test functions (conventional and well-behaved functions) into the set of real numbers $\mathbb{R}$.

In the simplest case, the space of test functions considered is $\mathcal{D}(\mathbb{R}^n, K)$, which is the set

of functions $\varphi: \mathbb{R}^n \to \mathbb{R}$ having two properties:

(i) $\varphi$ is smooth (infinitely differentiable);

(ii) $\varphi$ has compact support (is identically zero outside some compact set $K \subset \mathbb{R}^n$).

**Definition 1.10.5.** Schwartz distribution with compact support (Schwartz generalized functions) $T$ is a linear mapping : $\mathcal{D}(\mathbb{R}^n, K) \to \mathbb{R}$. Instead of writing $T(\varphi)$, it is conventional

to write $\langle T, \varphi \rangle$ for the value of $T$ acting on a test function $\varphi$. A simple example of a distribution is the Dirac delta $\delta$, defined by $\langle \delta, \varphi \rangle = \varphi(0)$, meaning that $\delta$ evaluates a test function at 0. Its physical interpretation is as the density of a point source.

**Definition 1.10.6.** Suppose that $f: K \subset \mathbb{R}^n \to \mathbb{R}$ is a locally integrable function. Then a corresponding distribution $T_f \in \mathcal{D}'(\mathbb{R}^n, K)$ may be defined by

$$ \langle T_f, \varphi \rangle = \int_K f(\hat{x}) \varphi(\hat{x}) d^n x \qquad (1.10.27) $$

for $\varphi \in \mathcal{D}(\mathbb{R}^n, K)$.

**Definition 1.10.7.** We chose now the a space of test functions $\mathcal{D}(\mathbb{R}^4, K, \hat{x}^0)$ :

$$ \mathcal{D}(\mathbb{R}^4, K, \hat{x}^0) = \{\psi(\hat{x})\} = \{\Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x}) | \varphi(\hat{x}) \in \mathcal{D}(\mathbb{R}^4, K, \hat{x}^0)\}, $$

$$ \text{where } \Phi(\hat{x} - \hat{x}^0) = \prod_{i=0}^{4} (x_i - x_i^0)^{2\delta}, \delta \ge 1, \varphi(\hat{x}) \in \mathcal{D}(\mathbb{R}^4, K), \hat{x}^0 \in K. $$

Let us introduce now similarly to canonical Definition 1.8.5 the formula for the *regularized* (or generalized) change $\widetilde{\Delta A_k}(\varphi)$ in a vector $A_i(\hat{x})$ after parallel displacement

around infinitesimal closed contour $\Gamma$ (see Fig.1.10.3). This regularized change $\widetilde{\Delta A_k}$ can clearly be written in the form

$$ \widetilde{\Delta A_k}(\varphi) = \langle \delta A_k(\hat{x}), \Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x}) \rangle = \oint_{\Gamma} \Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x})\delta A_k(\hat{x}), \quad (1.10.28) $$

$$ \text{where } \Phi(\hat{x} - \hat{x}^0) = \prod_{i=0}^{4} (x_i - x_i^0)^{2\delta}, \delta \ge 1, \varphi(\hat{x}) \in \mathcal{D}(\mathbb{R}^4, K), \hat{x}^0 \in K \text{ and where the integral is} $$

taken over the given contour $\Gamma \subset K, \hat{x}^0 \notin \Gamma$. If $\hat{x}^0 \in \Gamma$ (see Fig.1.10.4) the regularized change $\widetilde{\Delta A_k}$ can clearly be written in the form

$$ \widetilde{\Delta A_k}(\varphi) = \langle \delta A_k(\hat{x}), \Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x}) \rangle = \oint_{\Gamma \setminus \{\hat{x}^0\}} \Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x})\delta A_k(\hat{x}). \quad (1.10.29) $$

Substituting in place of $\delta A_k$ the canonical expression $\delta A_k = \Gamma_{kl}^i(\hat{x})A_k dx^l$ (see [4], Eq. (85.5)) we obtain

$$ \widetilde{\Delta A_k}(\varphi) = \langle \Gamma_{kl}^i(\hat{x}), \Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x})A_k \rangle = \oint_{\Gamma} \Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x})\delta A_k = \oint_{\Gamma} \Gamma_{kl}^i(\hat{x})\Phi(\hat{x} - \hat{x}^0)\varphi(\hat{x})A_k dx^l, \quad (1.10.30) $$
where

$$ \frac{\partial A_i}{\partial x^l} = \Gamma^i_{kl}(\hat{x}) A_k. \qquad (1.10.31) $$

**Remark 1.10.18.** Note that: (i) Eq.(1.10.31) holds since $\hat{x}^0 \notin \Gamma$. (ii) In any neighborhood $O(\hat{x}^0, \varepsilon) = \{\hat{x} : \|\hat{x} - \hat{x}^0\| \le \varepsilon\}, \varepsilon > 0, \varepsilon \ll 1$ of the singular point $\hat{x}^0$ the functions $\Gamma^i_{kl}(\hat{x})\Phi(\hat{x} - \hat{x}^0)$ is regular. (iii) At singular point $\hat{x}^0$ the quantities $\Gamma^i_{kl}(\hat{x})\Phi(\hat{x} - \hat{x}^0)|_{\hat{x}=\hat{x}^0}$ are well defined by the limit:

$$ \Gamma^i_{kl}(\hat{x})\Phi(\hat{x} - \hat{x}^0)|_{\hat{x}=\hat{x}^0} = \lim_{\hat{x} \to \hat{x}^0} \Gamma^i_{kl}(\hat{x})\Phi(\hat{x} - \hat{x}^0), $$

since for any $i, k, l$ the limit $\lim_{\hat{x} \to \hat{x}^0} \Gamma^i_{kl}(\hat{x})\Phi(\hat{x} - \hat{x}^0)$ exists and finite by the choosing of the function $\Phi(\hat{x} - \hat{x}^0)$.

(iv) It follows from (i)-(iii) the classical Stokes' theorem (see [4], Eq.(6.19)) holds for the integral (1.10.30).

<figure><img src="image_20.png" /><figcaption>Fig.1.10.5. Infinitesimal closed contour $\Gamma$ and corresponding singular surface $\Sigma_{\Gamma} \ni \hat{x}^0$ spanning by $\Gamma$. Due to the degeneracy of the Schwarzschild metric (1.1) at point $r=0$, the Levi-Civita connection $\Gamma^{+l}_{kj}(\{\}) = \frac{1}{2}[g^{lm}(\{\})][(g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})]$ is not available on $\mathbb{R}_+^3 \cup \{0\}$.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Fig.1.10.6. Infinitesimal closed contour $\Gamma$ with a singularity at point $\hat{x}^0$ on Horizon and corresponding singular surface $\Sigma_{\Gamma} \ni \hat{x}^0$ spanning by $\Gamma$.</figcaption></figure>

Due to the degeneracy of the Schwarzschild metric field (1.1)

at $r = 2m$,

$$ \begin{aligned} \text{the classical Levi-Civit\'a connection } \Gamma_{kj}^{+l}(\{\}) &= \\ &= \frac{1}{2}[g^{lm}(\{\})][g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})] \end{aligned} $$

is not available on $\mathbb{R}_+^3 \cup \{r = 2m\}$ in classical sense

but is available on $\mathbb{R}_+^3 \cup \{r = 2m\} \cup \{r = 0\}$ in the sense of the generalized functions in $D'(\mathbb{R}^4, K, \hat{x}^0)$.

**Remark 1.10.19.** Note that: (i) by using the Eq. (1.10.30) the classical singular Levi-Civitá connection corresponding to the degenerate and singular Schwarzschild metric field (1.6) now is available on extended Schwarzschild spacetime $\overline{\mathcal{S}h}=(S^2 \times \{r \ge 2m\} \cup \{0 \le r \le 2m\}) \times \mathbb{R}$, since the singular Christoffel symbols (1.7) in Schwarzschild coordinates are well defined as generalized functions in $D'(\mathbb{R}^4, K, \hat{x}^0)$.

(ii) The same holds for the Schwarzschild metric in isotropic coordinates (1.11).

Now applying Stokes' theorem (see [4], Eq. (6.19)) to the integral (1.10.30) and considering that the area enclosed by the contour has the infinitesimal value $\Delta f^{lm}$, we get
$$
\begin{align*}
\widetilde{\Delta A_k} &= \oint_{\Gamma} \Phi(\hat{x} - \hat{x}^0) \Gamma_{kl}^i(\hat{x}) A_k dx^l \\
&= \frac{1}{2} \int_{\Sigma_{\Gamma}} \left[ \frac{\partial(\Gamma_{km}^i(\hat{x}) A_i \Phi(\hat{x} - \hat{x}^0))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(\hat{x}) A_i \Phi(\hat{x} - \hat{x}^0))}{\partial x^m} \right] d\hat{f}^{lm} \\
&\approx \left[ \frac{\partial(\Gamma_{km}^i(\hat{x}) A_i \Phi(\hat{x} - \hat{x}^0))}{\partial x^l} - \frac{\partial(\Gamma_{kl}^i(\hat{x}) A_i \Phi(\hat{x} - \hat{x}^0))}{\partial x^m} \right] \frac{\Delta f^{lm}}{2} \\
&= \left[ \Phi(\hat{x} - \hat{x}^0) \frac{\partial(\Gamma_{km}^i(\hat{x}) A_i)}{\partial x^l} + (\Gamma_{km}^i(\hat{x}) A_i) \frac{\partial \Phi(\hat{x} - \hat{x}^0)}{\partial x^l} - \right. \\
&\qquad \left. - \Phi(\hat{x} - \hat{x}^0) \frac{\partial(\Gamma_{kl}^i(\hat{x}) A_i)}{\partial x^m} - (\Gamma_{kl}^i(\hat{x}) A_i) \frac{\partial \Phi(\hat{x} - \hat{x}^0)}{\partial x^m} \right] \frac{\Delta f^{lm}}{2} \\
&= \left[ \Phi(\hat{x} - \hat{x}^0) \frac{\partial(\Gamma_{km}^i(\hat{x}) A_i)}{\partial x^l} - \Phi(\hat{x} - \hat{x}^0) \frac{\partial(\Gamma_{kl}^i(\hat{x}) A_i)}{\partial x^m} - \right. \\
&\qquad \left. A_i(\hat{x}) \Phi(\hat{x} - \hat{x}^0) \frac{2\delta\Gamma_{km}^i(\hat{x})}{x_l - x_l^0} - A_i(\hat{x}) \Phi(\hat{x} - \hat{x}^0) \frac{2\delta\Gamma_{kl}^i(\hat{x})}{x_m - x_m^0} \right] \frac{\Delta f^{lm}}{2}.
\end{align*}
$$

Substituting the values of the derivatives (1.10.31) into Eq.(1.10.32), we get finally:

$$
\widetilde{\Delta A}_k = \tilde{R}_{klm}^i \frac{A_i(\hat{x}) \Phi(\hat{x} - \hat{x}^0) \Delta f^{lm}}{2}, \quad (1.10.33)
$$

where $\tilde{R}_{klm}^i$, is a tensor of the fourth rank

$$
\tilde{R}_{klm}^i = R_{klm}^i + 2\delta \left[ \frac{\Gamma_{km}^i(\hat{x})}{x_l - x_l^0} - \frac{\Gamma_{kl}^i(\hat{x})}{x_m - x_m^0} \right]. \quad (1.10.34)
$$

Here $R_{klm}^i$ is the classical Riemann curvature tensor. That $\tilde{R}_{klm}^i$ is a tensor is clear from the fact that in (1.10.6) the left side is a vector-the difference $\widetilde{\Delta A}_k$ between the values of vectors at one and the same point.

**Definition1.10.8.** The tensor $\tilde{R}_{klm}^i$ is called the generalized curvature tensor or the generalized Riemann tensor. Note that for any $i,k,l,m : \tilde{R}_{klm}^i \in D'(\mathbb{R}^4, K, \hat{x}^0)$.

**Definition1.10.9.** The generalized Ricci curvature tensor $\tilde{R}_{km}$ is defined as

$$
\tilde{R}_{km} = \tilde{R}_{km}^{i}. \qquad (1.10.35)
$$

**Remark 1.10.20.** Note that for any $k,m : \tilde{R}_{km} \in D'(\mathbb{R}^4, K, \hat{x}^0)$.

**Definition1.10.10.** The generalized Ricci scalar $\tilde{R}$ is defined as

$$
\tilde{R} = g^{km} \tilde{R}_{km}. \qquad (1.10.36)
$$

**Remark 1.10.21.** Note that the generalized Ricci scalar $\tilde{R} \in D'(\mathbb{R}^4, K, \hat{x}^0)$.

**Definition1.10.11.** The generalized Einstein tensor for any $k,m : \tilde{G}_{km} \in D'(\mathbb{R}^4, K, \hat{x}^0)$ is defined as

$$
\tilde{G}_{km} = \tilde{R}_{km} - \frac{1}{2} g_{km} \tilde{R}. \qquad (1.10.37)
$$

Thus the revisited densitized Einstein field equations in $D'(\mathbb{R}^4, K, \hat{x}^0)$ reads:
$$ \int_K \tilde{G}_{km}(\hat{x}) \Phi(\hat{x} - \hat{x}^0) df^{km} = \int_K \left( \tilde{R}_{km}(\hat{x}) - \frac{1}{2} g_{km}(\hat{x}) \tilde{R}(\hat{x}) \right) \Phi(\hat{x} - \hat{x}^0) df^{km} = - \int_K T_{km}(\hat{x}) \Phi(\hat{x} - \hat{x}^0) df^{km}. \quad (1.10.38) $$

**Remark 1.10.22.** Note that Beyond any small neighborhood $O(\hat{x}^0, \varepsilon) = \{\hat{x} : \|\hat{x} - \hat{x}^0\| \le \varepsilon\}$, $\varepsilon > 0, \varepsilon \ll 1$ of the singular point $\hat{x}^0$ the equations (1.10.38) becomes to the following form

$$ \tilde{G}_{km}(\hat{x}) \Phi(\hat{x} - \hat{x}^0) = \tilde{R}_{km}(\hat{x}) - \frac{1}{2} g_{km} \tilde{R}(\hat{x}) = -T_{km}(\hat{x}), \quad (1.10.39) $$

where $\hat{x} \notin O(\hat{x}^0, \varepsilon)$. We rewrite now the Eq.(1.10.34) in the following form

$$ \tilde{R}_{klm}^i = R_{klm}^i + \Re_{klm}^i \quad (1.10.40) \\ \Re_{klm}^i = 2\delta \left[ \frac{\Gamma_{km}^i(\hat{x})}{x_l - x_l^0} - \frac{\Gamma_{kl}^i(\hat{x})}{x_m - x_m^0} \right]. $$

Thus the revisited densitized Einstein field equations (1.10.38) reads

$$ \begin{aligned} \int_K \tilde{R}_{km}(\hat{x}) - \frac{1}{2} \int_K g_{km}(\hat{x}) \tilde{R}(\hat{x}) = & \\ \int_K R_{km}(\hat{x}) \Phi(\hat{x} - \hat{x}^0) df^{km} + \int_K \Re_{km}(\hat{x}) \Phi(\hat{x} - \hat{x}^0) df^{km} - & \\ \frac{1}{2} \int_K g_{km}(\hat{x}) (R(\hat{x}) + \Re(\hat{x})) \Phi(\hat{x} - \hat{x}^0) df^{km} = & \\ \int_K \left( R_{km}(\hat{x}) - \frac{1}{2} g_{km}(\hat{x}) R(\hat{x}) \right) \Phi(\hat{x} - \hat{x}^0) df^{km} + & \\ \int_K \left( \Re_{km}(\hat{x}) - \frac{1}{2} g_{km}(\hat{x}) \Re(\hat{x}) \right) \Phi(\hat{x} - \hat{x}^0) df^{km} = - & \end{aligned} \quad (1.10.41) $$

We assume now that

$$ \int_K \left( \Re_{km}(\hat{x}) - \frac{1}{2} g_{km}(\hat{x}) \Re(\hat{x}) \right) \Phi(\hat{x} - \hat{x}^0) df^{km} = - \int_K T_{km}(\hat{x}) \Phi(\hat{x} - \hat{x}^0) df^{km}. \quad (1.10.42) $$

From the Eqs.(1.10.41)-(1.10.42) we get the (revisited) densitized "vacuum" equations:

$$ \int_K \left( R_{km}(\hat{x}) - \frac{1}{2} g_{km}(\hat{x}) R(\hat{x}) \right) \Phi(\hat{x} - \hat{x}^0) df^{km} = 0. \quad (1.10.43) $$

**Remark 1.10.23.** It follows that a metric field $g_{km}(\hat{x})$ has a distributional source in $D'(\mathbb{R}^4, K, \hat{x}^0)$ given by Eq.(1.10.42), since the "vacuum" equations (1.10.43) follows from the densitized Einstein field equations (1.10.38) under setting given by Eq.(1.10.42).

**Remark 1.10.24.** Beyond any neighborhood $O(\hat{x}^0, \varepsilon) = \{\hat{x} : \|\hat{x} - \hat{x}^0\| \le \varepsilon\}$, $\varepsilon > 0$, of the singular point $\hat{x}^0$ the equations (1.10.43) becomes to the following canonical "vacuum" form

$$ R_{km}(\hat{x}) - \frac{1}{2} g_{km}(\hat{x}) R(\hat{x}) = 0. \quad (1.10.44) $$

**Remark 1.10.25.** The "vacuum" equations (1.10.44) return the canonical Schwarzschild solutions (1.6) and (1.11) except any neighborhood $O(\{r-2m\} \cup \{r=0\}, \varepsilon)$, $\varepsilon > 0$ of
horizon and Schwarzschild singularity. But we ephazized that scalar curvature of the Schwarzschild spacetime is given exactly by the generalized Ricci scalar $\tilde{R} \neq R$.

**Remark 1.10.26.** It follows from the Eqs. (1.10.41) that the Möller metric field (1.10.9) has

a singular source $T_{km}$.

Beyond the neighborhood $O(x_{hor}, \epsilon) = \{x: |a+gx| \le \epsilon\}, \epsilon > 0$ of the Möller horizon $x_{hor} = -a/g$, but not on whole Möller spacetime, one obtains in accordance with classical

Möller's result, see Eqs. (1.10.15):

$$R_m^k(x) - \frac{1}{2} g_m^k(x) R(x) = 0. \qquad (1.10.45)$$

Inside the submanifold $O(x_{hor}, \epsilon)$ the corresponding densitized "vacuum" equations (1.10.43) reads

$$\int_{O(x_{hor}, \epsilon)} (R_m^k(x) - \frac{1}{2} g_m^k(x) R(x)) \Delta(x) df^{km} = 0 \qquad (1.10.46)$$

Therefore from the Eqs. (1.10.46) and Eqs. (1.10.10) one obtains

$$\int_{O(x_{hor}, \epsilon)} G_2^2(x) \Delta(x) dx = \int_{O(x_{hor}, \epsilon)} G_3^3(x) dx = - \int_{O(x_{hor}, \epsilon)} \frac{1}{2\Delta(x)} \left[ \Delta''(x) - \frac{(\Delta'(x))^2}{2\Delta(x)} \right] \Delta(x) dx = 0, \qquad (1.10.47)$$

since at Möller horizon $x_{hor} = -a/g$ the function

$$\Xi(x) = \frac{1}{\Delta(x)} \left[ \Delta''(x) - \frac{(\Delta'(x))^2}{2\Delta(x)} \right] \Delta(x) = \frac{1}{\Delta(x)} \left[ 2g^2 - \frac{4g^2\Delta(x)}{2\Delta(x)} \right] \qquad (1.10.48)$$

is well defined by taking the limit $x \to x_{hor}$ (see Remark 1.10.18) and therefore

$$\lim_{x \to x_{hor}} \Xi(x) = \lim_{x \to x_{hor}} \left( \frac{\Delta(x)}{\Delta(x)} \left[ 2g^2 - \frac{4g^2\Delta(x)}{2\Delta(x)} \right] \right) = \lim_{x \to x_{hor}} \left( \frac{\Delta(x)}{\Delta(x)} \right) \left[ 2g^2 - 2g^2 \lim_{x \to x_{hor}} \left( \frac{\Delta(x)}{\Delta(x)} \right) \right] = 2g^2 - 2g^2 = 0. \qquad (1.10.49)$$

**Remark 1.10.27.** Note that in contrast with abnormal Möller's calculation, see Remark 1.10.10-1.10.11, Eq. (1.10.16)

**Remark 1.10.28.** (I) Note that the Schwarzschild metric field (1.6) in classical sense is well defined only for $r > 2m$, The boundary of the manifold $\{r > 2m\}$ in $\mathbb{R}^3 \times \mathbb{R}$ is the submanifold $\{r = 2m\}$ of $\mathbb{R}^3 \times \mathbb{R}$, diffeomorphic to a product $S^2 \times \mathbb{R}$. This submanifold is called the event horizon, or simply the horizon [33], [34].

(II) The Schwarzschild metric (1.10.12) in canonical coordinates $(x^0, r, \theta, \phi)$, with $m > 0$, ceases to be a smooth Lorentzian metric for $r = 2m$, because for such a value of $r$ the coefficient $g_{00}$ becomes zero while $g_{11}$ becomes infinite. For $0 < r < 2m$ the metric (1.9.9) again a smooth Lorentzian metric but $t$ is a space coordinate while $r$ a time coordinate. Hence the metric cannot be said to be either spherically symmetric or static for $r < 2m$ [33].

(III) From consideration above obviously follows that on Schwarzschild spacetime
$$
\mathbf{Sh} = (S^2 \times \{r > 2m\} \cup \{0 < r < 2m\}) \times \mathbb{R} \text{ the Levi-Civita connection} \\
\left\{ \Gamma_{kj}^{+l}(\{\}) = \frac{1}{2}[g^{lm}(\{\})][g_{mk,j}(\{\}) + g_{mj,k}(\{\})] - g_{kj,m}(\{\})] \right\} \quad (1.10.50)
$$

is not available in classical sense and that is well known many years from
mathematical

literature, see for example [22] and Remark 1.10.19 above.

(IV) Note that [4] : (i) The determinat det($g_{lm}$(♦)) = -r⁴ sin²θ of the metric (1.6)
is regular on horizon,i.e., smooth and non-vanishing for r = 2m.

In addition:

(ii) The curvature scalar $\mathbf{R}(\{\}) = g^{\mu\nu}\mathbf{R}_{\mu\nu}(\{\})$ is zero for $r = 2m$.

(iii) The none of higher-order scalars such as $\mathbf{R}^{\mu\nu}(\{\})\mathbf{R}_{\mu\nu}(\{\})$, etc. blows up. For example

the quadratic scalar $\mathbf{R}^{\rho\sigma\mu\nu}(\{\})\mathbf{R}_{\rho\sigma\mu\nu}(\{\}) = 48m^2/r^6$ is regular on horizon,i.e.,smooth and

non-vanishing for $r = 2m$.

(V) Note that: (i) In physical literature (see for example [4],[33],[35,]) it was wrongly assumed that a property (i)-(iii) is enough to convince us that $r = 2m$ represent the non honest physical singularity but only coordinate singularity.

(VI) Such assumption based only on *wrong* formal extensions

$$
\hat{\mathbf{R}}(\{\}), \hat{\mathbf{R}}^{\mu\nu}(\{\})\hat{\mathbf{R}}_{\mu\nu}(\{\}), \dots,
$$

$$
\hat{\mathbf{R}}^{\rho\sigma\mu\nu}(\{\})\hat{\mathbf{R}}_{\rho\sigma\mu\nu}(\{\}) \text{ of the curvature scalar } \mathbf{R}(\{\}) \text{ and higher-order scalars such as}
$$

$$
\mathbf{R}^{\mu\nu}(\{\})\mathbf{R}_{\mu\nu}(\{\}), \dots, \mathbf{R}^{\rho\sigma\mu\nu}(\{\})\mathbf{R}_{\rho\sigma\mu\nu}(\{\}) \text{ on horizon } r = 2m \text{ and on origin } r = 0 \text{ by}
$$

formulae

$$
\hat{\mathbf{R}}(r) \Big|_{r=2m} = \lim_{r \to 2m} \mathbf{R}(r) = 0, \hat{\mathbf{R}}(r) \Big|_{r=0} = \lim_{r \to 0} \mathbf{R}(r) = 0
$$

$$
\hat{\mathbf{R}}^{\rho\sigma\mu\nu}(r) \hat{\mathbf{R}}_{\rho\sigma\mu\nu}(r) \Big|_{r=2m} = \lim_{r \to 2m} (\mathbf{R}^{\rho\sigma\mu\nu}(r) \mathbf{R}_{\rho\sigma\mu\nu}(r)) = \lim_{r \to 2m} \frac{48m^2}{r^6} = \frac{48m^2}{r^6} \Big|_{r=2m}, \quad (1.10.51)
$$

$$
\hat{\mathbf{R}}^{\rho\sigma\mu\nu}(r) \hat{\mathbf{R}}_{\rho\sigma\mu\nu}(r) \Big|_{r=0} = \lim_{r\to 0} (\mathbf{R}^{\rho\sigma\mu\nu}(r) \mathbf{R}_{\rho\sigma\mu\nu}(r)) = \lim_{r\to 0} \frac{48m^2}{r^6} = \infty.
$$

However in the limit $r \to 2m$ the Levi-Civita connection $\Gamma_{kj}^{+l}(\{\})$ becomes infinite [4]:

$$
\Gamma_{00}^{1}(r)|_{r=2m} = \lim_{r \to 2m} \frac{m(r-2m)}{r^3} = 0, \Gamma_{11}^{1}(r)|_{r=2m} = \lim_{r \to 2m} \frac{-m}{r(r-2m)} = \infty,
$$

$$
\Gamma_{01}^{0}(r)|_{r=2m} = \lim_{r \to 2m} \frac{m}{r(r-2m)} = \infty,
$$

$$
\Gamma_{12}^{2}(r)|_{r=2m} = \lim_{r \to 2m} \frac{1}{r} = 2^{-1}m^{-1}, \Gamma_{22}^{1}|_{r=2m} = -\lim_{r \to 2m} (r-2m) = 0,
$$

$$
\Gamma_{13}^{3}|_{r=2m} = \lim_{r \to 2m} \frac{1}{r} = 2^{-1}m^{-1}, \Gamma_{33}^{1}|_{r=2m} = -\lim_{r \to 2m} (r-2m)\sin^2\theta = 0, \quad (1.10.52)
$$

$$
\Gamma_{00}^{1}(r)|_{r=0} = \lim_{r \to 0} \frac{m(r-2m)}{r^3} = \infty \Gamma_{11}^{1}(r)|_{r=0} = \lim_{r \to 0} \frac{-m}{r(r-2m)} = \infty,
$$

$$
\Gamma_{33}^{2} = -\sin\theta \cos\theta, \Gamma_{23}^{3} = \frac{\cos\theta}{\sin\theta}.
$$
Thus obviously by consideration above (see Remark 1.1-Remark 1.10.2) this extension given by Eq.(1.10.15) has no any sense in respect of the canonical Riemannian geometry.

