau obviously was wrong, since the equality $\det(g_{ij}^{Sch}(r)) = -r^4\sin\theta$ without any doubts holds if and only if $r \neq r_s$. Obviously at $r = r_s$ one obtains uncertainty $\det(g_{ij}^{Sch}(r_s)) = 0/0$ and nothing more. Such uncertainty can be deleted only by taking the limit $\lim_{r \to r_s} \det(g_{ij}^{Sch}(r)) = -r_s^4\sin\theta$. However, in the limit $r \to r_s$ the Christoffel symbols $\Gamma_{tr}^t$ and $\Gamma_{rr}^r$ become infinite, since the Levi-Cività connection is not available at Schwarzschild horizon {$r = r_s$} and therefore L.D. Landau argumentation gives nothing rigorous mathematical information about physical nature of the singularity at Schwarzschild horizon.

Remark 17. We studied Schwarzschild solution using Colombeau distributional geometry, thus _without leaving_ Schwarzschild coordinates $(t, r, \theta, \varphi)$. The full nonlinear distributional Schwarzschild geometry at horizon is obtained in sect V.

2. The _singular_ transformation of the Schwarzschild metric (12) from the usual Schwarzschild time coordinate $t$ to the advanced null coordinate $v$ with

$$cv = ct + r + r_s \ln(r - r_s) \quad (13)$$

leads to the ingoing Eddington-Finkelstein metric (9) with coordinates $(v, r, \theta, \varphi)$

The Schwarzschild metric (14) in Eddington-Finkelstein coordinates $(v, r, \theta, \varphi)$ reads

$$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dv^2 + 2cdvdr + r^2 d\Omega^2. \quad (14)$$

The Christoffel symbols in Eddington-Finkelstein coordinates $(v, r, \theta, \varphi)$, read$^{8,9,27}$

$$\Gamma_{vv}^v = \frac{cr_s}{2r^2}, \quad \Gamma_{vv}^r = \frac{c^2 r_s (r-r_s)}{2r^3}, \quad \Gamma_{vr}^r = -\frac{cr_s}{2r^2}, \quad \Gamma_{r\theta}^\varphi = \frac{1}{r}, \quad \Gamma_{r\varphi}^\varphi = \frac{1}{r}, \quad \Gamma_{\theta\theta}^v = -\frac{r}{c}, \\ \Gamma_{\theta\theta}^r = -(r-r_s), \quad \Gamma_{\theta\varphi}^\varphi = \cot\theta, \quad \Gamma_{\varphi\varphi}^v = \frac{-r \sin^2\theta}{c}, \quad \Gamma_{\varphi\varphi}^r = -(r-r_s)[\sin\theta]^2, \\ \Gamma_{\varphi\varphi}^\varphi = -\sin\theta \cos\theta.$$

Remark 18. The Eddington-Finkelstein space-time is the classical semi Riemannian
manifold except any finite neighborhood of singularity {$r = 0$}.

Remark 19. Note, that Eddington-Finkelstein space-time is not equivalent with Schwarzschild space-time, since the transform (13) is singular on Schwarzschild horizon {$r = r_s$}, see Example 1.

3. The Schwarzschild metric (12) in isotropic coordinates $(t, \rho, \theta, \varphi)$ reads$^{8,9,27}$

$$ds^2 = - \left(\frac{1-r_s/4\rho}{1+r_s/4\rho}\right)^2 c^2 dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 \left[d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\varphi^2)\right], \quad (15)$$

where

$$r = \left(1 + \frac{r_s}{4\rho}\right)^2 \quad (16)$$

is the singular coordinate transformation between the Schwarzschild radial coordinate $r$ and the isotropic radial coordinate $\varphi$.

The Christoffel symbols read$^{27}$

$$\Gamma_{tt}^{\rho} = 2048 \frac{(4\rho-r_s)\rho^4 r_s c^2}{(4\rho+r_s)^7}, \quad \Gamma_{t\rho}^{t} = \frac{8r_s}{16\rho^2-r_s^2}, \quad \Gamma_{\rho\rho}^{\rho} = -\frac{2r_s}{(4\rho+r_s)\rho}, \\ \Gamma_{\rho\theta}^{\theta} = \frac{4\rho-r_s}{(4\rho+r_s)\rho}, \quad \Gamma_{\rho\varphi}^{\varphi} = \frac{4\rho-r_s}{(4\rho+r_s)\rho}, \quad \Gamma_{\theta\theta}^{\rho} = -\rho \frac{4\rho-r_s}{(4\rho+r_s)}, \quad \Gamma_{\theta\varphi}^{\varphi} = \cot\theta,$$

$$\Gamma_{\varphi\varphi}^{\rho} = -\frac{(4-r_s)\rho \sin^2 \theta}{4\rho+r_s}, \quad \Gamma_{\varphi\varphi}^{\theta} = -\sin \theta \cos \theta$$

Remark 20. Note that (1) the metric (15) is _degenerated_ on Schwarzschild horizon {$\rho = r_s/4$}, (2) the metric (15) is singular only on Schwarzschild singularity {$\rho = 0$}, (3) the Christoffel symbol $\Gamma_{t\rho}^t$ is infinite on Schwarzschild horizon $\rho = r_s/4$. Thus, the full Schwarzschild space-time is the nonclassical semi Riemannian manifold.

Remark 21. We studied Schwarzschild solution (15) in isotropic coordinates using Colombeau distributional geometry, thus _without leaving isotropic coordinates_ $(t, \rho, \theta, \varphi)$. The full nonlinear distributional Schwarzschild geometry at horizon in isotropic coordinates $(t, \rho, \theta, \varphi)$ is obtained in sect VI.

4. The Möller metric line element reads$^{17}$

$$ds^2 = -(a + gx)^2 dt^2 + dx^2 + dy^2 + dz^2. \quad (17)$$

The Möller metric with $a = 0$ is called the Rindler metric.

The Rindler metric reads

$$ds^2 = -(gx)^2 dt^2 + dx^2 + dy^2 + dz^2.$$

The Möller's metric (17) is degenerates at Möller horizon $x_{hor} = -a/g$. Note that formally corresponding to the Möller's metric (12) classical Levi-Cività connection read$^{17}$

$$\Gamma_{44}^{1}(x) = g(a + gx), \quad \Gamma_{14}^{4}(x) = \Gamma_{41}^{4}(x) = g(a + gx)^{-1}. \quad (18)$$

Remark 22. The Möller metric (17) is degenerated on Möller horizon $x_{hor} = -a/g$. and therefore classical Levi-Cività connection (18) of course is not available at Möller horizon since at horizon formal expression (18) becomes infinity: $\Gamma_{14}^4(-a/g) = \Gamma_{14}^4(-a/g) = \infty$. Thus, the Möller space-time is a nonclassical semi Riemannian manifold.
<figure><img src="image_8.png" /><figcaption>Fig.6. Parallel transportation of a vector $A_k, k = 0,1$ around a contour $\Gamma_{sing}$ in Möller spacetime. (1) Singular surface $\Sigma^* = \Sigma_{\Gamma}^{sing}$ spanned by singular contour $\Gamma_{sing}$ (solid curve). (2) Singular contour $\Gamma_{sing}$ with singular point $(0, x_{hor})$ on Möller horizon, where $x_{hor} = -\frac{a}{g}$. (3) $\Delta A_k(\Gamma_{sing}) = \infty$.</figcaption></figure>

Remark 23. Note that in physical literature Möller spacetime mistakenly considered as classical semi Riemannian manifold. The full nonlinear distributional Möller geometry¹⁷ is obtained in sect. VIII.

We rewrite now the Moller metric (17) in the following equivalent form

$$ds^2 = -(1+\frac{gx}{a^2})^2 a^2 dt^2 + dx^2 + dy^2 + dz^2. \quad (19)$$

We set now

$$1 + \frac{gx}{a^2} = \exp\left(\frac{gu}{a^2}\right) \quad (20)$$

From Eq. (20)

one obtains

$$dx = (du)\exp\left(\frac{gu}{a^2}\right) \quad (21)$$

From Eq. (18) and Eq. (21) one obtains
$$ds^2 = -(a^2 dt^2 - du^2)\exp\left(\frac{2gu}{a^2}\right) + dy^2 + dz^2. \quad (22)$$

Remark 24. Obviously, the space-time is endowed with metric (22) in contrast with original Möller spacetime, that is the classical semi Riemannian manifold.

## II. 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 distributions¹⁴. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras¹⁸⁻²⁶ provide a rigorous framework for this.

Remark 25. 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. Colombeau¹⁸.

As a mathematical tool, Colombeau algebras¹⁸ can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity²¹ so far.

**Definition 10.** The algebra moderate functions $\mathcal{E}_M^\infty(\mathbb{R}^n)$ on $\mathbb{R}^n$ is the algebra of families of smooth functions $(f_\epsilon(x))_\epsilon \triangleq (f_\epsilon(x))_{\epsilon, x \in \mathbb{R}^n, \epsilon \in (0, \delta], \delta \le 1}$ (smooth $\epsilon$-regularisations, where $\epsilon$ is the regularization parameter), such that: for all compact subsets $K$ of $\mathbb{R}^n$ and all multi indices $\alpha$ there is an $N > 0$ such that

$$\sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_\epsilon(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\epsilon^{-N}), \epsilon \to 0, \quad (23)$$

with addition and multiplication defined by natural way:

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

$$(f_{\epsilon}(x))_{\epsilon} \times (g_{\epsilon}(x))_{\epsilon} = (f_{\epsilon}(x) \times g_{\epsilon}(x))_{\epsilon}. \quad (24)$$

**Definition 11.** The ideal $\mathcal{N}_\delta(\mathbb{R}^n)$ of negligible functions is defined in the same way but with the partial derivatives instead bounded by $O(\epsilon^N)$ for all $N > 0$, i.e.

$$\sup_{x \in K} \left| \frac{\partial^{|\alpha|} f_{\epsilon}(x)}{(\partial x_1)^{\alpha_1} \cdots (\partial x_n)^{\alpha_n}} \right| = O(\epsilon^N), \epsilon \to 0. \quad (25)$$

**Definition 12.** The Colombeau Algebra $G_\delta(\mathbb{R}^n)$ is defined as the quotient¹⁸ algebra $G_\delta(\mathbb{R}^n) = \mathcal{E}_M^\infty(\mathbb{R}^n)/\mathcal{N}_\delta(\mathbb{R}^n)$. Elements of $G_\delta(\mathbb{R}^n)$ are denoted by:

$$u = \mathrm{cl}[(u_{\epsilon})_{\epsilon}] \triangleq (u_{\epsilon})_{\epsilon} + N_{\delta}(\mathbb{R}^n).$$
## 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 26.** 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 3.** 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 $\epsilon \to 0$

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

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

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

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

**Remark 27.** 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 has well defined value at any point $x \in \mathbb{R}^n$, and these point values of the Colombeau generalized functions is the Colombeau generalized numbers$^{18,26}$.

**Example 4.** Delta function $\delta(x)$ 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

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

Designation 1. (1) We denote by $\tilde{\mathbb{R}}_\delta, \delta \ll 1$ the ring of real Colombeau generalized numbers$^{18,26}$. Recall that by definition $\tilde{\mathbb{R}}_\delta = \varepsilon_\delta(\mathbb{R})/\mathcal{N}_\delta(\mathbb{R})$ where

$$ \varepsilon_\delta(\mathbb{R}) = \{ (x_\varepsilon)_\varepsilon \in \mathbb{R}^{(0,\delta)} \mid (\exists a \in R_+) (\exists \varepsilon_0 \in (0,1)) (\forall \varepsilon \le \varepsilon_0) [|x_\varepsilon| \le \varepsilon^{-a}] \}, $$

$$ \mathcal{N}_\delta(\mathbb{R}) = \{ (x_\varepsilon)_\varepsilon \in \mathbb{R}^{(0,\delta)} \mid (\forall a \in R_+) (\exists \varepsilon_0 \in (0,1)) (\forall \varepsilon \le \varepsilon_0) [|x_\varepsilon| \le \varepsilon^a] \}. $$

(2) Notice that the ring $\tilde{\mathbb{R}}_\delta$ arises naturally as the ring of constants of the Colombeau algebras $\tilde{G}_\delta(\Omega)$. 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 13. We say that $r$ is 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 14. (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 $\check{\delta} \approx_{\tilde{\mathbb{R}}} 0$ 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 often abbreviated $\check{\delta} \approx 0$ instead $\check{\delta} \approx_{\tilde{\mathbb{R}}} 0$

(3) 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 $\Delta^{-1} \approx_{\tilde{\mathbb{R}}_\delta} 0$.

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

(5) 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 $\tilde{\mathbb{R}}_{\delta\pm\infty} = \varepsilon_\delta(\mathbb{R}_{\pm;\infty})/\mathcal{N}_\delta(\mathbb{R}_{\pm;\infty})$. (6) We define the set of finite Colombeau numbers $\mathcal{F}(\tilde{\mathbb{R}}_\delta)$ as follows: $\mathcal{F}(\tilde{\mathbb{R}}_\delta) = \{x \in \tilde{\mathbb{R}}_\delta \mid |x| < n \text{ for some } n \in \mathbb{N}\}$.

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

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

**Definition 15. (Standard Part Mapping).** (1) The standard part mapping st: $\mathcal{F}(\tilde{\mathbb{R}}_{\delta}) \to \mathbb{R}$ is defined by the formula:

$$ \mathrm{st}(x) = \sup\{r \in \mathbb{R} \mid r \le \tilde{\mathbb{R}}_{\delta} x\}. \quad (26) $$

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 $\mathrm{st}(y) \ne 0$.

**Definition 16.** 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 17. (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

$$ \text{p.v.} \{\mathrm{cl}[(f_{\varepsilon}(x))_{\varepsilon}]\} = \sup_{\varepsilon \in (0,1]} |f_{\varepsilon}(x)|. \quad (27) $$

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

Let $u = (u_{\varepsilon}(x))_{\varepsilon}$ be Colombeau generalized function such that $u \in \mathcal{E}_{\mathbb{R}}^{\infty}$ 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

$$ \tilde{u}[(\mu_{\varepsilon})_{\varepsilon}] = (u_{\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} \in \mathcal{E}_{\delta}(\mathbb{R}) \quad (28) $$

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 $u_1, u_2 \in \mathcal{E}_{\mathbb{R}}^{\infty}$. The addition $\tilde{u}_1 + \tilde{u}_2$ and multiplication $\tilde{u}_1 \cdot \tilde{u}_2$ by using Eq. (28) is defined in natural way by formulas

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

$$ (\tilde{u}_1 \cdot \tilde{u}_2)[(\mu_{\varepsilon})_{\varepsilon}] = (u_{1,\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} \cdot (u_{\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} = (u_{1,\varepsilon}(\mu_{\varepsilon}) \cdot u_{2,\varepsilon}(\mu_{\varepsilon}))_{\varepsilon}. \quad (29) $$

**Definition 18.** The algebra moderate functions $\mathcal{E}_{\mathbb{R}}^{\infty}(\tilde{\mathbb{R}})$ 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. (28) 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 $\varepsilon \le \varepsilon_0$
$$ \left( \sup_{\mu_{\varepsilon} \in \mathbb{K}} \left| \frac{d^m u_{\varepsilon}(\mu_{\varepsilon})}{d\mu_{\varepsilon}^m} \right| \right)_{\varepsilon} = O((\varepsilon^{-N})_{\varepsilon}), \quad (30) $$

and with addition and multiplication defined by Eq. (29).

**Definition 19.** The ideal $N_{\delta}(\tilde{\mathbb{R}})$ 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.,

$$ \left( \sup_{\mu_{\varepsilon} \in \mathbb{K}} \left| \frac{d^m u_{\varepsilon}(\mu_{\varepsilon})}{d\mu_{\varepsilon}^m} \right| \right)_{\varepsilon} = O((\varepsilon^N)_{\varepsilon}). \quad (31) $$

for all $\varepsilon \le \varepsilon_0$.

**Definition 20.** The Colombeau Algebra $G_{\delta}(\tilde{\mathbb{R}})$ is defined as the quotient algebra $G_{\delta}(\tilde{\mathbb{R}}) = \varepsilon_{\mathcal{M}}^{\infty}(\tilde{\mathbb{R}})/N_{\delta}(\tilde{\mathbb{R}})$. 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 21.** The Colombeau Algebra $G_{\delta}(\tilde{\mathbb{R}}^n)$ is defined as the quotient algebra $G_{\delta}(\tilde{\mathbb{R}}) = \varepsilon_{\mathcal{M}}^{\infty}(\tilde{\mathbb{R}}^n)/N_{\delta}(\tilde{\mathbb{R}}^n)$ where $\varepsilon_{\mathcal{M}}^{\infty}(\tilde{\mathbb{R}}^n)$ and $N_{\delta}(\tilde{\mathbb{R}}^n)$ is defined similarly as $\varepsilon_{\mathcal{M}}^{\infty}(\tilde{\mathbb{R}})$ and $N_{\delta}(\tilde{\mathbb{R}})$ by obvious way. (1) Let $u = (u_{\varepsilon}(x))_{\varepsilon}$ be Colombeau generalized function such that $u \in \varepsilon_{\mathcal{M}}^{\infty}(\tilde{\mathbb{R}}^n)$ 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 $\mu_i \in \tilde{\mathbb{R}}_{\delta}$ with representatives $(\mu_{i,\varepsilon})_{\varepsilon} \in \varepsilon_{\delta}(\tilde{\mathbb{R}})$. Thus we have a mapping $\tilde{u}: \varepsilon_{\delta}(\tilde{\mathbb{R}}^n) \to \varepsilon_{\delta}(\tilde{\mathbb{R}})$ is defined by the formula

$$ \tilde{u}[(\mu_{\varepsilon})_{\varepsilon}] = (u_{\varepsilon}(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon}))_{\varepsilon} \in \varepsilon_{\delta}(\tilde{\mathbb{R}}). \quad (32) $$

(2) The algebra moderate functions $\varepsilon_{\mathcal{M}}^{\infty}(\tilde{\mathbb{R}}^n)$ on $\tilde{\mathbb{R}}^n$ is the algebra of a functions $\tilde{u}: \varepsilon_{\delta}(\tilde{\mathbb{R}}^n) \to \varepsilon_{\delta}(\tilde{\mathbb{R}})$ 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 $\varepsilon \le \varepsilon_0$

$$ \left( \sup_{x \in \mathbb{K}} \left| \frac{\partial^{|\alpha|} u_{\varepsilon}(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon})}{(\partial \mu_{1,\varepsilon})^{\alpha_1} \dots (\partial \mu_{n,\varepsilon})^{\alpha_n}} \right| \right)_{\varepsilon} = O((\varepsilon^{-N})_{\varepsilon}), \quad (33) $$

and with addition and multiplication defined by natural way by formulas

$$ \begin{align} (\tilde{u}_1 + \tilde{u}_2)[(\mu_{\varepsilon})_{\varepsilon}] &= (u_{1,\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} + (u_{\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} = (u_{1,\varepsilon}(\mu_{\varepsilon}) + u_{2,\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} \nonumber \\ (\tilde{u}_1 \cdot \tilde{u}_2)[(\mu_{\varepsilon})_{\varepsilon}] &= (u_{1,\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} \cdot (u_{\varepsilon}(\mu_{\varepsilon}))_{\varepsilon} = (u_{1,\varepsilon}(\mu_{\varepsilon}) \cdot u_{2,\varepsilon}(\mu_{\varepsilon}))_{\varepsilon}. \tag{34} \end{align} $$

(3) The ideal $N_{\delta}(\tilde{\mathbb{R}})$ 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.,

$$ \left( \sup_{x \in \mathbb{K}} \left| \frac{\partial^{|\alpha|} u_{\varepsilon}(\mu_{1,\varepsilon}, \dots, \mu_{i,\varepsilon}, \dots, \mu_{n,\varepsilon})}{(\partial \mu_{1,\varepsilon})^{\alpha_1} \dots (\partial \mu_{n,\varepsilon})^{\alpha_n}} \right| \right)_{\varepsilon} = O((\varepsilon^N)_{\varepsilon}), \quad (34) $$

for all $\varepsilon \le \varepsilon_0$.

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

### III. 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$^{18-24,28-36}$ and contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry$^{12,28,29}$. The regularizations of singularities present in some
solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions such that Schwarzschild solution etc. of the Einstein field equations recognized in the sense of Colombeau generalized functions and not classically. Note that in physical literature these singular solutions many years were mistakenly considered as vacuum solutions of the Einstein field equations$^{8,9}$.

Remark 28. Let $(R_{bcd,\epsilon}^a) \in G_\delta(\mathbb{R}^4)$ be Colombeau generalized function obtained using the standard definition of the Riemann curvature in a coordinate basis, i.e.

$$ (R_{bcd,\epsilon}^a) = (\Gamma_{db,c,\epsilon}^a) - (\Gamma_{cb,d,\epsilon}^a) + (\Gamma_{cf,\epsilon}^a \Gamma_{ab,\epsilon}^f) - (\Gamma_{df,\epsilon}^a \Gamma_{cb,\epsilon}^f), \quad (35) $$

connection coefficients in terms of the regularized metric $(g_{ab,\epsilon})_\epsilon$, where $\epsilon \in (0, \delta]$ such that $(g_{ab,\epsilon})_\epsilon, (g_{ab,\epsilon}^{ab})_\epsilon \in G_\delta(\mathbb{R}^4), (\det(g_{ab,\epsilon}))_\epsilon \neq 0_{\mathbb{R}}$. It has been shown by many authors that under appropriate regularization using the Eq. (35) one can defines the curvature scalar as a classical Schwartz distribution in $D'(\mathbb{R}^4)$.