(VII) From consideration above (see Remark 1.10.1-Remark 1.10.2) obviously follows that the scalars such as $\hat{\mathbf{R}}(\{\})$, $\hat{\mathbf{R}}^{\mu\nu}(\{\})\tilde{\mathbf{R}}_{\mu\nu}(\{\})$, ..., $\hat{\mathbf{R}}^{\rho\sigma\mu\nu}(\{\})\hat{\mathbf{R}}_{\rho\sigma\mu\nu}(\{\})$ has no any rigorous sense in respect to the canonical Levi-Civita connection (1.10.11) and therefore cannot be said to be either honest physical singularity or only coordinate singularity in respect of the canonical Riemannian geometry.

**Remark 1.10.29.** Note that in physical literature the spacetime singularity usually is defined as location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the classical scalar invariant curvatures of singular spacetime, which includes a measure of the density of matter.

**Remark 1.10.30.** In general relativity, many investigations have been derived with regard to singular exact vacuum solutions of the Einstein equation and the singularity structure of space-time. Such solutions have been formally derived under condition $T^\nu_\mu(x) = 0$, where $T^\nu_\mu(x)$ represent the energy-momentum densities of the gravity source. This for example is the case for the well-known Schwarzschild solution, which is given by, in the Schwarzschild coordinates $(x^0, r, \theta, \phi)$,

$$ds^2 = -h(r)(dx^0)^2 + h^{-1}(r)(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], h(r) = 1 - \frac{r_s}{r}, \quad (1.10.52)$$

where, $r_s$ is the Schwarzschild radius, $r_s = 2GM/c^2$ with $G, M$ and $c$ being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. The metric (1.10.44) describe the gravitational field produced by a point-like particle located at $r = 0$, see [30].

**Remark 1.10.31.** Note that when we say, on the basis of the canonical expression of the curvature square

$$\mathbf{R}^{\rho\sigma\mu\nu}(r)\mathbf{R}_{\rho\sigma\mu\nu}(r) = \frac{12r_s^2}{r^6} \qquad (1.10.53)$$

formally obtained from the metric (1.10.44), that $r = 0$ is a singularity of the Schwarzschild space-time, the source is considered to be point-like and this metric is regarded as *meaningful everywhere* in space-time.

**Remark 1.10.32.** From the metric (1.10.44), the calculation of the canonical Einstein tensor proceeds in a straightforward manner gives for $r \neq 0$

$$\begin{aligned} G'_{t}(r) &= G'_{r}(r) = -\frac{h'(r)}{\hat{r}} - \frac{1+h(r)}{\hat{r}^2} \equiv 0, \\ G'_{\theta}(r) &= G'_{\phi}(r) = -\frac{h''(r)}{2} - \frac{h(r)}{\hat{r}^2} \equiv 0, \end{aligned} \qquad (1.10.54)$$

where $h(r) = -1 + r_s/r$. Using Eq.(1.10.54) one formally obtains a boundary conditions

$$\left\{ \begin{array}{l} G'_{t}(0) \triangleq \lim_{r \to 0} G'_{t}(r) = 0, G'_{r}(0) \triangleq \lim_{r \to 0} G'_{r}(r) = 0, \\ G'_{\theta}(0) \triangleq \lim_{r \to 0} G'_{\theta}(r) = 0, G'_{\phi}(0) \triangleq \lim_{r \to 0} G'_{\phi}(r) = 0. \end{array} \right. \qquad (1.10.55)$$

However as pointed out above the canonical expression of the Einstein tensor in a sufficiently small neighborhood $\Omega$ of the point $r = 0$ and must be replaced by the
generalized Einstein tensor $\tilde{G}_{km}$ (1.10.37). By simple calculation easy to see that

$$ \left\{ \begin{array}{l} \tilde{G}_t^t(0) \triangleq \lim_{r \to 0} \tilde{G}_t^t(r) = -\infty, \tilde{G}_r^r(0) \triangleq \lim_{r \to 0} \tilde{G}_r^r(r) = -\infty, \\ \tilde{G}_\theta^\theta(0) \triangleq \lim_{r \to 0} \tilde{G}_\theta^\theta(r) = -\infty, \tilde{G}_\phi^\phi(0) \triangleq \lim_{r \to 0} \tilde{G}_\phi^\phi(r) = -\infty. \end{array} \right. \qquad (1.10.56) $$

and therefore the boundary conditions (1.10.56) are completely wrong. But other hand as pointed out by many authors [5]-[17] that the canonical representation of the Einstein tensor, valid only in a weak (distributional) sense,i.e. [12]:

$$ G_b^a(\hat{x}) = -8\pi m \delta_0^a \delta_b^0 \delta^3(\hat{x}) \qquad (1.10.57) $$

and therefore again we obtain $G_b^a(0) = -\infty \times (\delta_0^a \delta_b^0)$. Thus canonical property fo the Einstein tensor: $G_b^a(\hat{x}) = 0$, is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin $r = 0$.

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

Note that the $\epsilon$-regularization related to Colombeau geometry originally considered in A. Einstein and N. Rosen paper [32]. A. Einstein emphasized that:

"The solution (1.10.1) (see [32],eq.1) naturally has no deeper physical significance insofar as it extends into spatial infinity. It allows one to see however to what extent the regularization of the hypersurfaces $g = 0$ leads to a theoretical representation of matter, regarded from the standpoint of the original theory. Thus, in the framework of the original theory one has the gravitational equations (see [32],eq.4)

$$ R_{ik} - \frac{1}{2} g_{ik} R = -T_{ik}, \qquad (1.11.1) $$

where $T_{ik}$ is the tensor of mass or energy density. To interpret (1.11.1) in the framework of this theory we must approximate the line element by a slightly different one which avoids the singularity $g = 0$. Accordingly we introduce a small constant $\sigma$ and let (see [32],eq.1a)

$$ ds^2 = -dx_1^2 - dx_2^2 - dx_3^2 + (\alpha^2 x_1^2 + \sigma) dx_4^2 \qquad (1.11.2) $$

the smaller $\sigma$ ($> 0$) is chosen, the nearer does this gravitational field come to that of (1.11.2). If one calculates from this the (fictitious) (see Remark 1.11.1) energy tensor $T_{ik}$ one obtains as nonvanishing components

$$ T_{22} = T_{23} = \frac{\alpha^2 \sigma}{(\alpha^2 x_1^2 + \sigma)^2}. \qquad (1.11.3) $$

We see then that the smaller one takes $\sigma$ the more is the tensor concentrated in the neighborhood of the hypersurface $x_1 = 0$. From the standpoint of the original theory the solution (1.11.1) contains a singularity which corresponds to an energy or mass concentrated in the surface $x_1 = 0$; from the standpoint of the modified theory, however, (1.11.2) is a solution of (1.10.3) (see [32],eq.3a), free from singularities, which describes the "field-producing mass," without requiring for this the introduction of any new field
quantities".

**Remark 1.11.1.** Note that the energy tensor $T_{ik}$ (1.11.3) A. Einstein mistakenly has been

considered as "fictitious". It is clear that A. Einstein thought that by using any fixed semi

Riemannian metric and thus in particular by the fixed metric field mentioned in his paper

[32]:

$$ds^2 = -dx_1^2 - dx_2^2 - dx_3^2 + \alpha^2 x_1^2 dx_4^2 \quad (1.11.4)$$

one obtains full and unique semi Riemannian structure corresponding to uniformly accelerated reference system, etc. This principle mistake holds in physical community from A. Einstein time until nowadays. However even in A. Einstein time it were well known

that for non classical semi Riemannian metric fields this opinion completely wrong and one needs something more than only simply formula (1.11.4), see [4]-[6] in References B.

Actually as we know the imbedding $ds^2 \hookrightarrow (ds_\sigma^2)_\sigma$ in Colombeau generalized object $(ds_\sigma^2)_\sigma$

is necessary.

**Remark 1.11.2.** Note that the regularization mentioned in A. Einstein and N. Rosen paper

[32]. by introducing $\epsilon$-regularization $\epsilon = \sqrt{\sigma}, \epsilon \in (0, \delta], \delta \ll 1$ in contemporary Colombeau

notations reads

$$(ds_{\epsilon}^{2})_{\epsilon} = -dx_{1}^{2} - dx_{2}^{2} - dx_{3}^{2} + ((\alpha^{2}x_{1}^{2} + \epsilon^{2})_{\epsilon}) dx_{4}^{2} \quad (1.11.5)$$

Thus in accordance with the Colombeau framework the classical metric field (1.1.1) embedded in Colombeau object $(ds_{\epsilon}^{2})_{\epsilon}$.

**Remark 1.11.3.** (i) Note that in contemporary Colombeau notations A. Einstein "fictitious"

energy tensor $T_{ik}(\sigma)$ obviously reads [5]:

$$(T_{22}(\epsilon))_{\epsilon} = T_{23}(\epsilon) = \alpha^2 \left( \frac{\epsilon^2}{(\alpha^2 x_1^2 + \epsilon^2)^2} \right)_{\epsilon}, \quad (1.11.6)$$

where $\epsilon = \sqrt{\sigma}, \epsilon \in (0, \delta], \delta \ll 1$. The generalized Einstein field equations (1.9.17) corresponding to distributional energy tensor (1.11.5.) reads

$$(R_{ik,\epsilon})_{\epsilon} - \frac{1}{2}g_{ik,\epsilon}(R_{\epsilon})_{\epsilon} = -(T_{ik}(\epsilon))_{\epsilon}. \quad (1.11.7)$$

(ii) Note that in any classical point $(x_1, x_2, x_3, x_4)$ with $x_1 \in \mathbb{R} \setminus \{0\}$ the energy tensor (1.10.)

becomes to infinite small values, i.e. $(T_{ik}(\epsilon))_{\epsilon} \approx \tilde{\mathbb{R}} 0$

(iii) Note that in any classical point $(x_1, x_2, x_3, x_4)$ with $x_1 = 0$ the energy tensor (1.11.6) has well defined point value $\epsilon \in \tilde{\mathbb{R}}_{\inf} \subset \tilde{\mathbb{R}}$.

(iv) Note that in accordance with result obtained above in subsection 1.10.3 the

generalized Einstein field equations meant that metric field (1.11.1) has a non trivial
Colombeau distributional source

$$ \left( \frac{\varepsilon^2}{(\alpha^2 x_1^2 + \varepsilon^2)^2} \right)_\varepsilon \in \mathcal{G}(\mathbb{R}). \qquad (1.11.8) $$

(v) Note that it follows from Eq.() the Colombeau curvature scalar $(R_\varepsilon)_\varepsilon$ in any classical point $(x_1, x_2, x_3, x_4)$ with $x_1 \in \mathbb{R} \setminus \{0\}$
We consider now the Möller's metric (1.10.4)

$$ ds^2 = -\Delta(x)dt^2 + dx^2 + dy^2 + dz^2, \qquad (1.11.9) $$

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

In order to avoid difficultness with degeneracy of the classical metric (1.11.9) mentioned

above in subsection 1.10.2, we replace now Möller's line element (1.11.8) by corresponding Colombeau line element

$$ (d_\varepsilon s^2)_\varepsilon = -[(\Delta_\varepsilon(x))_\varepsilon]dt^2 + dx^2 + dy^2 + dz^2, \qquad (1.11.10) $$

where $\Delta_\varepsilon(x) = [(a + gx)^2 + \varepsilon^2], \varepsilon \in (0, 1]$.

**Remark 1.11.4.** Note that in contrast with a wrong formal expression (1.10.6) the distributional Levi-Cività connection corresponding Colombeau line element (1.11.10) reads

$$ (\Gamma^1_{44,\varepsilon}(x))_\varepsilon = ((a+gx))_\varepsilon, \quad (\Gamma^4_{14,\varepsilon}(x)) = \Gamma^4_{41,\varepsilon}(x) = \frac{a+gx}{((a+gx)^2 + \varepsilon^2)_\varepsilon} \qquad (1.11.11) $$

Notice that the distributional Levi-Cività connection (1.11.11) is well defined and even regular on whole Möller's space-time.

Let $(G^k_{i,\varepsilon}(x))_\varepsilon$ be the distributional Einstein tensor $(G^k_i(x))_\varepsilon \triangleq (R^k_i(x))_\varepsilon - \frac{1}{2}\delta^k_i(R_\varepsilon(x))_\varepsilon$, where $(R^k_{i,\varepsilon}(x))_\varepsilon$ is the contracted distributional Riemann-Christoffel tensor calculated

by using distributional Levi-Civita connection (1.11.1) corresponding to Colombeau line element (1.11.10) [5],[35] and $(R_\varepsilon(x))_\varepsilon = (R^i_\varepsilon(\varepsilon))_\varepsilon$. Therefore for the case of the Colombeau line element (1.11.10) we get

$$ (G_2^2(x; \varepsilon))_\varepsilon = (G_3^3(x; \varepsilon))_\varepsilon = -\frac{1}{2(\Delta_\varepsilon(x))_\varepsilon} \left\{ (\Delta_\varepsilon''(x))_\varepsilon - \frac{[(\Delta_\varepsilon'(x))_\varepsilon]^2}{2(\Delta_\varepsilon(x))_\varepsilon} \right\} = -\left( \frac{g^2\varepsilon^2}{\Delta_\varepsilon^2(x)} \right)_\varepsilon. \qquad (1.11.12) $$

Notice that $(\Im(x; \varepsilon))_\varepsilon \triangleq -\left(\frac{g^2\varepsilon^2}{\Delta_\varepsilon^2(x)}\right)_\varepsilon, \varepsilon \in (0, 1]$ is Colombeau generalized function such

that $cl[(\Im(x; \varepsilon))_\varepsilon] \in G(\mathbb{R})$ and $cl[(\Im(-a/g)^{-1}; \varepsilon))_\varepsilon] = cl[(\varepsilon^{-2})_\varepsilon] \in \tilde{\mathbb{R}}_{\inf} \subset \tilde{\mathbb{R}}$.

Thus Colombeau generalized fundamental tensor $(g_{ik}(\varepsilon))_\varepsilon$ corresponding to Colombeau

metric (1.11.10) that is non vacuum Colombeau solution of the of the generalized Einstein's field equations (1.11.7) with Colombeau generalized source. For Colombeau scalars $(R_\varepsilon(x))_\varepsilon$ and $(R^{\mu\nu}(x, \varepsilon)R_{\mu\nu}(x, \varepsilon))_\varepsilon$ we get [5]:

$$ (R_\varepsilon(x))_\varepsilon = \left( \frac{g^2 \varepsilon^2}{\Delta_\varepsilon^2(x)} \right)_\varepsilon \qquad (1.11.13) $$

and
$$ (R^{\mu\nu}(x, \varepsilon) R_{\mu\nu}(x, \varepsilon))_{\varepsilon} = g^4 \varepsilon^4 \left[ (a + gx)^2 + \varepsilon^2 \right]^{-4} \quad (1.11.14) $$

correspondingly.

## 1.12. The distributional Schwarzschild geometry by using the linear L.Schwartz distributions and by using the full algebra of the Colombeau generalized functions.

General relativity as a physical theory is governed by particular physical equations; the focus of interest is the breakdown of physics which need not coincide with the breakdown of geometry. It has been suggested to describe singularity at the origin as *internal* point of the Schwarzschild spacetime, where the Einstein field equations are satisfied in a weak sense [5]-[22].

### 1.12.1. The distributional Schwarzschild geometry at the origin. The smooth regularization of the singularity at the origin.

The two singular functions we will work with throughout this paper (namely the singular components of the Schwarzschild metric) are $\frac{1}{r}$ and $\frac{1}{r-r_s}$, $r_s \ge 0$. Since $\frac{1}{r} \in L_{loc}^1(\mathbb{R}^3)$, it obviously gives the regular distribution $\frac{1}{r} \in D'(\mathbb{R}^3)$. By convolution with a mollifier $\rho(x)$ (adapted to the symmetry of the spacetime, i.e. chosen radially symmetric) we embed it into the Colombeau algebra $G(\mathbb{R}^3)$ [22]:

$$ \frac{1}{r} \xrightarrow{\text{i}} \iota\left(\frac{1}{r}\right) \triangleq \left(\frac{1}{r}\right) * \rho_{\varepsilon} \triangleq \left(\frac{1}{r}\right)_{\varepsilon}, \rho_{\varepsilon} = \frac{1}{\varepsilon^3} \rho\left(\frac{r}{\varepsilon}\right), \varepsilon \in (0,1]. \quad (1.12.1) $$

Inserting (1.12.1) into Schwarzschild metric (1.10.44) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:

$$ \left\{ \begin{array}{l} (ds_{\varepsilon}^2)_{\varepsilon} = (h_{\varepsilon}(r)(dt)^2)_{\varepsilon} - (h_{\varepsilon}^{-1}(r)(dr)^2)_{\varepsilon} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], \\ h_{\varepsilon}(r) = -1 + r_s\left(\frac{1}{r}\right)_{\varepsilon}, \varepsilon \in (0,1]. \end{array} \right. \quad (1.12.2) $$

**Remark 1.12.1.** Note that under regularization (1.12.1) for any $\varepsilon \in (0,1]$ the metric

$$ ds_{\varepsilon}^2 = h_{\varepsilon}(r)(dt)^2 - h_{\varepsilon}^{-1}(r)(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2] $$

obviously is a classical Riemannian object and there no exist an the breakdown of canonical formalism of Riemannian geometry for these metrics, even at origin $r = 0$. It has been suggested by many authors to describe singularity at the origin as an *internal point*, where the Einstein field equations are satisfied in a distributional sense [5]-[22]. From the Colombeau metric (1.12.2) one obtains in a distributional sense [22]:

$$ (R_2^2(r, \varepsilon))_{\varepsilon} = (R_3^3(r, \varepsilon))_{\varepsilon} = \left( \frac{h_{\varepsilon}^\prime(r)}{r} + \frac{1 + h_{\varepsilon}(r)}{r^2} \right)_{\varepsilon} = 8\pi m \frac{\delta(r)}{r^2}, \quad (1.12.3) $$

$$ (R_0^0(r, \varepsilon))_{\varepsilon} = (R_1^1(r, \varepsilon))_{\varepsilon} = \frac{1}{2} \left( \frac{h_{\varepsilon}^{\prime\prime}(r)}{2} + \frac{h_{\varepsilon}^{\prime}(r)}{r} \right)_{\varepsilon} = -4\pi m \delta \frac{\delta(r)}{r^2}. $$

Hence, the Colombeau distributional Ricci tensor and the Colombeau distributional
curvature scalar $(R_\epsilon(r))_\epsilon$ are of $\delta$-type, i.e. $(R_\epsilon(r))_\epsilon = \pi m \frac{\delta(r)}{r^2} \in \mathcal{D}(\mathbb{R}^3)$.

**Remark 1.12.2.** Note that the formulae (1.12.3) should be contrasted with what is the expected result $G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x)$ given by Eq.(1.10.49). However the equations (1.12.3) are obviously given in spherical coordinates and therefore strictly speaking this is not correct, because the basis fields $\{\frac{\partial}{\partial r}, \frac{\partial}{\partial \varphi}, \frac{\partial}{\partial \theta}\}$ are not globally defined.

Representing distributions concentrated at the origin requires a basis regular at the origin. Transforming the formulae for $(R_{ij}(\epsilon))_\epsilon$ into Cartesian coordinates associated with the spherical ones, i.e., $\{r, \theta, \varphi\} \leftrightarrow \{x^i\}$, we obtain, e.g., for the Einstein tensor the expected result $G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x)$ given by Eq.(1.10.49), see [22].

## 1.12.2. The nonsmooth regularization of the singularity at the origin.

The nonsmooth regularization of the Schwarzschild singularity at the origin $r = 0$ is considered by N. R. Pantoja and H. Rago in paper [12]. Pantoja non smooth regularization regularization of the Schwarzschild singularity reads

$$ (h_{\epsilon}(r))_{\epsilon} = -1 + \left(\frac{r_s}{r}\Theta(r-\epsilon)\right)_{\epsilon}, \quad \epsilon \in (0,1], r < r_s. \qquad (1.12.4) $$

Here $\Theta(u)$ is the Heaviside function and the limit $\epsilon \to 0$ is understood in a distributional sense. Equation (1.12.2) with $h_\epsilon$ as given in (1.12.4) can be considered as an regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.12.4), the calculation of the distributional Einstein tensor proceeds in a straightforward manner. By simple calculation it gives [12]:

$$ \left\{ \begin{aligned} (G_t^t(r, \epsilon))_{\epsilon} &= (G_r^r(r, \epsilon))_{\epsilon} = -\left(\frac{h_{\epsilon}^t(r)}{r}\right)_{\epsilon} - \left(\frac{1+h_{\epsilon}(r)}{r^2}\right)_{\epsilon} = \\ &= -r_s \left(\frac{\delta(r-\epsilon)}{r^2}\right)_{\epsilon} = -r_s \frac{\delta(r)}{r^2} \end{aligned} \right. \qquad (1.12.5) $$

and

$$ \left\{ \begin{aligned} (G_{\theta}^{\theta}(r, \epsilon))_{\epsilon} &= (G_{\phi}^{\phi}(r, \epsilon))_{\epsilon} = -\left(\frac{h_{\epsilon}''(r)}{2}\right)_{\epsilon} - \left(\frac{h_{\epsilon}(r)}{r^2}\right)_{\epsilon} = \\ &r_s \left(\frac{\delta(r-\epsilon)}{r^2}\right)_{\epsilon} - r_s \left(\frac{\epsilon}{r^2} \frac{d}{dr} \delta(r-\epsilon)\right)_{\epsilon} = -r_s \frac{\delta(r)}{r^2}. \end{aligned} \right. \qquad (1.12.6) $$

which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function $\Theta(r-\epsilon)$. Transforming now the formulae for $(G_b^a(r, \epsilon))_\epsilon$ into Cartesian coordinates associated with the spherical ones, i.e., $\{r, \theta, \varphi\} \leftrightarrow \{x^i\}$, we obtain for the generalized Einstein tensor the expected result given by Eq.(1.10.49)

$$ G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x), \qquad (1.12.7) $$

see Remark 1.12.2.

## 1.12.3. The smooth regularization via Horizon in
Schwarzschild coordinates and the smooth regularization at horizon in isotropic coordinates.

### 1.12.3.1. The smooth regularization via Horizon in Schwarzschild coordinates

The smooth regularization via Horizon is considered by J.M.Heinzle and R.Steinbauer in paper [22]. Note that $\frac{1}{r-r_s} \notin L_{loc}^1(\mathbb{R}^3)$. An canonical regularization is the principal value $\mathbf{vp}(\frac{1}{r-r_s}) \in D'(\mathbb{R}^3)$ which can be embedded into $\mathcal{G}(\mathbb{R}^3)$ [22]:

$$ \frac{1}{r-r_s} \xrightarrow{\mathbf{vp}} \mathbf{vp}(\frac{1}{r-r_s}) \xrightarrow{\iota} \iota[\rho_{\varepsilon} * \mathbf{vp}(\frac{1}{r-r_s})] \triangleq (\frac{1}{r-r_s})_{\varepsilon} \in \mathcal{G}(\mathbb{R}^3). \quad (1.12.8) $$

Inserting now(1.12.8) into Schwarzschild metric (1.10.44) we obtain a generalized Colombeau object modeling the singular Schwarzschild space-time [22]:

$$ (ds_{\varepsilon}^2)_{\varepsilon} = (h(r)(dt)^2)_{\varepsilon} - (h_{\varepsilon}^{-1}(r)(dr)^2)_{\varepsilon} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], \quad (1.12.9) $$

where

$$ h(r) = -1 + \frac{r_s}{r}, h_{\varepsilon}^{-1}(r) = -1 - r_s \left(\frac{1}{r-r_s}\right)_{\varepsilon}, \varepsilon \in (0,1]. \quad (1.12.10) $$

**Remark 1.12.3.** Note that obviously Colombeau object (1.12.9) is degenerate at $r = r_s$, because $h(r)$ is zero at the horizon. However, this does not come as a surprise. Both $h(r)$ and $h^{-1}(r)$ are positive outside of the black hole and negative in the interior. As a consequence any smooth regularization of $h(r)$ (or $h^{-1}$) must pass through zero somewhere and, additionally, this zero must converge to $r = r_s$ as the regularization parameter goes to zero.

**Remark 1.12.4.** Note that due to the degeneracy of Colombeau object (1.10.26), even the distributional Levi-Civitá connection obviously is not available by using the smooth regularization via horizon [22].

**Remark 1.12.5.** Note that the smooth regularization (1.12.8) doesn't make any sense, since Colombeau object (1.12.9) again is degenerate at $r = r_s$. However in isotropic coordinates $(t, \rho, \theta, \phi)$ the smooth regularization at horizon obviously possible.