Remark 29. 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

$$ ds^2 = -(1 - \frac{a}{\hat{r}})(d\hat{x}^0)^2 + (1 - \frac{a}{\hat{r}})^{-1}(d\hat{r})^2 + \hat{r}^2[(d\theta)^2 + \sin^2 \theta (d\varphi)^2]. \quad (36) $$

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 is degenerate or singular: (i) at $\hat{r} = 0$ and (ii) at $\hat{r} = a$.

Remark 30. Note that: (1) in canonical papers$^{18-26}$ the Colombeau distributional metric tensor $(g_{ab,\epsilon})_\epsilon \in G_\delta(\mathbb{R}^4)$ related to $(R_{bcd,\epsilon}^a) \in G_\delta(\mathbb{R}^4)$ by Eq. (35) never is not considered as the Colombeau solution of the Einstein field equations, (2) Colombeau nonlinear distributional geometry never is not considered as the rigorous mathematical model related to really physical spacetime but only as useful purely mathematical tools in order to obtain related to $(R_{bcd,\epsilon}^a) \in G_\delta(\mathbb{R}^4)$ classical Schwartz distributions in $D'(\mathbb{R}^4)$, (3) there is no any important physical applications of the classical linear distributional geometry were obtained.

Remark 31. Originally fundamental physical applications of the Colombeau nonlinear distributional geometry has been obtained originally in authors papers$^{14,25}$. By using now the Cartesian coordinates $(\hat{x}^0, \hat{x}^1, \hat{x}^2, \hat{x}^3)$, which are related to $(\hat{x}^0, \hat{r}, \theta, \varphi)$ through the canonical relations:

$$ \begin{aligned} \hat{x}^1 &= \hat{r} \cos \varphi \sin \theta, \quad \hat{x}^2 = \hat{r} \sin \varphi \sin \theta, \quad \hat{x}^3 = \hat{r} \cos \theta, \quad \text{the metric (36) reads} \\ ds^2 &= \hat{g}_{\mu\nu} d\hat{x}^\mu d\hat{x}^\nu, \quad \text{where at points } \hat{r} \neq 0, \hat{r} \neq a \quad \text{the metric } \hat{g}_{\mu\nu} \text{ is given by} \\ \hat{g}_{00} &= -(1-h), \quad \hat{g}_{0\alpha} = 0, \\ \hat{g}_{\alpha\beta} &= \delta^{\alpha\beta} + h(1-h)^{-1} \frac{\hat{x}^\alpha \hat{x}^\beta}{\hat{r}^2}, \quad \alpha, \beta = 1,2,3 \quad (37) \end{aligned} $$

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

$$ \kappa \hat{T}_0^0 = -\frac{h'}{\hat{r}} - \frac{h}{\hat{r}^2}, $$

$$ \kappa \hat{T}_0^\alpha = 0, $$
$$ \kappa \hat{T}_{\alpha}^{0} = 0, \kappa \hat{T}_{\alpha}^{\beta} = \delta_{\alpha}^{\beta} \left(-\frac{h''}{2} - \frac{h'}{\hat{r}}\right) + \frac{\hat{x}^{\alpha} \hat{x}^{\beta}}{\hat{r}^2} \left(\frac{h''}{2} - \frac{h}{\hat{r}^2}\right), \quad (38) $$

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 32. We extend now the quantity (37)-(38) 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 (37) by the well defined Colombeau generalized metric

$$ ds^2 = (\hat{g}_{\mu\nu,\varepsilon} d\hat{x}^{\mu} d\hat{x}^{\nu})_{\varepsilon}, \quad (39) $$

where

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

$$ (\hat{g}_{\alpha\beta,\varepsilon})_{\varepsilon} = \delta^{\alpha\beta} + ((h_{\varepsilon}(1 - h_{\varepsilon})^{-1})_{\varepsilon}) \left(\frac{\hat{x}^{\alpha}\hat{x}^{\beta}}{\hat{r}^2 + \varepsilon^2}\right)_{\varepsilon}, \quad \alpha, \beta = 1,2,3 \quad (40) $$

and therefore

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

$$ \kappa(\hat{T}_{\alpha}^{\beta}(\hat{x}; \varepsilon))_{\varepsilon} = \delta_{\alpha}^{\beta} \left( \frac{3a\varepsilon^2}{2(\hat{r}^2 + \varepsilon^2)^{5/2}} \right)_{\varepsilon} - \left( \left( \frac{\hat{x}^{\alpha}\hat{x}^{\beta}}{(\hat{r}^2 + \varepsilon^2)} \right)_{\varepsilon} \left( \frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}} \right)_{\varepsilon} \left( \frac{5}{2} + \left( \frac{\varepsilon^2}{\hat{r}^2 + \varepsilon^2} \right)_{\varepsilon} \right) \right). \quad (41) $$

Note that from Eq. (41) one obtains$^{37}$

$$ \hat{T}_{\mu}^{\nu}(\hat{x}) \triangleq w \lim_{\varepsilon \to 0} \hat{T}_{\mu,\varepsilon}^{\nu}(\hat{x}) \approx -Mc^2 \delta_{\mu}^{0} \delta_{0}^{\nu} \delta^{(3)}(\hat{x}). \quad (42) $$

**Remark 33.** 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$^{25}$ related to Eq. (40)- Eq. (42) in Colombeau notations read:

$$ (\hat{g}_{\mu,\varepsilon}^{\nu})_{\varepsilon} = (\hat{R}_{\mu,\varepsilon}^{\nu})_{\varepsilon} - \frac{1}{2} \delta_{\mu}^{\nu} (\hat{R}_{\varepsilon})_{\varepsilon} = \kappa (\hat{T}_{\mu,\varepsilon}^{\nu})_{\varepsilon}, \quad (43) $$

where

$$ \begin{aligned} (\hat{R}_{\varepsilon}(\hat{r}))_{\varepsilon} = (\hat{R}^{\mu}_{\mu,\varepsilon}(\hat{r}))_{\varepsilon} = & -\left(\frac{3a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} + \left(\frac{2a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} = -\left(\frac{a\varepsilon^2}{(\hat{r}^2 + \varepsilon^2)^{5/2}}\right)_{\varepsilon} \end{aligned} \quad (43) $$

**Remark 34.** Note that the regularized scalar curvature $\hat{R}_{\varepsilon}$ has the well-defined weak limit $\hat{R}_{w}$ in $D'(\mathbb{R}^4)$

$$ \hat{R}_{w}(\hat{x}) \triangleq w \lim_{\varepsilon \to 0} \hat{R}_{\varepsilon}(\hat{x}) = -\frac{4}{3} \pi a \delta^{(3)}(\hat{x}). \quad (44) $$

**Remark 35.** 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 15) where $\hat{r}_{\text{fin}} \in \mathbb{R}$ from Eq. (44) it follows

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

(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 14) from Eq. (43) it follows

$$ (\hat{R}_{\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon} \approx -\infty, $$

(3) at origin $(\hat{r}_{\varepsilon}^{0})_{\varepsilon} = 0_{\tilde{\mathbb{R}}}$ (see Definition 12) 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 obtains³⁷

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

$$\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}) = \frac{4a^2}{\hat{r}^2+\varepsilon^2}\left[\frac{3}{(\hat{r}^2+\varepsilon^2)^2}\right] - \frac{12a^2\varepsilon^2}{(\hat{r}^2+\varepsilon^2)^4} + \frac{9a^2\varepsilon^4}{(\hat{r}^2+\varepsilon^2)^5}. \quad (46)$$

**Remark 36.** 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

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

$$\frac{12a^2}{\hat{r}^6} + \frac{16\pi a^2}{3} \frac{1}{\hat{r}^3} \delta^{(3)}(\hat{x}) + 16\pi^2 a^2 [\delta^{(3)}(\hat{x})]^2. \quad (47)$$

**Remark 37.** However Colombeau quadratic scalars $(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ and $(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon}$ well defined as Colombeau generalized functions in $G_{\delta}(\tilde{\mathbb{R}}^3)$, since

$$(\hat{R}_{\varepsilon}^{\mu\nu}(\hat{r}_{\varepsilon})\hat{R}_{\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}))_{\varepsilon} = \frac{1}{2}\left\{\left[\left(\frac{3a\varepsilon^2}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon}\right]^2 + 2\left[\left(\frac{a\varepsilon^2}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^{5/2}}\right)_{\varepsilon}\right]^2\right\} \in G_{\delta}(\tilde{\mathbb{R}}^3), \\ (\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}))_{\varepsilon} = \left\{\frac{12a^2}{((\hat{r}_{\varepsilon}^2+\varepsilon^2)^3)_{\varepsilon}} - \left(\frac{12a^2\varepsilon^2}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^4}\right)_{\varepsilon} + \left(\frac{9a^2\varepsilon^4}{(\hat{r}_{\varepsilon}^2+\varepsilon^2)^5}\right)_{\varepsilon}\right\} \in G_{\delta}(\tilde{\mathbb{R}}^3). \quad (48)$$

**Remark 38.** 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 treating 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

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

$$(\hat{R}_{\varepsilon}^{\rho\sigma\mu\nu}(\hat{r}_{\varepsilon}^{0})\hat{R}_{\rho\sigma\mu\nu,\varepsilon}(\hat{r}_{\varepsilon}^{0}))_{\varepsilon} = \left\{\frac{12a^2}{(\varepsilon^6)_{\varepsilon}} - \left(\frac{12a^2\varepsilon^2}{(\varepsilon^8)_{\varepsilon}}\right)_{\varepsilon} + \left(\frac{9a^2\varepsilon^4}{(\varepsilon^{10})_{\varepsilon}}\right)_{\varepsilon}\right\} \in \tilde{\mathbb{R}}_{\delta}. \quad (49)$$

**Remark 39.** In the canonical Schwarzschild coordinates $(t, r, \theta, \varphi)$, $r \neq a$ the Schwarzschild metric (29) takes the form (1) above horizon: $r > 2m$:

$$ds^{+2} = h^{+}(r)dt^{2} - [h^{+}(r)]^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (50)$$

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

and (2) below horizon $r < 2m$:
$$ds^{-2} = h^{-}(r)dt^{2} - h^{-}(r)^{-1}dr^{2} + r^{2}d\Omega^{2}, \quad (51)$$
$$h^{-}(r) = -1 + \frac{a}{r} = \frac{a-r}{r}$$

Examining the Schwarzschild metric (39) 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 horizon $r > 2m$ and below horizon $r < 2m$ correspondingly$^{14,25}$

$$r \ge 2m : \quad r - 2m \mapsto \sqrt{(r - 2m)^2 + \varepsilon^2}, \quad (52)$$

$$r < 2m : \quad 2m - r \mapsto \sqrt{(2m - r)^2 + \varepsilon^2}. \quad (53)$$

Inserting Eq. (52) and Eq. (53) into Eq. (50) and Eq. (51) correspondingly, we obtain Colombeau generalized object $\{(ds_{\varepsilon}^{+2})_{\varepsilon}, (ds_{\varepsilon}^{-2})_{\varepsilon}\}$ modeling the singular Schwarzschild metric above (below) horizon, i.e.,

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

$$h_{\varepsilon}^{+}(r) = -\frac{\sqrt{(r-2m)^2 + \varepsilon^2}}{r}, \quad (54)$$

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

$$h_{\varepsilon}^{-}(r) = \frac{\sqrt{(2m-r)^2 + \varepsilon^2}}{r}. \quad (55)$$

The generalized Ricci tensor above horizon $([R_{\varepsilon}^{+}]_{\alpha}^{\beta})_{\varepsilon}$ may now be calculated componentwise using the classical formulae$^{14,25}$

$$([R_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2}((h_{\varepsilon}^{+''})_{\varepsilon} + \frac{2}{r}(h_{\varepsilon}^{+'})_{\varepsilon}), \quad (56)$$

$$([R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{(h_{\varepsilon}^{+'})_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+})_{\varepsilon}}{r^2}, \quad (57)$$

where

$$h_{\varepsilon}^{+'}(r) = -\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2}, \quad (58)$$

$$\begin{aligned} h_{\varepsilon}^{+''}(r) = & -\left(\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}}\right)' + \left(\frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2}\right)' \\ = & -\frac{1}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2+\varepsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} \\ & + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} - \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r^3}. \end{aligned} \quad (59)$$

From Eq. (56), Eq. (57), Eq. (58) and Eq. (59) we obtain $(r^2[R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = (r^2[R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} =$

$$r(h_{\varepsilon}^{+'}(r))_{\varepsilon} + 1 + (h_{\varepsilon}^{+}(r))_{\varepsilon} =$$

$$r\left\{-\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2}\right\} + 1 - \frac{\sqrt{(r-2m)^2+\varepsilon^2}}{r} =$$
$$ \begin{aligned} & -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r} + 1 - \frac{\sqrt{(r-2m)^2+\varepsilon^2}}{r} = \\ & -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} + 1 \end{aligned} \quad (60) $$

and

$$ \begin{aligned} (2r^2[R_{\varepsilon}^{+}(r)]_0^0)_{\varepsilon} = (2r^2[R_{\varepsilon}^{+}(r)]_1^1)_{\varepsilon} = & \\ r^2(h_{\varepsilon}^{++}(r))_{\varepsilon} + 2r(h_{\varepsilon}^{++}(r))_{\varepsilon} = & \\ r^2\left\{-\frac{1}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2+\varepsilon^2]^{3/2}} + \right. & \\ & \left. + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} - \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r^3}\right\} + & \\ +2r\left\{-\frac{r-2m}{r[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{[(r-2m)^2+\varepsilon^2]^{1/2}}{r^2}\right\} = & \\ -\frac{r}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r^2[(r-2m)^2+\varepsilon^2]^{3/2}} + & \\ \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2+\varepsilon^2]^{1/2}} - & \\ \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r} - & \\ -\frac{2(r-2m)}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{2[(r-2m)^2+\varepsilon^2]^{1/2}}{r} = & \\ -\frac{r}{[(r-2m)^2+\varepsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2+\varepsilon^2]^{3/2}}. & \end{aligned} \quad (61) $$

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

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

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

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

$$ \begin{aligned} & \int_{2m}^{R} \left( r \left( h_{\varepsilon}^{++}(r) \right)_{\varepsilon} + 1 + (h_{\varepsilon}^{++}(r))_{\varepsilon} \right) \hat{\Phi}(r) dr = \\ & \int_{2m}^{R} \left\{ -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr. \end{aligned} \quad (62) $$

By replacement $r - 2m = u$, from Eq. (62) we obtain

$$ \begin{aligned} & \int_{2m}^{R} \left\{ -\frac{r-2m}{[(r-2m)^2+\varepsilon^2]^{1/2}} \right\} \hat{\Phi}(r) dr + \int_{2m}^{R} \hat{\Phi}(r) dr = \\ & - \int_{0}^{R-2m} \frac{u \hat{\Phi}(u+2m) du}{(u^2+\varepsilon^2)^{1/2}} + \int_{0}^{R-2m} \hat{\Phi}(u+2m) du. \end{aligned} \quad (63) $$

By replacement $u = \varepsilon\eta$, from Eq. (63) we obtain the expression

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

From Eq. (63) we obtain
$$
I_3^+(\varepsilon) = I_2^+(\varepsilon) = -\varepsilon \frac{\hat{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] -
$$

$$
-\frac{\varepsilon^2}{1!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta =
$$

$$
-\varepsilon\hat{\Phi}(2m)\left[\sqrt{\left(\frac{R-2m}{\varepsilon}\right)^2 + 1 - \left(\frac{R-2m}{\varepsilon} - 1\right)}\right] -
$$

$$
-\frac{\varepsilon^2}{1} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta, \quad (64)
$$

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

$$
\hat{\Phi}(\varepsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\hat{\Phi}^{(l)}(2m)}{l!} (\varepsilon\eta)^l + \frac{1}{n!} (\varepsilon\eta)^n \hat{\Phi}^{(n)}(\xi),
$$
$$
\xi \triangleq \theta\varepsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 \quad (65)
$$

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

The Eq. (64) - (Eq. (65) gives

$$
\lim_{\epsilon \to 0} I_3^+(\epsilon) = \lim_{\epsilon \to 0} I_2^+(\epsilon) =
$$

$$
\lim_{\epsilon \to 0} \left\{ -\epsilon \hat{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1 + 1 - \frac{R-2m}{\epsilon}} \right] + \right. \\
\left. + \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} d\eta \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \hat{\Phi}^{(1)}(\xi)\eta \right\} = 0. \right\} \quad (66)
$$

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

$$
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(r)]_{3}^{3} = \lim_{\epsilon \to 0} I_{3}^{+}(\epsilon) = 0,
$$

$$
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}(r)]_{2}^{2} = \lim_{\epsilon \to 0} I_{2}^{+}(\epsilon) = 0. \quad (67)
$$

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

$$
2 \int_K ([R_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} \Phi(x) d^{3}x = 2 \int_K ([R_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} \Phi(x) d^{3}x =
$$

$$
\int_{2m}^{R} \left( r^2 \left( h_{\epsilon}^{+''} \right)_{\epsilon} + 2r \left( h_{\epsilon}^{+'} \right)_{\epsilon} \right) \hat{\Phi}(r) dr =
$$
$$
= \int_{2m}^{R} \left\{ -\frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \hat{\Phi}(r) dr. \quad (68)
$$

By replacement $r - 2m = u$, from Eq. (68) we obtain

$$
I_1^+(\epsilon) = 2 \int_K \left([R_\epsilon^+(r)]_1^1\right)_\epsilon \Phi(x) d^3 x = I_2^+(\epsilon) = 2 \int_K \left([R_\epsilon^+(r)]_0^0\right)_\epsilon \Phi(x) d^3 x
$$

$$
= \int_{2m}^{R} \left( r^2 (h_{\epsilon}^{+''})_{\epsilon} + 2r (h_{\epsilon}^{+'})_{\epsilon} \right) \hat{\Phi}(r) dr = \\
= \int_{0}^{R-2m} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{1/2}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \hat{\Phi}(u+2m) du. \quad (69)
$$

By replacement $u = \epsilon\eta$, from Eq. (69) we obtain

$$
2 \int_K \left( [R_{\epsilon}^{+}(r)]_{1}^{1} \right)_{\epsilon} \Phi(x) d^3 x = 2 \int_K \left( [R_{\epsilon}^{+}(r)]_{0}^{0} \right)_{\epsilon} \Phi(x) d^3 x =
$$

$$
= \int_{2m}^{R} \left( r^2 (h_{\epsilon}^{+''})_{\epsilon} + 2r (h_{\epsilon}^{+'})_{\epsilon} \right) \hat{\Phi}(r) dr =
$$
$$
= \epsilon \int_{0}^{R-2m} \frac{\epsilon \eta + 2m}{\epsilon^2 \eta^2 + \epsilon^2} \frac{d\eta}{(\epsilon^2 \eta^2 + \epsilon^2)^{1/2}} + \frac{\epsilon^2 \eta^2 (\epsilon \eta + 2m)}{(\epsilon^2 \eta^2 + \epsilon^2)^{3/2}} \hat{\Phi}(\epsilon \eta + 2m) =
$$
$$
- \int_{0}^{R-2m} \frac{\epsilon^2 \eta \hat{\Phi}(\epsilon \eta + 2m)}{(\epsilon^2 \eta^2 + \epsilon^2)^{1/2}} \frac{d\eta}{(\epsilon^2 \eta^2 + \epsilon^2)^{1/2}} - 2m \int_{0}^{R-2m} \frac{\epsilon \hat{\Phi}(\epsilon \eta + 2m)}{(\epsilon^2 \eta^2 + \epsilon^2)^{1/2}} \frac{d\eta}{(\epsilon^2 \eta^2 + \epsilon^2)^{3/2}} +
$$

$$
\int_{0}^{R-2m} \frac{\epsilon^4 \eta^3 \hat{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2 \eta^2 + \epsilon^2)^{3/2}} + 2m \int_{0}^{R-2m} \frac{\epsilon^3 \eta^2 \hat{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2 \eta^2 + \epsilon^2)^{3/2}} =
$$

$$
\epsilon \left[ - \int_{0}^{R-2m} \frac{\eta \hat{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} + \int_{0}^{R-2m} \frac{\eta^3 \hat{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right] + \\
2m \left[ - \int_{0}^{R-2m} \frac{\hat{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} + \int_{0}^{R-2m} \frac{\eta^2 \hat{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right]. (70)
$$
From Eq. (70) we obtain

$$
\begin{align*}
I_0^+(\varepsilon) = I_1^+(\varepsilon) = {}& 2m \frac{\hat{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] + \\
& + \frac{\varepsilon}{1!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta + \\
& + \frac{\varepsilon \hat{\Phi}(2m)}{0!} \int_{\frac{R-2m}{\varepsilon}}^{\frac{R-2m}{\varepsilon}} d\eta \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] + \\
& + \frac{\varepsilon^2}{1!} \int_0^{\frac{R-2m}{\varepsilon}} d\eta \hat{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta, \quad (71)
\end{align*}
$$

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

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

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

with $\hat{\Phi}^{(l)}(\xi) \triangleq d^l \hat{\Phi}(\xi)/d\xi^l$. The Eq. (71) gives

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

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

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

$$
2m\hat{\Phi}(2m)\lim_{S\to\infty} \left[ \int_0^S \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^S \frac{d\eta}{(\eta^2+1)^{1/2}} \right] = -2m\hat{\Phi}(2m). \quad (72)
$$

where use is made of the relation

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

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

$$
w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}]_{1}^{0} = w - \lim_{\epsilon \to 0} [R_{\epsilon}^{+}]_{0}^{0} = -2m\hat{\Phi}(2m). \quad (73)
$$

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

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

$$
([R_{\epsilon}^{-}]_{2}^{2})_{\epsilon} = ([R_{\epsilon}^{-}]_{3}^{3})_{\epsilon} = \frac{(h_{\epsilon}^{-})_{\epsilon}}{r} + \frac{1+(h_{\epsilon}^{-})_{\epsilon}}{r^2}. \quad (74)
$$

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:

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

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

From Eq. (74) similarly as above we obtain

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

$$
\int_{K^{-}} ([R_{\epsilon}^{-}(r)]_{1}^{0})_{\epsilon} \Phi(x) d^{3}x = m\hat{\Phi}(2m), \quad (75)
$$
$$ \int_{K^{-}} ([R_{\varepsilon}^{-}(r)]_{2})_{\varepsilon} \Phi(x) d^{3}x = \\ \int_{K^{-}} ([R_{\varepsilon}^{-}(r)]_{3})_{\varepsilon} \Phi(x) d^{3}x = 0 \quad (76) $$

## IV. GENERALIZED COLOMBEAU CALCULUS

We use ¹⁸⁻²⁴ as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper Ω will denote an open subset of ℝⁿ. 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 functions¹⁸ 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 a separable, smooth orientable Hausdorff manifold of dimension n.

**Definition 22.** The classical Colombeau algebra of generalized functions on M is defined as the quotient:

$$ G_{\delta}(M) = \mathcal{E}_{M}(M)/\mathcal{N}(M) \quad (77) $$

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 $M$:

$$ \mathcal{E}_M(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subset \subset M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\ \qquad \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \} \quad (78) $$

$$ \mathcal{N}(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K(K \subset \subset M), \forall k(k \in N_0) \forall q(q \in \mathbb{N}) \\ \qquad \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \}. \quad (79) $$

**Remark 40.** 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 23.** Elements of $G_\delta(M)$ are denoted by:

$$ u = \mathrm{cl}[(u_{\epsilon})_{\epsilon}] \triangleq (u_{\epsilon})_{\epsilon} + \mathcal{N}(M). \quad (80) $$

**Remark 41.** 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 42. 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

$$G_s^r(M) = G_\delta(M) \otimes T_s^r(M), \quad (81)$$

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 setting¹² 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

$$G_s^r(M) \cong L_{(M)}(G_1^0(M)^r, G_0^1(M)^s; G_\delta(M)). \quad (82)$$

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

$\forall K \subset M \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^m]$ 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 $G_0^1(M)$ 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-Civita 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 construction¹³,¹⁴. 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<sup>13,14</sup> 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) $L_{\xi_1} \dots L_{\xi_k} u_\epsilon \in \mathcal{D}'(M)$, for all $\epsilon \in (0, \delta]$, where $M$ a separable, smooth orientable Hausdorff manifold of dimension $n$.

Definition 24. 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 $\mathcal{X}(M\setminus\Sigma)$ denoting the space of smooth vector fields on $M\setminus\Sigma$):

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

$$ \forall K(K \not\subseteq M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \forall f(f \in C^\infty(M)) \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma)) \\ \left[ \| L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon \| = \left( \sup_{f \in C^\infty(M\setminus\Sigma)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^{-N}), \epsilon \to 0 \right] \quad (83) $$

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

$$ \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q), \epsilon \to 0 \right] \& $$ 
$\forall K(K \not\subseteq M) \forall k(k \in N_0) \exists N(N \in \mathbb{N}) \forall f(f \in C^\infty(M)) \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathcal{X}(M\setminus\Sigma))$

$$ \left[ \| L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon \| = \left( \sup_{f \in C^\infty(M\setminus\Sigma)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^q), \epsilon \to 0 \right] \quad (84) $$

where $L_{\xi_k}^w$ denoting 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 25. Elements of $G_\delta(M, \Sigma)$ are denoted by
u = cl[(u_ε)_ε] ≜ (u_ε)_ε + N(M, Σ). (85)

Remark 43. With component wise operations (·, ±) $G_δ(M,Σ)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_ξ u \triangleq \text{cl}[(L_ξ u_ε)_ε]$. The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in G_δ(M,Σ)$ if $u \circ \psi_α \in G_δ(\psi_α(V_α))$ for all charts $(V_α, \psi_α)$, where on the open set $\psi_α(V_α) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 44.** Smooth functions $f \in C^\infty(M\setminus\Sigma)$ are embedded into $G_δ(M,Σ)$ simply by the constant embedding $\sigma$, i.e., $\sigma(f) = \text{cl}[(f)_ε]$, hence $C^\infty(M\setminus\Sigma)$ is a faithful subalgebra of $G_δ(M,Σ)$.

**Definition 26.** The $G_δ(M,Σ)$-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 $G_s^r(M,Σ) = G_δ(M,Σ) \otimes T_s^r(M\setminus\Sigma)$, 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 $G_δ(M,Σ)$-multilinear maps taking generalized vector and co vector fields to generalized functions, i.e., as $G_δ(M,Σ)$-modules we have

$$G_s^r(M, \Sigma) \cong L_{(M)}(G_1^0(M, \Sigma)^r, G_0^1(M, \Sigma)^s; G_δ(M, \Sigma)) \quad (86)$$

In particular a super generalized metric is defined to be a symmetric, super generalized (0,2)-tensor field $g_{ab} = [(g_ε)_{ab})_ε]$ (with its index independent of $ε$ and) whose determinant $\det(g_{ab})$ is invertible in $G_δ(M,Σ)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_ε)_{ab})_ε$ 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 $((g_ε)_{ab})_ε$ of $g_{ab}$ such that for fixed $\varepsilon$ (small enough) $(g_ε)_{ab} = g_{ab}(\varepsilon)$ (resp. $(g_ε)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $G(M,Σ)$-linear isomorphism from $G_0^1(M,Σ)$ to $G_1^0(M,Σ)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\varepsilon))_ε]$ is a well defined element of $G_0^2(M,Σ)$ (i.e., independent of the representative $((g_ε)_{ab})_ε$). 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.
## Distributional Derivatives in $\mathbb{R}$

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:

$$g(x) = f(x) - [f]_{x_0} H(x - x_0), (87)$$

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. (87) and therefore one obtains:

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

Thus, finally we get:

$$\bar{f}'(x) = \{f'(x)\} + [f]_{x_0} \delta(x - x_0). \quad (88)$$

or

$$\bar{f}'(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta(x - x_0) & x < x_0 \\ f_2'(x) + [f]_{x_0} \delta(x - x_0) & x \ge x_0 \end{cases} \quad (89)$$

Eq. (89) 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:

$$\bar{f}''(x) = \{f''(x)\} + \sum_{i=0}^{k} [f]_{x_i} \delta(x - x_i). \quad (90)$$

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. (88) 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

$$\bar{f}''(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0) & x < x_0 \\ f_2'(x) + [f]_{x_0} \delta'(x - x_0) + [f]_{x_0} \delta(x - x_0) & x \ge x_0 \end{cases} \quad (91)$$

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 $\bar{f}^{(m)}(x)$:

$$\bar{f}^{(m)}(x) = \{f^{(m)}(x)\} + \sum_{j=1}^{l} a_j [\delta^{(m-1)}(x - x_j) + b_j \delta^{(m-2)}(x - x_j) + \dots + f_j \delta(x - x_j)], (92)$$

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 $x_j$.

# V. 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, θ, φ).

Note that the Colombeau generalized object {($ds_{\epsilon}^{+2}$)$_\epsilon$, ($ds_{\epsilon}^{-2}$)$_\epsilon$}, given by Eq. (54)-Eq. (55) 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 $G_0^2(\mathbb{R}^4, \{r = 2m\})$.

Thus the full super generalized Ricci tensor reads

$$ ([R_{\epsilon}(r)]_{0}^{0})_{\epsilon} = ([R_{\epsilon}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2}((\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} + \frac{2}{r}(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}), \quad (92) $$

$$ ([R_{\epsilon}(r)]_{2}^{2})_{\epsilon} = ([R_{\epsilon}(r)]_{3}^{3})_{\epsilon} = \frac{(\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon}}{r} + \frac{1+(h_{\epsilon}(r))_{\epsilon}}{r^{2}}, \quad (93) $$

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. (89) and by Eq. (91) correspondingly.

From Eq. (54)-Eq. (55) we obtain

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

$$ -\left(\frac{\epsilon}{2m}\right)_{\epsilon} - \left(\frac{\epsilon}{2m}\right)_{\epsilon} = -(\epsilon)_{\epsilon}/m \quad (94) $$

From Eq. (94) and Eq. (89) we obtain

$$ (\bar{h}_{\epsilon}^{\prime}(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon}\delta(r - 2m) & r < 2m \end{cases} \quad (95) $$

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

$$ \bar{h}_{\epsilon}^{\prime}(r) = \begin{cases} (h_{\epsilon}^{+\prime}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m}\delta(r - 2m) & r \ge 2m \\ (h_{\epsilon}^{-\prime}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m}\delta(r - 2m) & r < 2m \end{cases} \quad (96) $$

From Eq. (58) we obtain

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

$$ \frac{(\epsilon)_{\epsilon}}{4m^2} - \left(-\frac{(\epsilon)_{\epsilon}}{4m^2}\right) = \frac{(\epsilon)_{\epsilon}}{2m^2}. \quad (97) $$

From Eq. (91) we obtain

$$ (\bar{h}_{\epsilon}^{\prime\prime}(r))_{\epsilon} = \begin{cases} h_{\epsilon}^{+\prime\prime}(r) + [h_{\epsilon}]_{2m}\delta'(r - 2m) + [h_{\epsilon}']_{2m}\delta(r - 2m) & r \ge 2m \\ h_{\epsilon}^{-\prime\prime}(r) + [h_{\epsilon}]_{2m}\delta(r - 2m) + [h_{\epsilon}']_{x_0}\delta(r - 2m) & r < 2m \end{cases} \quad (98) $$

where $[h_{\epsilon}']_{2m} = (\epsilon)_{\epsilon}/2m^2$ and therefore
$$ \bar{h}_{\varepsilon}''(r) = \begin{cases} (h_{\varepsilon}^{+}''(r))_{\varepsilon} - \frac{\langle \varepsilon \rangle_{\varepsilon}}{m} \delta'(r-2m) + \frac{\langle \varepsilon \rangle_{\varepsilon}}{2m^2} \delta(r-2m) & r \ge 2m \\ h_{\varepsilon}^{-}''(r) - \frac{\langle \varepsilon \rangle_{\varepsilon}}{m} \delta'(r-2m) + \frac{\langle \varepsilon \rangle_{\varepsilon}}{2m^2} \delta(r-2m) & r < 2m \end{cases} \quad (99) $$

Inserting Eq. (96) and Eq. (99) into Eq. (92) and into Eq. (93) finally we get for

$$ \begin{aligned} r \ge 2m : \quad & ([R_{\varepsilon}^{+}(r)]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2} \left( \left( \bar{h}_{\varepsilon}''(r) \right)_{\varepsilon} + \frac{2}{r} \left( \bar{h}_{\varepsilon}'(r) \right)_{\varepsilon} \right) = \\ & \frac{1}{2} \left( \left( h_{\varepsilon}^{+}''(r) \right)_{\varepsilon} + \frac{2}{r} \left( h_{\varepsilon}^{+}'(r) \right)_{\varepsilon} \right) - \\ & - \frac{\langle \varepsilon \rangle_{\varepsilon}}{2m} \delta'(r-2m) + \frac{\langle \varepsilon \rangle_{\varepsilon}}{2m^2} \delta(r-2m) - \frac{2\langle \varepsilon \rangle_{\varepsilon}}{rm} \delta(r-2m), \end{aligned} $$

$$ \begin{aligned} & ([R_{\varepsilon}^{+}(r)]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(r)]_{3}^{3})_{\varepsilon} = \frac{\left(\bar{h}'_{\varepsilon}(r)\right)_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}(r))_{\varepsilon}}{r^2} = \\ & \frac{\left(h_{\varepsilon}^{+}(r)\right)_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{+}(r))_{\varepsilon}}{r^2} - \frac{\langle \varepsilon \rangle_{\varepsilon}}{rm} \delta(r-2m). \quad (100) \end{aligned} $$

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

$$ \begin{aligned} & ([R_{\varepsilon}^{-}(r)]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{-}(r)]_{1}^{1})_{\varepsilon} = \frac{1}{2} \left( \left( \bar{h}_{\varepsilon}''(r) \right)_{\varepsilon} + \frac{2}{r} \left( \bar{h}_{\varepsilon}'(r) \right)_{\varepsilon} \right) = \\ & \frac{1}{2} \left( \left( h_{\varepsilon}^{-}''(r) \right)_{\varepsilon} + \frac{2}{r} \left( h_{\varepsilon}^{-}'(r) \right)_{\varepsilon} \right) - \\ & - \frac{\langle \varepsilon \rangle_{\varepsilon}}{2m} \delta'(r-2m) + \frac{\langle \varepsilon \rangle_{\varepsilon}}{2m^2} \delta(r-2m) - \frac{2\langle \varepsilon \rangle_{\varepsilon}}{rm} \delta(r-2m), \end{aligned} $$

$$ \begin{aligned} & ([R_{\varepsilon}^{-}(r)]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{-}(r)]_{3}^{3})_{\varepsilon} = \frac{\left(\bar{h}'_{\varepsilon}(r)\right)_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}(r))_{\varepsilon}}{r^2} = \\ & \frac{\left(h_{\varepsilon}^{-}(r)\right)_{\varepsilon}}{r} + \frac{1+(h_{\varepsilon}^{-}(r))_{\varepsilon}}{r^2} - \frac{\langle \varepsilon \rangle_{\varepsilon}}{rm} \delta(r-2m) . \quad (101) \end{aligned} $$

$$ \text{Note that in } \mathcal{D}'([2m, \infty)) : \quad w - \lim_{\varepsilon \to 0} \frac{2\pi^{-1}\varepsilon}{(r-2m)^2 + \varepsilon^2} = \delta(r-2m), \quad (102) $$

$$ \text{and therefore in } \mathcal{D}'([2m, \infty)) : \quad w - \lim_{\varepsilon \to 0} \frac{\mathrm{d}}{\mathrm{d}r} \frac{2\pi^{-1}\varepsilon}{(r-2m)^2 + \varepsilon^2} = \\ w - \lim_{\varepsilon \to 0} \frac{4\pi^{-1}\varepsilon(r-2m)}{[(r-2m)^2 + \varepsilon^2]^2} = \delta'(r-2m) . \quad (103) $$

By using Eq. (102) and Eq. (103) one obtains the natural imbedding

$$ \delta(r-2m) \mapsto \mathcal{G}_{\delta}(\mathbb{R}) : \quad \delta(r-2m) \mapsto (\delta(r-2m; \varepsilon))_{\varepsilon} = 2\pi^{-1} \left( \frac{\varepsilon}{(r-2m)^2 + \varepsilon^2} \right)_{\varepsilon} \in \mathcal{G}_{\delta}(\mathbb{R}) \quad (104) $$
and

$$
\begin{gather*}
\delta'(r-2m) \hookrightarrow G_{\delta}(\mathbb{R}) : \\
\delta'(r-2m) \hookrightarrow (\delta'(r-2m; \varepsilon))_{\varepsilon} = 4\pi^{-1} \left( \frac{\varepsilon(r-2m)}{[(r-2m)^2+\varepsilon^2]^2} \right)_{\varepsilon} \in G_{\delta}(\mathbb{R}) \quad (105)
\end{gather*}
$$

correspondingly.

**Remark 45.** Note, that: in $D'([2m, \infty))$ :

$$
w - \lim_{\epsilon \to 0} \frac{4\pi^{-1}\epsilon^2(r-2m)}{[(r-2m)^2+\epsilon^2]^2} = c_1\delta(r-2m). \quad (106)
$$

Inserting Eq. (104) and Eq. (105) into Eq. (100) and into Eq. (101) we get for

$$
\begin{align*}
& r \ge 2m \text{ :} \\
& ([R_{\epsilon}^{+}(r)]_{0}^{0})_{\epsilon} = ([R_{\epsilon}^{+}(r)]_{1}^{1})_{\epsilon} = \frac{1}{2}((\tilde{h}_{\epsilon}''(r))_{\epsilon} + \frac{2}{r}(\tilde{h}_{\epsilon}'(r))_{\epsilon}) = \\
& \frac{1}{2}((h_{\epsilon}^{+''}(r))_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{+'}(r))_{\epsilon}) - \\
& - \frac{(\epsilon)_{\epsilon}}{2m}(\delta'(r-2m; \epsilon))_{\epsilon} + \left[\frac{(\epsilon)_{\epsilon}}{2m^2} - \frac{2(\epsilon)_{\epsilon}}{rm}\right](\delta(r-2m; \epsilon))_{\epsilon}, \quad (107)
\end{align*}
$$

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

**Remark 46.** Note that the distributional semi Riemannian manifold $(\widetilde{\text{Sch}}, (g_{ij}^{\text{Sch}}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon})$ is endowed point free geometry$^{41,42,43}$ in natural way, since $(g_{ij}^{\text{Sch}}(t_{\epsilon}, r_{\epsilon}, \theta_{\epsilon}, \varphi_{\epsilon}, \epsilon))_{\epsilon} \in G_{\delta}(\widetilde{\mathbb{R}}_{\delta}^4)$, $i, j = 0,1,2,3$.

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

$$
h_{\epsilon}^{+}(r_{\epsilon}) = - \frac{\sqrt{(r_{\epsilon}-2m)^2 + \epsilon^2}}{r_{\epsilon}}, \quad (108)
$$

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

where
$$h_{\epsilon}^{-}(r_{\epsilon}) = \frac{\sqrt{(2m-r_{\epsilon})^{2}+\epsilon^{2}}}{r_{\epsilon}}. \quad (109)$$

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

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

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

$$\frac{1}{2}((h_{\epsilon}^{+''}(r_{\epsilon}))_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{+'}(r_{\epsilon}))_{\epsilon}) \quad (110)$$

and

$$([R_{\epsilon}^{+}(r_{\epsilon})]_{2}^{2})_{\epsilon} = ([R_{\epsilon}^{+}(r_{\epsilon})]_{3}^{3})_{\epsilon} =$$

$$\frac{(h_{\epsilon}^{+'(r_{\epsilon})})_{\epsilon}}{(r_{\epsilon})_{\epsilon}} + \frac{1+(h_{\epsilon}^{+}(r_{\epsilon}))_{\epsilon}}{(r_{\epsilon}^{2})_{\epsilon}} - \frac{(\epsilon)_{\epsilon}}{rm} \delta(r-2m). \quad (111)$$

From Eq. (110) and Eq. (107) we obtain

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

$$-\frac{1}{2(r_{\epsilon})_{\epsilon}}\left(\frac{1}{\left[\left(r_{\epsilon}-2m\right)^{2}+\epsilon^{2}\right]^{1/2}}\right)_{\epsilon} +$$

$$\frac{1}{2(r_{\epsilon})_{\epsilon}}\left(\frac{(r_{\epsilon}-2m)^{2}}{\left[\left(r_{\epsilon}-2m\right)^{2}+\epsilon^{2}\right]^{3/2}}\right)_{\epsilon} =$$

$$-\frac{1}{2(r_{\epsilon})_{\epsilon}}\left[\left(\left(r_{\epsilon}-2m\right)^{2}+\epsilon^{2}\right)_{\epsilon}\right]^{1/2} +$$

$$\frac{((r_{\epsilon})_{\epsilon}-2m)^{2}}{2(r_{\epsilon})_{\epsilon}\left[\left((r_{\epsilon})_{\epsilon}-2m\right)^{2}+(\epsilon^{2})_{\epsilon}\right]^{3/2}} \quad (112)$$

For any $(r_{\epsilon})_{\epsilon}$ such that $(r_{\epsilon})_{\epsilon} - 2m \neq 0_{\tilde{\mathbb{R}}}$ from Eq. (112) one obtains

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

$$-\frac{1}{2(r_{\epsilon}(r_{\epsilon}-2m))_{\epsilon}\left[1+(r_{\epsilon}-2m)_{\epsilon}^{-2}(\epsilon^{2})_{\epsilon}\right]^{1/2}} +$$

$$\frac{1}{2(r_{\epsilon}(r_{\epsilon}-2m))_{\epsilon}\left[1+(r_{\epsilon}-2m)_{\epsilon}^{-2}(\epsilon^{2})_{\epsilon}\right]^{3/2}} \quad (113)$$

For any $(r_{\epsilon})_{\epsilon}$ such that $(r_{\epsilon})_{\epsilon} - 2m \in \mathcal{F}(\mathbb{R}_{+})$ and $\delta[(r_{\epsilon})_{\epsilon} - 2m]^{-2} < 1$ from Eq. (113) one obtains

$$([R_{\epsilon}^{+}(r_{\epsilon})]_{0}^{0})_{\epsilon} = ([R_{\epsilon}^{+}(r_{\epsilon})]_{1}^{1})_{\epsilon} =$$
$$ \frac{1}{2(r_{\varepsilon}(r_{\varepsilon}-2m))_{\varepsilon}} \left[ - \frac{1}{[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^2)_{\varepsilon}]^{1/2}} + \frac{1}{[1+(r_{\varepsilon}-2m)_{\varepsilon}^{-2}(\varepsilon^2)_{\varepsilon}]^{3/2}} \right] \approx - \frac{(\varepsilon^2)_{\varepsilon}}{(r_{\varepsilon}(r_{\varepsilon}-2m)_{\varepsilon})^3} \quad (114) $$

and therefore

$$ ([R_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1})_{\varepsilon} \approx 0, \quad (115) $$

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

It follows from Eq. (114) for any $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_+)$