### 1.12.3.2. The smooth regularization at horizon in isotropic coordinates.

The Schwarzschild metric in isotropic coordinates $(t, \rho, \theta, \phi)$ reads [3]:

$$ ds^2 = -\frac{(4\rho - r_s)^2}{4\rho + r_s} dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (1.12.11) $$

where we let $c = 1$. The canonical simple regularization at horizon reads

$$ (4\rho - r_s)^2 \rightarrow (4\rho - r_s)^2 + \varepsilon^2. \quad (1.12.12) $$

Inserting now(1.12.12) into Schwarzschild metric (1.12.11) we obtain a generalized Colombeau object modeling the degenerate Schwarzschild space-time in isotropic coordinates

$$ (ds_{\varepsilon}^2)_{\varepsilon} = -\frac{((4\rho - r_s)^2 + \varepsilon^2)_{\varepsilon}}{4\rho + r_s} dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\phi^2). \quad (1.12.13) $$
1.12.4. The nonsmooth regularization via Gorizon

In this paper we leave the neighborhood of the singularity at the origin and turn to the
singularity at the horizon. The question we are aiming at is the following: using
distributional geometry (thus without leaving Schwarzschild coordinates), is it possible to
show that the horizon singularity of the Schwarzschild metric is not merely a coordinate
singularity. In order to investigate this issue we calculate the distributional curvature at
the horizon in Schwarzschild coordinates.

The main focus of this work is a (nonlinear) superdistributional description of the Schwarzschild spacetime. Although the nature of the Schwarzschild singularity is much “worse” than the quasi-regular conical singularity, there are several distributional treatments in the literature [8]-[29], mainly motivated by the following considerations: the physical interpretation of the Schwarzschild metric is clear as long as we consider it merely as an exterior (vacuum) solution of an extended (sufficiently large) massive spherically symmetric body. Together with the interior solution it describes the entire spacetime. The concept of point particles—well understood in the context of linear field theories—suggests a mathematical idealization of the underlying physics: one would like to view the Schwarzschild solution as defined on the entire spacetime and regard it as generated by a point mass located at the origin and acting as the gravitational source.

This of course amounts to the question of whether one can reasonably ascribe
distributional curvature quantities to the Schwarzschild singularity at the horizon.

The emphasis of the present work lies on mathematical rigor. We derive the
“physically expected” result for the distributional energy momentum tensor of the
Schwarzschild geometry, i.e., $T_0^0 = 8\pi m\delta^{(3)}(\vec{x})$, in a conceptually satisfactory way.
Additionally, we set up a unified language to comment on the respective merits of some
of the approaches taken so far. In particular, we discuss questions of differentiable
structure as well as smoothness and degeneracy problems of the regularized metrics,
and present possible refinements and workarounds. These aims are accomplished using
the framework of nonlinear supergeneralized functions (superseneralized Colombeau
algebras $\mathcal{G}(\mathbb{R}^3, \Sigma)$). Examining the Schwarzschild metric (1.12) in a neighborhood of the
horizon, we see that, whereas $h(r)$ is smooth, $h^{-1}(r)$ is not even $L_{loc}^1$ (note that the origin
is now always excluded from our considerations; the space we are working on is $\mathbb{R}^3\setminus\{0\}$).
Thus, regularizing the Schwarzschild metric amounts to embedding $h^{-1}$ into $\mathcal{G}(\mathbb{R}^3, \Sigma)$ (as
done in (3.2)). Obviously, (3.1) is degenerate at $r = 2m$, because $h(r)$ is zero at the
horizon. However, this does not come as a surprise. Both $h(r)$ and $h^{-1}(r)$ are positive
outside of the black hole and negative in the interior. As a consequence any (smooth)
regularization $h_\epsilon^+(r)$ ($h_\epsilon^-(r)$) [above (below) horizon] of $h(r)$ must pass through small
enough vicinity $O_\epsilon^+(2m) = \{\vec{x} \in \mathbb{R}^3 \|\|\vec{x}\| > 2m, \|\vec{x} - 2m\| \le \epsilon\}$
($O_\epsilon^-(2m) = \{\vec{x} \in \mathbb{R}^3 \|\|\vec{x}\| < 2m, \|\vec{x} - 2m\| \le \epsilon\}$) of zeros set $O_0(2m) = \{\vec{y} \in \mathbb{R}^3 \|\|\vec{y}\| = 2m\}$
somewhere and, additionally, this vicinity $O_\epsilon^+(2m)$ ($O_\epsilon^-(2m)$) must converge to $O_0(2m)$ as
the regularization parameter $\epsilon$ goes to zero. Due to the degeneracy of the Schwarzschild
metric (1.10.44), the Levi-Cività connection is not available. By aprioriate nonsmooth
regularization (see sect. 3) we obtain a Colombeau generalized object modeling the
singular Schwarzschild metric above and below horizon, i.e.,

$$ \left\{ \begin{array}{l} (ds_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (ds_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ \epsilon \in (0,1]. \end{array} \right. \qquad (1.12.14) $$

Consider corresponding distributional connections $(\Gamma_{kj}^{+l}(\epsilon))_{\epsilon} = (\Gamma_{kj}^{+l}[h_{\epsilon}^{+}])_{\epsilon} \in \mathcal{G}(\mathbb{R}^3, \Sigma)$ and $(\Gamma_{kj}^{-l}(\epsilon))_{\epsilon} = (\Gamma_{kj}^{-l}[h_{\epsilon}^{+}])_{\epsilon} \in \mathcal{G}(\mathbb{R}^3, \Sigma):$

$$ \begin{aligned} (\Gamma_{kj}^{+l}(\epsilon))_{\epsilon} &= \frac{1}{2}((g_{\epsilon}^{+lm})[(g_{\epsilon}^{+})_{mk,j} + (g_{\epsilon}^{+})_{mj,k} - (g_{\epsilon}^{+})_{kj,m}])_{\epsilon}, \\ (\Gamma_{kj}^{-l}(\epsilon))_{\epsilon} &= \frac{1}{2}((g_{\epsilon}^{-lm})[(g_{\epsilon}^{-})_{mk,j} + (g_{\epsilon}^{-})_{mj,k} - (g_{\epsilon}^{-})_{kj,m}])_{\epsilon}. \end{aligned} \qquad (1.12.15) $$

Obviously $(\Gamma_{kj}^{+l}[h_{\epsilon}^{+}])_{\epsilon}, (\Gamma_{kj}^{-l}[h_{\epsilon}^{+}])_{\epsilon}$ coincides with the corresponding Levi-Civita connection on $\mathbb{R}^3\setminus\{(r=0) \cup (r=2m)\}$, as $(h_{\epsilon}^{+})_{\epsilon} = h_{0}^{+}, (h_{\epsilon}^{-})_{\epsilon} = h_{0}^{-}$, and $(g_{\epsilon}^{+lm})_{\epsilon} = g_{0}^{+lm}, (g_{\epsilon}^{-lm})_{\epsilon} = g_{0}^{-lm}$ there. Clearly, connections $\Gamma_{kj}^{+l}(\epsilon), \Gamma_{kj}^{-l}(\epsilon), \epsilon \in (0,1]$ in respect the regularized metric $g_{\epsilon}^{\pm}, \epsilon \in (0,1]$, i.e., $(g_{\epsilon}^{\pm})_{ij,k} = 0$. Proceeding in this manner, we obtain the nonstandard result

$$ \left\{ \begin{array}{l} ([\mathbf{R}_{\epsilon}^{+}]_{1})_{\epsilon} = ([\mathbf{R}_{\epsilon}^{+}]_{0})_{\epsilon} \approx -m\tilde{\Phi}(2m), \\ ([\mathbf{R}_{\epsilon}^{-}]_{1})_{\epsilon} = ([\mathbf{R}_{\epsilon}^{-}]_{0})_{\epsilon} \approx m\tilde{\Phi}(2m). \end{array} \right. \qquad (1.12.16) $$

Investigating the weak limit of the angular components of the generalized Ricci tensor using the abbreviation

$$ \tilde{\Phi}(r) = \int_{0}^{\pi} \sin\theta d\theta \int_{0}^{2\pi} d\phi \Phi(x) $$

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

(i) **supp**($\Phi(x)$) $\subset \{x||x|| \ge 2m\}$ (supp($\Phi(x)$) $\subset \{x||x|| \le 2m\}$) (ii) $\tilde{\Phi}(r) \in C^{\infty}(\mathbb{R})$. Then for any function $\Phi(x) \in S_{2m}^{\pm}(\mathbb{R}^3)$ we get:

$$ \left\{ \begin{array}{l} w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} = w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} = m\langle \tilde{\delta} | \Phi \rangle = -m\tilde{\Phi}(2m), \\ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} = w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} = m\langle \tilde{\delta} | \Phi \rangle = m\tilde{\Phi}(2m), \end{array} \right. \qquad (1.12.17) $$

i.e., the Schwarzschild spacetime is weakly Ricci-nonflat (the origin was excluded from our considerations). Furthermore, the Tolman formula [3],[4] for the total energy of a static and asymptotically flat spacetime with $g$ the determinant of the four dimensional

metric and $d^3x$ the coordinate volume element, gives

$$ E_T = \int (\mathbf{T}_r^r + \mathbf{T}_{\theta}^{\theta} + \mathbf{T}_{\phi}^{\phi} + \mathbf{T}_t^t) \sqrt{-g} \, d^3x = m, \qquad (1.12.18) $$

as it should be.

The paper is organized in the following way: in section II we discuss the conceptual as well as the mathematical prerequisites. In particular we comment on geometrical matters (differentiable structure, coordinate invariance) and recall the basic facts of nonlinear superdistributional geometry in the context of algebras $\mathcal{G}(M, \Sigma)$ of supergeneralized
functions. Moreover, we derive sensible nonsmooth regularizations of the singular functions to be used throughout the paper. Section III is devoted to these approach to the problem. We present a new conceptually satisfactory method to derive the main result. In these final section III we investigate the horizon and describe its distributional curvature. Using nonlinear superdistributional geometry and supergeneralized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.

### 1.12.5. Distributional Eddington-Finkelstein spacetime.

In physical literature many years exist belief that Schwarzschild spacetime ($S^2 \times \{r > 2m\} \times \mathbb{R}$) is extendible, in the sense that it can be immersed in a larger spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with $r > 2m$. In physical literature [4],[33], [34],[35] one considers the formal change of coordinates obtained by replacing the canonical Schwarzschild time by "retarded time" above horizon $v^+$ given when $r > 2m$ by

$$v^+ = t + r + 2m \ln\left(\frac{r}{2m} - 1\right). \qquad (1.12.19)$$

From (1.12.19) it follows for $r > 2m$

$$dt = -\frac{dr}{1 - \frac{2m}{r}} + dv^+. \qquad (1.12.20)$$

The Schwarzschild metric (1.10.44) above horizon $ds^2$ (see section 3) in this coordinate obviously takes the form

$$ds^{+2} = -\left(1 - \frac{2m}{r}\right)dv^{+2} + 2drdv^{-} + r^2\left[(d\theta)^2 + \sin^2\theta(d\phi)^2\right]. \qquad (1.12.21)$$

When $r < 2m$ we replace (1.12.19) below horizon by

$$v^{-} = t + r + 2m \ln\left(1 - \frac{r}{2m}\right) \qquad (1.12.22)$$

From (1.12.22) it follows for $r < 2m$

$$dt = \frac{dr}{\frac{2m}{r} - 1} + dv^{-}. \qquad (1.12.23)$$

The Schwarzschild metric (1.10.44) below horizon $ds^2$ (see section 3) in this coordinate obviously takes the form

$$ds^{-2} = \left(\frac{2m}{r} - 1\right)dv^{-2} - 2drdv^{-} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2]. \qquad (1.12.24)$$

**Remark 1.12.6. (i)** Note that the metric (1.12.21) is well defined on the manifold $S^2 \times (r > 0) \times \mathbb{R}$ and obviously it is regular Lorentzian metric: its coefficients are smooth.

(ii) The term $2drdv$ ensures its non-degeneracy for $r = 2m$.

(iii) Due to the nondegeneracy of the metric (1.12.24) the Levi-Civita connection

$$\{ \Gamma_{kj}^{+l}(\{\}) = \frac{1}{2}[g^{lm}(\{\})][g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})] \qquad (1.12.25)$$

obviously now available and therefore nonsingular on horizon in contrast with Schwarzschild metric one obtains [3]:
$$
\begin{align}
\Gamma_{vv}^{v} &= \frac{r_s}{2r^2}, \quad \Gamma_{vv}^{r} = \frac{r_s(r-r_s)}{2r^3}, \quad \Gamma_{vr}^{r} = -\frac{r_s}{2r^2}, \quad \Gamma_{r\theta}^{\theta} = \frac{1}{r}, \nonumber \\
\Gamma_{r\phi}^{\phi} &= \frac{1}{r}, \quad \Gamma_{\theta\theta}^{v} = -r, \quad \Gamma_{\theta\theta}^{r} = -r(r-r_s), \quad \Gamma_{\theta\phi}^{\phi} = \cot\theta, \tag{1.12.26} \\
\Gamma_{\phi\phi}^{v} &= -r \sin^2\theta, \quad \Gamma_{\phi\phi}^{r} = -r(r-r_s) \sin^2\theta, \quad \Gamma_{\phi\theta}^{\phi} = -\sin\theta \cos\theta. \nonumber
\end{align}
$$

(iv) In physical literature [3],[4] by using properties (i)-(iii) this spacetime wrongly convicted as an rigorous mathematical extension of the Schwarzschild spacetime.

**Remark 1.12.7.** Let us consider now the coordinates: (i) $v^+, r' = r, \theta' = \theta, \phi' = \phi$ and (ii) $v^-, r' = r, \theta' = \theta, \phi' = \phi$. Obviously both transformations given by Eq.(1.12.20) and Eq.(1.12.23) are singular because the both Jacobian of these transformations is singular at $r = 2m$:

$$ \left( \begin{array}{cc} \frac{\partial v^+}{\partial t} & \frac{\partial v^+}{\partial r} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & \frac{r}{r-2m} \\ 0 & 1 \end{array} \right) \qquad (1.12.27) $$

and

$$ \left( \begin{array}{cc} \frac{\partial v^{-}}{\partial t} & \frac{\partial v^{-}}{\partial r} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & -\frac{r}{2m-r} \\ 0 & 1 \end{array} \right). \qquad (1.12.28) $$

**Remark 1.12.8.** Note first (i) such singular transformations not allowed in conventional Lorentzian geometry and second (ii) both Eddington-Finkelstein metrics given by Eq.(1.12.21) and by Eq.(1.12.24) well defined in rigorous mathematical sense at $r = 2m$.

**Remark 1.12.9.** (I) From consideration above follows that Schwarzschild spacetime $(S^2 \times \{r > 2m\}) \times \mathbb{R}$ is not extendible, in the sense that it can be immersed in a larger spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with $r > 2m$. Thus Eddington-Finkelstein spacetime cannot be considered as an extension of the Schwarzschild spacetime in natural way in respect with conventional Lorentzian geometry. Such "extension" are the extension by abnormal definition and nothing more. (II) However distributional Eddington-Finkelstein spacetime (1.10.53) is equivalent of the distributional Schwarzschild spacetime in natural way.

**Remark 1.12.10.** From consideration above follows that it is necessary an regularization of the Eq.(1.12.20) and Eq.(1.12.23) on horizon. However obviously only nonsmooth regularization via horizon $r = 2m$ possible. Under nonsmooth regularization (see section 3) Eq.(1.12.20) and Eq.(1.12.23) takes the form

$$ dt = -\frac{dr}{r \sqrt{(r-2m)^2 + \epsilon^2}} + dv_{\epsilon}^{+}, \qquad (1.12.29) \\ \epsilon \in (0,1] $$

and

$$ dt = \frac{dr}{r \sqrt{(2m-r)^2 + \epsilon^2}} + dv_{\epsilon}^{-}, \qquad (1.12.30) \\ \epsilon \in (0,1] $$

correspondingly. Therefore Eq.(1.12.27)-Eq.(1.12.28) takes the form
$$ \begin{pmatrix} \frac{\partial v_{\epsilon}^{+}}{\partial t} & \frac{\partial v_{\epsilon}^{+}}{\partial r} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} = \begin{pmatrix} 1 & \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \\ 0 & 1 \end{pmatrix} \qquad (1.12.32) $$

and

$$ \begin{pmatrix} \frac{\partial v_{\epsilon}^{-}}{\partial t} & \frac{\partial v_{\epsilon}^{-}}{\partial r} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} = \begin{pmatrix} 1 & -\frac{r}{\sqrt{(2m-r)^2 + \epsilon^2}} \\ 0 & 1 \end{pmatrix}. \qquad (1.12.33) $$

From Eq.(1.12.29)-Eq.(1.12.30) one obtain generalized Eddington-Finkelstein transformations

$$ dt = - \frac{rdr}{\left(\sqrt{(r-2m)^2 + \epsilon^2}\right)_{\epsilon}} + (dv_{\epsilon}^{+})_{\epsilon}, \qquad (1.12.34) \\ \epsilon \in (0,1] $$

and

$$ dt = \frac{rdr}{\left(\sqrt{(2m-r)^2 + \epsilon^2}\right)_{\epsilon}} + (dv_{\epsilon}^{-})_{\epsilon}, \qquad (1.12.35) \\ \epsilon \in (0,1]. $$

Therefore Eq.(1.12.32)-Eq.(1.12.33) takes the form

$$ \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{+}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{+}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} = \begin{pmatrix} 1 & \frac{r}{\left(\sqrt{(r-2m)^2 + \epsilon^2}\right)_{\epsilon}} \\ 0 & 1 \end{pmatrix} \qquad (1.12.36) $$

and

$$ \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{-}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{-}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} = \begin{pmatrix} 1 & -\frac{r}{\left(\sqrt{(2m-r)^2 + \epsilon^2}\right)_{\epsilon}} \\ 0 & 1 \end{pmatrix}. \qquad (1.12.37) $$

At point $r = 2m$ one obtain

$$ \left. \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{+}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{+}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} \right|_{r=2m} = \begin{pmatrix} 1 & r(\epsilon^{-1})_{\epsilon} \\ 0 & 1 \end{pmatrix} \qquad (1.12.38) $$

and

$$ \left. \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{-}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{-}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} \right|_{r=2m} = \begin{pmatrix} 1 & -r(\epsilon^{-1})_{\epsilon} \\ 0 & 1 \end{pmatrix}, \qquad (1.12.39) $$

where $(\epsilon^{-1})_{\epsilon} \in \mathcal{R}$. Thus generalized Eddington-Finkelstein transformations (1.12.34)-(1.12.35) well defined in the sense of Colombeau generalized functions. From the
Eq.(1.12.34) one obtains

$$
\begin{align}
dt^2 = & \left[ -\frac{rdr}{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon} + (dv_\epsilon^+) \right]^2 = \frac{r^2 dr^2}{((r-2m)^2 + \epsilon^2)_\epsilon} - \nonumber \\
& -\frac{2rdr}{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon} + (dv_\epsilon^{+2})_\epsilon, \tag{1.12.40} \\
dt^2(h_\epsilon^+(r))_\epsilon = & -\frac{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon}{r} dt^2 = \nonumber \\
& -\frac{rdr^2}{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon} + 2dr(dv_\epsilon^+) - \frac{1}{r}\sqrt{(r-2m)^2 + \epsilon^2} (dv_\epsilon^{+2})_\epsilon. \nonumber
\end{align}
$$

From the Eq.(1.12.35) one obtains

$$
\begin{align}
dt^2 = & \left[ \frac{rdr}{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon} + (dv_\epsilon^-) \right]^2 = \frac{r^2 dr^2}{((r-2m)^2 + \epsilon^2)_\epsilon} + \nonumber \\
& + \frac{2rdr}{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon} + (dv_\epsilon^{-2})_\epsilon, \tag{1.12.41} \\
dt^2(h_\epsilon^-(r))_\epsilon = & \frac{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon}{r} dt^2 = \nonumber \\
& - \frac{rdr^2}{(\sqrt{(r-2m)^2 + \epsilon^2})_\epsilon} + 2dr(dv_\epsilon^-) + \frac{1}{r}\sqrt{(r-2m)^2 + \epsilon^2} (dv_\epsilon^{-2})_\epsilon. \nonumber
\end{align}
$$

Substituting Eqs.(1.12.40)-(1.12.41) into distributional Schwarzschild metric (1.12.42)
above (below) horizon (see subsect.3.1)

$$
\begin{align}
(ds_{\epsilon}^{+2})_{\epsilon} &= (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} + ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \nonumber \\
(ds_{\epsilon}^{-2})_{\epsilon} &= (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \tag{1.12.42} \\
(h_{\epsilon}^{+}(r))_{\epsilon} &= -\frac{(\sqrt{(r-2m)^{2} + \epsilon^{2}})_{\epsilon}}{r}, \quad (h_{\epsilon}^{-}(r))_{\epsilon} = \frac{(\sqrt{(r-2m)^{2} + \epsilon^{2}})_{\epsilon}}{r} \nonumber
\end{align}
$$

one obtains

$$
\begin{align}
(ds_{\epsilon}^{+2})_{\epsilon} &= -\frac{1}{r} \sqrt{(r-2m)^2 + \epsilon^2} dv_{\epsilon}^{+2} + 2dr dv_{\epsilon}^{+} + r^2 \left[ (d\theta)^2 + \sin^2\theta (d\phi)^2 \right]. \tag{1.12.43} \\
(ds_{\epsilon}^{-2})_{\epsilon} &= \frac{1}{r} \sqrt{(2m-r)^2 + \epsilon^2} dv_{\epsilon}^{-2} + 2dr dv_{\epsilon}^{-} + r^2 \left[ (d\theta)^2 + \sin^2\theta (d\phi)^2 \right]. \nonumber
\end{align}
$$

Therefore Colombeau generalized object modeling the classical Eddington-Finkelstein
metric given Eq.(1.12.21) and Eq.(1.10.24) above and below horizon takes the form
$$ (ds_{\epsilon}^{+2})_{\epsilon} = h_{\epsilon}^{+}(r)dv_{\epsilon}^{+2} + 2dr dv_{\epsilon}^{+} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], \qquad (1.12.44) \\ (ds_{\epsilon}^{-2})_{\epsilon} = h_{\epsilon}^{-}(r)dv_{\epsilon}^{-2} + 2dr dv_{\epsilon}^{-} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2]. $$

It easily to verify by using formula A.2 (see appendix) that the distributional curvature scalar $(\mathbf{R}(\epsilon))_{\epsilon}$ again singular at $r = 2m$ as in the case of the distributional Schwarzschild spacetime given by Eq.(1.12.42). However this is not surprising because the classical Eddington- Finkelstein spacetime and generalized Eddington-Finkelstein spacetime given by Eq.(1.12.44) that is essentially different geometrical objects.

## 2. Generalized Colombeau Calculus

### 2.1. Notation and basic notions from standard Colombeau theory.

We use [1],[2],[7] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper $\Omega$ will denote an open subset of $\mathbb{R}^n$. Stanford Colombeau generalized functions on $\Omega$ are defined as equivalence classes $u = [(u_{\epsilon})_{\epsilon}]$ of nets of smooth functions $u_{\epsilon} \in C^{\infty}(\Omega)$ (regularizations) subjected to asymptotic norm conditions with respect to $\epsilon \in (0, 1]$ for their derivatives on compact sets.

The basic idea of classical Colombeau's theory of nonlinear generalized functions [1],[2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter $\epsilon$. Let $(u_{\epsilon})_{\epsilon \in (0,1]}$ with $(u_{\epsilon})_{\epsilon} \in C^{\infty}(M)$ for all $\epsilon \in \mathbb{R}_{+}$, where $M$ a separable, smooth orientable Hausdorff manifold of dimension $n$.

**Definition 2.1.1.** The classical Colombeau's algebra of generalized functions on $M$ is defined as the quotient:

$$ \mathcal{G}(M) \triangleq \mathcal{E}_M(M)/\mathcal{N}(M) \qquad (2.1.1) $$

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

$$ \mathcal{E}_M(M) \triangleq \{ (u_\epsilon)_\epsilon \mid \forall K(K \subsetneq M) \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\ \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}, \qquad (2.1.2) $$

$$ \mathcal{N}(M) \triangleq \{ (u_\epsilon)_\epsilon \mid \forall K(K \subsetneq M), \forall k(k \in \mathbb{N}_0) \forall q(q \in N) \\ \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \}. \qquad (2.1.3) $$

**Remark 2.1.1.** 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 2.1.2.** Elements of $\mathcal{G}(M)$ are denoted by:

$$ u = \mathbf{cl}[(u_{\epsilon})_{\epsilon}] \triangleq (u_{\epsilon})_{\epsilon} + \mathcal{N}(M). \qquad (2.1.4) $$

**Remark 2.1.2.** With componentwise operations $(\cdot, \pm) \mathcal{G}(M)$ is a fine sheaf of differential
algebras with respect to the Lie derivative defined by $L_{\xi}u \triangleq \mathbf{cl}[(L_{\xi}u_{\epsilon})_{\epsilon}]$.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be

characterized locally, i.e., $u \in \mathcal{G}(M)$ iff $u \circ \psi_{\alpha} \in \mathcal{G}(\psi_{\alpha}(V_{\alpha}))$ 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., $u \in \mathcal{G}(M)$ iff $u \circ \psi_{\alpha} \in \mathcal{G}(\psi_{\alpha}(V_{\alpha}))$ 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 2.1.3.** Smooth functions $f \in \mathcal{C}^{\infty}(M)$ are embedded into $\mathcal{G}(M)$ simply by the "constant" embedding $\sigma$, i.e., $\sigma(f) = \mathbf{cl}[(f)_{\epsilon}]$, hence $\mathcal{C}^{\infty}(M)$ is a faithful subalgebra of $\mathcal{G}(M)$.

## 2.2. Point Values of a Generalized Functions on $M$. Generalized Numbers.

Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field $\mathbb{R}$ or $\mathbb{C}$. Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".

**Definition 2.2.1.** Inserting $p \in M$ into $u \in \mathcal{G}(M)$ yields a well defined element of the ring of constants (also called generalized numbers) $\mathcal{K}$ (corresponding to $\mathbf{K} = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers $((r_{\epsilon})_{\epsilon} \in \mathbf{K}^{(0,1]}$ with $|r_{\epsilon}| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_{\epsilon}| = O(\epsilon^{m})$ for each $m$); componentwise insertion of points of $M$ into elements of $\mathcal{G}(M)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ \mathcal{K} = \mathcal{E}_{\mathbf{c}}(M)/\mathcal{N}_{\mathbf{c}}(M) \qquad (2.2.1) $$

(with $\mathcal{K} = \tilde{\mathbb{R}}$ or $\mathcal{K} = \tilde{\mathbb{C}}$ for $\mathbf{K} = \mathbb{R}$ or $\mathbf{K} = \mathbb{C}$), where

$$ \left\{ \begin{array}{l} \mathcal{E}_{\mathbf{c}}(M) = \{(r_{\epsilon})_{\epsilon} \in \mathbf{K}^I | \exists n(n \in \mathbb{N}) [ |r_{\epsilon}| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0 ] \} \\ \mathcal{N}_{\mathbf{c}}(M) = \{(r_{\epsilon})_{\epsilon} \in \mathbf{K}^I | \forall m(m \in \mathbb{N}) [ |r_{\epsilon}| = O(\epsilon^m) \text{ as } \epsilon \to 0 ] \} \\ I = (0,1]. \end{array} \right. \qquad (2.2.2) $$

Generalized functions on $M$ are characterized by their generalized point values, i.e., by their values on points in $\tilde{M}_{\epsilon}$, the space of equivalence classes of compactly supported nets $(p_{\epsilon})_{\epsilon} \in M^{(0,1]}$ with respect to the relation $p_{\epsilon} \sim p'_{\epsilon} :\Leftrightarrow d_{h}(p_{\epsilon}, p'_{\epsilon}) = O(\epsilon^{m})$ for all $m$, where $d_{h}$ denotes the distance on $M$ induced by any Riemannian metric.

**Definition 2.2.2.** For $u \in \mathcal{G}(M)$ and $x_{0} \in M$, the point value of $u$ at the point $x_{0}, u(x_{0})$, is defined as the class of $(u_{\epsilon}(x_{0}))_{\epsilon}$ in $\mathcal{K}$.

**Definition 2.2.3.** We say that an element $r \in \mathcal{K}$ is strictly nonzero if there exists a
representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

Treating the elements of Colombeau algebras as a generalization of classical functions,

the question arises whether the definition of point values can be extended in such a way

that each element is characterized by its values. Such an extension is indeed possible.

**Definition 2.2.4.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. On a set $\hat{\Omega}$:

$$ \hat{\Omega} = \{(x_\epsilon)_\epsilon \in \Omega^l \exists p(p>0) [x_\epsilon] = O(\epsilon^p)]\} = \{(x_\epsilon)_\epsilon \in \Omega^l \exists p(p>0) \exists \epsilon_0 (\epsilon_0 > 0) [x_\epsilon] \le \epsilon^p, \text{ for } 0 < \epsilon < \epsilon_0]\} \quad (2.2.3) $$

we introduce an equivalence relation:

$$ (x_\epsilon)_\epsilon \sim (y_\epsilon)_\epsilon \Leftrightarrow \forall q(q > 0) \forall \epsilon(\epsilon > 0) [|x_\epsilon - y_\epsilon| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0] \quad (2.2.4) $$

and denote by $\tilde{\Omega} = \hat{\Omega}/\sim$ the set of generalized points. The set of points with compact support is

$$ \tilde{\Omega}_c = \{ \tilde{x} = \text{cl}[(x_\epsilon)_\epsilon] \in \tilde{\Omega} | \exists K \subset \Omega) \exists \epsilon_0 (\epsilon_0 > 0) [x_\epsilon \in K \text{ for } 0 < \epsilon < \epsilon_0] \} \quad (2.2.5) $$

**Definition 2.2.5.** A generalized function $u \in \mathcal{G}(M)$ is called associated to zero, $u \approx 0$ on

$\Omega \subseteq M$ in L.Schwartz sense if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly,i.e.

$$ w - \lim_{\epsilon \to 0} u_\epsilon = 0 \quad (2.2.6) $$

We shall often write:

$$ u \approx 0. \quad (2.2.7) $$

The $\mathcal{G}(M)$-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

$$ \mathcal{G}_s^r(M) = \mathcal{G}(M) \otimes T_s^r(M), \quad (2.2.8) $$

where $T_s^r(M)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $C^\infty(M)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [1]-[2], in particular 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 $\mathcal{G}(M)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $\mathcal{G}(M)$-modules we have

$$ \mathcal{G}_s^r(M) \cong L_{(M)}(\mathcal{G}_1^0(M)^r, \mathcal{G}_0^1(M)^s; \mathcal{G}(M)). \quad (2.2.9) $$
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 $\mathcal{G}(M)$. 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 $\mathcal{G}(M)$-linear isomorphism from $\mathcal{G}_0^1(M)$ to $\mathcal{G}_0^0(M)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_{\epsilon}]$ is a well defined element of $\mathcal{G}_0^2(M)$ (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 representatives.

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

There exist two different types of distributional Colombeau solution of the Generalized Einstein's Field Equations (1.9.18) corresponding to classical Schwarzschild solution. That is: (i) full distributional Schwarzschild blackhole geometry, given by Colombeau generalized object $\{(ds_{\epsilon}^{+2})_{\epsilon}, (ds_{\epsilon}^{-2})_{\epsilon}\}$, given by Equations (1.10.28), see Figure 2.3.1(a) and (ii) the truncated distributional Schwarzschild space-time given by Colombeau generalized object $(ds_{\epsilon}^{+2})_{\epsilon}$, i.e. in this case distributional space-time ends just on the Schwarzschild horizon, see Figure 2.3.1(b).

<figure><img src="image_22.png" /><figcaption>Fig.2.3.1.</figcaption></figure>

Fig.2.3.1.(a) The picture of a distributional Schwarzschild blackhole, given by the full Colombeau generalized object (1.10.28). Distributional spacetime ends just on the Schwarzschild singularity. (b) The truncated Schwarzschild distributional geometry, given

by Colombeau generalized object. Distributional spacetime ends just on the Schwarzschild horizon.

The Colombeau generalized Ricci tensor above horizon $[R^+(r)]^{\beta}_{\alpha}$ (see Eq.3.1.4) reads
$$ \left\{ \begin{aligned} \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{0}^{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{1}^{1} \right)_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{+\prime\prime}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{+\prime}(r))_{\epsilon} \right) \\ \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{2}^{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{3}^{3} \right)_{\epsilon} = \frac{(h_{\epsilon}^{+\prime}(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{+}(r))_{\epsilon}}{r^2}. \end{aligned} \right. \qquad (2.3.1) $$

From Eq.(3.1.5) we obtain

$$ \begin{aligned} \left( 2r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{0}^{0} \right)_{\epsilon} = \left( 2r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{1}^{1} \right)_{\epsilon} = r^2(h_{\epsilon}^{+\prime\prime}(r))_{\epsilon} + 2r(h_{\epsilon}^{+\prime}(r))_{\epsilon} = & \\ - \left( \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} \right)_{\epsilon} + \left( \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right)_{\epsilon} = & \\ - \frac{r}{[(r-2m)^2 + (\epsilon^2)_{\epsilon}]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + (\epsilon^2)_{\epsilon}]^{3/2}} & = \end{aligned} \qquad (2.3.2) $$

and

$$ \begin{aligned} \left( r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{2}^{2} \right)_{\epsilon} = \left( r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{3}^{3} \right)_{\epsilon} = r(h_{\epsilon}^{+\prime}(r))_{\epsilon} + 1 + (h_{\epsilon}^{+}(r))_{\epsilon} = & \\ - \left( \frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} \right)_{\epsilon} + 1 = - \frac{r-2m}{[(r-2m)^2 + (\epsilon^2)_{\epsilon}]^{1/2}} + 1. & \end{aligned} \qquad (2.3.3) $$

For any $r - 2m \in \mathbb{R}_+$ from Eq.(2.3.2)-Eq.(2.3.3) one obtains

$$ \begin{aligned} \left( 2r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{0}^{0} \right)_{\epsilon} = \left( 2r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{1}^{1} \right)_{\epsilon} = r^2(h_{\epsilon}^{+\prime\prime}(r))_{\epsilon} + 2r(h_{\epsilon}^{+\prime}(r))_{\epsilon} = & \\ - \frac{r}{[(r-2m)^2 + (\epsilon^2)_{\epsilon}]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + (\epsilon^2)_{\epsilon}]^{3/2}} = & \\ - \frac{r}{(r-2m)[1 + (r-2m)^{-2}(\epsilon^2)_{\epsilon}]^{1/2}} + \frac{r(r-2m)^2}{(r-2m)^3[1 + (r-2m)^{-2}(\epsilon^2)_{\epsilon}]^{3/2}} = & \\ - \frac{r}{(r-2m)} + \frac{r}{(r-2m)} + O((\epsilon^2)_{\epsilon}) = O((\epsilon^2)_{\epsilon}) & \end{aligned} \qquad (2.3.4) $$

and

$$ \begin{aligned} \left( r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{2}^{2} \right)_{\epsilon} = \left( r^2 [\mathbf{R}_{\epsilon}^{+}(r)]_{3}^{3} \right)_{\epsilon} = r(h_{\epsilon}^{+\prime}(r))_{\epsilon} + 1 + (h_{\epsilon}^{+}(r))_{\epsilon} = & \\ - \frac{r-2m}{[(r-2m)^2 + (\epsilon^2)_{\epsilon}]^{1/2}} + 1 = - \frac{r-2m}{(r-2m)[1 + (r-2m)^{-2}(\epsilon^2)_{\epsilon}]^{1/2}} + 1 = & \\ -1 + 1 + O((\epsilon^2)_{\epsilon}) = O((\epsilon^2)_{\epsilon}), & \end{aligned} \qquad (2.3.5) $$

where $\epsilon \in (0, \delta]$, $\delta \ll 1$.

Thus for any $r - 2m \in \mathbb{R}_+$ the nonlinear distributional Schwarzschild geometry returns the classical result.

For any $r - 2m = 0 \approx \mathbb{R}$ from Eq.(2.3.2)-Eq.(2.3.3) in contrast with ubnormal canonical result

one obtains
$$ \begin{aligned} \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{0}^{0} \right)_{\epsilon} \Big|_{r=2m} &= \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{1}^{1} \right)_{\epsilon} \Big|_{r=2m} = \\ & \quad - \frac{r}{4m^2 \left[ (r-2m)^2 + (\epsilon^2)_{\epsilon} \right]^{1/2}} \left. \right|_{r=2m} + \frac{r(r-2m)^2}{4m^2 \left[ (r-2m)^2 + (\epsilon^2)_{\epsilon} \right]^{3/2}} \left. \right|_{r=2m} = \\ &= -2^{-1}m^{-1}(\epsilon^{-1})_{\epsilon} \end{aligned} \quad (2.3.6) $$

and

$$ \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{2}^{2} \right)_{\epsilon} \Big|_{r=2m} = \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{3}^{3} \right)_{\epsilon} \Big|_{r=2m} = \\ - \frac{r-2m}{4m^2 \left[ (r-2m)^2 + (\epsilon^2)_{\epsilon} \right]^{1/2}} \left. \right|_{r=2m} + 4^{-1}m^{-2} = 4^{-1}m^{-2}. \quad (2.3.7) $$

It follows from Eq.(2.3.7)

$$ \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{2}^{2} \right)_{\epsilon} = \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{3}^{3} \right)_{\epsilon} = 4^{-1}m^{-2}\psi_{+}(r, 2m), \quad (2.3.8) $$

where

$$ \psi_{+}(r, 2m) = \begin{cases} 0 & \text{if } r > 2m \\ 1 & \text{if } r = 2m \end{cases} \quad (2.3.9) $$

The Colombeau generalized Ricci tensor below horizon $[\mathbf{R}_{\epsilon}^{-}]_{\alpha}^{\beta} = [\mathbf{R}_{\epsilon}^{-}]_{\alpha}^{\beta}$ (see Eq.3.1.21) reads

$$ \begin{aligned} \left( [\mathbf{R}_{\epsilon}(r)]_{0}^{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon^{-}}(r)]_{1}^{1} \right)_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon''}^{-}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon'}^{-}(r))_{\epsilon} \right), \\ \left( [\mathbf{R}_{\epsilon^{-}}(r)]_{2}^{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon^{-}}(r)]_{3}^{3} \right)_{\epsilon} = \frac{(h_{\epsilon'}^{-}(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon^{-}}(r))_{\epsilon}}{r^2}. \end{aligned} \quad (2.3.10) $$

From Eq.(3.1.22) we obtain

$$ \left( 2r^2 [\mathbf{R}_{\epsilon}^{-}(r)]_{0}^{0} \right)_{\epsilon} = \left( 2r^2 [\mathbf{R}_{\epsilon}^{-}(r)]_{1}^{1} \right)_{\epsilon} = \frac{r}{\left[ (r-2m)^2 + (\epsilon^2)_{\epsilon} \right]^{1/2}} - \frac{r(r-2m)^2}{\left[ (r-2m)^2 + (\epsilon^2)_{\epsilon} \right]^{3/2}} \quad (2.3.11) $$

and

$$ \left( r^2 [\mathbf{R}_{\epsilon}^{-}(r)]_{2}^{2} \right)_{\epsilon} = \left( r^2 [\mathbf{R}_{\epsilon}^{-}(r)]_{3}^{3} \right)_{\epsilon} = \frac{r-2m}{\left[ (r-2m)^2 + (\epsilon^2)_{\epsilon} \right]^{1/2}} + 1. \quad (2.3.12) $$

For any $r - 2m \in \mathbb{R}_-$ from Eq.(2.3.11)-Eq.(2.3.12) one obtains
$$ (2r^2[\mathbf{R}_\epsilon^-(r)]_0^0)_\epsilon = (2r^2[\mathbf{R}_\epsilon^-(r)]_1^1)_\epsilon = \frac{r}{[(r-2m)^2 + (\epsilon^2)_\epsilon]^{1/2}} - \frac{r(r-2m)^2}{[(r-2m)^2 + (\epsilon^2)_\epsilon]^{3/2}} = \frac{r}{(r-2m)[1 + (r-2m)^{-2}(\epsilon^2)_\epsilon]^{1/2}} - \frac{r(r-2m)^2}{(r-2m)^3[1 + (r-2m)^{-2}(\epsilon^2)_\epsilon]^{3/2}} = \frac{r}{(r-2m)} - \frac{r}{(r-2m)} + O((\epsilon^2)_\epsilon) = O((\epsilon^2)_\epsilon) \quad (2.3.13) $$

and

$$ (r^2[\mathbf{R}_\epsilon^-(r)]_2^2)_\epsilon = (r^2[\mathbf{R}_\epsilon^-(r)]_3^3)_\epsilon = \frac{r-2m}{[(r-2m)^2 + (\epsilon^2)_\epsilon]^{1/2}} + 1 = O((\epsilon^2)_\epsilon) \quad (2.3.14) $$

where $\epsilon \in (0, \delta]$, $\delta \ll 1$.

Thus for any $r-2m \in \mathbb{R}_-$ the nonlinear distributional Schwarzschild geometry returns the classical result.

For any $r-2m = 0_{\mathbb{R}}$ from Eq.(2.3.11)-Eq.(2.3.12) in contrast with ubnormal canonical result one obtains

$$ (2r^2[\mathbf{R}_\epsilon^-(r)]_0^0)_\epsilon \Big|_{r=2m} = (2r^2[\mathbf{R}_\epsilon^-(r)]_1^1)_\epsilon \Big|_{r=2m} = \frac{r}{[(r-2m)^2 + (\epsilon^2)_\epsilon]^{1/2}} - \frac{r(r-2m)^2}{[(r-2m)^2 + (\epsilon^2)_\epsilon]^{3/2}} \Bigg|_{r=2m} = 2^{-1}m^{-1}(\epsilon^{-1})_\epsilon \quad (2.3.15) $$

and

$$ ([\mathbf{R}_\epsilon^-(r)]_2^2)_\epsilon \Big|_{r=2m} = ([\mathbf{R}_\epsilon^-(r)]_3^3)_\epsilon \Big|_{r=2m} = \frac{r-2m}{4m^2[(r-2m)^2 + (\epsilon^2)_\epsilon]^{1/2}} + 4^{-1}m^{-2} = 4^{-1}m^{-2}. \quad (2.3.14) $$

It follows from Eq.(2.3.14)

$$ ([\mathbf{R}_\epsilon^-(r)]_2^2)_\epsilon = ([\mathbf{R}_\epsilon^-(r)]_3^3)_\epsilon = 4^{-1}m^{-2}\psi_-(r,2m), \quad (2.3.15) $$

where

$$ \psi_-(r, 2m) = \begin{cases} 0 & \text{if } r < 2m \\ 1 & \text{if } r = 2m \end{cases} \quad (2.3.16) $$

**Remark 2.3.1.** Note that any sequence $[\mathbf{R}_\epsilon^+(r)]_0^0, [\mathbf{R}_\epsilon^+(r)]_1^1, [\mathbf{R}_\epsilon^+(r)]_2^2, [\mathbf{R}_\epsilon^+(r)]_3^3$ and any sequence $[\mathbf{R}_\epsilon^-(r)]_0^0, [\mathbf{R}_\epsilon^-(r)]_1^1, [\mathbf{R}_\epsilon^-(r)]_2^2, [\mathbf{R}_\epsilon^-(r)]_3^3$ has a weak limit in $D'(\mathbb{R}^3)$ : $w$-lim$_{\epsilon \to 0}[\mathbf{R}_\epsilon^+(r)]_0^0 \in D'(\mathbb{R}^3)$, ..., $w$-lim$_{\epsilon \to 0}[\mathbf{R}_\epsilon^-(r)]_3^3 \in D'(\mathbb{R}^3)$, see Section 3.

## 2.4. The nonlinear distributional Schwarzschild geometry in isotropic coordinates.

The Schwarzschild metric in isotropic coordinates $(t, \rho, \theta, \phi)$ reads [3]:
$$ds^2 = -\left(\frac{1-r_s/4\rho}{1+r_s/4\rho}\right)^2 dt^2 + \left(1+\frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (2.4.1)$$

see Eq.(1.1) where we let $c = 1$, or in the equivalent form

$$ds^2 = -\frac{(4\rho - r_s)^2}{4\rho + r_s} dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\phi^2). \quad (2.4.2)$$

In order to obtain Colombeau solutions of the generalized Einstein field equations (1.9.18), corresponding to the nonclassical semi Riemannian metric (2.4.2) we apply the regularization $(4\rho - r_s)^2 \rightarrow (4\rho - r_s)^2 + \epsilon^2$ and embed (2.4.2) in the Colombeau object

$$(ds_{\varepsilon}^2)_{\varepsilon} = - \frac{((4\rho - r_s)^2 + \varepsilon^2)_{\varepsilon}}{4\rho + r_s} dt^2 + \left(1 + \frac{r_s}{4\rho}\right)^4 d\rho^2 + \rho^2(d\theta^2 + \sin^2\theta d\phi^2). \quad (2.4.3)$$

We rewrite now Eq.(2.4.3) in the following form (see Appendix A,Eq.(A.1))

$$(ds_{\varepsilon}^2)_{\varepsilon} = -[(A_{\varepsilon}(\rho))_{\varepsilon}]dt^2 + [B(\rho) + C(\rho)]d\rho^2 + B(\rho)\rho^2(d\theta^2 + \sin^2\theta d\phi^2), \quad (2.4.4)$$

where

$$A_{\varepsilon}(\rho) = \frac{(4\rho - r_s)^2 + \varepsilon^2}{4\rho + r_s}, \quad B(\rho) = 1, \quad B(\rho) + C(\rho) = \left(1 + \frac{r_s}{4\rho}\right)^4 = G(\rho). \quad (2.4.5)$$

The Colombeau generalized curvature scalar $(R_{\varepsilon}(\rho))_{\varepsilon}$ reads (see Appendix A,Eq.(A.2))

$$(R_{\varepsilon}(\rho))_{\varepsilon} = G(\rho) \left( \left[ \frac{2}{\rho} \left( -\frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} + \frac{\Delta'_{\varepsilon}(\rho)}{\Delta_{\varepsilon}(\rho)} \right) + \frac{2C(\rho)}{\rho^2} - \frac{A''_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} \right. \right. \\ \left. \left. + \frac{1}{2} \frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} \frac{\Delta'_{\varepsilon}(\rho)}{\Delta_{\varepsilon}(\rho)} \right] \right)_{\varepsilon}, \quad (2.4.6)$$

where

$$\begin{aligned} \Delta_{\varepsilon}(\rho) &= A_{\varepsilon}(\rho)G(\rho), & \Delta'_{\varepsilon}(\rho) &= A'_{\varepsilon}(\rho)G(\rho) + A_{\varepsilon}(\rho)G'(\rho) \\ \frac{\Delta'_{\varepsilon}(\rho)}{\Delta_{\varepsilon}(\rho)} &= \frac{A'_{\varepsilon}(\rho)G(\rho) + A_{\varepsilon}(\rho)G'(\rho)}{A_{\varepsilon}(\rho)G(\rho)} &= \frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} + \frac{G'(\rho)}{G(\rho)}. \end{aligned} \quad (2.4.7)$$

Substituting Eq.(2.4.7) into Eq.(2.4.3) we get

$$(R_{\varepsilon}(\rho))_{\varepsilon} = G(\rho) \left( \left[ \frac{2}{\rho} \left( -\frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} + \frac{\Delta'_{\varepsilon}(\rho)}{\Delta_{\varepsilon}(\rho)} \right) + \frac{2C(\rho)}{\rho^2} - \frac{A''_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} \right. \right. \\ \left. \left. + \frac{1}{2} \frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} \frac{\Delta'_{\varepsilon}(\rho)}{\Delta_{\varepsilon}(\rho)} \right] \right)_{\varepsilon} = \\ G(\rho) \left( \left[ \frac{2}{\rho} \frac{G'(\rho)}{G(\rho)} + \frac{2C(\rho)}{\rho^2} - \frac{A''_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} + \frac{1}{2} \frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} \left( \frac{A'_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} + \frac{G'(\rho)}{G(\rho)} \right) \right] \right)_{\varepsilon}, \quad (2.4.8)$$

where
$$
\begin{align}
A_{\varepsilon}(\rho) &= \frac{(4\rho - r_s)^2 + \varepsilon^2}{4\rho + r_s}, & A'_{\varepsilon}(\rho) &= \frac{8(4\rho - r_s)}{4\rho + r_s} - \frac{(4\rho - r_s)^2 + \varepsilon^2}{(4\rho + r_s)^2}, \nonumber \\
A''_{\varepsilon}(\rho) &= \frac{64r_s}{(4\rho + r_s)^2} + \frac{8(\varepsilon^2 + 2r_s^2 - 8\rho r_s)}{(4\rho + r_s)^3}, \tag{2.4.9} \\
\frac{A''_{\varepsilon}(\rho)}{A_{\varepsilon}(\rho)} &= \frac{64r_s}{[(4\rho - r_s)^2 + \varepsilon^2](4\rho + r_s)} + \frac{8}{(4\rho + r_s)^2} \nonumber
\end{align}
$$

and

$$
\frac{A'_{\epsilon}(\rho)}{A_{\epsilon}(\rho)} = \frac{8(4\rho - r_s)}{(4\rho - r_s)^2 + \epsilon^2} - \frac{1}{4\rho + r_s}. \quad (2.4.10)
$$

From Eq.(2.4.9)-Eq.(2.4.10) we obtain

$$
\begin{align*}
\left. \frac{A'_{\epsilon}(\rho)}{A_{\epsilon}(\rho)} \right|_{4\rho=r_s} &= \left. \frac{8(4\rho-r_s)}{(4\rho-r_s)^2 + \epsilon^2} \right|_{4\rho=r_s} - \left. \frac{1}{4\rho+r_s} \right|_{4\rho=r_s} = -\frac{1}{2r_s} \\
\left. \frac{A''_{\epsilon}(\rho)}{A_{\epsilon}(\rho)} \right|_{4\rho=r_s} &= \left. \frac{8r_s(4\rho+r_s)}{[(4\rho-r_s)^2 + \epsilon^2]} \right|_{4\rho=r_s} + \left. \frac{8}{(4\rho+r_s)^2} \right|_{4\rho=r_s} = \\
&\qquad \frac{8r_s(4\rho+r_s)}{[(4\rho-r_s)^2 + \epsilon^2](4\rho+r_s)} = \frac{16r_s}{\epsilon^2} + \frac{2}{r_s^2}.
\end{align*}
$$

From Eq.(2.4.11) and Eq.(2.4.8) finally we obtain

$$
(R_{\epsilon}(\rho))|_{\epsilon \atop 4\rho=r_s} = \frac{16r_s}{(\epsilon^2)_{\epsilon}} + \text{const.} \qquad (2.4.11)
$$

2.5.Generalized Colombeau Calculus.

2.5.1. Super Generalized Functions.

We briefly recall the basic generalized Colombeau construction. Colombeau
supergeneralized functions on Ω ⊆ Rⁿ, where dim(Ω) = n are defined as equivalence
classes u = [(u_ε)ε] of nets of functions u_ε ∈ C∞(ΩΣ), ε ∈ (0,δ] such that any u_ε is a net
of functions smooth on Ω\Σ and has a discontinuity on a subset Σ ⊂ Ω, where
dim(Σ) < n. We assume that for any ε ∈ (0,δ] the derivative ∂⁸u_ε / ∂x¹k₁ ... ∂xⁿkₙ exists in the
sense of the theory of generalized functions and ∂⁸u_ε / ∂x¹k₁ ... ∂xⁿkₙ ∈ D'(Ω).