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

$$ \mathrm{st}(( [R_{\varepsilon}^{+}(r_{\varepsilon})]_{0}^{0} )_{\varepsilon}) = \mathrm{st}(( [R_{\varepsilon}^{+}(r_{\varepsilon})]_{1}^{1} )_{\varepsilon}) = 0. \quad (116) $$

Thus for any $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_+)$ 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. (112) one obtains

$$ ([R_{\varepsilon}^{+}(2m)]_{0}^{0})_{\varepsilon} = ([R_{\varepsilon}^{+}(2m)]_{1}^{1})_{\varepsilon} = - \frac{(\varepsilon^{-1})_{\varepsilon}}{2m} \approx \infty. \quad (117) $$

From Eq. (111) and Eq. (100) we obtain

$$ ([R_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = - \frac{(r_{\varepsilon})_{\varepsilon} - 2m}{(r_{\varepsilon}^{2})_{\varepsilon} [((r_{\varepsilon})_{\varepsilon} - 2m)^{2} + (\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}}. \quad (118) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $(r_{\varepsilon})_{\varepsilon} - 2m \neq 0_{\tilde{\mathbb{R}}}$ from Eq. (118) one obtains

$$ ([R_{\varepsilon}^{+}(r_{\varepsilon})]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(r_{\varepsilon})]_{3}^{3})_{\varepsilon} = - \frac{(r_{\varepsilon})_{\varepsilon} - 2m}{(r_{\varepsilon}^{2})_{\varepsilon} ((r_{\varepsilon})_{\varepsilon} - 2m)[1 + ((r_{\varepsilon})_{\varepsilon} - 2m)^{-2}(\varepsilon^{2})_{\varepsilon}]^{1/2}} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} = \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}[1 + ((r_{\varepsilon})_{\varepsilon} - 2m)^{-2}(\varepsilon^{2})_{\varepsilon}]} + \frac{1}{(r_{\varepsilon}^{2})_{\varepsilon}} \quad (119) $$

For any $(r_{\varepsilon})_{\varepsilon}$ such that $(r_{\varepsilon})_{\varepsilon} - 2m \in \mathcal{F}(\tilde{\mathbb{R}}_+)$ and $\delta^2[(r_{\varepsilon})_{\varepsilon} - 2m]^{-2} < 1$ from Eq. (118) one obtains

For any $(r_{\varepsilon})_{\varepsilon}$ such that $((r_{\varepsilon})_{\varepsilon} - 2m) = 0_{\tilde{\mathbb{R}}}$ from Eq. (118) one obtains

$$ ([R_{\varepsilon}^{+}(2m)]_{2}^{2})_{\varepsilon} = ([R_{\varepsilon}^{+}(2m)]_{3}^{3})_{\varepsilon} = \frac{1}{2m}. \quad (121) $$

### 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 horizon reads

$$ (d\tilde{s}_{\varepsilon}^{+2})_{\varepsilon} = (h_{\varepsilon}^{+}(r_{\varepsilon})dt_{\varepsilon}^{2})_{\varepsilon} - ([h_{\varepsilon}^{+}(r_{\varepsilon})]^{-1}dr_{\varepsilon}^{2})_{\varepsilon} + (r_{\varepsilon}^{2}d\Omega_{\varepsilon}^{2})_{\varepsilon} \quad (122) $$
where

$$h_{\epsilon}^{+}(r_{\epsilon}) = -\frac{\sqrt{(r_{\epsilon}-2m)^{2}+\epsilon^{2}}}{r_{\epsilon}}, (123)$$

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

$$h_{\epsilon}^{-}(r_{\epsilon}) = \epsilon. (124)$$

By super generalized object $\{(d\tilde{s}_{\epsilon}^{+2})_{\epsilon}, (d\tilde{s}_{\epsilon}^{-2})_{\epsilon}\}$ non-spinning gravitational singularity presented, see Fig.7. Similarly, as above one obtains that the distributional Ricci tensor and the curvature scalar at horizon are of $\delta$-type, i.e.

$$(R_{0,\epsilon}^{0}(r))_{\epsilon} = (R_{1,\epsilon}^{1}(r))_{\epsilon} \approx -2m\delta(r - 2m), \\ (R_{2,\epsilon}^{2}(r))_{\epsilon} = (R_{3,\epsilon}^{3}(r))_{\epsilon} \approx 0. \quad (125)$$

$$(R_{\epsilon}^{0}(r))_{\epsilon} \approx -2m\delta(r - 2m) \quad (126)$$

Remark 47. It follows from Eq. (111) and the generalized Einstein equations (162) has a singular source with support on horizon.

<figure><img src="image_9.png" /><figcaption>Fig.7. Non-spinning gravitational singularity.</figcaption></figure>

## VI. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY IN ISOTROPIC COORDINATES

In this section we studied Schwarzschild solution in isotropic coordinates using Colombeau distributional geometry, thus without leaving isotropic coordinates $(t, \rho, \theta, \varphi)$. The classical Schwarzschild metric in isotropic coordinates $(t, \rho, \theta, \varphi)$ reads

$$ds^2 = - \left(\frac{1-r_s/4\rho}{1+r_s/4\rho}\right)^2 dt^2 + \left(1+\frac{r_s}{4\rho}\right)^4 d^2\rho + \rho^2(d\theta^2 + \sin^2\theta d\varphi^2), (127)$$

see Eq. (15), where we let $c = 1$, or in the equivalent form
$$ds^2 = -\frac{(4\rho - r_s)^2}{(4\rho + r_s)^2} dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 d^2 + \rho^2(d\theta^2 + \sin^2\theta d^2\varphi) . \quad (128)$$

The metric (108) is nonclassical semi Riemannian metric, since the metric (108) is degenerated on horizon $\rho = r_s/4$. In order to obtain Colombeau solutions of the generalized Einstein field equations (141), corresponding to the nonclassical semi Riemannian metric (108) we apply the $\epsilon$-regulariation: $(4\rho - r_s)^2 \to (4\rho - r_s)^2 + \epsilon^2$ and embed the metric (108) into the following Colombeau object

$$(ds_{\epsilon}^2)_{\epsilon} = - \frac{((4\rho - r_s)^2 + \epsilon^2)_{\epsilon}}{(4\rho + r_s)^2} dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d^2\varphi) \quad (129)$$

Let $(\tilde{t}, \tilde{r}, \tilde{\theta}, \tilde{\varphi})$ be the generalized Colombeau coordinates (see Appendix B) on

$\tilde{\mathbb{R}}_{\delta}^{4}$, with $(\tilde{r}, \tilde{\theta}, \tilde{\varphi})$ the usual spherical polar coordinates on $\tilde{\mathbb{R}}_{\delta}^{3}$. Here we abbreviate for short $\tilde{t} = (t_{\epsilon})_{\epsilon}$, $\tilde{\rho} = (\rho_{\epsilon})_{\epsilon}$, $\tilde{\theta} = (\theta_{\epsilon})_{\epsilon}$, $\tilde{\varphi} = (\varphi_{\epsilon})_{\epsilon}$. We replace now the generalized metric (109) by the general Colombeau object

$$(d\tilde{S}_{\epsilon}^2)_{\epsilon} = - \frac{((4\rho_{\epsilon}-r_s)^2+\epsilon^2)_{\epsilon}}{(4(\rho_{\epsilon})_{\epsilon}+r_s)^2} (dt_{\epsilon}^2)_{\epsilon} + \left(1+\frac{r_s}{4(\rho_{\epsilon})_{\epsilon}}\right)^4 (d\rho_{\epsilon}^2)_{\epsilon} + (\rho_{\epsilon}^2)_{\epsilon}((d\theta_{\epsilon}^2)_{\epsilon} + (\sin^2\theta_{\epsilon})_{\epsilon}(d\varphi_{\epsilon}^2)_{\epsilon}) \quad (130)$$

We rewrite now Eq. (130) in the following form (see Appendix A, Eq. (A1))

$$(d\tilde{S}_{\epsilon}^2)_{\epsilon} = - \left[ (A_{\epsilon}(\rho_{\epsilon}))_{\epsilon} \right] dt^2 + \left[ B(\rho_{\epsilon}) + C(\rho_{\epsilon}) \right] (d\rho_{\epsilon}^2)_{\epsilon} + (B(\rho_{\epsilon}))_{\epsilon} (\rho_{\epsilon}^2)_{\epsilon} ((d\theta_{\epsilon}^2)_{\epsilon} + (\sin^2\theta_{\epsilon})_{\epsilon} (d\varphi_{\epsilon}^2)_{\epsilon}), \quad (131)$$

where

$$A_{\epsilon}(\rho_{\epsilon}) = \frac{(4\rho_{\epsilon}-r_s)^2+\epsilon^2}{(4\rho_{\epsilon}+r_s)^2}, \quad B(\rho_{\epsilon}) = 1, \quad B(\rho_{\epsilon}) + C(\rho_{\epsilon}) = \left(1+\frac{r_s}{4\rho_{\epsilon}}\right)^4 = G(\rho_{\epsilon}). \quad (132)$$

The Colombeau generalized curvature scalar $(R_{\epsilon}(\rho_{\epsilon}))_{\epsilon}$ reads (see Appendix A,

Eq. (A2))

$$(R_{\epsilon}(\rho_{\epsilon}))_{\epsilon} = (G(\rho_{\epsilon}))_{\epsilon} \left( \left[ \frac{2}{\rho_{\epsilon}} \left( -\frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} + \frac{\Delta'_{\epsilon}(\rho_{\epsilon})}{\Delta_{\epsilon}(\rho_{\epsilon})} \right) + \frac{2C(\rho_{\epsilon})}{\rho_{\epsilon}^2} - \frac{A''_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} \right. \right. \\
\left. \left. + \frac{1}{2} \frac{A'_{\epsilon}(\rho_{\epsilon}) \Delta'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon}) \Delta_{\epsilon}(\rho_{\epsilon})} \right] \right)_{\epsilon}, \quad (133)$$

where

$$\Delta_{\epsilon}(\rho_{\epsilon}) = A_{\epsilon}(\rho_{\epsilon})G(\rho_{\epsilon}), \quad \Delta'_{\epsilon}(\rho_{\epsilon}) = A'_{\epsilon}(\rho_{\epsilon})G(\rho_{\epsilon}) + A_{\epsilon}(\rho_{\epsilon})G'(\rho_{\epsilon}), \\ \frac{\Delta'_{\epsilon}(\rho_{\epsilon})}{\Delta_{\epsilon}(\rho_{\epsilon})} = \frac{A'_{\epsilon}(\rho_{\epsilon})G(\rho_{\epsilon})+A_{\epsilon}(\rho_{\epsilon})G'(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})G(\rho_{\epsilon})} = \frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} + \frac{G'(\rho_{\epsilon})}{G(\rho_{\epsilon})}. \quad (134)$$

Substituting Eq. (134) into Eq. (133) we get

$$(R_{\epsilon}(\rho_{\epsilon}))_{\epsilon} = G(\rho_{\epsilon}) \left( \left[ \frac{2}{\rho_{\epsilon}} \left( -\frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} + \frac{\Delta'_{\epsilon}(\rho_{\epsilon})}{\Delta_{\epsilon}(\rho_{\epsilon})} \right) + \frac{2C(\rho_{\epsilon})}{\rho_{\epsilon}^2} - \frac{A''_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} \right. \right. \\
\left. \left. + \frac{1}{2} \frac{A'_{\epsilon}(\rho_{\epsilon}) \Delta'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon}) \Delta_{\epsilon}(\rho_{\epsilon})} \right] \right)_{\epsilon} = \\
G(\rho_{\epsilon}) \left( \left[ \frac{2}{\rho_{\epsilon}} \frac{G'(\rho_{\epsilon})}{G(\rho_{\epsilon})} + \frac{2C(\rho_{\epsilon})}{\rho_{\epsilon}^2} - \frac{A''_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} + \frac{1}{2} \frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} \left( \frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} + \frac{G'(\rho_{\epsilon})}{G(\rho_{\epsilon})} \right) \right] \right)_{\epsilon}, \quad (135)$$

where

$$A_{\epsilon}(\rho_{\epsilon}) = \frac{(4\rho_{\epsilon}-r_s)^2+\epsilon^2}{(4\rho_{\epsilon}+r_s)^2}, \quad (136)$$
$$A'_{\epsilon}(\rho_{\epsilon}) = - \frac{8(\epsilon^2 + 2r_s(r_s - 4\rho_{\epsilon}))}{(4\rho_{\epsilon} + r_s)^3}, \quad (137)$$

$$A''_{\epsilon}(\rho_{\epsilon}) = \frac{32(3\epsilon^2 + 8r_s(r_s - 2\rho_{\epsilon}))}{(4\rho_{\epsilon} + r_s)^4}, \quad (138)$$

$$\frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} = - \frac{(8\epsilon^2 + 16r_s(r_s - 4\rho_{\epsilon}))}{((r_s - 4\rho_{\epsilon})^2 + \epsilon^2)(4\rho_{\epsilon} + r_s)}, \quad (139)$$

$$\frac{A''_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} = \frac{(96\epsilon^2 + 256r_s(r_s - 2\rho_{\epsilon}))}{((r_s - 4\rho_{\epsilon})^2 + \epsilon^2)(4\rho_{\epsilon} + r_s)^2} \quad (140)$$

From Eq. (139)-Eq. (140) we obtain

$$\left. \frac{A'_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} \right|_{4\rho_{\epsilon}=r_s} = - \left. \frac{(8\epsilon^2 + 16r_s(r_s - 4\rho_{\epsilon}))}{((r_s - 4\rho_{\epsilon})^2 + \epsilon^2)(4\rho_{\epsilon} + r_s)} \right|_{4\rho=r_s} = -\frac{4}{r_s}, \quad (141)$$

$$\left. \frac{A''_{\epsilon}(\rho_{\epsilon})}{A_{\epsilon}(\rho_{\epsilon})} \right|_{4\rho_{\epsilon}=r_s} = \left. \frac{(96\epsilon^2 + 256r_s(r_s - 2\rho_{\epsilon}))}{((r_s - 4\rho_{\epsilon})^2 + \epsilon^2)(4\rho_{\epsilon} + r_s)^2} \right|_{4\rho_{\epsilon}=r_s} = \frac{(96\epsilon^2 + 128r_s^2)}{4\epsilon^2 r_s^2} = \frac{32}{\epsilon^2} + \frac{24}{r_s^2} \quad (142)$$

From Eq. (135)-Eq. (142) finally we get

$$(R_{\epsilon}(\rho_{\epsilon}))_{\epsilon}|_{4\rho=r_s} = \frac{c_1}{(\epsilon^2)_{\epsilon}} + c_2, \quad (143)$$

where $c_1, c_2 \in \mathbb{R}$.

Remark 48. Note that curvature scalar (143) essentially more singular then curvature scalar (126) related to distributional geometry considered in Sect. V above.

## VII. THE GENERALIZED GRAVITATIONAL FIELD EQUATIONS RELATED TO POINT FREE DISTRIBUTIONAL GEOMETRY

We now proceed to the derivation of the generalized equations of the gravitational field. These equations are obtained from the principle of least action

$$\delta((S_{m,\epsilon})_{\epsilon} + (S_{g,\epsilon})_{\epsilon}) = 0_{\mathbb{R}}, \quad (144)$$

where $(S_{m,\epsilon})_{\epsilon}$ and $(S_{g,\epsilon})_{\epsilon}$ are the distributional actions of the gravitational field and matter respectively, see Appendix G. We now subject the gravitational Colombeau metric field, that is, the quantities $(g_{ik,\epsilon})_{\epsilon}$ to variation. Calculating the variation

$\delta(S_{g,\epsilon})_{\epsilon}$, we get

$$\delta(\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon})_{\epsilon} = (\delta \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon})_{\epsilon} = (\delta \int g_{\epsilon}^{ik} R_{ik,\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon})_{\epsilon} =$$

$$\int \left\{ (R_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik})_{\epsilon} + (R_{ik,\epsilon} g_{\epsilon}^{ik} \delta \sqrt{-g_{\epsilon}})_{\epsilon} + (g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon})_{\epsilon} (d\Omega_{\epsilon})_{\epsilon} \right\} = (\int R_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega_{\epsilon})_{\epsilon} + (\int R_{ik,\epsilon} g_{\epsilon}^{ik} \delta \sqrt{-g_{\epsilon}} d\Omega_{\epsilon})_{\epsilon} +$$
$$ \left( \int g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} d\Omega_{\epsilon} \right)_{\epsilon} \quad (145) $$

From the equality $(dg_{\epsilon})_{\epsilon} = (g_{\epsilon}g_{\epsilon}^{ik}dg_{ik,\epsilon})_{\epsilon} = -(g_{\epsilon}g_{ik,\epsilon}dg_{\epsilon}^{ik})_{\epsilon}$, where $(g_{\epsilon})_{\epsilon} =$

$(\det(g_{ik,\epsilon}))_{\epsilon}$, we obtain

$$ \begin{aligned} (\delta \sqrt{-g_{\epsilon}})_{\epsilon} = & - \left\{ \left( \frac{1}{2\sqrt{-g_{\epsilon}}} \right)_{\epsilon} \right\} (\delta g_{\epsilon})_{\epsilon} = \\ & - \frac{1}{2} \left\{ (\sqrt{-g_{\epsilon}})_{\epsilon} \right\} (g_{ik,\epsilon} \delta g_{\epsilon}^{ik})_{\epsilon}. \end{aligned} \quad (146) $$

Substituting Eq. (146) into Eq. (145) we obtain

$$ \begin{aligned} \delta \left( \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon} \right)_{\epsilon} = & \left( \int \left\{ R_{ik,\epsilon} - \frac{1}{2} g_{ik,\epsilon} R_{\epsilon} \right\} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega_{\epsilon} \right)_{\epsilon} \\ & + \left( \int g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} d\Omega_{\epsilon} \right)_{\epsilon}. \end{aligned} \quad (147) $$

Remark 49. Note that although the quantities $(\Gamma^i_{kl,\epsilon})_\epsilon$ don't constitute a tensor and their variations $\delta(\Gamma^i_{kl,\epsilon})_\epsilon$ do form a generalized tensor, for $(\Gamma^i_{kl,\epsilon}A_k dx_\epsilon^l)_\epsilon$ is the change in a vector under parallel displacement from some point $P$ to an infinitesimally separated point $P'$ and therefore $(\Gamma^i_{kl,\epsilon}A_k dx_\epsilon^l)_\epsilon$ is the difference between the two vectors, obtained as the result of two parallel displacements (one with the unvaried, the other with the varied $(\Gamma^i_{kl,\epsilon})_\epsilon$ from the point $P$ to one and the same point $P'$. The difference between two generalized vectors at the same point is a vector, and therefore $\delta(\Gamma^i_{kl,\epsilon})_\epsilon = (\delta\Gamma^i_{kl,\epsilon})_\epsilon$ is a generalized tensor.