The basic idea of generalized *Colombeau's theory of nonlinear supergeneralized
functions* [33] is regularization by sequences (nets) of nonsmooth functions with
derivatives in *D'(Ω)* and the use of asymptotic estimates in terms of a regularization
parameter *ε*. Let (*u*<sub>*ε*</sub>)<sub>*ε*∈(0,1]</sub> with *u*<sub>*ε*</sub> such that: (i) *u*<sub>*ε*</sub> ∈ *C*<sup>∞</sup>(*M*\*Σ) and (ii)
*L*<sub>*ε*₁</sub>...*L*<sub>*ε*ₖ</sub>*u*<sub>*ε*</sub> ∈ *D*(*M*), for all *ε* ∈ (0, *δ*], where *M* a separable, smooth orientable Hausdorff
manifold of dimension *n*.

**Definition 2.5.1.** The supergeneralized Colombeau’s algebra $\tilde{\mathcal{G}} = \tilde{\mathcal{G}}(M, \Sigma)$ of supergeneralized functions on $M$, where $\Sigma \subset M$, $\dim(M) = n$, $\dim(\Sigma) < n$, is defined as the quotient:

$$
\tilde{\mathcal{G}}(M, \Sigma) \triangleq \mathcal{E}_M(M, \Sigma) / \mathcal{N}(M, \Sigma) \quad (2.5.1)
$$
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 $\mathcal{X}(M\Sigma)$ denoting the space of smooth vector fields on $M\Sigma$):

$$ \mathcal{E}_M(M, \Sigma) \triangleq \{(u_\varepsilon)_\varepsilon | \forall K(K \not\subseteq M\Sigma), \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \\ \qquad \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M\Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\varepsilon(p)| = O(\varepsilon^{-N}), \varepsilon \to 0 \right] \& \\ \qquad \forall K(K \not\subseteq M), \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \forall (f \in \mathcal{C}^\infty(M)) \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \\ \qquad \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\varepsilon\| = \left( \sup_{f \in \mathcal{C}^\infty(M)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\varepsilon(f)| \right) = O(\varepsilon^{-N}), \varepsilon \to 0 \right] \}, $$

$$ \mathcal{N}(M, \Sigma) \triangleq \{(u_\varepsilon)_\varepsilon | \forall K(K \not\subseteq M\Sigma), \forall k(k \in \mathbb{N}_0) \forall q(q \in N) \\ \qquad \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M\Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\varepsilon(p)| = O(\varepsilon^q), \varepsilon \to 0 \right] \& \\ \qquad \forall K(K \not\subseteq M), \forall k(k \in \mathbb{N}) \exists N(N \in \mathbb{N}) \forall (f \in \mathcal{C}^\infty(M)) \forall \xi_1, \dots, \xi_k(\xi_1, \dots, \xi_k \in \mathcal{X}(M)) \\ \qquad \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\varepsilon\| = \left( \sup_{f \in \mathcal{C}^\infty(M)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\varepsilon(f)| \right) = O(\varepsilon^q), \varepsilon \to 0 \right] \}, $$

where $L_{\xi_k}^w$ denoting the weak Lie derivative in L.Schwartz sense.In the definition the Landau symbol $a_\varepsilon = O(\psi(\varepsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \varepsilon_0(\varepsilon_0 \in (0,1]) \forall \varepsilon(\varepsilon < \varepsilon_0)[a_\varepsilon \le C\psi(\varepsilon)]$.

**Definition 2.5.2.** Elements of $\mathcal{G}(M, \Sigma)$ are denoted by:

$$ u = \mathbf{cl}[(u_\varepsilon)_\varepsilon] \triangleq (u_\varepsilon)_\varepsilon + \mathcal{N}(M, \Sigma). \qquad (2.5.4) $$

**Remark 2.5.1.** With componentwise operations $(\cdot, \pm) \mathcal{G}(M, \Sigma)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_{\xi}u \triangleq \mathbf{cl}[(L_{\xi}u_\varepsilon)_\varepsilon]$.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \mathcal{G}(M, \Sigma)$ iff $u \circ \psi_\alpha \in \mathcal{G}(\psi_\alpha(V_\alpha))$ 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., $u \in \mathcal{G}(M, \Sigma)$ iff $u \circ \psi_\alpha \in \mathcal{G}(\psi_\alpha(V_\alpha))$ 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 2.5.2.** Smooth functions $f \in \mathcal{C}^\infty(M\Sigma)$ are embedded into $\mathcal{G}(M, \Sigma)$ simply by the "constant" embedding $\sigma$, i.e., $\sigma(f) = \mathbf{cl}[(f)_\varepsilon]$, hence $\mathcal{C}^\infty(M\Sigma)$ is a faithful subalgebra of $\mathcal{G}(M, \Sigma)$.

## 2.5.2. Point Values of a Supergeneralized Functions on $M$. Supergeneralized Numbers

Within the classical distribution theory, distributions cannot be characterized by their
point values in any way similar to classical functions. On the other hand, there is a very
natural and direct way of obtaining the point values of the elements of Colombeau's
algebra: points are simply inserted into representatives. The objects so obtained are
sequences of numbers, and as such are not the elements in the field $\mathbb{R}$ or $\mathbb{C}$. Instead,
they are the representatives of Colombeau's generalized numbers. We give the exact
definition of these "numbers".

**Definition 2.5.3.** Inserting $p \in M$ into $u \in \tilde{\mathcal{G}}(M, \Sigma)$ yields a well defined element of the ring of constants (also called generalized numbers) $\tilde{\mathcal{K}}$ (corresponding to $\mathbf{K} = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers $((r_\epsilon)_\epsilon \in \mathbf{K}^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $\tilde{\mathcal{G}}(M, \Sigma)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$
\tilde{\mathcal{K}}_{\Sigma} = \mathcal{E}_c(M, \Sigma) / \mathcal{N}_c(M, \Sigma) \qquad (2.5.5)
$$

(with $\tilde{\mathcal{K}}_{\Sigma} = \tilde{\mathbb{R}}_{\Sigma}$ or $\mathcal{K} = \tilde{\mathbb{C}}_{\Sigma}$ for $\mathbf{K} = \mathbb{R}$ or $\mathbf{K} = \mathbb{C}$), where

$$
\left\{
\begin{array}{l}
\mathcal{E}_c(M, \Sigma) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \exists n(n \in \mathbb{N}) [ |r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0 ] \}, \\
\mathcal{N}_c(M, \Sigma) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \forall m(m \in \mathbb{N}) [ |r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0 ] \}
\end{array}
\right.
\qquad (2.5.6)
$$
$$
I = (0, 1].
$$

Supergeneralized functions on $M$ are characterized by their generalized point values, i.e., by their values on points in $\tilde{M}_c$, the space of equivalence classes of compactly supported nets $(p_\epsilon)_\epsilon \in (M\Sigma)^{(0,1]}$ with respect to the relation

$p_{\epsilon} \sim p'_{\epsilon} :\Leftrightarrow d_h(p_{\epsilon}, p'_{\epsilon}) = O(\epsilon^m)$ for all $m$, where $d_h$ denotes the distance on $M\Sigma$ induced by any Riemannian metric.

**Definition 2.5.4.** For $u \in \tilde{\mathcal{G}}(M, \Sigma)$ and $x_0 \in M$, the point value of $u$ at the point $x_0$, $u(x_0)$, is

defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $\tilde{\mathcal{K}}$.

**Definition 2.5.5.** We say that an element $r \in \tilde{\mathcal{K}}$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

Treating the elements of Colombeau algebras as a generalization of classical
functions, the question arises whether the definition of point values can be extended in
such a way that each element is characterized by its values. Such an extension is
indeed possible.

**Definition 2.5.6.** Let $\Omega$ be an open subset of $\mathbb{R}^n \setminus \Sigma$. On a set $\hat{\Omega}_\Sigma$ :

$$
\left\{
\begin{array}{l}
\hat{\Omega}_{\Sigma} = \{(x_{\epsilon})_{\epsilon} \in (\Omega \setminus \Sigma)^{I} | \exists p(p > 0) [ |x_{\epsilon}| = O(\epsilon^{p}) ] \} = \\
\quad \{(x_{\epsilon})_{\epsilon} \in (\Omega \setminus \Sigma)^{I} | \exists p(p > 0) \exists \epsilon_{0} (\epsilon_{0} > 0) [ |x_{\epsilon}| \le \epsilon^{p}, \text{ for } 0 < \epsilon < \epsilon_{0} ] \}
\end{array}
\right.
\qquad (2.5.7)
$$

we introduce an equivalence relation:

$$
(x_{\epsilon})_{\epsilon} \sim (y_{\epsilon})_{\epsilon} \Leftrightarrow \forall q(q > 0) \forall \epsilon(\epsilon > 0) [ |x_{\epsilon} - y_{\epsilon}| \le \epsilon^{q}, \text{ for } 0 < \epsilon < \epsilon_{0} ] \quad (2.5.8)
$$

and denote by $\tilde{\Omega}_{\Sigma} = \hat{\Omega}_{\Sigma} / \sim$ the set of supergeneralized points. The set of points with
compact support is

$$ \tilde{\Omega}_{\Sigma, \epsilon} = \{ \tilde{x} = \mathbf{cl}[(x_{\epsilon})_{\epsilon}] \in \tilde{\Omega}_{\Sigma} | \exists K (K \subset \Omega \setminus \Sigma) \exists \epsilon_{0} (\epsilon_{0} > 0) [x_{\epsilon} \in K \text{ for } 0 < \epsilon < \epsilon_{0}] \} \quad (2.5.9) $$

**Definition 2.5.7.** A supergeneralized function $u \in \tilde{\mathcal{G}}(M, \Sigma)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in L. *Schwartz's sense* if one (hence any) representative $(u_{\epsilon})_{\epsilon}$ converges to zero weakly,i.e.

$$ w \lim_{\epsilon \to 0} u_{\epsilon} = 0 \quad (2.5.10) $$

We shall often write:

$$ u \approx 0. \qquad (2.5.11) $$

**Definition 2.5.8.** The $\tilde{\mathcal{G}}(M, \Sigma)$-module of supergeneralized sections in vector bundles-especially the space of generalized tensor fields $T_s^r(M\Sigma)$-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

$$ \tilde{\mathcal{G}}_s^r(M, \Sigma) = \tilde{\mathcal{G}}(M, \Sigma) \otimes T_s^r(M\Sigma), \quad (2.5.12) $$

where $T_s^r(M\Sigma)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $C^\infty(M\Sigma)$. 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, in particular 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 $\tilde{\mathcal{G}}(M, \Sigma)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $\tilde{\mathcal{G}}(M, \Sigma)$-modules we have

$$ \tilde{\mathcal{G}}_s^r(M, \Sigma) \cong L_{(M)}(\tilde{\mathcal{G}}_1^0(M, \Sigma)^r, \tilde{\mathcal{G}}_0^1(M, \Sigma)^s; \tilde{\mathcal{G}}(M, \Sigma)). \quad (2.5.13) $$

In particular a supergeneralized metric is defined to be a symmetric, supergeneralized (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 $\tilde{\mathcal{G}}(M, \Sigma)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_{\epsilon})_{ab})_{\epsilon}$ of $\det(g_{ab})$ we have $\forall K \subset M\Sigma \exists m \in \mathbb{N}[\inf_{p \in K}|\det(g_{ab}(\epsilon))| \ge \epsilon^q]$ for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric 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 $\tilde{\mathcal{G}}(M, \Sigma)$-linear isomorphism from $\tilde{\mathcal{G}}_0^1(M, \Sigma)$ to $\tilde{\mathcal{G}}_1^0(M, \Sigma)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_{\epsilon}]$ is a well defined element of $\tilde{\mathcal{G}}_0^2(M, \Sigma)$ (i.e., independent of the representative $((g_{\epsilon})_{ab})_{\epsilon}$). Also the supergeneralized Levi-Civita connection as well as the supergeneralized Riemann, Ricci and Einstein tensor of a supergeneralized metric are defined simply by the usual coordinate formulae on the level of representatives.
### 2.5.3. Remarks on distributional derivatives in $D'(\mathbb{R})$

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 $x = x_0$ (see Fig.2.5.1) i.e.,

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

<figure><img src="image_23.png" /><figcaption>Fig.2.5.1.</figcaption></figure>

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:

$$g(x) = f(x) - [f]_{x_0} H(x - x_0), \qquad (2.5.14)$$

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.(2.5.14) and therefore one obtains

$$\{f'(x)\} = \bar{f}'(x) - [f]_{x_0} \delta(x - x_0). \qquad (2.5.15)$$

Thus finally we get

$$\bar{f}'(x) = \{f'(x)\} + [f]_{x_0} \delta(x - x_0). \qquad (2.5.16)$$

or

$$\bar{f}'(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta(x - x_0) & x < x_0 \\ f_2'(x) + [f]_{x_0} \delta(x - x_0) & x \ge x_0 \end{cases} \qquad (2.5.17)$$

Equation (2.5.17) is easily generalized to a function $f(x)$ that has jumps of magnitude $[f]_{x_0}, [f]_{x_1}, \dots, [f]_{x_k}$ at $x_0, x_1, \dots, x_k$. The result is

$$\bar{f}'(x) = \{f'(x)\} + \sum_{i=0}^{k} [f]_{x_i} \delta(x - x_i). \qquad (2.5.18)$$

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 (2.5.14) and immediately one obtains
$$
\bar{f}''(x) = \{f''(x)\} + [f]_{x_0} \delta'(x - x_0) + [f']_{x_0} \delta(x - x_0), \quad (2.5.19)
$$

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

$$
\bar{f}''(x) = \begin{cases} f_1'(x) + [f]_{x_0} \delta'(x-x_0) + [f']_{x_0} \delta(x-x_0) & x < x_0 \\ f_2'(x) + [f]_{x_0} \delta(x-x_0) + [f']_{x_0} \delta(x-x_0) & x \ge x_0 \end{cases} \quad (2.5.20)
$$

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

$$
\bar{f}^{(m)}(x) = \{f^{(m)}(x)\} + \sum_{j=1}^{l} [a_j \delta^{(m-1)}(x-x_j) + b_j \delta^{(m-2)}(x-x_j) + \dots + f_j \delta(x-x_j)], \quad (2.5.21)
$$

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

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

Note that the Colombeau generalized object $\{(ds_{\epsilon}^{+2})_{\epsilon}, (ds_{\epsilon}^{-2})_{\epsilon}\}$, given by Equations (1.10.28) that is a natural way is a supergeneralized (0,2)-tensor field $g_{ab} = [((g_{\epsilon})_{ab})_{\epsilon}]$ whose determinant $\det(g_{ab})$ is invertible in $\mathcal{G}(\mathbb{R}^4, \{r=2m\})$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_{\epsilon}]$ is a well defined element of $\mathcal{G}_0^2(\mathbb{R}^4, \{r=2m\})$. Thus the full super generalized Ricci tensor reads

$$
\begin{align}
([\mathbf{R}_{\epsilon}(r)]_0^0)_{\epsilon} &= ([\mathbf{R}_{\epsilon}(r)]_1^1)_{\epsilon} = \frac{1}{2} \left( (\bar{h}_{\epsilon}''(r))_{\epsilon} + \frac{2}{r} (\bar{h}_{\epsilon}'(r))_{\epsilon} \right), \tag{2.5.22} \\
([\mathbf{R}_{\epsilon}(r)]_2^2)_{\epsilon} &= ([\mathbf{R}_{\epsilon}(r)]_3^3)_{\epsilon} = \frac{(\bar{h}_{\epsilon}'(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}(r))_{\epsilon}}{r^2}, \nonumber
\end{align}
$$

where $(h_{\epsilon})_{\epsilon} = \{(h_{\epsilon}^{-})_{\epsilon}, (h_{\epsilon}^{+})_{\epsilon}\}$ and the istributional derivatives $\bar{h}_{\epsilon}'(r)$ and $\bar{h}_{\epsilon}''(r)$ is defined
by Eq.(2.5.17) and by Eq.(2.5.20) correspondingly.

From Eq.(3.1.2) we obtain

$$
([h_{\epsilon}]_{2m})_{\epsilon} = (h_{\epsilon}^{+}(2m+0))_{\epsilon} - (h_{\epsilon}^{-}(2m-0))_{\epsilon} = (h_{\epsilon}^{+}(2m) - h_{\epsilon}^{-}(2m))_{\epsilon} = \\
-\frac{\epsilon}{2m} - \frac{\epsilon}{2m} = -\epsilon/m. \tag{2.5.23}
$$

From Eq.(2.5.23) and Eq.(2.5.17) we obtain

$$
(\bar{h}_{\epsilon}'(r))_{\epsilon} = \begin{cases} (h_{\epsilon}^{+}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon} \delta(r-2m) & r \ge 2m \\ (h_{\epsilon}^{-}(r))_{\epsilon} + ([h_{\epsilon}]_{2m})_{\epsilon} \delta(r-2m) & r < 2m \end{cases} \quad (2.5.24)
$$

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

$$
\bar{h}_{\epsilon}'(r) = \begin{cases} (h_{\epsilon}^{+}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m} \delta(r-2m) & r \ge 2m \\ (h_{\epsilon}^{-}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m} \delta(r-2m) & r < 2m \end{cases} \quad (2.5.25)
$$

From Eq.(3.1.5) and Eq.(3.1.22) we obtain
$$ 
\begin{aligned}
\left[ h'_{\epsilon} \right]_{2m} &= (h_{\epsilon}^{+'}(2m+0))_{\epsilon} - (h_{\epsilon}^{-'}(2m-0))_{\epsilon} = (h_{\epsilon}^{+'}(2m))_{\epsilon} - (h_{\epsilon}^{-'}(2m))_{\epsilon} = \\
&\qquad \frac{(\epsilon)_{\epsilon}}{4m^2} - \left( -\frac{(\epsilon)_{\epsilon}}{4m^2} \right) = \frac{(\epsilon)_{\epsilon}}{2m^2}.
\end{aligned}
\quad (2.5.26) $$

From Eq.(3.1.5),Eq.(3.1.22) and Eq.(2.5.7) we obtain

$$ \left( \bar{h}_{\epsilon}''(r) \right)_{\epsilon} = \begin{cases} h_{\epsilon}^{+''}(r) + [h_{\epsilon}]_{2m} \delta'(r-2m) + \left[ h'_{\epsilon} \right]_{2m} \delta(r-2m) & r \ge 2m \\ h_{\epsilon}^{-''}(r) + [h_{\epsilon}]_{2m} \delta(r-2m) + \left[ h'_{\epsilon} \right]_{x_0} \delta(r-2m) & r < 2m \end{cases} \quad (2.5.27) $$

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

$$ \bar{h}_{\epsilon}''(r) = \begin{cases} (h_{\epsilon}^{+''}(r))_{\epsilon} - \frac{(\epsilon)_{\epsilon}}{m} \delta'(r-2m) + \frac{(\epsilon)_{\epsilon}}{2m^2} \delta(r-2m) & r \ge 2m \\ h_{\epsilon}^{-''}(r) - \frac{(\epsilon)_{\epsilon}}{m} \delta(r-2m) + \frac{(\epsilon)_{\epsilon}}{2m^2} \delta(r-2m) & r < 2m \end{cases} \quad (2.5.28) $$

Inserting Eqs.(2.5.26)-(2.5.28) into Eqs.(2.5.22) finally we get for $r \ge 2m$:

$$ 
\begin{aligned}
\left( [\mathbf{R}_{\epsilon}^{+}(r)]_{0}^{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{1}^{1} \right)_{\epsilon} = \frac{1}{2} \left( (\bar{h}_{\epsilon}''(r))_{\epsilon} + \frac{2}{r} (\bar{h}_{\epsilon}'(r))_{\epsilon} \right) = \\
&\quad \frac{1}{2} \left( (h_{\epsilon}^{+''}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{+'}(x))_{\epsilon} \right) - \\
&\quad - \frac{(\epsilon)_{\epsilon}}{2m} \delta'(r-2m) + \frac{(\epsilon)_{\epsilon}}{2m^2} \delta(r-2m) - \frac{2(\epsilon)_{\epsilon}}{rm} \delta(r-2m), \\
\left( [\mathbf{R}_{\epsilon}^{+}(r)]_{2}^{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}(r)]_{3}^{3} \right)_{\epsilon} = \frac{(\bar{h}_{\epsilon}'(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}(r))_{\epsilon}}{r^2} = \\
&\quad \frac{(h_{\epsilon}^{+'}(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{+}(r))_{\epsilon}}{r^2} - \frac{(\epsilon)_{\epsilon}}{rm} \delta(r-2m).
\end{aligned}
\quad (2.5.29) $$

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

$$ 
\begin{aligned}
\left( [\mathbf{R}_{\epsilon}^{-}(r)]_{0}^{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{-}(r)]_{1}^{1} \right)_{\epsilon} = \frac{1}{2} \left( (\bar{h}_{\epsilon}''(r))_{\epsilon} + \frac{2}{r} (\bar{h}_{\epsilon}'(r))_{\epsilon} \right) = \\
&\quad \frac{1}{2} \left( (h_{\epsilon}^{-''}(r))_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{-'}(x))_{\epsilon} \right) - \\
&\quad - \frac{(\epsilon)_{\epsilon}}{2m} \delta'(r-2m) + \frac{(\epsilon)_{\epsilon}}{2m^2} \delta(r-2m) - \frac{2(\epsilon)_{\epsilon}}{rm} \delta(r-2m), \\
\left( [\mathbf{R}_{\epsilon}^{-}(r)]_{2}^{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{-}(r)]_{3}^{3} \right)_{\epsilon} = \frac{(\bar{h}_{\epsilon}'(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}(r))_{\epsilon}}{r^2} = \\
&\quad \frac{(h_{\epsilon}^{-'}(r))_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{-}(r))_{\epsilon}}{r^2} - \frac{(\epsilon)_{\epsilon}}{rm} \delta(r-2m).
\end{aligned}
\quad (2.5.30) $$

Note that in $D'(\mathbb{R})$:

$$ \text{w-lim}_{\epsilon \to 0} \frac{\pi^{-1}\epsilon}{(r-2m)^2 + \epsilon^2} = \delta(r-2m), \quad (2.5.31) $$
and therefore in $\mathcal{D}'(\mathbb{R})$ :

$$ \text{w-lim}_{\epsilon \to 0} \frac{d}{dr} \frac{\epsilon}{(r-2m)^2 + \epsilon^2} = \text{w-lim}_{\epsilon \to 0} \frac{2\pi^{-1}\epsilon(r-2m)}{[(r-2m)^2 + \epsilon^2]^2} = \delta'(r-2m). \quad (2.5.32) $$

By using Eq.(2.5.31) and Eq.(2.5.32) one obtains the natural imbeddings

$$ \delta(r-2m) \hookrightarrow \mathcal{G}(\mathbb{R}) : \delta(r-2m) \triangleq \pi^{-1} \left( \frac{\epsilon}{(r-2m)^2 + \epsilon^2} \right)_{\epsilon} \in \mathcal{G}(\mathbb{R}) \quad (2.5.33) $$

and

$$ \delta'(r-2m) \hookrightarrow \mathcal{G}(\mathbb{R}) : \delta'(r-2m) \triangleq 2\pi^{-1} \left( \frac{\epsilon(r-2m)}{[(r-2m)^2 + \epsilon^2]^2} \right)_{\epsilon} \in \mathcal{G}(\mathbb{R}) \quad (2.5.34) $$

correspondingly.

**Remark 2.5.3**

**Remark 2.5.4.** Note that in $\mathcal{D}'([2m, \infty))$ :

$$ \text{w-lim}_{\epsilon \to 0} \frac{\epsilon^2(r-2m)}{[(r-2m)^2 + \epsilon^2]^2} = c_1 \delta(r-2m). \quad (2.5.35) $$

## 2.6. Superdistributional general relativity

We briefly summarize the basics of superdistributional general relativity, as a preliminary to latter discussion. In the classical theory of gravitation one is led to consider the Einstein field equations which are, in general, quasilinear partial differential equations involving second order derivatives for the metric tensor. Hence, continuity of the first fundamental form is expected and at most, discontinuities in the second fundamental form, the coordinate independent statements appropriate to consider 3-surfaces of discontinuity in the spacetime manifold of General Relativity.

In standard general relativity, the space-time is assumed to be a four-dimensional differentiable manifold $M$ endowed with the Lorentzian metric $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ ($\mu, \nu = 0, 1, 2, 3$). At each point $p$ of space-time $M$, the metric can be diagonalized as

$ds_p^2 = \eta_{\mu\nu}(dX^\mu)_p(dX^\nu)_p$ with $\eta_{\mu\nu} \triangleq (-1, 1, 1, 1)$, by choosing the coordinate system $\{X^\mu; \mu = 0, 1, 2, 3\}$ appropriately.

In superdistributional general relativity the space-time is assumed to be a four-dimensional differentiable manifold $M\Sigma$, where $\dim(M) = 4$, $\dim(\Sigma) \le 3$ endowed with the Lorentzian supergeneralized metric

$$ (ds_{\epsilon}^2)_{\epsilon} = (g_{\mu\nu}(\epsilon)dx^{\mu}dx^{\nu})_{\epsilon}; \mu, \nu = 0, 1, 2, 3. \quad (2.27) $$

At each point $p \in M\Sigma$, the metric can be diagonalized as

$$ (ds_p^2(\epsilon))_{\epsilon} = (\eta_{\mu\nu}(dX_{\epsilon}^{\mu})_p(dX_{\epsilon}^{\nu})_p)_{\epsilon} \text{ with } \eta_{\mu\nu} \triangleq (-1, 1, 1, 1), \quad (2.28) $$

by choosing the generalized coordinate system $\{ (X_{\epsilon}^{\mu})_{\epsilon}; \mu = 0, 1, 2, 3 \}$ appropriately.