Let us use now a locally geodesic system of a generalized coordinates. Then at that point all the $(\Gamma^i_{kl,\epsilon})_\epsilon = 0_{\bar{R}}$. Using now the expression for the $(R_{ik,\epsilon})_\epsilon$, we obtain (we remind that the first derivatives of the $(g^i_{\epsilon})_\epsilon$ are now equal to $0_{\mathbb{R}}$)

$$ \begin{aligned} (g_{\epsilon}^{ik} \delta R_{ik,\epsilon})_{\epsilon} = & \left\{ (g_{\epsilon}^{ik})_{\epsilon} \right\} \left\{ \left( \frac{\partial}{\partial x_{\epsilon}^{l}} \delta \Gamma_{ik,\epsilon}^{l} \right)_{\epsilon} - \left( \frac{\partial}{\partial x_{\epsilon}^{k}} \delta \Gamma_{il,\epsilon}^{l} \right)_{\epsilon} \right\} = \\ & \left( g_{\epsilon}^{ik} \frac{\partial}{\partial x_{\epsilon}^{l}} \delta \Gamma_{ik,\epsilon}^{l} \right)_{\epsilon} - \left( g_{\epsilon}^{il} \frac{\partial}{\partial x_{\epsilon}^{l}} \delta \Gamma_{ik,\epsilon}^{k} \right)_{\epsilon} = \left( \frac{\partial w_{\epsilon}^{l}}{\partial x_{\epsilon}^{l}} \right)_{\epsilon}, \end{aligned} \quad (148) $$

where

$$ (w_{\epsilon}^{l})_{\epsilon} = (g_{\epsilon}^{ik} \delta \Gamma_{ik,\epsilon}^{l})_{\epsilon} - (g_{\epsilon}^{il} \delta \Gamma_{ik,\epsilon}^{k})_{\epsilon}. \quad (149) $$

Note that the generalized divergence $(A^i_{;i,\epsilon})_\epsilon$ of a generalized vector $(A^i_\epsilon)_\epsilon$ in distributional curvilinear coordinates reads

$$ (A^i_{;i,\epsilon})_{\epsilon} = \frac{1}{\sqrt{-(g_{\epsilon})_{\epsilon}}} \left( \frac{\partial (A^i_{\epsilon} \sqrt{-(g_{\epsilon})_{\epsilon}})}{\partial x^i_{\epsilon}} \right)_{\epsilon}. \quad (150) $$

Since $(w^i_\epsilon)_\epsilon$ is a generalized vector, replacing $(dw^i_\epsilon/dx^l)_\epsilon$ by $(w^l_{;l,\epsilon})_\epsilon$ and using Eq. (150), we may write the relation we have obtained above, in any generalized coordinate system, in the form

$$ (g_{\epsilon}^{ik} \delta R_{ik,\epsilon})_{\epsilon} = \left\{ \left( \frac{1}{\sqrt{-g_{\epsilon}}} \right)_{\epsilon} \right\} \frac{\partial}{\partial x_{\epsilon}^{l}} (\sqrt{-g_{\epsilon}} w_{\epsilon}^{l})_{\epsilon}. \quad (151) $$

Therefore the second Colombeau integral on the right side of Eq. (126) is equal to

$$ \left( \int g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} d\Omega_{\epsilon} \right)_{\epsilon} = \left( \int \frac{\partial \sqrt{-g_{\epsilon}} w_{\epsilon}^{l}}{\partial x_{\epsilon}^{l}} d\Omega_{\epsilon} \right)_{\epsilon} \quad (152) $$

and by generalized Gauss' theorem [see Appendix B, Eq. (28)], can be transformed into the Colombeau integral of $(w^l_\epsilon)_\epsilon$ over the hypersurface surrounding the whole four-volume. Since the variations of the field are zero at the integration limits, this term drops out. Thus, the variation $\delta S[(g_\epsilon)_\epsilon]$ is equal to
$$ \delta S[(g_{\varepsilon})_{\varepsilon}] = -\frac{c^3}{16\pi\kappa} \left( \int \left\{ R_{ik,\varepsilon} - \frac{1}{2} g_{ik,\varepsilon} R_{\varepsilon} \right\} \sqrt{-g_{\varepsilon}} \delta g_{\varepsilon}^{ik} d\Omega_{\varepsilon} \right)_{\varepsilon}. \quad (153) $$

Remark 50. Note that if we had started from the expression

$$ \delta S[(g_{\varepsilon})_{\varepsilon}] = -\frac{c^3}{16\pi\kappa} \left( \delta \int G_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega_{\varepsilon} \right)_{\varepsilon} $$

for the action of the field, then we get

$$ \begin{aligned} \delta S[(g_{\varepsilon})_{\varepsilon}] = & -\frac{c^3}{16\pi\kappa} \int (g_{\varepsilon}^{ik})_{\varepsilon} \{(d\Omega_{\varepsilon})_{\varepsilon}\} \left\{ \left( \frac{\partial\{G_{\varepsilon}\sqrt{-g_{\varepsilon}}\}}{\partial g_{\varepsilon}^{ik}} \right)_{\varepsilon} - \right. \\ & \left. \left( \frac{\partial}{\partial x_{\varepsilon}^{l}} \frac{\partial\{G_{\varepsilon}\sqrt{-g_{\varepsilon}}\}}{\partial \frac{\partial g_{\varepsilon}^{ik}}{\partial x_{\varepsilon}^{l}}} \right)_{\varepsilon} \right\} \end{aligned} \quad (154) $$

Comparing Eq. (154) with Eq. (153) we get

$$ \begin{aligned} (R_{ik,\varepsilon})_{\varepsilon} - \frac{1}{2} (g_{ik,\varepsilon} R_{\varepsilon})_{\varepsilon} = & \left\{ \left( \frac{1}{\sqrt{-g_{\varepsilon}}} \right)_{\varepsilon} \right\} \left\{ \left( \frac{\partial\{G_{\varepsilon}\sqrt{-g_{\varepsilon}}\}}{\partial g_{\varepsilon}^{ik}} \right)_{\varepsilon} - \right. \\ & \left. \left( \frac{\partial}{\partial x_{\varepsilon}^{l}} \frac{\partial\{G_{\varepsilon}\sqrt{-g_{\varepsilon}}\}}{\partial \frac{\partial g_{\varepsilon}^{ik}}{\partial x_{\varepsilon}^{l}}} \right)_{\varepsilon} \right\}. \end{aligned} \quad (155) $$

For the variation of the action of the matter we can write immediately from Eq. (H4)

$$ (\delta S_{m,\varepsilon})_{\varepsilon} = \frac{1}{2c} \left( \int T_{ik,\varepsilon} \sqrt{-g_{\varepsilon}} \delta g_{\varepsilon}^{ik} d\Omega_{\varepsilon} \right)_{\varepsilon}, \quad (156) $$

where $(T_{ik,\varepsilon})_{\varepsilon} \in G_{\delta}(\mathbb{R}^4)$ is the generalized energy-momentum tensor of the matter

fields and thus, from the principle of least action

$$ \delta \{ S[(g_{\varepsilon})_{\varepsilon}] + (S_{m,\varepsilon})_{\varepsilon} \} = 0_{\mathbb{R}} \quad (157) $$

one obtains

$$ -\frac{c^3}{16\pi\kappa} \left( \int \left\{ R_{ik,\varepsilon} - \frac{1}{2} g_{ik,\varepsilon} R_{\varepsilon} - \frac{8\pi\kappa}{c^4} T_{ik,\varepsilon} \right\} \sqrt{-g_{\varepsilon}} \delta g_{\varepsilon}^{ik} d\Omega_{\varepsilon} \right) = 0_{\mathbb{R}}. \quad (158) $$

From Eq. (137), since of the arbitrariness of the $(\delta g_{\varepsilon}^{ik})_{\varepsilon} \in G_{\delta}(\mathbb{R}^4)$ finally we get

$$ (G_{ik,\varepsilon})_{\varepsilon} = (R_{ik,\varepsilon})_{\varepsilon} - \frac{1}{2} (g_{ik,\varepsilon} R_{\varepsilon})_{\varepsilon} = \frac{8\pi\kappa}{c^4} (T_{ik,\varepsilon})_{\varepsilon} \quad (159) $$

or, in mixed components,

$$ (G_{i,\varepsilon}^{k})_{\varepsilon} = (R_{i,\varepsilon}^{k})_{\varepsilon} - \frac{1}{2} \delta_{i}^{k} (R_{\varepsilon})_{\varepsilon} = \frac{8\pi\kappa}{c^4} (T_{i,\varepsilon}^{k})_{\varepsilon}. \quad (160) $$

They are called the generalized Einstein equations.

Contracting Eq. (160) on the indices i and k, we get

$$ (R_{\varepsilon})_{\varepsilon} = -\frac{8\pi\kappa}{c^4} (T_{i,\varepsilon}^{i})_{\varepsilon} = -\frac{8\pi\kappa}{c^4} (T_{\varepsilon})_{\varepsilon}. \quad (161) $$

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

$$ (R_{ik,\varepsilon})_{\varepsilon} = \frac{8\pi\kappa}{c^4} \left\{ (T_{ik,\varepsilon})_{\varepsilon} - \frac{1}{2} (g_{ik,\varepsilon} T_{\varepsilon})_{\varepsilon} \right\}. \quad (162) $$

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

# VIII. THE MÖLLER DISTRIBUTIONAL GEOMETRY
As an important example of the geometry with the degenerate fundamental tensor, we consider now Möller's uniformly accelerated frame given by Möller's line element (163). Recall that Möller dealing with the following line element⁴¹

$$ds^2 = -\Delta(x)dt^2 + dx^2 + dy^2 + dz^2, \quad (163)$$

where $\Delta(x) = (a + gx)^2$. Einstein⁷ considered the similar line element

$$ds^2 = \alpha^2 x_1^2 dx_1^2 - dx_1^2 - dx_2^2 - dx_3^2 \quad (164)$$

Note that the transformation that converts Minkowski line element

$$ds^2 = d\xi_4^2 - d\xi_1^2 - d\xi_2^2 - d\xi_3^2 \quad (165)$$

into Eq. (143) is

$$\xi_1 = x_1 \cosh \alpha x_4, \xi_2 = x_2,$$

$$\xi_3 = x_3, \xi_4 = x_1 \sinh \alpha x_4. \quad (166)$$

Remark 51. Note that the Jacobian of the transformation (166) reads

$$J(\xi_1, \xi_2, \xi_3, \xi_4) = x_1 \alpha.$$

Obviously, the metric (163) is degenerated on the event horizon $\{x_1 = 0\}$.

Remark 52. A. Einstein⁷ emphasized that the metric tensor $g_{\mu\nu}$ corresponding to (164) are regular for all points $x = (x_1, x_2, x_3, x_4)$ with $x_1 \neq 0$, but one cannot assert that Eq. (167)

$$R_{kl} = 0 \quad (167)$$

are satisfied by (164) for all values of $x_1, x_2, x_3, x_4$ since the determinant $\det(g_{\mu\nu})$ vanishes for $x_1 = 0$.

<figure><img src="image_10.png" /><figcaption>Fig.8. Rindler spacetime with (1) singular surface $\Sigma^* = \Sigma_{\Gamma}^{\text{sing}}$ spanned by singular contour $\Gamma_{\text{sing}}$ (2) truncated singular contour $\Gamma^{\#} = \Gamma_{\text{sing}}\setminus\{\hat{x}_0\}$ (solid curve) with truncated singular point $\hat{x} = (0,0)$ on Rindler horizon such that $\Delta A_k(\Gamma^{\#}) = \infty$.</figcaption></figure>

Remark 53. Note that in contrary with A. Einstein⁷, we emphasized before, (see Sect. I) that the Rindler line element (164) cannot be treated classically even for the patch of spacetime which contains the all points $x = (x_1, x_2, x_3, x_4)$ with $x_1 \neq 0$, since there exists truncated singular contour $\Gamma^{\#} = \Gamma_{\text{sign}}\setminus\{\hat{x}_0\}$ with deleted singular point $\hat{x}_0$ on Rindler horizon, such that $(\Delta A_k(\Gamma_{\text{sing}}^{\#}) = \infty$.
Remark 54. Of course, Möller's metric (163) degenerated at Möller horizon {$x = -a/g$}. However, in contrast with Einstein⁷, Möller⁴¹ mistakenly argued that metric field (163) is a global vacuum solution of the canonical A. Einstein field equations, i.e. the $g_{\mu\nu}$ of this field for all values of $t, x, y, z$ satisfy the equations

$$R_{ik} - \frac{1}{2} g_{ik} R = 0. \quad (168)$$

Remark 55. In physical literature this Möller's mistake holds from Möller's time until nowadays.

Remark 55. Note that formally corresponding to the Möller's metric (163) classical Levi-Civita connection reads

$$\begin{aligned} \Gamma_{44}^{1}(x) &= g(a + gx), \\ \Gamma_{14}^{4}(x) &= \Gamma_{41}^{4}(x) = g(a + gx)^{-1} \quad (169) \end{aligned}$$

and therefore, classical Levi-Civita connection (169) of course is not available at Möller horizon since at horizon formal expressions (169) becomes infinite:

$$\Gamma_{14}^{4} \left(-\frac{a}{g}\right) = \Gamma_{41}^{4} \left(-\frac{a}{g}\right) = \infty. \quad (170)$$

Remark 56. Note that Möller dealing with Einstein's field equations in the following form⁴¹

$$G_i^k = R_i^k - \frac{1}{2} \delta_i^k R = 0, \quad (171)$$

where $R_i^k$ is the contracted Riemann-Christoffel tensor, formally calculated by canonical way by using classical Levi-Civita connection (148) and where $R = R_i^i$. By using the following ansatz

$$ds^2 = -\Delta(x)dt^2 + dx^2 + dy^2 + dz^2, \quad (172)$$

Möller finally obtain

$$\begin{aligned} G_2^2(x) = G_3^3(x) = & -\frac{1}{2\Delta(x)} \left[ \Delta''(x) - \frac{(\Delta'(x))^2}{2\Delta(x)} \right] = \\ & -\frac{(\Delta^{1/2}(x))^2}{\Delta^{1/2}(x)}. \end{aligned} \quad (173)$$

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

Remark 57. From Eq. (173) and Eq. (171) Möller obtain the following ordinary differential equation

$$(\Delta^{1/2}(x))^2 = 0, \quad (174)$$

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

$$G_2^2(x) = G_3^3(x) \equiv 0. \quad (175)$$

The equation (175) obviously has the following trivial general solution

$$\Delta(x) = (a + gx)^2. \quad (176)$$

Remark 58. Note, that at Möller horizon $x_{hor} = -a/g$ the functions $G_2^2(x)$ and $G_3^3(x)$ of course is not zero identically but becomes uncertainty, since

$$\begin{aligned} G_2^2(-a/g) = G_3^3(-a/g) = & -\frac{([\Delta(-a/g)]^{1/2})^2}{[\Delta(-a/g)]^{1/2}} = \frac{0}{0}. \end{aligned} \quad (177)$$

Remark 59. Note that at any point $x \in \mathbb{R}$ such that $x \neq -a/g$ obviously
$G_2^2(x) = G_3^3(x) \equiv 0$ since at these points one obtains

$$G_2^2(x) = G_3^3(x) =$$

$$-\frac{1}{2\Delta(x)} \left[ \Delta''(x) - \frac{(\Delta'(x))^2}{2\Delta(x)} \right] =.$$

$$-\frac{1}{2(a+gx)^2} \left[ 2g^2 - \frac{4g^2(a+gx)^2}{2(a+gx)^2} \right] =$$

$$-\frac{1}{2(a+gx)^2} [2g^2 - 2g^2] \equiv 0. \quad (178)$$

Remark 60. Note that at point $x = -a/g$ the quantity $G_2^2(-a/g)$ and $G_3^3(-a/g)$ well defined only by using the formal limit

$$G_2^2(-a/g) = G_3^3(-a/g) = \lim_{x \to -a/g} -\frac{0}{2(a+gx)^2} = 0. \quad (179)$$

Remark 61. However, in the limit $x \to -a/g$ the Christoffel symbols (169) become infinite

$$\lim_{x \to -a/g} \Gamma_{14}^4(x) = \lim_{x \to -a/g} \Gamma_{41}^4(x) =$$

$$\lim_{x \to -a/g} g(a + gx)^{-1} = \infty. \quad (180)$$

It follows from (180) that at horizon $x_{hor} = -a/g$ the canonical expression for the contracted Riemann-Christoffel tensor $R_i^k$ obviously no longer holds, since due to the degeneracy of the Möller's metric (163), the Levi-Civita connection is not available at Möller horizon $x_{hor} = -a/g$.

In order to avoid difficulty with degeneracy of the classical Möller's metric (163) we replace now Möller's line element (163) by corresponding Colombeau line element

$$(d_{\epsilon}s^2)_{\epsilon} = - \left[ (\Delta_{\epsilon}(x_{\epsilon}))_{\epsilon} \right] (dt_{\epsilon}^2)_{\epsilon} + (dx_{\epsilon}^2)_{\epsilon} + (dy_{\epsilon}^2)_{\epsilon} + (dz_{\epsilon}^2)_{\epsilon} \quad (181)$$

$$\text{where } \Delta_{\epsilon}(x_{\epsilon}) = [(a + gx_{\epsilon})^2 + \epsilon^2], \epsilon \in (0, \delta], (\Delta_{\epsilon}(x_{\epsilon}))_{\epsilon} \in \mathcal{G}_{\delta}(\tilde{\mathbb{R}}_{\delta}^{4}).$$

Remark 62. Let $(g_{\mu\nu,\epsilon}(x_{\epsilon}))_{\epsilon}$ be the distributional metric tensor corresponding to (181). This tensor is regular for all points $(x_{\epsilon})_{\epsilon} \in \tilde{\mathbb{R}}_{\delta}$ since $(\det(g_{\mu\nu,\epsilon}(x_{\epsilon})))_{\epsilon} \neq 0_{\mathbb{R}}$ for all $(x_{\epsilon})_{\epsilon} \in \tilde{\mathbb{R}}_{\delta}$.

Remark 63. Note that in contrast with a wrong formal expression (169) the distributional Levi-Civita connection corresponding Colombeau line element (181) reads

$$(\Gamma_{44,\epsilon}^1(x_{\epsilon}))_{\epsilon} = g((a+gx_{\epsilon}))_{\epsilon},$$

$$(\Gamma_{14,\epsilon}^4(x_{\epsilon}))_{\epsilon} = (\Gamma_{41,\epsilon}^4(x_{\epsilon}))_{\epsilon} = \frac{g(a+gx_{\epsilon})_{\epsilon}}{((a+gx_{\epsilon})^2+\epsilon^2)_{\epsilon}} \quad (182)$$

Note that the distributional Levi-Civita connection (182) is well defined and regular on whole Möller's space-time.

Let $(G_{i,\epsilon}^k(x_{\epsilon}))_{\epsilon}$ be the distributional Einstein tensor [see Sect. VII, Eq. (160)]

$$(G_i^k(x_{\epsilon}))_{\epsilon} \triangleq (R_i^k(x_{\epsilon}))_{\epsilon} - \frac{1}{2} \delta_i^k (R_{\epsilon}(x_{\epsilon}))_{\epsilon}, \quad (183)$$

where $(R_{i,\epsilon}^k(x_{\epsilon}))_{\epsilon}$ is the contracted distributional Riemann-Christoffel tensor calculated by using distributional Levi-Civita connection (182) corresponding to Colombeau line element (181) and

$$(R_{\epsilon}(x_{\epsilon}))_{\epsilon} = (R_i^i(x_{\epsilon}, \epsilon))_{\epsilon}. \quad (184)$$
Therefore, for the case of the Colombeau line element (181) we get

$$ (G_2^2(x_\epsilon; \epsilon))_\epsilon = (G_3^3(x_\epsilon; \epsilon))_\epsilon = -\frac{1}{2(\Delta_\epsilon(x_\epsilon))_\epsilon} \left\{ (\Delta''_\epsilon(x_\epsilon))_\epsilon - \frac{[(\Delta'_\epsilon(x_\epsilon))_\epsilon]^2}{2(\Delta_\epsilon(x_\epsilon))_\epsilon} \right\} = -\left(\frac{g^2\epsilon^2}{\Delta_\epsilon^2(x_\epsilon)}\right)_\epsilon. \quad (185) $$

Note that $(\mathfrak{I}(x_\epsilon; \epsilon))_\epsilon \triangleq -\left(\frac{g^2\epsilon^2}{\Delta_\epsilon^2(x_\epsilon)}\right)_\epsilon$, $\epsilon \in (0, \delta]$ is Colombeau generalized function such that $\mathrm{cl}[(\mathfrak{I}(x_\epsilon; \epsilon))_\epsilon] \in G_\delta(\tilde{\mathbb{R}}_\delta)$ and $\mathrm{cl}[(\mathfrak{I}(-(a/g)^{-1}; \epsilon))_\epsilon] = \mathrm{cl}[(\epsilon^{-2})_\epsilon] \approx \infty$.

Thus, Colombeau generalized fundamental tensor $(g_{\mu\nu,\epsilon}(x_\epsilon))_\epsilon$ corresponding to Colombeau metric (181) that is non vacuum Colombeau solution of the generalized Einstein's field equations with Colombeau generalized source $(\mathfrak{I}(x_\epsilon; \epsilon))_\epsilon$ since

$$ (G_i^k(x_\epsilon))_\epsilon = (R_i^k(x_\epsilon))_\epsilon - \frac{1}{2}\delta_i^k(R_\epsilon(x_\epsilon))_\epsilon = (\mathfrak{I}(x_\epsilon; \epsilon))_\epsilon = -\left(\frac{g^2\epsilon^2}{\Delta_\epsilon^2(x_\epsilon)}\right)_\epsilon \neq 0_{\mathbb{R}}. \quad (186) $$

For Colombeau scalars $(R_\epsilon(x_\epsilon))_\epsilon$ and $(R^{\mu\nu}(x_\epsilon, \epsilon)R_{\mu\nu}(x_\epsilon, \epsilon))_\epsilon$ we get
$$ (R_\epsilon(x_\epsilon))_\epsilon = \left(\frac{g^2\epsilon^2}{\Delta_\epsilon^2(x_\epsilon)}\right)_\epsilon \in G_\delta(\tilde{\mathbb{R}}_\delta)\setminus\mathcal{D}'(\mathbb{R}) \quad (187) $$

and

$$ (R^{\mu\nu}(x_\epsilon, \epsilon)R_{\mu\nu}(x_\epsilon, \epsilon))_\epsilon = g^4(\epsilon^4[(a + gx_\epsilon)^2 + \epsilon^2]^{-4})_\epsilon \in G_\delta(\tilde{\mathbb{R}}_\delta)\setminus\mathcal{D}'(\mathbb{R}) \quad (188) $$

correspondingly.

From Eq. (186)-Eq. (187) one obtains that for any $(x_\epsilon)_\epsilon \in \tilde{\mathbb{R}}_\delta$ such that $\mathrm{st}((a + gx_\epsilon)_\epsilon) \neq 0$, the following equalities hold

$$ \mathrm{st}((G_i^k(x_\epsilon))_\epsilon) = \mathrm{st}((R_i^k(x_\epsilon))_\epsilon) - \frac{1}{2}\delta_i^k \mathrm{st}((R_\epsilon(x_\epsilon))_\epsilon) = -\mathrm{st}\left(\left(\frac{g^2\epsilon^2}{\Delta_\epsilon^2(x_\epsilon)}\right)_\epsilon\right) = 0. \quad (189) $$

It follows from Eq. (189) that for any classical point $x \in \mathbb{R}\setminus\{-a/g\}$ the generalized Einstein's field equations (186) returns the classical Möller's metric (163).