The classical smooth curvature tensor is given by

$$ R^{\rho}_{\sigma\mu\nu} \triangleq \partial_{\mu} \left\{ \frac{\rho}{\sigma\nu} \right\} - \partial_{\nu} \left\{ \frac{\rho}{\sigma\mu} \right\} + \left\{ \frac{\rho}{\lambda\mu} \right\} \left\{ \frac{\lambda}{\sigma\nu} \right\} - \left\{ \frac{\rho}{\lambda\nu} \right\} \left\{ \frac{\lambda}{\sigma\mu} \right\} \quad (2.29) $$
with $\{\frac{\rho}{\sigma v}\}$ being the smooth Christoffel symbol. The supergeneralized nonsmooth
curvature tensor is given by

$$
\left\{
\begin{aligned}
(R^\rho \sigma_{\mu\nu} \epsilon)_\epsilon &\triangleq \partial_\mu \left( \left\{ \frac{\rho}{\sigma v} \right\}_\epsilon \right)_\epsilon - \partial_v \left( \left\{ \frac{\rho}{\sigma \mu} \right\}_\epsilon \right)_\epsilon + \left( \left\{ \frac{\rho}{\lambda \mu} \right\}_\epsilon \right)_\epsilon \left( \left\{ \frac{\lambda}{\sigma v} \right\}_\epsilon \right)_\epsilon - \\
&\qquad - \left( \left\{ \frac{\rho}{\lambda v} \right\}_\epsilon \right)_\epsilon \left( \left\{ \frac{\lambda}{\sigma \mu} \right\}_\epsilon \right)_\epsilon
\end{aligned}
\right.
\qquad (2.30)
$$

with $\left(\left\{\frac{\rho}{\sigma v}\right\}_{\epsilon}\right)$ being the supergeneralized Christoffel symbol. The fundamental
classical action integral $\mathbb{I}$ is

$$
\mathbb{I} = \frac{1}{c} \int (\bar{\mathbb{L}}_G + \mathbb{L}_M) d^4 x, \qquad (2.31)
$$

where $\mathbb{L}_M$ is the Lagrangian density of a gravitational source and $\bar{\mathbb{L}}_G$ is the gravitational
Lagrangian density given by

$$
\bar{\mathbb{L}}_G = \frac{1}{2\kappa} \mathbb{G}. \qquad (2.32)
$$

Here $\kappa$ is the Einstein gravitational constant $\kappa = 8\pi G/c^4$ and $\mathbb{G}$ is defined by

$$
\mathbb{G} = \sqrt{-g} g^{\mu\nu} \left( \left\{ \frac{\lambda}{\mu\rho} \right\} \left\{ \frac{\rho}{\nu\lambda} \right\} - \left\{ \frac{\lambda}{\mu\nu} \right\} \left\{ \frac{\rho}{\lambda\rho} \right\} \right) \quad (2.33)
$$

with $g = \det(g_{\mu\nu})$. There exists the relation

$$
\sqrt{-g} R = \mathcal{G} + \partial_{\mu} \mathcal{D}^{\mu}, \qquad (2.34)
$$

with

$$
\mathbb{D}^{\mu} = -\sqrt{-g} \left( g^{\mu\nu} \left\{ \frac{\lambda}{\nu\lambda} \right\} - g^{\nu\lambda} \left\{ \frac{\mu}{\nu\lambda} \right\} \right). \quad (2.35)
$$

Thus the supergeneralized fundamental action integral $(\mathbb{I}_{\epsilon})_{\epsilon}$ is

$$
(\mathbb{I}_{\epsilon})_{\epsilon} = \frac{1}{c} \int ((\bar{\mathbb{L}}_{G}(\epsilon))_{\epsilon} + (\mathbb{L}_{M}(\epsilon))_{\epsilon}) d^{4}x, \qquad (2.36)
$$

where $(\mathbb{L}_M(\epsilon))_\epsilon$ is the supergeneralized Lagrangian density of a gravitational source and
$(\bar{\mathbb{L}}_G(\epsilon))_\epsilon$ is the supergeneralized gravitational Lagrangian density given by

$$
(\bar{\mathbb{L}}_G(\epsilon))_\epsilon = \frac{1}{2\kappa} (\mathbb{G}_\epsilon)_\epsilon . \qquad (2.37)
$$

Here $\kappa$ is the Einstein gravitational constant $\kappa = 8\pi G/c^4$ and $(\mathbb{G}_\epsilon)_\epsilon$ is defined by

$$
(\mathbb{G}_{\epsilon})_{\epsilon} = \sqrt{-(g_{\epsilon})_{\epsilon}} (g^{\mu\nu})_{\epsilon} \left( \left( \left\{ \frac{\lambda}{\mu\rho} \right\}_{\epsilon} \right) \left( \left\{ \frac{\rho}{\nu\lambda} \right\}_{\epsilon} \right) - \left( \left\{ \frac{\lambda}{\mu\nu} \right\}_{\epsilon} \right) \left( \left\{ \frac{\rho}{\lambda\rho} \right\}_{\epsilon} \right) \right) \quad (2.38)
$$

with $g_{\epsilon} = \det[(g_{\mu\nu}(\epsilon))_{\epsilon}]$. There exists the relation

$$
\sqrt{-(g_{\epsilon})_{\epsilon}} (\mathbf{R}_{\epsilon})_{\epsilon} = (\mathbb{G}_{\epsilon})_{\epsilon} + \partial_{\mu} (\mathbb{D}_{\epsilon}^{\mu})_{\epsilon}, \qquad (2.39)
$$

with

$$
(\mathbb{D}_{\epsilon}^{\mu})_{\epsilon} = - \sqrt{-(g_{\epsilon})_{\epsilon}} \left( (g_{\epsilon}^{\mu\nu})_{\epsilon} \left( \left\{ \frac{\lambda}{\nu\lambda} \right\}_{\epsilon} \right)_{\epsilon} - (g_{\epsilon}^{\nu\lambda})_{\epsilon} \left( \left\{ \frac{\mu}{\nu\lambda} \right\}_{\epsilon} \right)_{\epsilon} \right). \quad (2.40)
$$

Also, we have defined the classical scalar curvature by

$$
\mathbf{R} = R^{\mu}_{\mu} \qquad (2.41)
$$

with the smooth Ricci tensor

$$
R_{\mu\nu} = R^{\lambda}_{\mu\lambda\nu}. \qquad (2.42)
$$
From the action $\mathcal{I}$, the classical Einstein equation

$$G_{\mu}{}^{\nu} = R_{\mu}{}^{\nu} - \frac{1}{2} \delta_{\mu}{}^{\nu} R = \kappa T_{\mu}{}^{\nu}, \qquad (2.43)$$

follows, where $T_{\mu}{}^{\nu}$ is defined by

$$T_{\mu}{}^{\nu} = \frac{\tilde{\mathbf{T}}_{\mu}{}^{\nu}}{\sqrt{-g}} \qquad (2.44)$$

with

$$\tilde{\mathbf{T}}_{\mu}{}^{\nu} \triangleq 2g_{\mu\lambda} \frac{\delta \mathcal{L}_{M}}{\delta g_{\lambda\nu}} \qquad (2.45)$$

being the energy-momentum density of the classical gravity source. Thus we have defined the supergeneralized scalar curvature by

$$(\mathbf{R}_{\epsilon})_{\epsilon} = (R^{\mu}_{\mu}(\epsilon))_{\epsilon} \qquad (2.46)$$

with the supergeneralized Ricci tensor

$$(\mathbf{R}_{\mu\nu}(\epsilon))_{\epsilon} = (R^{\lambda}_{\mu\lambda\nu}(\epsilon))_{\epsilon}. \qquad (2.47)$$

From the action $(\mathcal{I}_{\epsilon})_{\epsilon}$, the generalized Einstein equation

$$(G_{\mu}{}^{\nu}(\epsilon))_{\epsilon} = (R_{\mu}{}^{\nu}(\epsilon))_{\epsilon} - \frac{1}{2} \delta_{\mu}{}^{\nu} (\mathbf{R}_{\epsilon})_{\epsilon} = \kappa (T_{\mu}{}^{\nu}(\epsilon))_{\epsilon}, \qquad (2.48)$$

follows, where $(T_{\mu}{}^{\nu}(\epsilon))_{\epsilon}$ is defined by

$$(T_{\mu}{}^{\nu}(\epsilon))_{\epsilon} = \frac{(\tilde{\mathbf{T}}_{\mu}{}^{\nu}(\epsilon))_{\epsilon}}{\sqrt{-(g_{\epsilon})_{\epsilon}}} \qquad (2.49)$$

with

$$(\tilde{\mathbf{T}}_{\mu}{}^{\nu}(\epsilon))_{\epsilon} \triangleq 2(g_{\mu\lambda}(\epsilon))_{\epsilon} \frac{\delta(\mathcal{L}_{M}(\epsilon))_{\epsilon}}{\delta(g_{\lambda\nu}(\epsilon))_{\epsilon}} \qquad (2.50)$$

being the supergeneralized energy-momentum density of the supergeneralized gravity source. The classical energy-momentum pseudo-tensor density $\tilde{\mathbf{t}}_{\mu}{}^{\nu}$ of the gravitational field is defined by

$$\tilde{\mathbf{t}}_{\mu}{}^{\nu} = \delta_{\mu}{}^{\nu} \tilde{\mathcal{L}}_{G} - \frac{\partial \tilde{\mathcal{L}}_{G}}{\partial g_{\sigma\tau,\nu}} g_{\sigma\tau,\mu} \qquad (2.51)$$

with $g_{\sigma\tau,\nu} = \partial g_{\sigma\tau}/\partial x^{\nu}$. The supergeneralized energy-momentum pseudo-tensor density $\tilde{\mathbf{t}}_{\mu}{}^{\nu}$ of the gravitational field is defined by

$$(\tilde{\mathbf{t}}_{\mu}{}^{\nu}(\epsilon))_{\epsilon} = \delta_{\mu}{}^{\nu} (\tilde{\mathcal{L}}_{G}(\epsilon))_{\epsilon} - \left( \frac{\partial \tilde{\mathcal{L}}_{G}(\epsilon)}{\partial g_{\sigma\tau,\nu}(\epsilon)} \right)_{\epsilon} (g_{\sigma\tau,\mu}(\epsilon))_{\epsilon} \qquad (2.52)$$

with $(g_{\sigma\tau,\nu}(\epsilon))_{\epsilon} = (\partial g_{\sigma\tau}(\epsilon)/\partial x^{\nu})_{\epsilon}$.

## 3. Distributional Schwarzschild Geometry from nonsmooth regularization via Horizon

### 3.1. Calculation of the stress-tensor by using nonsmooth regularization via Horizon

In this section we leave the neighborhood of the singularity at the origin and turn to the
singularity at the horizon. The question we are aiming at is the following: using distributional geometry (thus without leaving Schwarzschild coordinates), is it possible to show that the horizon singularity of the Schwarzschild metric is not merely only a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at horizon in Schwarzschild coordinates. In the usual Schwarzschild coordinates $(t, r, \theta, \phi)$, $r \neq 2m$ the Schwarzschild metric (1.12) takes the form above horizon $r > 2m$ and below horizon $r < 2m$ correspondingly

$$ \left\{ \begin{array}{ll} \text{above horizon } r > 2m: & \\ ds^{+2} = h^{+}(r)dt^{2} - [h^{+}(r)]^{-1}dr^{2} + r^{2}d\Omega^{2}, & \\ h^{+}(r) = -1 + \frac{2m}{r} = -\frac{r-2m}{r} & (3.1.1) \\ \text{below horizon } r < 2m: & \\ ds^{-2} = h^{-}(r)dt^{2} - [h^{-}(r)]^{-1}dr^{2} + r^{2}d\Omega^{2}, & \\ h^{-}(r) = -1 + \frac{2m}{r} = \frac{2m-r}{r} & \end{array} \right. $$

**Remark 3.1.1.** Following the above discussion we consider the metric coefficients $h^{+}(r)$, $[h^{+}(r)]^{-1} h^{-}(r)$, and $[h^{-}(r)]^{-1}$ as an element of $\mathcal{D}'(\mathbb{R}^3)$ and embed it into $(\mathcal{G}(\mathbb{R}^3))$ by replacements above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly

$$ \begin{aligned} r \ge 2m: & r - 2m \mapsto \sqrt{(r - 2m)^2 + \epsilon^2}, \\ r \le 2m: & 2m - r \mapsto \sqrt{(2m - r)^2 + \epsilon^2}. \end{aligned} $$

**Remark 3.1.2.** Note that, accordingly, we have fixed the differentiable structure of the manifold: the usual Schwarzschild coordinates and the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1.1) are extended on $r = 2m$ through the horizon. Therefore we have above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly
$$h(r) = \begin{cases} -\frac{r-2m}{r} & \text{if } r \ge 2m \\ 0 & \text{if } r \le 2m \end{cases} \mapsto (h_{\epsilon}^{+}(r))_{\epsilon} = \left( -\frac{\sqrt{(r-2m)^2 + \epsilon^2}}{r} \right)_{\epsilon},$$

where $(h_{\epsilon}^{+}(r))_{\epsilon} \in \mathcal{G}(\mathbb{R}^3, B^{+}(2m, R)), B^{+}(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}$.

$$h^{-1}(r) = \begin{cases} -\frac{r}{r-2m}, & r > 2m \\ \infty, & r = 2m \end{cases} \mapsto (h_{\epsilon}^{+})^{-1}(r) \qquad (3.1.2)$$
$$h^{-}(r) = \begin{cases} -\frac{r-2m}{r} & \text{if } r \le 2m \\ 0 & \text{if } r \ge 2m \end{cases} \mapsto h_{\epsilon}^{-}(r) = \frac{\sqrt{(2m-r)^2 + \epsilon^2}}{r}$$

where $h^{-}(r) \in \mathcal{G}(\mathbb{R}^3, B^{-}(0,2m)), B^{-}(0,2m) = \{x \in \mathbb{R}^3 | 0 < \|x\| \le 2m\}$

$$h^{-1}(r) = \begin{cases} -\frac{r}{r-2m}, & r < 2m \\ \infty, & r = 2m \end{cases} \mapsto (h_{\epsilon}^{-})^{-1}(r) = \\ = \left( \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \right)_{\epsilon} \in \mathcal{G}(\mathbb{R}^3, B^{-}(0,2m))$$

Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) horizon, i.e.,

$$\left\{ \begin{aligned} (ds_{\epsilon}^{+2})_{\epsilon} &= (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (ds_{\epsilon}^{-2})_{\epsilon} &= (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2} \end{aligned} \right. \qquad (3.1.3)$$

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

$$\left\{ \begin{aligned} ([\mathbf{R}_{\epsilon}^{+}]_{0}^{0})_{\epsilon} &= ([\mathbf{R}_{\epsilon}^{+}]_{1}^{1})_{\epsilon} = \frac{1}{2}((h_{\epsilon}^{+''})_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{+'})_{\epsilon}) \\ ([\mathbf{R}_{\epsilon}^{+}]_{2}^{2})_{\epsilon} &= ([\mathbf{R}_{\epsilon}^{+}]_{3}^{3})_{\epsilon} = \frac{(h_{\epsilon}^{+'})_{\epsilon}}{r} + \frac{1+(h_{\epsilon}^{+})_{\epsilon}}{r^2}. \end{aligned} \right. \qquad (3.1.4)$$

From (3.1.2) we obtain
$$h_{\epsilon}^{+\prime}(r) = -\frac{r-2m}{r\left[(r-2m)^2 + \epsilon^2\right]^{1/2}} + \frac{\left[(r-2m)^2 + \epsilon^2\right]^{1/2}}{r^2},$$

$$r(h_{\epsilon}^{+\prime})_{\epsilon} + 1 + (h_{\epsilon}^{+})_{\epsilon} =$$

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

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

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

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

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

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

$$r^2(h_{\epsilon}^{+\prime\prime})_{\epsilon} + 2r(h_{\epsilon}^{+\prime})_{\epsilon} = \qquad (3.1.5)$$

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

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

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

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

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

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

Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation
$$ \tilde{\Phi}(r) = \int_{0}^{\pi} \sin \theta d\theta \int_{0}^{2\pi} d\phi \Phi(x) $$

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:

(i) **supp**($\Phi(x)$) $\subset \{x \mid \|x\| \ge 2m\}$ (ii) $\tilde{\Phi}(r) \in C^{\infty}(\mathbb{R})$.

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

$$ \int_K ([\mathbf{R}_\epsilon^+]_2^2)_\epsilon \Phi(\vec{x}) d^3x = \int_K ([\mathbf{R}_\epsilon^+]_3^3)_\epsilon \Phi(\vec{x}) d^3x = \\ \int_{2m}^R (r(h_\epsilon^+) + 1 + (h_\epsilon^+)_\epsilon) \tilde{\Phi}(r) dr = \int_{2m}^R \left\{ -\frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} \right\} \tilde{\Phi}(r) dr + \int_{2m}^R \tilde{\Phi}(r) dr. \quad (3.1.6) $$

By replacement $r - 2m = u$, from (3.1.6) we obtain

$$ \int_K ([\mathbf{R}_\epsilon^+]_2^2)_\epsilon \Phi(x) d^3x = \int_K ([\mathbf{R}_\epsilon^+]_3^3)_\epsilon \Phi(x) d^3x = \\ - \int_0^{R-2m} \frac{u \tilde{\Phi}(u+2m) du}{(u^2+\epsilon^2)^{1/2}} + \int_0^{R-2m} \tilde{\Phi}(u+2m) du. \quad (3.1.7) $$

By replacement $u = \epsilon\eta$, from (3.1.7) we obtain the expression

$$ \left\{ \begin{aligned} \mathbf{I}_3^+(\epsilon) &= \int_K ([\mathbf{R}_\epsilon^+]_3^3)_\epsilon \Phi(x) d^3x = \mathbf{I}_2^+(\epsilon) = \int_K ([\mathbf{R}_\epsilon^+]_2^2)_\epsilon \Phi(\vec{x}) d^3x = \\ &\quad -\epsilon \times \left( \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta \tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} - \int_0^{\frac{R-2m}{\epsilon}} \tilde{\Phi}(\epsilon\eta + 2m) d\eta \right). \end{aligned} \right. \quad (3.1.8) $$

From Eq.(3.1.8) we obtain

$$ \begin{aligned} \mathbf{I}_3^+(\epsilon) = \mathbf{I}_2^+(\epsilon) = &-\epsilon \frac{\tilde{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] d\eta \\ &- \frac{\epsilon^2}{1!} \int_0^{\frac{R-2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta \\ &- \epsilon \tilde{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1} - \left(\frac{R-2m}{\epsilon} - 1\right) \right] \\ &- \frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta, \end{aligned} \quad (3.1.9) $$

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

$$ \left\{ \begin{array}{l} \tilde{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\Phi^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{(n)}(\xi), \quad (3.1.10) \\ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 \end{array} \right. $$

with $\tilde{\Phi}^{(l)}(\xi) \triangleq d^l \tilde{\Phi}/d\xi^l$. Equations (3.1.9)-(3.1.10) gives
$$
\begin{equation}
\begin{aligned}
\lim_{\epsilon \to 0} \mathbf{I}_3^+(\epsilon) &= \lim_{\epsilon \to 0} \mathbf{I}_2^+(\epsilon) = \\
&\quad \lim_{\epsilon \to 0} \left\{ -\epsilon \tilde{\Phi}(2m) \left[ \sqrt{\left( \frac{R-2m}{\epsilon} \right)^2 + 1} + 1 - \frac{R-2m}{\epsilon} \right] \right\} + \\
&\quad + \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1} \int_0^{\frac{R-2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta \, d\eta \right\} = 0.
\end{aligned}
\tag{3.1.11}
\end{equation}
$$

Thus in $S'_{2m}(B_R^+(2m)) \subset S'_{2m}(\mathbb{R}^3) \subset \mathcal{D}'(\mathbb{R}^3)$, where $B^+(2m,R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}$
from Eq.(3.1.11) we obtain

$$
\begin{equation}
\begin{aligned}
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{3}^{3} &= \lim_{\epsilon \to 0} \mathbf{I}_{3}^{+}(\epsilon) = 0, \\
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{2}^{2} &= \lim_{\epsilon \to 0} \mathbf{I}_{2}^{+}(\epsilon) = 0.
\end{aligned}
\tag{3.1.12}
\end{equation}
$$

For $([\mathbf{R}_{\epsilon}^{+}]_{1}^{1})_{\epsilon}$, $([\mathbf{R}_{\epsilon}^{+}]_{0}^{0})_{\epsilon}$ we get:

$$
\begin{equation}
\begin{aligned}
& 2 \int_K ([\mathbf{R}_\epsilon^+]_1^1)_\epsilon \Phi(x) d^3x = 2 \int_K ([\mathbf{R}_\epsilon^+]_0^0)_\epsilon \Phi(x) d^3x = \\
& \qquad \int_{2m}^R (r^2(h_\epsilon^{++})_\epsilon + 2r(h_\epsilon^{+\prime})_\epsilon) \tilde{\Phi}(r) dr = \\
& = \int_{2m}^R \left\{ -\frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \tilde{\Phi}(r) dr.
\end{aligned}
\tag{3.1.13}
\end{equation}
$$

By replacement $r - 2m = u$, from (3.1.13) we obtain

$$
\left\{
\begin{aligned}
\mathbf{I}_1^+(\epsilon) &= 2 \int_K ([\mathbf{R}_\epsilon^+]_1^1)_\epsilon \Phi(x) d^3x = \mathbf{I}_2^+(\epsilon) = 2 \int_K ([\mathbf{R}_\epsilon^+]_0^0)_\epsilon \Phi(x) d^3x \\
&= \int_{2m}^R (r^2(h_\epsilon^{++})_\epsilon + 2r(h_\epsilon^{+\prime})_\epsilon) \tilde{\Phi}(r) dr = \\
&= \int_0^{R-2m} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{1/2}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \tilde{\Phi}(u+2m) du.
\end{aligned}
\right.
\tag{3.1.14}
$$

By replacement $u = \epsilon\eta$, from (3.1.14) we obtain
$$
\begin{align*}
& 2 \int_K (\mathbf{R}_\epsilon^+)^1_1 \epsilon \Phi(x) d^3x = 2 \int_K (\mathbf{R}_\epsilon^+)^0_0 \epsilon \Phi(x) d^3x = \\
&= \int_{2m}^{R} (r^2(h_\epsilon^{++}) + 2r(h_\epsilon^{+\prime})_\epsilon) \tilde{\Phi}(r) dr = \\
&= \epsilon \int_0^{\frac{R-2m}{\epsilon}} \left\{ -\frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \tilde{\Phi}(\epsilon\eta + 2m) d\eta = \\
&\quad - \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^2\eta\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \\
&\quad \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^4\eta^3\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} + 2m \int_0^{\frac{R-2m}{\epsilon}} \frac{\epsilon^3\eta^2\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = \\
&\quad \epsilon \left[ - \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta^3\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{3/2}} \right] + \\
&\quad 2m \left[ - \int_0^{\frac{R-2m}{\epsilon}} \frac{\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \int_0^{\frac{R-2m}{\epsilon}} \frac{\eta^2\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{3/2}} \right]. \tag{3.1.15}
\end{align*}
$$

From Eq.(3.1.15) we obtain

$$
\begin{equation}
\begin{aligned}
I_0^+(\epsilon) = I_1^+(\epsilon) = {}& 2m \frac{\tilde{\Phi}(2m)}{0!} \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \\
& + \frac{\epsilon}{1!} \int_0^{\frac{R-2m}{\epsilon}} \tilde{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta + \\
& + \frac{\epsilon \tilde{\Phi}(2m)}{0!} \int_{\frac{-2m}{\epsilon}}^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \\
& + \frac{\epsilon^2}{1!} \int_0^{\frac{R-2m}{\epsilon}} \tilde{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta,
\end{aligned}
\tag{3.1.16}
\end{equation}
$$

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

$$
\left\{
\begin{array}{l}
\tilde{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\Phi^{\alpha\beta(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{\alpha\beta(n)}(\xi), \quad (3.1.17) \\
\xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1
\end{array}
\right.
$$

with $\tilde{\Phi}^{(l)}(\xi) \triangleq d^l \tilde{\Phi}/d\xi^l$. Equation (3.1.17) gives
$$
\begin{align}
w - \lim_{\epsilon \to 0} \mathbf{I}_0^\dagger(\epsilon) &= w - \lim_{\epsilon \to 0} \mathbf{I}_1^\dagger(\epsilon) = \nonumber \\
&\quad 2m \tilde{\Phi}(2m) \lim_{\epsilon \to 0} \left\{ \int_0^{\frac{R-2m}{\epsilon}} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta \right\} = \tag{3.1.18} \\
&\quad 2m \tilde{\Phi}(2m) \lim_{s \to \infty} \left[ \int_0^s \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - \int_0^s \frac{d\eta}{(\eta^2+1)^{1/2}} \right] = \nonumber \\
&\quad = -2m \tilde{\Phi}(2m). \nonumber
\end{align}
$$

where use is made of the relation

$$
\lim_{s \to \infty} \left[ \int_0^s \frac{\eta^2 d\eta}{(\eta^2 + 1)^{3/2}} - \int_0^s \frac{d\eta}{(u^2 + 1)^{1/2}} \right] = -1 \quad (3.1.19)
$$

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

$$
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{1}^{0} = w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} = -m\tilde{\Phi}(2m). \quad (3.1.20)
$$

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

$$
\left\{
\begin{array}{l}
([\mathbf{R}_{\epsilon}^{-}]_{0}^{0})_{\epsilon} = ([\mathbf{R}_{\epsilon}^{-}]_{1}^{1})_{\epsilon} = \frac{1}{2}((h_{\epsilon}^{-\prime\prime})_{\epsilon} + \frac{2}{r}(h_{\epsilon}^{-\prime})_{\epsilon}), \\
\\
([\mathbf{R}_{\epsilon}^{-}]_{2}^{2})_{\epsilon} = ([\mathbf{R}_{\epsilon}^{-}]_{3}^{3})_{\epsilon} = \frac{(h_{\epsilon}^{-\prime})_{\epsilon}}{r} + \frac{1+(h_{\epsilon}^{-})_{\epsilon}}{r^2}.
\end{array}
\right.
\qquad (3.1.21)
$$

From Eq.(3.1.21) we obtain

$$
\begin{align*}
h_{\epsilon}^{-}(r) &= -\frac{r-2m}{r} \mapsto h_{\epsilon}^{-}(r) = \left( \frac{\sqrt{(2m-r)^2 + \epsilon^2}}{r} \right) = -h_{\epsilon}^{+}(r), r < 2m. \\
h_{\epsilon}^{-\prime}(r) &= -h_{\epsilon}^{+\prime}(r) = \frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2}, \\
r(h_{\epsilon}^{-})_{\epsilon} + 1 + (h_{\epsilon}^{-})_{\epsilon} &= -r(h_{\epsilon}^{+\prime})_{\epsilon} + 1 - (h_{\epsilon}^{+})_{\epsilon} = \\
&\quad \frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} + 1. \tag{3.1.22} \\
h_{\epsilon}^{-\prime\prime}(r) &= -h_{\epsilon}^{+\prime\prime}(r) = \\
&\quad -\frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r^3}. \\
r^2(h_{\epsilon}^{-\prime\prime})_{\epsilon} + 2r(h_{\epsilon}^{-\prime})_{\epsilon} &= -r^2(h_{\epsilon}^{+\prime\prime})_{\epsilon} - 2r(h_{\epsilon}^{+\prime})_{\epsilon} = \\
&\quad \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}}.
\end{align*}
$$

Investigating the weak limit of the angular components of the Ricci tensor (using the
abbreviation $\tilde{\Phi}(r) = \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\phi \Phi(x)$ where $\Phi(\vec{x}) \in C^\infty(\mathbb{R}^3)$, $\Phi(x)$ is a function with
compact support K such that $K \subseteq B^-(0,2m) = \{x \in \mathbb{R}^3 | 0 \le \|x\| \le 2m\}$ we get:

$$ \int_K (\mathbf{[R}_\epsilon^-]_2^2)_\epsilon \Phi(\vec{x}) d^3x = \int_K (\mathbf{[R}_\epsilon^-]_3^3)_\epsilon \Phi(\vec{x}) d^3x = \\ \int_0^{2m} (r(h_\epsilon^{'-})_\epsilon + 1 + (h_\epsilon^-)_\epsilon) \tilde{\Phi}(r) dr = \int_0^{2m} \left\{ \frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} \right\} \tilde{\Phi}(r) dr + \int_0^{2m} \tilde{\Phi}(r) dr. \quad (3.1.23) $$

By replacement $r - 2m = u$, from Eq.(3.1.23) we obtain

$$ \int_K (\mathbf{[R}_\epsilon^-]_2^2)_\epsilon \Phi(x) d^3x = \int_K (\mathbf{[R}_\epsilon^-]_3^3)_\epsilon \Phi(x) d^3x = \\ \int_{-2m}^{0} \frac{u\tilde{\Phi}(u+2m)du}{(u^2+\epsilon^2)^{1/2}} + \int_{-2m}^{0} \tilde{\Phi}(u+2m)du. \quad (3.1.24) $$

By replacement $u = \epsilon\eta$, from (3.1.23) we obtain

$$ \mathbf{I}_3^-(\epsilon) = \int_K (\mathbf{[R}_\epsilon^-]_3^3)_\epsilon \Phi(x) d^3x = \mathbf{I}_2^-(\epsilon) = \int_K (\mathbf{[R}_\epsilon^-]_2^2)_\epsilon \Phi(\vec{x}) d^3x = \\ \epsilon \times \left( \int_{-\frac{2m}{\epsilon}}^{0} \frac{\eta \tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \int_{-\frac{2m}{\epsilon}}^{0} \tilde{\Phi}(\epsilon\eta + 2m)d\eta \right), \quad (3.1.25) $$

which is calculated to give

$$ \mathbf{I}_3^-(\epsilon) = \mathbf{I}_2^-(\epsilon) = \epsilon \frac{\tilde{\Phi}(2m)}{0!} \int_{-\frac{2m}{\epsilon}}^{0} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] d\eta + \\ + \frac{\epsilon^2}{1!} \int_{-\frac{2m}{\epsilon}}^{0} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta = \quad (3.1.26) $$

$$ \epsilon \tilde{\Phi}(2m) \left[ 1 - \sqrt{\left(\frac{2m}{\epsilon}\right)^2 + 1} + \frac{2m}{\epsilon} \right] + \epsilon \frac{\epsilon^2}{1} \int_{-\frac{2m}{\epsilon}}^{0} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta, $$

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

$$ \tilde{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\Phi^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{(n)}(\xi), \quad (3.1.27) \\ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n = 1 $$

with $\tilde{\Phi}^{(l)} \triangleq d^l \tilde{\Phi}/dr^l$. Equation (3.1.27) gives

$$ \lim_{\epsilon \to 0} \mathbf{I}_3^-(\epsilon) = \lim_{\epsilon \to 0} \mathbf{I}_2^-(\epsilon) = \\ \lim_{\epsilon \to 0} \left\{ \epsilon \tilde{\Phi}(2m) \left[ 1 - \sqrt{\left(\frac{2m}{\epsilon}\right)^2 + 1} + \frac{2m}{\epsilon} \right] \right\} + \\ + \lim_{\epsilon \to 0} \left\{ \epsilon^2 \frac{\epsilon^2}{2} \int_{-\frac{2m}{\epsilon}}^{0} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta \right\} = 0. \quad (3.1.28) $$
Thus in $S'_{2m}(B_R^-(2m)) \subset S'_{2m}(\mathbb{R}^3)$, where $B^-(0,2m) = \{x \in \mathbb{R}^3 | 0 \le \|x\| \le 2m\}$ from Eq.(3.1.28) we obtain

$$ 
\begin{aligned}
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{3}^{3} &= \lim_{\epsilon \to 0} \mathbf{I}_{3}^{-}(\epsilon) = 0. \\
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{2}^{2} &= \lim_{\epsilon \to 0} \mathbf{I}_{2}^{-}(\epsilon) = 0.
\end{aligned} 
\qquad (3.1.29) $$

For $([\mathbf{R}_{\epsilon}^{-}]_{1}^{1})_{\epsilon}$, $([\mathbf{R}_{\epsilon}^{-}]_{0}^{0})_{\epsilon}$ we get:

$$ 
\begin{aligned}
& 2 \int_K ([\mathbf{R}_{\epsilon}^{-}]_{1}^{1})_{\epsilon} \Phi(x) d^3x = 2 \int_K ([\mathbf{R}_{\epsilon}^{-}]_{0}^{0})_{\epsilon} \Phi(x) d^3x = \\
& \qquad \int_0^{2m} (r^2(h_{\epsilon}^{\prime\prime})_{\epsilon} + 2r(h_{\epsilon}^{\prime})_{\epsilon}) \tilde{\Phi}(r) dr = \\
& = \int_0^{2m} \left\{ \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \tilde{\Phi}(r) dr.
\end{aligned} 
\qquad (3.1.30) $$

By replacement $r - 2m = u$, from (3.1.30) we obtain

$$ 
\begin{aligned}
I_1^+(\epsilon) &= 2 \int ([\mathbf{R}_{\epsilon}^{-}]_1^1)_{\epsilon} \Phi(x) d^3x = I_2^+(\epsilon) = 2 \int ([\mathbf{R}_{\epsilon}^{-}]_0^0)_{\epsilon} \Phi(x) d^3x \\
&= \int_0^{2m} (r^2(h_{\epsilon}^{\prime\prime})_{\epsilon} + 2r(h_{\epsilon}^{\prime})_{\epsilon}) \tilde{\Phi}(r) dr = \\
&= \int_{-2m}^{0} \left\{ \frac{u+2m}{(u^2+\epsilon^2)^{1/2}} - \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \tilde{\Phi}(u+2m) du.
\end{aligned} 
\qquad (3.1.31) $$

By replacement $u = \epsilon\eta$, from (3.1.31) we obtain
$$
\begin{align*}
& 2 \int_K (\mathbf{[R}_\epsilon^-])_1^1 \epsilon \Phi(x) d^3x = 2 \int_K (\mathbf{[R}_\epsilon^-})_0^0 \epsilon \Phi(x) d^3x = \\
& \qquad \int_{-\frac{2m}{\epsilon}}^0 (r^2(h_\epsilon^{\prime\prime})_\epsilon + 2r(h_\epsilon^{\prime})_\epsilon) \tilde{\Phi}(r) dr = \\
&= \epsilon \int_{-\frac{2m}{\epsilon}}^0 \left\{ \frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \tilde{\Phi}(\epsilon\eta + 2m) d\eta = \\
& \qquad - \epsilon \int_{-\frac{2m}{\epsilon}}^0 \frac{\epsilon^4\eta^3\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} - 2m \int_{-\frac{2m}{\epsilon}}^0 \frac{\epsilon^3\eta^2\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = \\
& \qquad \epsilon \int_{-\frac{2m}{\epsilon}}^0 \frac{\eta\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} - \epsilon \int_{-\frac{2m}{\epsilon}}^0 \frac{\eta^3\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{3/2}} + \\
& \qquad + 2m \left[ \int_{-\frac{2m}{\epsilon}}^0 \frac{\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} - \int_{-\frac{2m}{\epsilon}}^0 \frac{\eta^2\tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{3/2}} \right].
\end{align*}
$$

which is calculated to give

$$
\begin{equation}
\begin{split}
\mathbf{I}_0^-(\epsilon) = \mathbf{I}_1^-(\epsilon) = {}& 2m \frac{\tilde{\Phi}(2m)}{0!} \epsilon^l \int_{-\frac{2m}{\epsilon}}^0 \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \\
& + \frac{\epsilon}{1!} \int_0^{\frac{2m}{\epsilon}} \tilde{\Phi}^{(1)}(\xi) \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta + O(\epsilon^2).
\end{split}
\tag{3.33}
\end{equation}
$$

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

$$
\begin{align}
\tilde{\Phi}(\epsilon\eta + 2m) &= \sum_{l=0}^{n-1} \frac{\Phi^{\alpha\beta(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{\alpha\beta(n)}(\xi), \tag{3.34} \\
\xi &\triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 \notag
\end{align}
$$

with $\tilde{\Phi}^{(l)}(\xi) \triangleq d^l \tilde{\Phi}/d\xi^l$. Equation (3.34) gives

$$
\begin{align}
\lim_{\epsilon \to 0} \mathbf{I}_0^-(\epsilon) &= \lim_{\epsilon \to 0} \mathbf{I}_1^-(\epsilon) = \nonumber \\
&\quad 2m \lim_{\epsilon \to 0} \left\{ \frac{\tilde{\Phi}(2m)}{0!} \int_{-\frac{2m}{\epsilon}}^{0} \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta \right\} = \tag{3.35} \\
&\quad 2m \tilde{\Phi}(2m) \lim_{s \to 0} \left[ \int_{-s}^{0} \frac{d\eta}{(\eta^2+1)^{1/2}} - \int_{-s}^{0} \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} \right] = \nonumber \\
&= 2m \tilde{\Phi}(2m). \nonumber
\end{align}
$$

where use is made of the relation
$$ \lim_{s \to \infty} \left[ \int_{-s}^{0} \frac{d\eta}{(u^2 + 1)^{1/2}} - \int_{-s}^{0} \frac{\eta^2 d\eta}{(\eta^2 + 1)^{3/2}} \right] = 1. \qquad (3.36) $$

Thus in $S'(B^-(0,2m)) \subset S'(\mathbb{R}^3)$ we obtain

$$ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} = w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} = m\tilde{\Phi}(2m). \qquad (3.37) $$

Using Egs. (3.12),(3.20),(3.29),(3.37) we obtain

$$ \int \left[ \left( \mathbf{T}_r^{+r} + \mathbf{T}_\theta^{+\theta} + \mathbf{T}_\phi^{+\phi} + \mathbf{T}_t^{+t} \right) + \left( \mathbf{T}_r^{-r} + \mathbf{T}_\theta^{-\theta} + \mathbf{T}_\phi^{-\phi} + \mathbf{T}_t^{-t} \right) \right] \sqrt{-g} d^3x = 0 \qquad (3.38) $$

Thus the Tolman formula [3],[4] for the total energy of a static and asymptotically flat spacetime with $g$ the determinant of the four dimensional metric and $d^3x$ the coordinate

volume element, gives

$$ E_T = \int (\mathbf{T}_r^r + \mathbf{T}_\theta^\theta + \mathbf{T}_\phi^\phi + \mathbf{T}_t^t) \sqrt{-g} d^3x = m, \qquad (3.39) $$

We rewrite now the Schwarzschild metric (3.3) in the form

$$ \left\{ \begin{array}{l} (ds_{\epsilon}^{\pm 2})_{\epsilon} = (h_{\epsilon}^{\pm}(r)dt^{2})_{\epsilon} - ((1 + C_{\epsilon}^{\pm}(r))dr^{2})_{\epsilon} + r^{2}d\Omega^{2} \\ C_{\epsilon}^{\pm}(r) = -1 + [h_{\epsilon}^{\pm}(r)]^{-1}. \end{array} \right. \qquad (3.40) $$

Using Eq.(A.5) from Eq.(3.40) one obtains for $r \approx 2m$

$$ \begin{aligned} (\mathbf{R}^{\pm\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}^{\pm}(\epsilon))_{\epsilon} = & \\ & \left( \left( \frac{1}{2}(h_{\epsilon}^{\pm})'' + \frac{1}{r}(h_{\epsilon}^{\pm})' \right)^2 \right)_{\epsilon} + 2 \left( \left[ -\frac{(h_{\epsilon}^{\pm})'}{r} + \frac{1}{r^2} \right]^2 \right)_{\epsilon} \times \\ & \left( \frac{\epsilon^4}{4m^4[\epsilon^2 + (r_{\epsilon} - 2m)^2]^3} \right)_{\epsilon}, \end{aligned} \qquad (3.41) $$

and

$$ [(\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_{\epsilon}] \propto K(r_s) + \left( \frac{\epsilon^4}{4m^4[\epsilon^2 + (r_{\epsilon} - 2m)^2]^3} \right)_{\epsilon}. \qquad (3.42) $$

## 3.2. Examples of distributional geometries. Calculation of the distributional quadratic scalars by using nonsmooth regularization via Horizon

Let us consider again the Schwarzschild metric (3.1)

$$ \left\{ \begin{array}{l} ds^2 = h(r)dt^2 - h(r)^{-1}dr^2 + r^2 d\Omega^2, \\ h(r) = -1 + \frac{2m}{r} = -\frac{r-2m}{r}, \\ h^{-1}(r) = -\frac{r}{r-2m}. \end{array} \right. \qquad (3.43) $$
We rewrite now the Schwarzschild metric (3.43) above Horizon ($r \ge 2m$) in the form

$$ egin{cases} ds^{+2} = -A^{+}(r)dt^{2} + (A^{+}(r))^{-1}(r)dr^{2} + r^{2}d\Omega^{2}, \\ A^{+}(r) = \frac{r-2m}{r}, \\ (A^{+}(r))^{-1} = \frac{r}{r-2m}. \end{cases} \quad (3.44) $$

Following the above discussion we consider the singular metric coefficient $A^{-1}(r)$ as an element of $\mathcal{D}'(\mathbb{R}^3)$ and embed it into $(\mathcal{G}(\mathbb{R}^3))$ by replacement

$$ r-2m \mapsto \sqrt{r^2 + \epsilon^2} - 2m. \quad (3.45) $$

Thus above Horizon ($r \ge 2m$) the corresponding distributional metric $(\tilde{ds}_{\epsilon}^{+2})_{\epsilon}$ takes the form

$$ egin{cases} (\tilde{ds}_{\epsilon}^{+2})_{\epsilon} = (-A_{\epsilon}^{+}(r)dt^{2} + (A_{\epsilon}^{+}(r))^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (A_{\epsilon}^{+}(r))_{\epsilon} = \left(\frac{\sqrt{r^{2} + \epsilon^{2}} - 2m}{r}\right)_{\epsilon}, \\ ((A_{\epsilon}^{+}(r))^{-1})_{\epsilon} = \left(\frac{r}{\sqrt{r^{2} + \epsilon^{2}} - 2m}\right)_{\epsilon}. \end{cases} \quad (3.46) $$

We rewrite now the Schwarzschild metric (3.43) below Horizon ($r < 2m$) in the form

$$ egin{cases} ds^{-2} = A^{-}(r)dt^{2} - (A^{-}(r))^{-1}dr^{2} + r^{2}d\Omega^{2}, \\ A^{-}(r) = \frac{2m-r}{r}, (A^{-}(r))^{-1} = \frac{r}{2m-r}. \end{cases} \quad (3.47) $$

Following the above discussion we consider the singular metric coefficient $A^{-1}(r)$ as an element of $\mathcal{D}'(\mathbb{R}^3)$ and embed it into $(\mathcal{G}(\mathbb{R}^3))$ by replacement

$$ 2m - r \mapsto 2m - \sqrt{r^2 + \epsilon^2}. \quad (3.48) $$

Thus below Horizon ($r < 2m$) the corresponding distributional metric $(\tilde{ds}_{\epsilon}^{-2})_{\epsilon}$ takes the form

$$ egin{cases} (d\tilde{s}_{\epsilon}^{-2})_{\epsilon} = (A_{\epsilon}^{-}(r)dt^{2} - (A_{\epsilon}^{-}(r))^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (A_{\epsilon}^{-}(r))_{\epsilon} = \left(\frac{2m - \sqrt{r^{2} + \epsilon^{2}}}{r}\right)_{\epsilon}, ((A_{\epsilon}^{-}(r))^{-1})_{\epsilon} = \left(\frac{r}{2m - \sqrt{r^{2} + \epsilon^{2}}}\right)_{\epsilon}. \end{cases} \quad (3.49) $$

From Eq.(3.46) one obtains

$$ egin{cases} (A_{\epsilon}^{+})' = \left(-\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r}\right)' = -\frac{1}{\sqrt{r^2 + \epsilon^2}} + \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^2}, \\ (A_{\epsilon}^{+})'' = \frac{r}{(r^2 + \epsilon^2)^{3/2}} - 2\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^3} + \frac{1}{r\sqrt{r^2 + \epsilon^2}} \end{cases} \quad (3.50) $$

From Eq.(3.46) using Eq.(A.5) one obtains
$$
\begin{equation}
\begin{split}
(\mathbf{R}(\epsilon))_{\epsilon} = \left(-\frac{4A'_{\epsilon}}{r} + \frac{2A_{\epsilon}C_{\epsilon}}{r^2} - A''_{\epsilon}\right)_{\epsilon} = & \\
& -\frac{4}{r}\left(-\frac{1}{\sqrt{r^2 + \epsilon^2}} + \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^2}\right)_{\epsilon} + \frac{2}{r^2} - \\
& -\left(\frac{r}{(r^2 + \epsilon^2)^{3/2}} - 2\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^3} + \frac{1}{r\sqrt{r^2 + \epsilon^2}}\right)_{\epsilon}.
\end{split}
\tag{3.51}
\end{equation}
$$

From Eq.(3.51) for $r = 2m$ one obtains

$$ (\mathbf{R}(\epsilon))_{\epsilon} = c_1 \delta(r - 2m), \quad (3.52) $$

see subsect. 2.5,Remark 2.5.4.

**Remark 3.3.** Note that curvature scalar $(\mathbf{R}(\epsilon))_{\epsilon}$ again nonzero but nonsingular.

Let us introduce now the general metric which has the form [11]:

$$ \left\{
\begin{array}{l}
ds^2 = -A(r)(dx^0)^2 - 2D(r)dx^0dr + (B(r) + C(r))(dr)^2 \\
\qquad + B(r)r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2],
\end{array}
\right.
\qquad (3.53) $$

where

$$
\left\{
\begin{array}{l}
A(r) = \Omega^2 \left(1 - \frac{a}{K(r)}\right), \quad B(r) = \frac{K^2(r)}{\rho^2(r)}, \\
C(r) = \left(1 - \frac{a}{K(r)}\right)^{-1} (K'(r))^2 - \frac{K^2(r)}{\rho^2(r)} - \left(1 - \frac{a}{K(r)}\right) (f')^2, \\
D(r) = \Omega \left(1 - \frac{a}{K(r)}\right) f', \quad K'(r) \triangleq dK(r)/dr, f'(r) \triangleq df(r)/dr, \\
K(r) = \rho(r) - |a|, \\
\phantom{K(r) =} a < 0.
\end{array}
\right.
\qquad (3.54)
$$

Note that the coordinates $t = x^0/c$ and $r$ are time and space coordinates, respectively, only if

$$ 1 - \frac{a}{K} > 0, \quad \left(1 - \frac{a}{K}\right)^{-1} (K')^2 - \left(1 - \frac{a}{K}\right) (f')^2 > 0. \qquad (3.55) $$

In the Cartesian coordinate system {$x^\mu, \mu = 0,1,2,3$} with

$$ x^1 = r \cos \phi \sin \theta, x^2 = r \sin \phi \sin \theta, x^3 = r \cos \theta, \qquad (3.56) $$

the metric (3.53)-(3.55) takes the form

$$ ds^2 = g_{\mu\nu}dx^\mu dx^\nu \qquad (3.57) $$

with $g_{\mu\nu}$ given by

$$ g_{00} = -A, \quad g_{0a} = -D \frac{x^a}{r}, \quad g_{a\beta} = B \delta^{a\beta} + C \frac{x^\alpha x^\beta}{r^2}. \qquad (3.58) $$

From Eq.(3.54) one obtain
$$
\begin{align}
A(r) &= \Omega^2 \left( \frac{\rho(r)}{\rho(r) - |a|} \right), \quad B = \frac{(\rho(r) - |a|)^2}{\rho^2(r)}, \nonumber \\
C(r) &= \left( \frac{\rho(r) - |a|}{\rho(r)} \right) - \frac{(\rho(r) - |a|)^2}{\rho^2(r)} - \left( \frac{\rho(r)}{\rho(r) - |a|} \right) (f'(r))^2, \tag{3.59} \\
D(r) &= \Omega \left( \frac{\rho(r)}{\rho(r) - |a|} \right) f'(r), f'(r) \triangleq \frac{df(r)}{dr}. \nonumber
\end{align}
$$

Regularizing the function $(\rho(r) - |a|)^{-1}$ above horizon (under condition $\rho(r) - |a| \ge 0$) such as

$$
\rho(r) - |a| \ge 0: \qquad (\rho(r) - |a|)^{-1} \mapsto (\rho_{\epsilon}(r) - |a|)^{-1} = (\sqrt{\rho^2(r) + \epsilon^2} - |a|)^{-1} \quad (3.60)
$$

with $\epsilon \in (0,1]$ from Eq.(3.59)-Eq.(3.60) one obtains

$$
\begin{align}
A_{\epsilon}^{+}(r) &= \Omega^2 \left( \frac{\rho_{\epsilon}(r)}{\rho_{\epsilon}(r) - |a|} \right), \quad B_{\epsilon}^{+}(r) = \frac{(\rho_{\epsilon}(r) - |a|)^2}{\rho_{\epsilon}^2(r)}, \nonumber \\
C_{\epsilon}^{+}(r) &= \left( \frac{\rho_{\epsilon}(r) - |a|}{\rho_{\epsilon}(r)} \right) - \frac{(\rho_{\epsilon}(r) - |a|)^2}{\rho_{\epsilon}^2(r)} - \left( \frac{\rho_{\epsilon}(r)}{\rho_{\epsilon}(r) - |a|} \right) (f'(r))^2, \tag{3.61} \\
D_{\epsilon}^{+}(r) &= \Omega \left( \frac{\rho_{\epsilon}(r)}{\rho_{\epsilon}(r) - |a|} \right) f'(r), f'(r) \triangleq \frac{df(r)}{dr}. \nonumber
\end{align}
$$

Regularizing the function $(|a| - \rho(r))^{-1}$ below horizon (under condition $|a| - \rho(r) \ge 0$) such as

$$
|a| - \rho(r) \ge 0: \qquad (|a| - \rho(r))^{-1} \mapsto (|a| - \rho_{\epsilon}(r)) = (|a| - \sqrt{r^2 + \epsilon^2})^{-1} \quad (3.62)
$$

with $\epsilon \in (0,1]$ from Eq.(3.59),Eq.(3.62) one obtains

$$
\begin{align}
A_{\epsilon}^{-}(r) &= -\Omega^2 \left( \frac{\rho_{\epsilon}(r)}{|a| - \rho_{\epsilon}(r)} \right), \quad B_{\epsilon}^{-}(r) = \frac{(|a| - \rho_{\epsilon}(r))^2}{\rho_{\epsilon}^2(r)}, \nonumber \\
C_{\epsilon}^{-}(r) &= -\left( \frac{|a| - \rho_{\epsilon}(r)}{\rho_{\epsilon}(r)} \right) - \frac{(|a| - \rho_{\epsilon}(r))^2}{\rho_{\epsilon}^2(r)} + \left( \frac{\rho_{\epsilon}(r)}{|a| - \rho_{\epsilon}(r)} \right) (f'(r))^2, \tag{3.63} \\
D_{\epsilon}^{-}(r) &= -\Omega \left( \frac{\rho_{\epsilon}(r)}{|a| - \rho_{\epsilon}(r)} \right) f'(r), f'(r) \triangleq \frac{df(r)}{dr}. \nonumber
\end{align}
$$

**Remark 3.4.** Finally the metric (3.57) becomes the Colombeau object of the form

$$
(ds_{\epsilon}^{2})_{\epsilon} = (g_{\mu\nu}^{\pm}(\epsilon)dx^{\mu}dx^{\nu})_{\epsilon} \qquad (3.64)
$$

with $g_{\mu\nu}(\epsilon)$ given by

$$
\begin{align}
g_{00}^{\pm}(\epsilon) &= -A_{\epsilon}^{\pm}(r), \quad g_{0\alpha}^{\pm}(\epsilon) = -D_{\epsilon}^{\pm}(r) \frac{x^{\alpha}}{r}, \nonumber \\
g_{\alpha\beta}^{\pm}(\epsilon) &= B_{\epsilon}^{\pm}(r) \delta^{\alpha\beta} + C_{\epsilon}^{\pm}(r) \frac{x^{\alpha} x^{\beta}}{r^2}. \tag{3.65}
\end{align}
$$

Using now Eq. A2 one obtains that the Colombeau curvature scalars $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon}$ in terms of
Colombeau generalized functions $(A_{\epsilon}^{\pm}(r))_{\epsilon}, (B_{\epsilon}^{\pm}(r))_{\epsilon}, (C_{\epsilon}^{\pm}(r))_{\epsilon}, (D_{\epsilon}^{\pm}(r))_{\epsilon}$ is expressed as

$$ (\mathbf{R}^{+}(\epsilon))_{\epsilon} = \left( \frac{r^2 + \epsilon^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^2} \left[ \frac{9a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} - \frac{2a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] \right)_{\epsilon} \quad (3.66) $$
$$ (\mathbf{R}^{-}(\epsilon))_{\epsilon} = - \left( \frac{r^2 + \epsilon^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^2} \left[ \frac{9a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} - \frac{2a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] \right)_{\epsilon} $$

**Remark 3.6.** Note that (i) on horizon $r = a$ Colombeau scalars $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon}$ well defined and becomes to infinite large Colombeau generalized numbers

$$ \left\{
\begin{aligned}
(\mathbf{R}^{+}(\epsilon))_{\epsilon} &= \left( \frac{a^2 + \epsilon^2}{(\sqrt{a^2 + \epsilon^2} - |a|)^2} \left[ \frac{9a\epsilon^2}{(a^2 + \epsilon^2)^{\frac{5}{2}}} - \frac{2a\epsilon^2}{a^2(a^2 + \epsilon^2)^{\frac{3}{2}}} \right] \right)_{\epsilon} = 7a^{-2}(\epsilon^{-2})_{\epsilon} \in \tilde{\mathbb{R}}, \\
(\mathbf{R}^{-}(\epsilon))_{\epsilon} &= -\left( \frac{a^2 + \epsilon^2}{(\sqrt{a^2 + \epsilon^2} - |a|)^2} \left[ \frac{9a\epsilon^2}{(a^2 + \epsilon^2)^{\frac{5}{2}}} - \frac{2a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] \right)_{\epsilon} = -7a^{-2}(\epsilon^{-2})_{\epsilon} \in \tilde{\mathbb{R}}
\end{aligned}
\right. \quad (3.67) $$

(ii) for $r \neq a$ Colombeau scalars $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon}$ well defined and becomes to infinite small Colombeau generalized numbers $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon} \approx \pm(\epsilon^2)_{\epsilon}$.