# IX. 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 impossible treated classically since, the Levi-Cività connection is not available on whole Schwarzschild spacetime $(\mathbf{Sch}, g_{ij}^{\mathbf{Sch}}(t, r, \theta, \varphi))$, where $\mathbf{Sch} = \mathbb{R} \times ((\{r \ge$
$2m\} \cup \{0 \le r \le 2m\}) \times S^2)$, 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 $(\text{Sch}, (g_{ij}^{Sch}(t, r, \theta, \varphi, \varepsilon))_\varepsilon)$ endowed on the tangent bundle with Colombeau distributional metric tensor $(g_{ij,\varepsilon}^{Sch})_\varepsilon \triangleq (g_{ij}^{Sch}(t, r, \theta, \varphi, \varepsilon))_\varepsilon, \varepsilon \in (0,\delta]$, $(g_{ij,\varepsilon}^{Sch})_\varepsilon \in G_\delta(\mathbb{R}^4, \Sigma)$. We obtain the distributional Ricci tensor and the curvature scalar are of $\delta$-type, i.e., $(R_{0,\varepsilon}^0(r))_\varepsilon = (R_{1,\varepsilon}^1(r))_\varepsilon \approx -2m\delta(r-2m), (R_{2,\varepsilon}^2(r))_\varepsilon = (R_{3,\varepsilon}^3(r))_\varepsilon \approx 0$. This provides a new physical interpretation for the distributional curvature scalar $(R_\varepsilon(r))_\varepsilon \in G(\mathbb{R}^4, \Sigma)$ and square scalars $(R_\varepsilon^{\mu\nu}(r)R_{\mu\nu,\varepsilon}(r))_\varepsilon \in G(\mathbb{R}^4, \Sigma)$ of this geometry$^{17}$.

## APPENDIX A. EXPRESSIONS FOR THE COLOMBEAU SCALARS $(R_\varepsilon(r_\varepsilon))_\varepsilon$, $(R^{\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon$, $(R^{\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon)R_{\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon$ IN TERMS OF COLOMBEAU GENERALIZED FUNCTIONS $(A_\varepsilon(r_\varepsilon))_\varepsilon, (B_\varepsilon(r_\varepsilon))_\varepsilon, (C_\varepsilon(r_\varepsilon))_\varepsilon$ AND $(D_\varepsilon(r_\varepsilon))_\varepsilon, \varepsilon \in (0, \delta]$.

Let us introduce now Colombeau generalized metric which has the form

$$ (ds^2)_\varepsilon = - \left[ (A_\varepsilon(r_\varepsilon))_\varepsilon \right] (dx^0)^2 - 2 \left[ (D_\varepsilon(r_\varepsilon)dr_\varepsilon)_\varepsilon \right] dx^0 + \left( (B_\varepsilon(r_\varepsilon) + C_\varepsilon(r_\varepsilon))(dr_\varepsilon)^2 \right)_\varepsilon + \\ + (B_\varepsilon(r_\varepsilon)r_\varepsilon^2[(d\theta)^2 + \sin^2\theta (d\varphi)^2])_\varepsilon. \quad (A1) $$

Colombeau generalized scalars $(R_\varepsilon(r_\varepsilon))_\varepsilon$, $(R^{\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon$ and $(R^{\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon)R_{\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon$ in terms of Colombeau generalized functions $(A_\varepsilon(r_\varepsilon))_\varepsilon, (B_\varepsilon(r_\varepsilon))_\varepsilon, (C_\varepsilon(r_\varepsilon))_\varepsilon$ and $(D_\varepsilon(r_\varepsilon))_\varepsilon$ are expressed as

$$ (R_\varepsilon(r_\varepsilon))_\varepsilon = \left( \frac{A_\varepsilon}{r_\varepsilon} \left[ -2 \frac{A'_\varepsilon}{A_\varepsilon} - 3 \frac{B'_\varepsilon}{B_\varepsilon} + \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} \right] + \frac{2}{r_\varepsilon^2} \frac{A_\varepsilon C_\varepsilon + D_\varepsilon^2}{A_\varepsilon B_\varepsilon} - \frac{A''_\varepsilon}{A_\varepsilon} - 2 \frac{B''_\varepsilon}{B_\varepsilon} \right. \\ \left. + \frac{1}{2} \left( \frac{B'_\varepsilon}{B_\varepsilon} \right)^2 - 2 \frac{A'_\varepsilon B'_\varepsilon}{A_\varepsilon B_\varepsilon} + \left( \frac{1}{2} \frac{A'_\varepsilon}{A_\varepsilon} + \frac{B'_\varepsilon}{B_\varepsilon} \right) \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} \right)_\varepsilon, \quad (A2) \\ \\ (R^{\mu\nu}(r_\varepsilon, \varepsilon)R_{\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon = \left( \frac{A_\varepsilon^2}{\Delta_\varepsilon^2} \left( \frac{1}{2} \frac{A''_\varepsilon}{A_\varepsilon} - \frac{1}{4} \frac{A'_\varepsilon \Delta'_\varepsilon}{A_\varepsilon \Delta_\varepsilon} + \frac{1}{2} \frac{A'_\varepsilon B'_\varepsilon}{A_\varepsilon B_\varepsilon} + \frac{1}{r_\varepsilon} \frac{A'_\varepsilon}{A_\varepsilon} \right)^2 \right)_\varepsilon + \\ + 2 \left( \frac{A_\varepsilon^2}{\Delta_\varepsilon^2} \left[ \frac{1}{r_\varepsilon} \left( \frac{1}{2} \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} - \frac{A'_\varepsilon}{A_\varepsilon} - 2 \frac{B'_\varepsilon}{B_\varepsilon} \right) + \frac{1}{r_\varepsilon^2} \frac{A_\varepsilon C_\varepsilon + D_\varepsilon^2}{A_\varepsilon B_\varepsilon} - \frac{1}{2} \frac{A'_\varepsilon B'_\varepsilon}{A_\varepsilon B_\varepsilon} - \right. \right. \\ \left. \left. - \frac{1}{2} \frac{B_\varepsilon''}{B_\varepsilon} + \frac{1}{4} \frac{B_\varepsilon' \Delta'_\varepsilon}{B_\varepsilon \Delta_\varepsilon} \right]^2 \right)_\varepsilon + \\ \left( \frac{A_\varepsilon^2}{\Delta_\varepsilon^2} \left[ \frac{1}{2} \frac{A_\varepsilon''}{A_\varepsilon} - \frac{1}{4} \frac{A'_\varepsilon \Delta'_\varepsilon}{A_\varepsilon \Delta_\varepsilon} + \frac{1}{2} \frac{A'_\varepsilon B'_\varepsilon}{A_\varepsilon B_\varepsilon} + \frac{B_\varepsilon''}{B_\varepsilon} - \frac{1}{2} \left( \frac{B_\varepsilon'}{B_\varepsilon} \right)^2 \right] \right)_\varepsilon $$
$$ -\left( \frac{1}{2} \frac{B'_{\varepsilon} \Delta'_{\varepsilon}}{B_{\varepsilon} \Delta_{\varepsilon}} + \frac{1}{r_{\varepsilon}} \left( \frac{A'_{\varepsilon}}{A_{\varepsilon}} - \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} + 2 \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right) \right)^2_{\varepsilon} , \quad (A3) $$

$$ (R^{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon) R_{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon} = $$

$$ \left( \frac{A_{\varepsilon}^2}{\Delta_{\varepsilon}^2} \left( \frac{A_{\varepsilon}''}{A_{\varepsilon}} - \frac{1}{2} \frac{A_{\varepsilon}' \Delta_{\varepsilon}'}{A_{\varepsilon} \Delta_{\varepsilon}} \right)^2 + 2 \frac{A_{\varepsilon}^2}{\Delta_{\varepsilon}^2} \left( \frac{1}{r_{\varepsilon}} \frac{A_{\varepsilon}'}{A_{\varepsilon}} + \frac{1}{2} \frac{A_{\varepsilon}' B_{\varepsilon}'}{A_{\varepsilon} B_{\varepsilon}} \right)^2 + 4 \frac{A_{\varepsilon}^2}{\Delta_{\varepsilon}^2} \left[ \frac{1}{r} \frac{B_{\varepsilon}'}{B_{\varepsilon}} - \frac{1}{r^2} \frac{A_{\varepsilon} C_{\varepsilon} + D_{\varepsilon}^2}{A_{\varepsilon} B_{\varepsilon}} + \frac{1}{4} \left( \frac{B_{\varepsilon}'}{B_{\varepsilon}} \right)^2 \right]^2 + 2 \frac{A_{\varepsilon}^2}{\Delta_{\varepsilon}^2} \left[ \frac{1}{r} \left( \frac{A_{\varepsilon}'}{A_{\varepsilon}} + 2 \frac{B_{\varepsilon}'}{B_{\varepsilon}} - \frac{\Delta_{\varepsilon}'}{\Delta_{\varepsilon}} \right) + \frac{1}{2} \frac{A_{\varepsilon}' B_{\varepsilon}'}{A_{\varepsilon} B_{\varepsilon}} + \frac{B_{\varepsilon}''}{B_{\varepsilon}} \right] - \frac{1}{2} \left( \frac{B_{\varepsilon}'}{B_{\varepsilon}} \right)^2 - \frac{1}{2} \frac{B_{\varepsilon}' \Delta_{\varepsilon}'}{B_{\varepsilon} \Delta_{\varepsilon}} \right]^2_{\varepsilon} . \quad (A4) $$

Here

$$ (\Delta_{\varepsilon})_{\varepsilon} = (A_{\varepsilon}(r)(B_{\varepsilon}(r) + C_{\varepsilon}(r)))_{\varepsilon} + (D_{\varepsilon}^2(r))_{\varepsilon}. \quad (A5) $$

$$ \text{Assume that } (\Delta_{\varepsilon}(r))_{\varepsilon} = 1, (B_{\varepsilon}(r))_{\varepsilon} = 1, (D_{\varepsilon}(r))_{\varepsilon} = 0. $$

From Eq. (A2)-Eq. (A5) one obtains

$$ (R_{\varepsilon}(r))_{\varepsilon} = \left( -\frac{4A_{\varepsilon}}{r} + \frac{2A_{\varepsilon}C_{\varepsilon}}{r^2} - A_{\varepsilon}'' \right)_{\varepsilon}, \quad (A6) $$

$$ (R^{\mu\nu}(r, \varepsilon) R_{\mu\nu}(r, \varepsilon))_{\varepsilon} = 2 \left( \left( \frac{1}{2} A''_{\varepsilon} + \frac{1}{r} A'_{\varepsilon} \right)^2 \right)_{\varepsilon} + 2 \left( \left[ -\frac{A'_{\varepsilon}}{r} + \frac{A_{\varepsilon} C_{\varepsilon}}{r^2} \right]^2 \right)_{\varepsilon}, \quad (A7) $$

$$ (R^{\rho\sigma\mu\nu}(r, \varepsilon) R_{\rho\sigma\mu\nu}(r, \varepsilon))_{\varepsilon} = \left( (A_{\varepsilon}^{\prime\prime})^2 + 4 \left( \frac{A_{\varepsilon}^{\prime}}{r} \right)^2 + 4 \frac{(A_{\varepsilon} C_{\varepsilon})^2}{r^4} \right)_{\varepsilon}. \quad (A8) $$

We set now

$$ B_{\varepsilon}(r_{\varepsilon}) = 1, C_{\varepsilon}(r_{\varepsilon}) = -1 + A_{\varepsilon}^{-1}(r_{\varepsilon}), D_{\varepsilon}(r_{\varepsilon}) = 0 \quad (A9) $$

and rewrite Colombeau generalized object (A1) corresponding to Schwarzschild metric above horizon in the following form

$$ (ds_{\varepsilon}^2)_{\varepsilon} = -(A_{\varepsilon}(r_{\varepsilon})dt^2)_{\varepsilon} + (A_{\varepsilon}^{-1}(r_{\varepsilon})dr_{\varepsilon}^2)_{\varepsilon} + (r_{\varepsilon}^2)_{\varepsilon}d\Omega^2, \quad (A10) $$

where

$$ A_{\varepsilon}(r_{\varepsilon}) = -r_{\varepsilon}^{-1} \sqrt{(r_{\varepsilon} - 2m)^2 + \varepsilon^2}, \quad r_{\varepsilon} \ge 2m. \quad (A11) $$

By differentiation we obtain

$$ \Delta_{\varepsilon} = A_{\varepsilon}(B_{\varepsilon} + C_{\varepsilon}) = 1, \Delta'_{\varepsilon} = 0, $$

$$ A'_{\varepsilon}(r_{\varepsilon}) = \frac{-2m(r_{\varepsilon}-2m)}{r_{\varepsilon}^2 \sqrt{(r_{\varepsilon}-2m)^2 + \varepsilon^2}}, \quad (A12) $$

$$ A''_{\varepsilon}(r) = \frac{2m(-16m^3 + 24m^2r_{\varepsilon} - 12mr_{\varepsilon}^2 - 4m\varepsilon^2 + 2r_{\varepsilon}^3 + r_{\varepsilon}\varepsilon^2)}{r_{\varepsilon}^3[(r_{\varepsilon}-2m)^2 + \varepsilon^2]^{3/2}} = $$
$$ \frac{4m(r_{\varepsilon}-2m)^3+(r_{\varepsilon}-4m)\varepsilon^2}{r_{\varepsilon}^3[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{3/2}}. \quad (A13) $$

From Eq. (A6)- Eq. (A8) and Eq. (A12)-Eq. (A13) we obtain

$$
\begin{align*}
(R(r, \varepsilon))_{\varepsilon} &= \left( \frac{A_{\varepsilon}}{\Delta_{\varepsilon}} \left[ \frac{2}{r_{\varepsilon}} \left( -2 \frac{A'_{\varepsilon}}{A_{\varepsilon}} - 3 \frac{B'_{\varepsilon}}{B_{\varepsilon}} + \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} \right) + \frac{2}{r_{\varepsilon}^2} \frac{A_{\varepsilon}C_{\varepsilon}+D_{\varepsilon}^2}{A_{\varepsilon}B_{\varepsilon}} - \frac{A''_{\varepsilon}}{A_{\varepsilon}} - 2 \frac{B''_{\varepsilon}}{B_{\varepsilon}} \right. \right. \\
&\qquad \left. \left. + \frac{1}{2} \left( \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right)^2 - 2 \frac{A'_{\varepsilon}B'_{\varepsilon}}{A_{\varepsilon}B_{\varepsilon}} + \left( \frac{1}{2} \frac{A'_{\varepsilon}}{A_{\varepsilon}} + \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right) \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} \right] \right)_{\varepsilon} = \\
&= (A_{\varepsilon} \left[ \frac{2}{r_{\varepsilon}} \left( -2 \frac{A'_{\varepsilon}}{A_{\varepsilon}} \right) - \frac{2A_{\varepsilon}(1-A_{\varepsilon}^{-1})}{r_{\varepsilon}^2} - \frac{A''_{\varepsilon}}{A_{\varepsilon}} \right])_{\varepsilon} = \\
&= \left( -\frac{4A'_{\varepsilon}}{r_{\varepsilon}} - \frac{2A_{\varepsilon}}{r_{\varepsilon}^2} + \frac{2}{r_{\varepsilon}^2} - A''_{\varepsilon} \right)_{\varepsilon} = \\
&= \left( \frac{8m(r_{\varepsilon}-2m)}{r_{\varepsilon}^3 \sqrt{(r_{\varepsilon}-2m)^2+\varepsilon^2}} \right)_{\varepsilon} + 2r_{\varepsilon}^{-3} \left( \sqrt{(r_{\varepsilon}-2m)^2+\varepsilon^2} \right)_{\varepsilon} - \frac{2}{(r_{\varepsilon}^2)_{\varepsilon}} - \\
&\qquad - \left( \frac{2m(-16m^3+24m^2r_{\varepsilon}-12mr_{\varepsilon}^2-4m\varepsilon^2+2r_{\varepsilon}^3+r_{\varepsilon}\varepsilon^2)}{r_{\varepsilon}^3[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{3/2}} \right)_{\varepsilon}. \quad (A14)
\end{align*}
$$

Finally, from Eq. (A14) one obtains the following expression for the distributional curvature scalar above horizon $(R^+(r, \varepsilon))_\varepsilon$

$$ (R^+(r_{\varepsilon}, \varepsilon))_{\varepsilon} = \left( \frac{8m(r_{\varepsilon}-2m)}{r_{\varepsilon}^3 \sqrt{(r_{\varepsilon}-2m)^2+\varepsilon^2}} \right)_{\varepsilon} + 2[(r_{\varepsilon}^{-3})_{\varepsilon}] \left( \sqrt{(r_{\varepsilon}-2m)^2+\varepsilon^2} \right)_{\varepsilon} - \frac{2}{(r_{\varepsilon}^2)_{\varepsilon}} - \left( \frac{4m(r_{\varepsilon}-2m)^3+(r_{\varepsilon}-4m)\varepsilon^2}{r_{\varepsilon}^3[(r_{\varepsilon}-2m)^2+\varepsilon^2]^{3/2}} \right)_{\varepsilon}. \quad (A15) $$

## APPENDIX B. THE POINT FREE COLOMBEAU GENERALIZED CURVILINEAR COORDINATES

Let us consider the Colombeau generalized transformation from one generalized coordinate system, $\{(x_{\varepsilon}^0)_{\varepsilon}, (x_{\varepsilon}^1)_{\varepsilon}, (x_{\varepsilon}^2)_{\varepsilon}, (x_{\varepsilon}^3)_{\varepsilon}\}$, where $\{(x_{\varepsilon}^0)_{\varepsilon}, (x_{\varepsilon}^1)_{\varepsilon}, (x_{\varepsilon}^2)_{\varepsilon}, (x_{\varepsilon}^3)_{\varepsilon}\} \in \mathbb{R}^4$ to another generalized coordinate system $\{(x_{\varepsilon}^{\prime 0})_{\varepsilon}, (x_{\varepsilon}^{\prime 1})_{\varepsilon}, (x_{\varepsilon}^{\prime 2})_{\varepsilon}, (x_{\varepsilon}^{\prime 3})_{\varepsilon}\} \in \mathbb{R}^4$ transform according to the relation

$$ (x_{\varepsilon}^{i})_{\varepsilon} = (f_{\varepsilon}^{i}(x_{\varepsilon}^{0}, x_{\varepsilon}^{\prime 1}, x_{\varepsilon}^{\prime 2}, x_{\varepsilon}^{\prime 3}))_{\varepsilon}, \quad (B1) $$

where the $(f_{\varepsilon}^i)_{\varepsilon} \in G_{\delta}(\mathbb{R}^4)$, $i = 1,2,3,4$ are certain Colombeau generalized functions and where $(J_{\varepsilon}(x^{0}, x^{\prime 1}, x^{\prime 2}, x^{\prime 3}))_{\varepsilon}$

$$ (J_{\varepsilon}(x_{\varepsilon}^{0}, x_{\varepsilon}^{\prime 1}, x_{\varepsilon}^{\prime 2}, x_{\varepsilon}^{\prime 3}))_{\varepsilon} = \left( \frac{\partial(x_{\varepsilon}^{0}, x_{\varepsilon}^{1}, x_{\varepsilon}^{2}, x_{\varepsilon}^{3})}{\partial(x_{\varepsilon}^{0}, x_{\varepsilon}^{\prime 1}, x_{\varepsilon}^{\prime 2}, x_{\varepsilon}^{\prime 3})} \right)_{\varepsilon} \neq 0_{\mathbb{R}} $$

is the Jacobian of the Colombeau generalized transformation is given by Eq. (B1). Remark B.1. When we transform the coordinates, their Colombeau differentials
$$ (dx_{\epsilon}^{i})_{\epsilon} \text{ transform according to the following equality } (dx_{\epsilon}^{i})_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}} dx_{\epsilon}^{k}\right)_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon} (dx_{\epsilon}^{k})_{\epsilon}. \quad (B2) $$

**Definition B.1.** Every tuple of four Colombeau point free quantities $(A_{\epsilon}^{i})_{\epsilon}$, $i = 0,1,2,3$, which under a transformation (B1) of coordinates, transform like the Colombeau coordinate differentials (B2), is called Colombeau contravariant four-vector:

$$ (A_{\epsilon}^{i})_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}} A_{\epsilon}^{k}\right)_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon} (A_{\epsilon}^{k})_{\epsilon}. \quad (B3) $$

Let $(\phi_{\epsilon})_{\epsilon}$ be the Colombeau scalar. Under a coordinate transformation (B1), the four Colombeau quantities $\left(\frac{\partial \phi_{\epsilon}}{\partial x_{\epsilon}^{i}}\right)_{\epsilon}$, $i = 0,1,2,3$ transform according to the formula

$$ \left(\frac{\partial \phi_{\epsilon}}{\partial x_{\epsilon}^{i}}\right)_{\epsilon} = \left(\frac{\partial \phi_{\epsilon}}{\partial x_{\epsilon}^{k}} \frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon} = \left(\frac{\partial \phi_{\epsilon}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon} \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon}. \quad (B4) $$

**Definition B.2.** Every tuple of four Colombeau generalized functions $(A_{i,\epsilon})_{\epsilon}$ which, under a coordinate transformation (B2), transform like the Colombeau derivatives of a scalar, is called Colombeau generalized covariant four-vector

$$ (A_{i,\epsilon})_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} A'_{k,\epsilon}\right)_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}}\right)_{\epsilon} (A'_{k,\epsilon})_{\epsilon}. \quad (B5) $$

**Definition B.3.** We call the Colombeau generalized contravariant tensor of the second rank, $(A_{\epsilon}^{ik})_{\epsilon}$, any tuple of sixteen Colombeau generalized functions which transform like the products of the components of two Colombeau generalized contravariant vectors, i.e. according to the law

$$ (A_{\epsilon}^{ik})_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} \frac{\partial x_{\epsilon}^{m}}{\partial x_{\epsilon}^{i}} A'_{im,\epsilon}\right)_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{k}}{\partial x_{\epsilon}^{i}} \frac{\partial x_{\epsilon}^{m}}{\partial x_{\epsilon}^{i}}\right)_{\epsilon} (A'_{im,\epsilon})_{\epsilon} \quad (B6) $$

and a mixed Colombeau generalized tensor transforms as follows

$$ (A_{k,\epsilon}^{i})_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{m}}{\partial x_{\epsilon}^{k}} A'_{m,\epsilon}\right)_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{m}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon} (A'_{m,\epsilon})_{\epsilon}. \quad (B7) $$

**Remark B.2.** Note that the scalar product of two four-vectors $(A_{\epsilon}^{i} B_{i,\epsilon})$ is invariant since

$$ (A_{\epsilon}^{i} B_{i,\epsilon})_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{m}}{\partial x_{\epsilon}^{i}} A'_{\epsilon} B'_{m,\epsilon}\right)_{\epsilon} = \left(\frac{\partial x_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \frac{\partial x_{\epsilon}^{m}}{\partial x_{\epsilon}^{l}} A'_{\epsilon} B'_{m,\epsilon}\right)_{\epsilon} = (A'_{\epsilon} B'_{l,\epsilon})_{\epsilon}. \quad (B8) $$

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

$$ (A_{\epsilon}^{k} \delta_{k}^{i})_{\epsilon} = (A_{\epsilon}^{i})_{\epsilon}, \quad (B9) $$

i.e. again, that is Colombeau generalized four-vector; this proves that $\delta_{k}^{i}$ is a tensor. Remark B.3. The square of the Colombeau generalized line element $(ds_{\epsilon}^{2})_{\epsilon}$ in curvilinear coordinates is a quadratic form in the Colombeau differentials $(dx_{\epsilon}^{i})_{\epsilon}$, $i = 0,1,2,3$

$$ (ds_{\epsilon}^{2})_{\epsilon} = (g_{ik,\epsilon} dx^{i} dx^{k})_{\epsilon} = \{(g_{ik,\epsilon})_{\epsilon}\} \{(dx_{\epsilon}^{i})_{\epsilon}\} \{(dx_{\epsilon}^{k})_{\epsilon}\}. \quad (B10) $$

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

$$ (g_{ik,\epsilon})_{\epsilon} = (g_{ki,\epsilon})_{\epsilon}. \quad (B11) $$
**Definition B.4.** Since the (contracted) product of $(g_{ik,\epsilon})_{\epsilon}$ and the contravariant tensor $dx^i dx^k$ is a scalar, the $(g_{ik,\epsilon})_{\epsilon}$ form a covariant tensor; it is called the Colombeau generalized metric tensor.

**Definition B.5.** Two tensors $(A_{ik,\epsilon})_{\epsilon}$ and $(B_{\epsilon}^{ik})_{\epsilon}$ are said to be reciprocal to each other if

$$ (A_{ik,\epsilon} B_{\epsilon}^{ik})_{\epsilon} = \{(A_{ik,\epsilon})_{\epsilon}\} \{(B_{\epsilon}^{ik})_{\epsilon}\} = \delta_{k}^{i}. \quad (B12) $$

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

$$ \{(g_{ik,\epsilon})_{\epsilon}\} \{(g_{\epsilon}^{ik})_{\epsilon}\} = \delta_{k}^{i}. \quad (B13) $$

The same physical quantity can be represented in contravariant or covariant components. It is obvious that the only quantities that can determine the connection between the different forms are the components of the metric tensor. Therefore, this connection is given by the formulas

$$ (A_{\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{ik} A_{k,\epsilon})_{\epsilon}, \quad (A_{i,\epsilon})_{\epsilon} = (g_{ik,\epsilon} A_{\epsilon}^{k})_{\epsilon}. \quad (B14) $$

These remarks also apply to Colombeau generalized tensors. The transition between the different forms of a given physical generalized tensors is accomplished by using the metric tensor according to the formulas:

$$ (A_{k,\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{il} A_{lk,\epsilon})_{\epsilon}, \quad (A_{\epsilon}^{ik})_{\epsilon} = (g_{\epsilon}^{il} g_{\epsilon}^{km} A_{lm,\epsilon})_{\epsilon}, \quad \text{etc.} \quad (B15) $$

The completely antisymmetric unit pseudo tensor in Galilean coordinates we denote by $e^{iklm}$. Let us transform it to an arbitrary system of Colombeau generalized coordinates, and now denote it by $(E_{\epsilon}^{iklm})_{\epsilon}$. We keep the notation $e^{iklm}$ for the quantities defined as before by $e^{0123} = 1$ (or $e_{0123} = -1$). Let the $x'^i, i = 0,1,2,3$ be Galilean, and the $(x_{\epsilon}^{i})_{\epsilon}, i = 0,1,2,3$ be arbitrary Colombeau generalized curvilinear coordinates. According to the general rules for transformation of Colombeau generalized tensors, we have

$$ (E_{\epsilon}^{iklm})_{\epsilon} = \left\{ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x'^{p}} \frac{\partial x_{\epsilon}^{k}}{\partial x'^{r}} \frac{\partial x_{\epsilon}^{l}}{\partial x'^{s}} \frac{\partial x_{\epsilon}^{m}}{\partial x'^{t}} \right) \right\} e^{prst}, \quad (B16) $$

or

$$ (E_{\epsilon}^{iklm})_{\epsilon} = \left\{ (J_{\epsilon}(x'^{0}, x'^{1}, x'^{2}, x'^{3}))_{\epsilon} \right\} e^{prst}, \quad (B17) $$

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

$$ \begin{gathered} (J_{\epsilon}(x'^{0}, x'^{1}, x'^{2}, x'^{3}))_{\epsilon} = \left( \frac{\partial x_{\epsilon}^{i}}{\partial x'^{p}} \frac{\partial x_{\epsilon}^{k}}{\partial x'^{r}} \frac{\partial x_{\epsilon}^{l}}{\partial x'^{s}} \frac{\partial x_{\epsilon}^{m}}{\partial x'^{t}} \right)_{\epsilon} = \\ \left( \frac{\partial(x_{\epsilon}^{0}, x_{\epsilon}^{1}, x_{\epsilon}^{2}, x_{\epsilon}^{3})}{\partial(x'^{0}, x'^{1}, x'^{2}, x'^{3})} \right)_{\epsilon}. \end{gathered} \quad (B18) $$

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

$$ (g_{\epsilon}^{ik})_{\epsilon} = \left\{ \left( \frac{\partial x_{\epsilon}^{i}}{\partial x'^{i}} \frac{\partial x_{\epsilon}^{k}}{\partial x'^{m}} \right) \right\} g^{(0)im}, \quad (B19) $$

where
$$g^{(0)im} = g_{im}^{(0)} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \\ 1 & 1 & 1 & -1 \end{pmatrix}, \quad (B20)$$

and equate the determinants of the two sides of this equation. The determinant of the reciprocal tensor $\det|(g_{\epsilon}^{ik})| = 1/(g_{\epsilon})$. The determinant $\det|g^{(0)im}| = -1$.

Thus we have $1/(g_{\epsilon}) = -(J_{\epsilon}^{2}(x'^{0}, x'^{1}, x'^{2}, x'^{3}))_{\epsilon}$, and so

$$(J_{\epsilon}^{2}(x'^{0}, x'^{1}, x'^{2}, x'^{3}))_{\epsilon} = 1/\sqrt{(g_{\epsilon})}. \quad (B21)$$

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

$$(E_{\epsilon}^{iklm})_{\epsilon} = \frac{1}{\sqrt{-(g_{\epsilon})}} e^{iklm} \quad (B22)$$

and its covariant components are

$$(E_{iklm,\epsilon})_{\epsilon} = \sqrt{-(g_{\epsilon})} e_{iklm}. \quad (B23)$$

In a Galilean coordinate system $x^i, i = 0,1,2,3$ the integral of a scalar with respect to $(d\Omega'_{\epsilon}) = (dx_{\epsilon}^{0} dx_{\epsilon}^{1} dx_{\epsilon}^{2} dx_{\epsilon}^{3})_{\epsilon}$ is also a scalar, since the element $(d\Omega'_{\epsilon})_{\epsilon}$ behaves as generalized scalar quantity. On transforming to Colombeau generalized curvilinear coordinates $(x_{\epsilon}^{i})$, $i = 0,1,2,3$, the element of integration $d(d\Omega'_{\epsilon})$ goes

$$(d\Omega'_{\epsilon}) = \{(J_{\epsilon}^{-1})_{\epsilon}\}(d\Omega_{\epsilon}) = \sqrt{-(g_{\epsilon})}(d\Omega_{\epsilon})_{\epsilon}, \quad (B24)$$

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

Thus, in Colombeau generalized curvilinear coordinates, when integrating over a four-volume the quantity $\sqrt{-(g_{\epsilon})}(d\Omega_{\epsilon})$ behaves like an invariant.

Remark B.4. The element of "area" of the Colombeau generalized hypersurface spanned by three infinitesimal Colombeau generalized displacements is the contravariant antisymmetric Colombeau generalized tensor $(dS_{\epsilon}^{ikl})_{\epsilon}$; the vector dual to it is gotten by multiplying by the tensor $\sqrt{-(g_{\epsilon})}e_{iklm}$, so it is equal to

$$\sqrt{-(g_{\epsilon})}(dS_{\epsilon,i})_{\epsilon} = -\frac{1}{6}\sqrt{-(g_{\epsilon})}e_{iklm}(dS_{\epsilon}^{kim})_{\epsilon}. \quad (B25)$$

Remark B.5. Let $(df_{\epsilon}^{ik})_{\epsilon}$ be the element of two-dimensional Colombeau generalized surface spanned by two infinitesimal Colombeau generalized displacements, the dual Colombeau generalized tensor is defined as

$$\sqrt{-(g_{\epsilon})}(df_{ik,\epsilon}^{*})_{\epsilon} = \frac{1}{2}\sqrt{-(g_{\epsilon})}e_{iklm}(df_{\epsilon}^{lm})_{\epsilon}. \quad (B26)$$

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

Remark B.6. Note that the canonical rules for transforming the various integrals into one another remain the same, since their derivation was formal in character and not related to the tensor properties of the different quantities. Of importance is the rule for transforming the integral over a hypersurface into an integral over a four-volume

(Gauss' theorem), which is accomplished by the substitution

$$(dS_{i,\epsilon})_{\epsilon} := \{(d\Omega_{\epsilon})_{\epsilon}\} \left(\frac{\partial}{\partial x_{\epsilon}^{i}}\right)_{\epsilon}. \quad (B27)$$
Remark B.7. Note that for the integral of Colombeau generalized vector $(A_{\epsilon}^i)_{\epsilon}$ we have

$$ (\oint A_{\epsilon}^i dS_{i,\epsilon})_{\epsilon} = \left( \int \frac{\partial A_{\epsilon}^i}{\partial x_{\epsilon}^i} d\Omega_{\epsilon} \right)_{\epsilon} = \int \left\{ \left( \frac{\partial A_{\epsilon}^i}{\partial x_{\epsilon}^i} \right)_{\epsilon} \right\} \{d\Omega_{\epsilon}\}_{\epsilon} \quad (B28) $$

This formula (B28) is the generalization of Gauss' theorem.

An integral over a two-dimensional surface is transformed into an integral over the hypersurface "spanning" it by replacing the Colombeau generalized element of integration $(df_{ik,\epsilon}^*)_{\epsilon}$ by the operator

$$ (df_{ik,\epsilon}^*)_{\epsilon} := \left( \int dS_{i,\epsilon} \frac{\partial}{\partial x_{\epsilon}^k} \right)_{\epsilon} - \left( \int dS_{k,\epsilon} \frac{\partial}{\partial x_{\epsilon}^i} \right)_{\epsilon} \quad (B29) $$

For example, for the integral of an antisymmetric Colombeau generalized tensor

$(A_{\epsilon}^{ik})_{\epsilon}$ we have

$$ \left( \int A_{\epsilon}^{ik} df_{ik,\epsilon}^{*} \right)_{\epsilon} = \left( \int dS_{i,\epsilon} \frac{\partial A_{\epsilon}^{ik}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} - $$

$$ \left( \int dS_{k,\epsilon} \frac{\partial A_{\epsilon}^{ik}}{\partial \partial x_{\epsilon}^{i}} \right)_{\epsilon} = 2 \left( \int dS_{i,\epsilon} \frac{\partial A_{\epsilon}^{ik}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} \quad (B30) $$

The integral over a four-dimensional Colombeau generalized closed curve is transformed into an integral over the surface spanning it by the substitution

$$ (dx_{\epsilon}^{i})_{\epsilon} := \left\{ (df_{\epsilon}^{ki})_{\epsilon} \right\} \left( \frac{\partial}{\partial x_{\epsilon}^{i}} \right)_{\epsilon} \quad (B31) $$

Thus, for the integral of Colombeau generalized vector, we have:

$$ (\oint A_{i,\epsilon} dx_{\epsilon}^{i})_{\epsilon} = \left( \int df_{\epsilon}^{ki} \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{i}} \right)_{\epsilon} = \frac{1}{2} \int \left( df_{\epsilon}^{ik} \frac{\partial A_{k,\epsilon}}{\partial x_{\epsilon}^{i}} - df_{\epsilon}^{ik} \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} \quad (B32) $$

which is the generalization of Stokes' theorem.

## APPENDIX C. COLOMBEAU GENERALIZED COVARIANT DERIVATIVES IN POINT FREE DISTRIBUTIONAL GEOMETRY

In Galilean coordinates the Colombeau generalized differentials $(dA_{i,\epsilon})_{\epsilon}$ of a vector $(A_{i,\epsilon})_{\epsilon}$ form the Colombeau generalized vector, and the derivatives $(\partial A_{i,\epsilon}/\partial x_{\epsilon}^k)_{\epsilon}$ of the components of a vector with respect to the coordinates form the Colombeau generalized tensor. In Colombeau generalized curvilinear coordinates this is not so; $(dA_{i,\epsilon})_{\epsilon}$ is not a vector, and $(\partial A_{i,\epsilon}/\partial x_{\epsilon}^k)_{\epsilon}$ is not the Colombeau generalized tensor.

This is due to the fact that $(dA_{i,\epsilon})_{\epsilon}$ is the difference of vectors located at different (infinitesimally separated) points of space; at different points in space vectors transform differently, since the coefficients in the formulas (B3)-(B4) are Colombeau generalized functions of the generalized coordinates. Thus, in order to compare two infinitesimally separated generalized vectors we must subject one of them to a parallel translation to the point where the second is located. Let us consider an arbitrary generalized contravariant vector ; if its value at the point $x^i$ is $(A_{\epsilon}^i)_{\epsilon}$, then at the neighboring point $(x_{\epsilon}^i)_{\epsilon} + (dx_{\epsilon}^i)_{\epsilon}$ it is equal to $(A_{\epsilon}^i)_{\epsilon} + (dA_{\epsilon}^i)_{\epsilon} = (A_{\epsilon}^i + dA_{\epsilon}^i)_{\epsilon}$. We subject the vector $(A_{\epsilon}^i)_{\epsilon}$ to an infinitesimal parallel displacement to the point $x^i + dx^i$; the change in the vector which results from this we denote by $(\delta A_{\epsilon}^i)_{\epsilon}$.
Then the difference $(DA_{\epsilon}^{i})_{\epsilon}$ between the two Colombeau generalized vectors which are now located at the same point is

$$ (DA_{\epsilon}^{i})_{\epsilon} = (dA_{\epsilon}^{i})_{\epsilon} - (\delta A_{\epsilon}^{i})_{\epsilon}. \quad (C1) $$

The change $(\delta A_{\epsilon}^{i})_{\epsilon}$ in the components of Colombeau generalized vector under an infinitesimal parallel displacement depends on the values of the components themselves, where the dependence must clearly be linear. This follows directly from the fact that the sum of two Colombeau generalized vectors must transform according to the same law as each of the constituents. Thus $(\delta A_{\epsilon}^{i})_{\epsilon}$ has the form

$$ (\delta A_{\epsilon}^{i})_{\epsilon} = -(\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k} dx_{\epsilon}^{l})_{\epsilon}, \quad (C2) $$

where $(\Gamma_{kl,\epsilon}^{i})_{\epsilon}$ are the certain Colombeau generalized functions of the coordinates. Their form depends, of course, on the coordinate system; for a Galilean coordinate system $(\Gamma_{kl,\epsilon}^{i})_{\epsilon} = 0_{\mathbb{R}}$. From this it is already clear that the quantities $(\Gamma_{kl,\epsilon}^{i})_{\epsilon}$ do not form Colombeau generalized tensor, since a tensor which is equal to zero in one coordinate system is equal to zero in every other one. In a curvilinear space it is, of course, impossible to make all the $(\Gamma_{kl,\epsilon}^{i})_{\epsilon}$ vanish over all of space. But we can choose a coordinate system for which the $(\Gamma_{kl,\epsilon}^{i})_{\epsilon}$ become $0_{\mathbb{R}}$ over a given infinitesimal region. The quantities $(\Gamma_{kl,\epsilon}^{i})_{\epsilon}$, are called generalized Christoffel symbols. In addition to the quantities $(\Gamma_{kl,\epsilon}^{i})_{\epsilon}$ we shall later also use Colombeau generalized quantities $(\Gamma_{i,kl,\epsilon})_{\epsilon}$ defined as follows

$$ (\Gamma_{i,kl,\epsilon})_{\epsilon} = (g_{im,\epsilon} \Gamma_{km,\epsilon}^{m})_{\epsilon}. \quad (C3) $$

Conversely,

$$ (\Gamma_{kl,\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{im} \Gamma_{m,kl,\epsilon})_{\epsilon}. \quad (C4) $$

It is also easy to relate the change in the components of a covariant vector under a parallel displacement to the Christoffel symbols. To do this we note that under a parallel displacement, a scalar is unchanged. In particular, the scalar product of two vectors does not change under a parallel displacement.

Let $(A_{i,\epsilon})_{\epsilon}$ and $(B_{\epsilon}^{i})_{\epsilon}$ be any covariant and contravariant vectors. Then from

$$ \delta(A_{i,\epsilon} B_{\epsilon}^{i})_{\epsilon} = 0_{\mathbb{R}}, \text{ we have} $$

$$ (B_{\epsilon}^{i} \delta A_{i,\epsilon})_{\epsilon} = -(A_{i,\epsilon} \delta B_{\epsilon}^{i})_{\epsilon} = (\Gamma_{kl,\epsilon}^{i} B_{\epsilon}^{k} A_{i,\epsilon} dx_{\epsilon}^{l})_{\epsilon} \quad (C5) $$

or, changing the indices,

$$ (B_{\epsilon}^{i} \delta A_{i,\epsilon})_{\epsilon} = (\Gamma_{il,\epsilon}^{k} B_{\epsilon}^{i} A_{k,\epsilon} dx_{\epsilon}^{l})_{\epsilon} \quad (C6) $$

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

$$ (\delta A_{i,\epsilon})_{\epsilon} = ((\Gamma_{il,\epsilon}^{k} A_{k,\epsilon})_{\epsilon}) (dx_{\epsilon}^{l})_{\epsilon} \quad (C7) $$

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

Substituting (C2) and $(dA_{\epsilon}^{i})_{\epsilon} = ((\partial A_{\epsilon}^{i}/\partial x^{l})_{\epsilon})(dx_{\epsilon}^{l})_{\epsilon}$ in (C1), we have $(DA_{\epsilon}^{i})_{\epsilon} =$

$$ \left[ \left( \frac{\partial A_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}} \right)_{\epsilon} + (\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k}) \right] (dx_{\epsilon}^{l})_{\epsilon}. \quad (C8) $$

Similarly, one finds for a covariant Colombeau generalized vector

$$ (DA_{i,\epsilon})_{\epsilon} = \left[ \left( \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{l}} \right)_{\epsilon} + (\Gamma_{il,\epsilon}^{k} A_{k,\epsilon}) \right] (dx_{\epsilon}^{l})_{\epsilon}. \quad (C9) $$

The expressions in parentheses in (C8) and (C9) are Colombeau generalized tensors, since when multiplied by the vector $dx_{\epsilon}^{l}$ they give a vector. Clearly, these are the Colombeau generalized tensors which give the desired generalization of the
concept of a derivative to Colombeau generalized curvilinear coordinates. These Colombeau generalized tensors are called the Colombeau generalized covariant derivatives of the vectors $(A_{\epsilon}^{i})_{\epsilon}$ and $(A_{i,\epsilon})_{\epsilon}$ respectively. We shall denote them by

$A_{i,k,\epsilon}^{i}$ and $A_{i,k,\epsilon}$. Thus,

$$ (DA_{\epsilon}^{i})_{\epsilon} = (A_{i,l,\epsilon}^{i} dx_{\epsilon}^{l})_{\epsilon}, (DA_{i,\epsilon})_{\epsilon} = (A_{i,l,\epsilon} dx_{\epsilon}^{l})_{\epsilon}, \quad (C10) $$

while the Colombeau generalized covariant derivatives themselves are

$$ (A_{;l,\epsilon}^{i})_{\epsilon} = \left(\frac{\partial A_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} + (\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k})_{\epsilon} \quad (C11) $$

and

$$ (A_{i; l, \epsilon})_{\epsilon} = \left(\frac{\partial A_{i, \epsilon}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} + (\Gamma_{il, \epsilon}^{k} A_{k, \epsilon})_{\epsilon}. \quad (C12) $$

It is also easy to calculate the covariant derivative of the Colombeau generalized tensor. We get the covariant derivative of the Colombeau generalized tensor $(A_{\epsilon}^{ik})_{\epsilon}$ in the form

$$ (A_{;l,\epsilon}^{ik})_{\epsilon} = \left(\frac{\partial A_{\epsilon}^{ik}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} + (\Gamma_{ml,\epsilon}^{i} A_{\epsilon}^{mk})_{\epsilon} + (\Gamma_{ml,\epsilon}^{k} A_{\epsilon}^{im})_{\epsilon}. \quad (C13) $$

The covariant derivative of the Colombeau generalized mixed tensor $(A_{k,\epsilon}^{i})_{\epsilon}$ and the Colombeau generalized covariant tensor $(A_{ik,\epsilon})_{\epsilon}$ has the form

$$ (A_{k; l, \epsilon})_{\epsilon} = \left(\frac{\partial A_{k, \epsilon}^{i}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} + (\Gamma_{kl, \epsilon}^{m} A_{m, \epsilon}^{i})_{\epsilon} + (\Gamma_{ml, \epsilon}^{i} A_{k, \epsilon}^{m})_{\epsilon} \quad (C14) $$

and

$$ (A_{ik; l, \epsilon})_{\epsilon} = \left(\frac{\partial A_{ik, \epsilon}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} - (\Gamma_{il, \epsilon}^{m} A_{mk, \epsilon})_{\epsilon} - (\Gamma_{kl, \epsilon}^{m} A_{im, \epsilon})_{\epsilon}. \quad (C15) $$

correspondingly. If in a covariant derivative we raise the index signifying the differentiation, we obtain the so-called contravariant derivative. Thus,

$$ (A_{i, \epsilon}^{ik})_{\epsilon} = (g_{\epsilon}^{kl} A_{i; l, \epsilon})_{\epsilon}; (A_{\epsilon}^{i; k})_{\epsilon} = (g_{\epsilon}^{kl} A_{; l, \epsilon}^{i})_{\epsilon}. \quad (C16) $$

## APPENDIX D. THE RELATIONS OF THE COLOMBEAU GENERALIZED CHRISTOFFEL SYMBOLS TO THE COLOMBEAU GENERALIZED METRIC TENSOR IN POINT FREE DISTRIBUTIONAL GEOMETRY

Let us show that the Colombeau generalized covariant derivative of the Colombeau Generalized metric tensor $(g_{ik,\epsilon})_{\epsilon}$ is $0_{\mathbb{R}}$. To do this we note that the relation

$$ (DA_{i,\epsilon})_{\epsilon} = (g_{ik,\epsilon}DA_{\epsilon}^{k})_{\epsilon} \quad (D1) $$

is valid for the generalized vector $(DA_{i,\epsilon})_{\epsilon}$ as for any generalized vector. On the other hand, $(A_{i,\epsilon})_{\epsilon} = (g_{ik,\epsilon}A_{\epsilon}^{k})_{\epsilon}$, so that

$$ (DA_{i,\epsilon})_{\epsilon} = D(g_{ik,\epsilon}A_{\epsilon}^{k})_{\epsilon} = (g_{ik,\epsilon}DA_{\epsilon}^{k})_{\epsilon} + (A_{\epsilon}^{k}Dg_{ik,\epsilon})_{\epsilon}. \quad (D2) $$

Comparing with $(DA_{i,\epsilon})_{\epsilon} = (g_{ik,\epsilon}DA_{\epsilon}^{k})_{\epsilon}$, and remembering that the vector $(A_{\epsilon}^{k})_{\epsilon}$ is arbitrary
$$ (D g_{ik,\varepsilon})_{\varepsilon} = 0_{\mathbb{R}}. \quad (\text{D3}) $$

Therefore, the covariant derivative

$$ (g_{ik,l,\varepsilon})_{\varepsilon} = 0_{\mathbb{R}}. \quad (\text{D4}) $$

Thus $(g_{ik,\varepsilon})_{\varepsilon}$ may be considered as a constant under covariant differentiation. The equation $(g_{ik,l,\varepsilon})_{\varepsilon} = 0_{\mathbb{R}}$ can be used to express the Colombeau generalized Christoffel symbols $(\Gamma^i_{kl,\varepsilon})_{\varepsilon}$ in terms of the Colombeau generalized metric tensor $(g_{ik,\varepsilon})_{\varepsilon}$. To do this we write in accordance with the general definition Eq. (C9)

$$ \begin{aligned} (g_{ik,l,\varepsilon})_{\varepsilon} &= \left(\frac{\partial g_{ik,\varepsilon}}{\partial x^{l}_{\varepsilon}}\right)_{\varepsilon} - (g_{mk,\varepsilon}\Gamma^{m}_{il,\varepsilon})_{\varepsilon} - (g_{im,\varepsilon}\Gamma^{m}_{kl,\varepsilon})_{\varepsilon} = \\ & \left(\frac{\partial g_{ik,\varepsilon}}{\partial x^{l}_{\varepsilon}}\right)_{\varepsilon} - (\Gamma_{k,il,\varepsilon})_{\varepsilon} - (\Gamma_{i,kl,\varepsilon})_{\varepsilon} = 0_{\mathbb{R}}. \quad (\text{D5}) \end{aligned} $$

Thus, the derivatives of $(g_{ik,\varepsilon})_{\varepsilon}$ are expressed in terms of the Colombeau generalized Christoffel symbols. We write the values of the derivatives of $(g_{ik,\varepsilon})_{\varepsilon}$, permuting the indices $i, k, l$:

$$ \left(\frac{\partial g_{ik,\varepsilon}}{\partial x^{l}_{\varepsilon}}\right)_{\varepsilon} = (\Gamma_{k,il,\varepsilon})_{\varepsilon} + (\Gamma_{i,kl,\varepsilon})_{\varepsilon}, \quad \left(\frac{\partial g_{il,\varepsilon}}{\partial x^{k}_{\varepsilon}}\right)_{\varepsilon} = (\Gamma_{i,kl,\varepsilon})_{\varepsilon} + (\Gamma_{l,ik,\varepsilon})_{\varepsilon}, $$

$$ \left(\frac{\partial g_{kl,\varepsilon}}{\partial x^{l}_{\varepsilon}}\right)_{\varepsilon} = -(\Gamma_{l,ki,\varepsilon})_{\varepsilon} + (\Gamma_{k,li,\varepsilon})_{\varepsilon}. $$

Taking half the sum of these equations, we find

$$ (\Gamma_{i,kl,\varepsilon})_{\varepsilon} = \frac{1}{2} \left[ \left( \frac{\partial g_{ik,\varepsilon}}{\partial x^{l}_{\varepsilon}} \right)_{\varepsilon} + \left( \frac{\partial g_{il,\varepsilon}}{\partial x^{k}_{\varepsilon}} \right)_{\varepsilon} - \left( \frac{\partial g_{kl,\varepsilon}}{\partial x^{l}_{\varepsilon}} \right)_{\varepsilon} \right], \quad (\text{D6}) $$

$$ \text{since } (\Gamma_{i,kl,\varepsilon})_{\varepsilon} = (\Gamma_{i,lk,\varepsilon})_{\varepsilon}. $$

From Eq. (D6) we have for the symbols $(\Gamma^i_{kl,\varepsilon})_{\varepsilon} = (g^i_{\varepsilon}\Gamma_{m,kl,\varepsilon})_{\varepsilon}$

$$ (\Gamma^i_{kl,\varepsilon})_{\varepsilon} = \frac{1}{2} \left( (g^i_{\varepsilon})_{\varepsilon} \left[ \left( \frac{\partial g_{mk,\varepsilon}}{\partial x^{l}_{\varepsilon}} \right)_{\varepsilon} + \left( \frac{\partial g_{ml,\varepsilon}}{\partial x^{k}_{\varepsilon}} \right)_{\varepsilon} - \left( \frac{\partial g_{kl,\varepsilon}}{\partial x^{m}_{\varepsilon}} \right)_{\varepsilon} \right] \right). \quad (\text{D7}) $$

# APPENDIX E. THE COLOMBEAU GENERALIZED CURVATURE TENSOR

In this Appendix we derive the general formula for the change in a vector after parallel displacement around any infinitesimal closed contour $(\gamma_{\varepsilon})_{\varepsilon}$. This generalized change $(\Delta A_{k,\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}$ can clearly be written in the form $(\oint \delta A_{k,\varepsilon})_{\varepsilon}$, where the Colombeau integral is taken over the given contour $\gamma_{\varepsilon}$. Substituting in place of $(\delta A_{k,\varepsilon})_{\varepsilon}$ the expression in Eq. (C7) we get

$$ (\Delta A_{k,\varepsilon})_{\varepsilon} = (\oint_{\gamma_{\varepsilon}} \Gamma^i_{kl,\varepsilon} (x_{\varepsilon}) A_{i,\varepsilon}(x_{\varepsilon}) dx^{l}_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}, \quad (\text{E1}) $$

where for any $i, k, l = 0,1,2,3$ : $(\Gamma^i_{kl,\varepsilon}(x_{\varepsilon}))_{\varepsilon} \in G(\tilde{\mathbb{R}}^4), x_{\varepsilon} = (x_{\varepsilon}^0, x_{\varepsilon}^1, x_{\varepsilon}^2, x_{\varepsilon}^3), A_{i,\varepsilon}(x_{\varepsilon}) \in \mathcal{D}(G), G \subseteq \mathbb{R}^4, \varepsilon \in (0, \delta]$ Note that the vector $(A_{i,\varepsilon})_{\varepsilon}$ which appears in the integrand obviously changes as we move along the contour $(\gamma_{\varepsilon})_{\varepsilon}$.

Definition E.1. We shall say that generalized change $(\Delta A_{k,\varepsilon})_{\varepsilon}$ exists in the sense of the Schwartz distributions if for any $A_i(x) \in \mathcal{D}(G)$ the limit: $\lim_{\varepsilon \to 0} \Delta A_{k,\varepsilon}$ exists, where $\Delta A_{k,\varepsilon} = \oint_{\gamma} \Gamma^i_{kl,\varepsilon} (x) A_i(x) dx^l$. Of course in this case obviously $\text{cl}[(\Gamma^i_{kl,\varepsilon}(x))_{\varepsilon}] \in \mathcal{D}'(G)$ and $\text{cl}[(\Delta A_{k,\varepsilon})_{\varepsilon}] \in \mathbb{R}$.
For the further transformation of this Colombeau integral, we must note the following. The values of the vector $(A_{i,\epsilon})_{\epsilon}$ at points inside the contour are not unique; they depend on the path along which we approach the given point. However, as we shall see from the result obtained below, this non-uniqueness is related to terms of second order. We may therefore, with the first-order accuracy which is sufficient for the transformation, regard the components of the vector $(A_{i,\epsilon})_{\epsilon}$ at points inside the infinitesimal contour $(\gamma_{\epsilon})_{\epsilon}$ as being uniquely determined by their values on the contour itself by the formulas $(\delta A_i(x_{\epsilon}))_{\epsilon} = (\Gamma_{il,\epsilon}^n(x_{\epsilon})A_{n,\epsilon}(x_{\epsilon})dx_{\epsilon}^l)_{\epsilon}$, i.e., by the derivatives

$$ \left( \frac{\partial A_{i,\epsilon}(x_{\epsilon})}{\partial x_{\epsilon}^{l}} \right)_{\epsilon} = \left( \Gamma_{il,\epsilon}^{n}(x_{\epsilon}) A_{n,\epsilon}(x_{\epsilon}) \right)_{\epsilon}. \quad (E2) $$

Now applying generalized Stokes' theorem (Theorem E.1) to the integral (E1) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f_{\epsilon}^{im})_{\epsilon}$, we get

$$ \begin{aligned} (\Delta A_{k,\epsilon})_{\epsilon} &= \frac{1}{2} \left[ \left( \frac{\partial (\Gamma_{km,\epsilon}^i(x_{\epsilon})A_{i,\epsilon}(x_{\epsilon}))}{\partial x_{\epsilon}^l} \right)_{\epsilon} - \left( \frac{\partial (\Gamma_{kl,\epsilon}^i(x_{\epsilon})A_{i,\epsilon}(x_{\epsilon}))}{\partial x_{\epsilon}^m} \right)_{\epsilon} \right] (\Delta f_{\epsilon}^{im})_{\epsilon} = \\ &= \frac{1}{2} \left[ A_i(x) \left( \frac{\partial (\Gamma_{km,\epsilon}^i(x_{\epsilon}))}{\partial x_{\epsilon}^l} \right)_{\epsilon} - A_i(x) \left( \frac{\partial (\Gamma_{kl,\epsilon}^i(x_{\epsilon}))}{\partial x_{\epsilon}^m} \right)_{\epsilon} + \right. \\ & \qquad \left. \left( \frac{\partial A_i(x_{\epsilon})}{\partial x_{\epsilon}^l} \right)_{\epsilon} (\Gamma_{km,\epsilon}^i(x_{\epsilon}))_{\epsilon} - \left( \frac{\partial A_i(x_{\epsilon})}{\partial x_{\epsilon}^m} \right)_{\epsilon} (\Gamma_{kl,\epsilon}^i(x_{\epsilon}))_{\epsilon} \right] (\Delta f_{\epsilon}^{im})_{\epsilon}. \quad (E3) \end{aligned} $$

Definition E.2. Colombeau generalized $k$-form $(\epsilon)_{\epsilon}$ on a differentiable manifold $M$ is a smooth section of the bundle of alternating Colombeau generalized $k$-tensors on $M$. Equivalently, $(\epsilon)_{\epsilon}$ associates to each $(x_{\epsilon})_{\epsilon} \in M$ an alternating Colombeau generalized $k$-tensor $(x_{\epsilon}, \epsilon)_{\epsilon}$, in such a way that in any chart for $M$, the coefficients $(i_1...i_k, \epsilon)_{\epsilon}$ are Colombeau generalized functions.

Theorem E.1. (Generalized Stokes' Theorem) Let $(\omega_{\epsilon})_{\epsilon}$ be Colombeau generalized differential form. Then the Colombeau integral of a differential form $(\omega_{\epsilon})_{\epsilon}$ over the boundary of some orientable Colombeau manifold $(\Sigma_{\epsilon})_{\epsilon} \subset M$ is equal to the integral of its exterior Colombeau derivative $(d\omega_{\epsilon})_{\epsilon}$ over the whole of $(\Sigma_{\epsilon})_{\epsilon}$, i.e.,

$$ \int_{(\partial \Sigma_{\epsilon})_{\epsilon}} (\omega_{\epsilon})_{\epsilon} = \left( \int_{\partial \Sigma_{\epsilon}} \omega_{\epsilon} \right)_{\epsilon} = \left( \int_{\Sigma_{\epsilon}} d\omega_{\epsilon} \right)_{\epsilon} = \int_{(\Sigma_{\epsilon})_{\epsilon}} (d\omega_{\epsilon})_{\epsilon}. \quad (E4) $$

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

Example E.1. For example, for the integral of Colombeau generalized vector

$$ (A_{i,\epsilon}(x_{\epsilon}))_{\epsilon} $$

we have

$$ \begin{aligned} \left( \oint_{\Gamma_{\epsilon}} A_{i,\epsilon} dx_{\epsilon}^{i} \right)_{\epsilon} &= \left( \int_{\Sigma_{\epsilon}} df_{\epsilon}^{ki} \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} = \\ & \frac{1}{2} \left( \int \left[ (df_{\epsilon}^{ki})_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x_{\epsilon}^{i}} - \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{k}} \right) \right)_{\epsilon} = \\ & \frac{1}{2} \int \left[ (df_{\epsilon}^{ki})_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x_{\epsilon}^{i}} - \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} = \\ & \frac{1}{2} \int \left[ (df_{\epsilon}^{ki})_{\epsilon} \right] \left[ \left( \frac{\partial A_{k,\epsilon}}{\partial x_{\epsilon}^{i}} \right)_{\epsilon} - \left( \frac{\partial A_{i,\epsilon}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} \right], \end{aligned} \quad (E5) $$

where $(\Gamma_{\epsilon})_{\epsilon} = (\partial \Sigma_{\epsilon})_{\epsilon}$ and $(df_{\epsilon}^{ki})_{\epsilon} = (dx_{\epsilon}^{i} dx_{\epsilon}^{k})_{\epsilon} - (dx_{\epsilon}^{k} dx_{\epsilon}^{i})_{\epsilon}$ is the infinitesimal
element of surface which is given by the antisymmetric tensor of second rank $(df_{\varepsilon}^{ki})_{\varepsilon}$. Substituting the values of the derivatives in Eq.(E2) into Eq.(E3), we get

$$ (\Delta A_{k,\varepsilon})_{\varepsilon} = \frac{1}{2} (R_{klm,\varepsilon}^{i}(x_{\varepsilon}) A_{i,\varepsilon}(x_{\varepsilon}) \Delta f_{\varepsilon}^{im})_{\varepsilon}, \quad (\text{E6}) $$

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

$$ \begin{aligned} (R_{klm,\varepsilon}^{i}(x_{\varepsilon}))_{\varepsilon} = & \left( \frac{\partial(\Gamma_{km,\varepsilon}^{i}(x_{\varepsilon}))}{\partial x_{\varepsilon}^{l}} \right)_{\varepsilon} - \left( \frac{\partial(\Gamma_{kl,\varepsilon}^{i}(x_{\varepsilon}))}{\partial x_{\varepsilon}^{m}} \right)_{\varepsilon} + \\ & (\Gamma_{ni,\varepsilon}^{i}(x_{\varepsilon}) \Gamma_{km,\varepsilon}^{n}(x_{\varepsilon}))_{\varepsilon} - (\Gamma_{nm,\varepsilon}^{i}(x_{\varepsilon}) \Gamma_{kl,\varepsilon}^{n}(x_{\varepsilon}))_{\varepsilon}. \end{aligned} \quad (\text{E7}) $$

Definition E.3. The tensor field $(R_{kim,\varepsilon}^{l}(x)_{\varepsilon})$ is called the distributional curvature tensor or the distributional Riemann tensor.

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

Definition E.4. We shall say that the distributional Riemann tensor $(R_{klm,\varepsilon}^{i}(x))_{\varepsilon}$ exists in the sense of the Schwartz distributions if for any $i, k, l = 0,1,2,3$ and for any $A_i(x) \in D(G)$ the limit:

$$ \lim_{\varepsilon \to 0} \int_G R_{klm,\varepsilon}^{i}(x) A_{i}(x) d^4 x \quad (\text{E8}) $$
exists.

Definition E.5. We shall say that the distributional Riemann tensor $(R_{klm,\varepsilon}^{i}(x_{\varepsilon}))_{\varepsilon}$ exists in the classical sense at point $(\tilde{x}_{\varepsilon})_{\varepsilon} \in \mathbb{R}^4$ if there exists standard part of point value of Colombeau generalized function $(R_{klm,\varepsilon}^{i}(x_{\varepsilon}))_{\varepsilon}$ at point $(\tilde{x}_{\varepsilon})_{\varepsilon} \in \mathbb{R}$, i.e.

$$ \text{st} \left( \text{cl} \left[ (R_{klm,\varepsilon}^{i}(\tilde{x}_{\varepsilon})) \right] \right) \neq 0. $$

Remark E.2. It is easy to obtain a similar formula to (E6) for a generalized contravariant vector $(A_{\varepsilon}^{k})_{\varepsilon}$. To do this we note, since under parallel displacement a scalar does not change, that $(\Delta(A_{\varepsilon}^{k}B_{k,\varepsilon}))_{\varepsilon} = 0_{\tilde{R}}$, where $(B_{k,\varepsilon})_{\varepsilon}$ is any Colombeau generalized covariant vector. Thus, we obtain

$$ \begin{aligned} (\Delta(A_{\varepsilon}^{k} B_{k,\varepsilon}))_{\varepsilon} = & (\Delta A_{\varepsilon}^{k} B_{k,\varepsilon})_{\varepsilon} + (A_{\varepsilon}^{k} \Delta B_{k,\varepsilon})_{\varepsilon} = \frac{1}{2} (A_{\varepsilon}^{k} B_{i,\varepsilon} R_{klm,\varepsilon}^{i} \Delta f_{\varepsilon}^{lm})_{\varepsilon} + (B_{k,\varepsilon} \Delta A_{\varepsilon}^{k})_{\varepsilon} = \\ & \{ (B_{k,\varepsilon})_{\varepsilon} \} \{ (\Delta A_{\varepsilon}^{k})_{\varepsilon} + \frac{1}{2} (A_{\varepsilon}^{l} R_{ilm,\varepsilon}^{k} \Delta f_{\varepsilon}^{lm})_{\varepsilon} \} = 0_{\tilde{R}} \end{aligned} \quad (\text{E9}) $$

or, in view of the arbitrariness of the vector $(B_{k,\varepsilon})_{\varepsilon}$ we get

$$ (\Delta A_{\varepsilon}^{k})_{\varepsilon} = -\frac{1}{2} (R_{ilm,\varepsilon}^{k} A_{\varepsilon}^{k} f_{\varepsilon}^{lm})_{\varepsilon}. \quad (\text{E10}) $$

If we now twice differentiate Colombeau generalized vector $(A_{i,\varepsilon})_{\varepsilon}$ covariantly with respect to $x_{\varepsilon}^{k}$ and $x_{\varepsilon}^{l}$, then the result obviously depends on the order of differentiation, contrary to the situation for ordinary differentiation. It turns out that the difference $(A_{i;k;l,\varepsilon})_{\varepsilon} - (A_{i;l;k,\varepsilon})_{\varepsilon}$ is given by the same Colombeau generalized curvature tensor which we introduced above. Namely, one finds the formula

$$ (A_{i;k;l,\varepsilon})_{\varepsilon} - (A_{i;l;k,\varepsilon})_{\varepsilon} = (A_{m,\varepsilon} R_{ikl,\varepsilon}^{m})_{\varepsilon}, \quad (\text{E11}) $$

which is easily verified by direct calculation in the locally geodesic coordinate system. Similarly, for a contravariant Colombeau generalized vector, one gets $(A_{;k;l,\varepsilon}^{i})_{\varepsilon} - (A_{;l;k,\varepsilon}^{i})_{\varepsilon} = (A_{\varepsilon}^{m} R_{mkl,\varepsilon}^{i})_{\varepsilon}$. (E12)