Using now Eq. A2 one obtains that the Colombeau scalars $(\mathbf{R}^{\pm\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}^{\pm}(\epsilon))_{\epsilon}$ in terms of Colombeau generalized functions $(A_{\epsilon}^{\pm}(r))_{\epsilon}, (B_{\epsilon}^{\pm}(r))_{\epsilon}, (C_{\epsilon}^{\pm}(r))_{\epsilon}, (D_{\epsilon}^{\pm}(r))_{\epsilon}$ is expressed as

$$ \left\{
\begin{aligned}
(\mathbf{R}^{\pm\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}^{\pm}(\epsilon))_{\epsilon} &= \pm\left(\frac{(r^2 + \epsilon^2)^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^4} \left\{ \left[ \frac{5}{2} \left[ \frac{3a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} \right]^2 - \frac{2a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] + 2 \left[ \frac{3a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} + \frac{a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \right\} \right)_{\epsilon} \quad (3.68)
\end{aligned}
\right. $$

**Remark 3.7.** Note that (i) on horizon $r = a$ Colombeau scalars $(\mathbf{R}^{\pm\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}^{\pm}(\epsilon))_{\epsilon}$ well defined and becomes to infinite large Colombeau generalized numbers, (ii) for $r \neq a$ Colombeau scalars $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon}$ well defined and becomes to infinite small Colombeau generalized numbers.

Using now Eq. A2 one obtains that the Colombeau scalars $(\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon}$ in terms of Colombeau generalized functions $(A_{\epsilon}^{\pm}(r))_{\epsilon}, (B_{\epsilon}^{\pm}(r))_{\epsilon}, (C_{\epsilon}^{\pm}(r))_{\epsilon}, (D_{\epsilon}^{\pm}(r))_{\epsilon}$ is expressed as
$$ (\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon} = \left( \frac{12a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^6} \left[ 1 + \frac{a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \mp \frac{4a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^5} \left[ 1 + \frac{a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \left[ \frac{2\epsilon^2}{r^2(r^2 + \epsilon^2)} + \frac{9\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} \right] + \frac{a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^4} \left[ \frac{4\epsilon^4}{r^4(r^2 + \epsilon^2)} + \frac{81\epsilon^4}{(r^2 + \epsilon^2)^3} \right] \right) \quad (3.69) $$

**Remark 3.8.** Note that (i) on horizon $r = a$ Colombeau scalars $(\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon}$ well defined and becomes to infinite large Colombeau generalized numbers, (ii) for $r \neq a$ Colombeau scalars finite

$$ (\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon} = \frac{12a^2}{(r^2 - |a|)^6} \quad (3.70) $$

and tends to zero in the limit $r \to \infty$.

**Remark 3.9.** Note that under generalized transformations such as

$$ dt = \left( d\left[ \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r} \right] v_{\epsilon}^{+} \right)_{\epsilon} + \left( \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r} dv_{\epsilon}^{+} \right)_{\epsilon}, \quad (3.71) $$

and

$$ dt = \left( d\left[ \frac{2m - \sqrt{r^2 + \epsilon^2}}{r} \right] v_{\epsilon}^{-} \right)_{\epsilon} + \left( \frac{2m - \sqrt{r^2 + \epsilon^2}}{r} dv_{\epsilon}^{-} \right)_{\epsilon}, \quad (3.72) $$

the metric given by Eq.(3.61)-Eq.(3.64) becomes to Colombeau metric of the form

$$ \left\{ \begin{aligned} ds_{\epsilon}^{\pm 2} = \mp A^{\pm}(r, \epsilon)(dv_{\epsilon}^{\pm})^{2} - 2v_{\epsilon}^{\pm}D_{2}^{\pm}(r, \epsilon)dv_{\epsilon}^{\pm}dr + [B^{\pm}(r, \epsilon) + C_{1}^{\pm}(v_{\epsilon}^{\pm}, r, \epsilon)](dr)^{2} + \\ + B^{\pm}(r, \epsilon)r^{2}[(d\theta)^{2} + \sin^{2}\theta(d\phi)^{2}]. \end{aligned} \right. \quad (3.73) $$

# 4. Quantum scalar field in curved distributional space-time

## 4.1 Canonical quantization in curved distributional space-time

Much of formalism can be explained with Colombeau generalized scalar field. The basic concepts and methods extend straightforwardly to distributional tensor and distributional spinor fields. To being with let's take a spacetime of arbitrary dimension $D$, with a metric $g_{\mu\nu}$ of signature(+---). The action for the Colombeau generalized scalar field $(\phi_{\epsilon})_{\epsilon} \in \mathcal{G}(M)$ is

$$ (S_{\epsilon})_{\epsilon} = \left( \int_{M} d^{D}x \frac{1}{2} \sqrt{|g_{\epsilon}|} (g_{\epsilon}^{\mu\nu} \partial_{\mu} \phi_{\epsilon} \partial_{\nu} \phi_{\epsilon}) - (m^2 + R_{\epsilon}) \phi_{\epsilon}^2 \right)_{\epsilon}. \quad (4.1) $$

The corresponding equation of motion is

$$ (\Box_{\epsilon} + m^2 + \xi R_{\epsilon})\phi_{\epsilon})_{\epsilon}, \epsilon \in (0, 1]. \quad (4.2) $$

Here

$$ (\Box_{\epsilon}\phi_{\epsilon})_{\epsilon} = (|g_{\epsilon}|^{-1/2} \partial_{\mu}|g_{\epsilon}|^{1/2} g_{\epsilon}^{\mu\nu} \partial_{\mu}\phi_{\epsilon})_{\epsilon}. \quad (4.3) $$

With $\hbar$ explicit, the mass $m$ should be replaced by $m/\hbar$. Separating out a time coordinate
$x^0, x^\mu = (x^0, x^i), i = 1, 2, 3$ we can write the action as

$$ (S_{\epsilon})_{\epsilon} = \left( \int dx^0 L_{\epsilon} \right)_{\epsilon}, (L_{\epsilon})_{\epsilon} = \left( \int d^{D-1}x \mathcal{L}_{\epsilon} \right)_{\epsilon}. \qquad (4.4) $$

The canonical momentum at a time $x^0$ is given by

$$ (\pi_{\epsilon}(\mathbf{x}))_{\epsilon} = (\delta L_{\epsilon} / \delta(\partial_{0}\varphi_{\epsilon}(\mathbf{x})))_{\epsilon} = (|h_{\epsilon}|^{1/2} n^{\mu} \partial_{\mu}\varphi_{\epsilon}(\mathbf{x}))_{\epsilon}, \qquad (4.5) $$

where $\mathbf{x}$ labels a point on a surface of constant $x^0$, the $x^0$ argument of $(\varphi_{\epsilon})_{\epsilon}$ is suppressed, $n^{\mu}$ is the unit normal to the surface, and $(|h_{\epsilon}|)_{\epsilon}$ is the determinant of the induced spatial metric $(h_{ij}(\epsilon))_{\epsilon}$. To quantize, the Colombeau generalized field $(\varphi_{\epsilon})_{\epsilon}$ and its conjugate momentum $(\pi_{\epsilon}(\mathbf{x}))_{\epsilon}$ are now promoted to hermitian operators and required to satisfy the canonical commutation relation,

$$ ( [ \varphi_{\epsilon}(\mathbf{x}), \pi_{\epsilon}(\mathbf{y}) ] )_{\epsilon} = i\hbar \delta^{D-1}(\mathbf{x}, \mathbf{y}), \epsilon \in (0,1]. \qquad (4.6) $$

Here $\int d^{D-1}y \delta^{D-1}(\mathbf{x}, \mathbf{y}) f(\mathbf{y}) = f(\mathbf{x})$ for any scalar function $f \in D(\mathbb{R}^3)$, without the use of a metric volume element. We form now a conserved bracket from two complex Colombeau solutions to the scalar wave equation (4.2) by

$$ (\langle\varphi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon} = \left( \int_{\Sigma} d\Sigma_{\mu} j_{\epsilon}^{\mu} \right)_{\epsilon}, \epsilon \in (0,1]. \qquad (4.7) $$

where

$$ (j_{\epsilon}^{\mu}(\varphi_{\epsilon}, \phi_{\epsilon}))_{\epsilon} = (i/h) (|g_{\epsilon}|^{1/2} g_{\epsilon}^{\mu\nu} (\bar{\phi}_{\epsilon} \partial_{\nu}\phi_{\epsilon} - \phi_{\epsilon} \partial_{\nu}\bar{\phi}_{\epsilon}))_{\epsilon}. \qquad (4.8) $$

This bracket is called the generalized Klein-Gordon inner product, and $(\langle\varphi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon}$ the generalized Klein Gordon norm of $(\varphi_{\epsilon})_{\epsilon}$. The generalized current density $(j_{\epsilon}^{\mu}(\varphi_{\epsilon}, \phi_{\epsilon}))_{\epsilon}$ is divergenceless,i.e. $(\partial_{\mu}j_{\epsilon}^{\mu}(\varphi_{\epsilon}, \phi_{\epsilon}))_{\epsilon} = 0$ when the Colombeau generalized functions $(\varphi_{\epsilon})_{\epsilon}$ and $(\phi_{\epsilon})_{\epsilon}$ satisfy the KG equation (4.2), hence the value of the integral in (4.7) is independent of the spacelike surface $\Sigma$ over which it is evaluated, provided the functions vanish at spatial infinity. The generalized KG inner product satisfies the relations

$$ (\overline{\langle\varphi_{\epsilon}, \phi_{\epsilon}\rangle})_{\epsilon} = -(\overline{\langle\bar{\phi}_{\epsilon}, \bar{\phi}_{\epsilon}\rangle})_{\epsilon} = (\langle\phi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon}, \epsilon \in (0,1]. \qquad (4.9) $$

We define now the annihilation operator associated with a complex Colombeau solution $(\varphi_{\epsilon})_{\epsilon}$ by the bracket of $(\phi_{\epsilon})_{\epsilon}$ with the generalized field operator $(\varphi_{\epsilon})_{\epsilon}$:

$$ (a(\varphi_{\epsilon}))_{\epsilon} = (\langle\varphi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon}. \qquad (4.10) $$

It follows from the hermiticity of $(\varphi_{\epsilon})_{\epsilon}$ that the hermitian conjugate of $(a(\varphi_{\epsilon}))_{\epsilon}$ is given by

$$ (a^{\dagger}(\varphi_{\epsilon}))_{\epsilon} = -(a(\bar{\phi}_{\epsilon}))_{\epsilon}. \qquad (4.11) $$

From Eq.(4.5) and CCR (4.6) one obtain

$$ ([a(\varphi_{\epsilon}), a^{\dagger}(\phi_{\epsilon})])_{\epsilon} = (\langle\varphi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon}. \qquad (4.12) $$

Note that from Eq.(4.11) follows

$$ ([a(\varphi_{\epsilon}), a(\phi_{\epsilon})])_{\epsilon} = -(\langle\varphi_{\epsilon}, \bar{\phi}_{\epsilon}\rangle)_{\epsilon}, ([a^{\dagger}(\varphi_{\epsilon}), a^{\dagger}(\phi_{\epsilon})])_{\epsilon} = -(\langle\bar{\phi}_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon} \qquad (4.13) $$

Note that if $(\varphi_{\epsilon})_{\epsilon}$ is a positive norm solution with unit norm and with, then $(a(\varphi_{\epsilon}))_{\epsilon}$ and $a^{\dagger}(\varphi_{\epsilon})$ satisfy the commutation relation $([a^{\dagger}(\varphi_{\epsilon}), a(\phi_{\epsilon})])_{\epsilon} = 1$. Suppose now that $|\Psi\rangle$ is a normalized quantum state satisfying $(a(\varphi_{\epsilon})|\Psi\rangle)_{\epsilon} = 1$, then for each $n$, the state
$|n, \Psi\rangle = ((1/\sqrt{n!})(a(\phi_\epsilon))^n|\Psi\rangle)_\epsilon$ is a normalized eigenstate of the number operator $(N[(\phi_\epsilon)])_\epsilon = (a^\dagger(\phi_\epsilon)a(\phi_\epsilon))_\epsilon$ with eigenvalue $n$. The span of all these states defines a Fock space of the distributional $(\phi_\epsilon)_\epsilon$-wavepacket "n-particle excitations" above the state $|\Psi\rangle$. If we want to construct the full Hilbert space of the field theory in curved distributional spacetime, how can we proceed? We should find a decomposition of the space of complex Colombeau solutions to the wave equation (4.2) into a direct sum of a positive norm subspace $S_p$ and its complex conjugate $\overline{S_p}$, such that all brackets between solutions from the two subspaces vanish. That is, we must find a direct sum decomposition:

$$S = S_p \oplus \overline{S_p} \qquad (4.14)$$

such that

$$(\langle\phi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon} > 0, \forall (\phi_{\epsilon})_{\epsilon} \in S_{p} \qquad (4.15)$$

and

$$(\langle\phi_{\epsilon}, \phi_{\epsilon}\rangle)_{\epsilon} > 0, \forall (\phi_{\epsilon})_{\epsilon}, (\phi_{\epsilon})_{\epsilon} \in S_{p}. \qquad (4.16)$$

The condition (4.15) implies that each $(\phi_\epsilon)_\epsilon$ in $S_p$ can be scaled to define its own harmonic oscillator sub-albegra. The second condition implies, according to (4.13), that the annihilators and creators for $(\phi_\epsilon)_\epsilon$ and $(\phi_\epsilon)_\epsilon$ in the subspace $S_p$ commute amongst themselves:

$$([a(\phi_{\epsilon}), a(\phi_{\epsilon})])_{\epsilon} = ([a^{\dagger}(\phi_{\epsilon}), a^{\dagger}(\phi_{\epsilon})])_{\epsilon} = 0. \qquad (4.17)$$

Given such a decomposition a total Hilbert space $\mathbf{H}$ for the field theory can be defined as the space of finite norm sums of possibly infinitely many states of the form

$$(a^{\dagger}(\phi_{1,\epsilon})...a^{\dagger}(\phi_{n,\epsilon})|0\rangle)_{\epsilon}, \qquad (4.18)$$

where $|0\rangle$ is a state such that $(a(\phi_{n,\epsilon})|0\rangle)_\epsilon = 0$ for all $(\phi_\epsilon)_\epsilon$ in $S_p$. The state $|0\rangle$, as in classical case, is called a Fock vacuum and Hilbert space $\mathbf{H}$ is called a Fock space. The representation of the field operator on this Fock space is hermitian and satisfies the canonical commutation relations in sense of Colombeau generalized function.

## 4.2 Defining distributional outgoing modes

For illustration we consider the non-rotating, uncharged $d$-dimensional SAdS BH with a distributional line element

$$(ds_{\epsilon}^{2})_{\epsilon} = \left(-f_{\epsilon} dt^{2} + f_{\epsilon}^{-1} dr^{2}\right)_{\epsilon} + r^{2} d\Omega_{d-2}^{2}, \epsilon \in (0,1], \qquad (4.19)$$

where

$$\left\{ \begin{array}{l} f_{\epsilon} \neq 0, \epsilon \in (0,1], \\ f_{\epsilon=0} = 1 + \frac{r^2}{L^2} - \frac{r_0^{d-3}}{r^{d-3}}, \end{array} \right. \qquad (4.20)$$

where $d\Omega_{d-2}^2$ is the metric of the $(d-2)$-sphere, and the AdS curvature radius squared $L^2$ is related to the cosmological constant by $L^2 = -(d-2)(d-1)/2\Lambda$. The parameter $r_0$ is proportional to the mass $M$ of the spacetime: $M = (d-2)A_{d-2}r_0^{d-3}/16\pi$, where $A_{d-2} = 2\pi^{(d-1)/2}/\Gamma[(d-1)/2]$. The distributional Schwarzschild geometry corresponds to $L \to \infty$. The corresponding equation of motion (4.2) for massless case are
$$ 
\begin{aligned}
(\nabla_{\mu}\nabla^{\mu}\varphi)_{\varepsilon} &= \frac{(d-2)\gamma}{4(d-1)}(R_{\varepsilon})_{\varepsilon}, \\
(G_{\varepsilon,\mu\nu})_{\varepsilon} + \Lambda(g_{\varepsilon,\mu\nu})_{\varepsilon} &= 8\pi G(T_{\varepsilon,\mu\nu})_{\varepsilon}, \\
(T_{\varepsilon,\mu\nu})_{\varepsilon} &\sim \delta(x).
\end{aligned} 
\quad (4.21) $$

The time-independence and the spherical symmetry of the metric imply the canonical decomposition

$$ (\varphi_{\varepsilon}(t, r, \theta))_{\varepsilon} = e^{-i\omega t} \left( \sum \frac{\Psi_{lm,\varepsilon}(r)Y_{lm}(\theta)}{r^{(d-2)/2} + \varepsilon} \right)_{\varepsilon}, \quad (4.22) $$

where $Y_{lm}(\theta)$ denotes the $d$-dimensional scalar spherical harmonics, satisfying

$$ \Delta\Omega_{d-2}Y_{lm}(\theta) = -l(l+d-3)Y_{lm}(\theta), \quad (4.23) $$

the Laplace-Beltrami operator. Substituting the decomposition (4.22) into Eq. (6) one gets a radial wave equation

$$ \left( f_{\varepsilon}^2 \frac{d^2 \Psi_{lm,\varepsilon}(r)}{dr^2} + f_{\varepsilon} f_{\varepsilon}^{\prime} \frac{d \Psi_{lm,\varepsilon}(r)}{dr} + (\omega^2 - V_{lm,\varepsilon}(r)) \Psi_{lm,\varepsilon}(r) \right)_{\varepsilon} = 0. \quad (4.24) $$

We define now a “tortoise” distributional coordinate $(r_{\varepsilon}^{*})_{\varepsilon} = (r_{\varepsilon}^{*}(r))_{\varepsilon}$ by the relation

$$ \left( \frac{dr_{\varepsilon}^{*}}{dr} \right)_{\varepsilon} = \left( f_{\varepsilon}^{-1}(r) \right)_{\varepsilon}. \quad (4.25) $$

By using a “tortoise” distributional coordinate the Eq. (4.24) can be written in the form of a Schrödinger equation with the potential $V_{lm,\varepsilon}(r)$

$$ \left( \frac{d\Psi_{\varepsilon}(r_{\varepsilon}^{*})}{dr_{\varepsilon}^{*}} \right)_{\varepsilon} + ((\omega^2 - V_{\varepsilon}(r_{\varepsilon}^{*}))\Psi_{\varepsilon}(r_{\varepsilon}^{*}))_{\varepsilon} = 0. \quad (4.26) $$

Note that the tortoise distributional coordinate $(r_{\varepsilon}^{*}(r))_{\varepsilon}$ becomes to infinite Colombeau constant $[(r_{\varepsilon}^{*}(r_{+}))_{\varepsilon}] = [(\ln \varepsilon)_{\varepsilon}]$ at the horizon, i.e. as $r \to r_{+}$, but its behavior at infinity is strongly dependent on the cosmological constant: $[(r_{\varepsilon}^{*}(r_{+}))_{\varepsilon}] = +\infty$ for asymptotically-flat spacetimes, and $[(r_{\varepsilon}^{*}(r_{+}))_{\varepsilon}] = \text{finite Colombeau constant for the SAdS}_d$ geometry.

### 4.2.1. Boundary conditions at the horizon of the distributional SAdS BH geometry.

For most spacetimes of interest the potential $(V_{\varepsilon}(r_{\varepsilon}^{*}(r)))_{\varepsilon} = 0$ as $r = r_{+}$, i.e. $(|r_{\varepsilon}^{*}(r)|)_{\varepsilon} = +\infty$, and in this limit solutions to the wave equation (4.26) behave as

$$ (\Psi_{\varepsilon}(t, r_{\varepsilon}^{*}))_{\varepsilon} \sim (\exp[-i\omega(t \pm r_{\varepsilon}^{*}(r))])_{\varepsilon}, \quad \text{as } r \sim r_{+}. \quad (4.27) $$

Note that classically nothing should leave the horizon and thus classically only ingoing modes (corresponding to a plus sign) should be present, i.e.

$$ (\Psi_{\varepsilon}(t, r_{\varepsilon}^{*}))_{\varepsilon} \sim (\exp[-i\omega(t + r_{\varepsilon}^{*}(r))])_{\varepsilon}, \quad \text{as } r \sim r_{+}. \quad (4.28) $$

Note that for non-extremal spacetimes, the tortoise coordinate tends to

$$ (r_{\varepsilon}^{*}(r))_{\varepsilon} = \left( \int f_{\varepsilon}^{-1}(r) dr \right)_{\varepsilon} \sim \left[ (f_{\varepsilon}^{\prime}(r_{+}))_{\varepsilon} \right]^{-1} (\ln(|r - r_{+}| + \varepsilon))_{\varepsilon} \quad \text{as } r \sim r_{+}, \quad (4.29) $$

where $(f_{\varepsilon}^{\prime}(r_{+}))_{\varepsilon} > 0$. Therefore near the horizon, outgoing modes behave as
$$ \left\{ \begin{aligned} & (\exp[-i\omega(t - r_{\varepsilon}^{*}(r))])_{\varepsilon} = \{(\exp[-i\omega v_{\varepsilon}^{*}(t, r)])_{\varepsilon}\}(\exp[-2i\omega r_{\varepsilon}^{*}(r)])_{\varepsilon} = \\ & = \{(\exp[-i\omega v_{\varepsilon}^{*}(t, r)])_{\varepsilon}\}\{(\lvert r - r_{+} \rvert + \varepsilon)^{2i\omega/f_{\varepsilon}(r_{+})}\}_{\varepsilon}, \end{aligned} \right. \quad (4.30) $$

where $(v_{\varepsilon}^{*}(t, r))_{\varepsilon} = t + (r_{\varepsilon}^{*}(r))_{\varepsilon}$. Now Eq. (4.30) shows that outgoing modes is Colombeau generalized function of class $G(\mathbb{R})$.

## 5. Energy-momentum tensor calculation by using Colombeau distributional modes

We shall assume now any distributional spacetime which is conformally static in both the asymptotic past and future. We will be considered distributional spacetime which is conformally flat in the asymptotic past, i.e.

$$ \left\{ \begin{aligned} ds_{\varepsilon}^2 &\sim (f_{\varepsilon,in}^2(-dt^2 + d\vec{x}^2))_{\varepsilon} \quad \text{asymp. past} \\ ds_{\varepsilon}^2 &\sim (f_{\varepsilon,out}^2(-dt^2 + h_{\varepsilon,ij}dx^i dx^j))_{\varepsilon}, \quad \text{asymp. future} \end{aligned} \right. \quad (5.1) $$

where $\varepsilon \in (0,1] \ (f_{\varepsilon,J})_{\varepsilon} = (f_{\varepsilon,J}(t,\vec{x}))_{\varepsilon} > 0, \mathbf{J} \in \{\text{in, out}\}$, are smooth functions and $h_{\varepsilon,ij} = h_{\varepsilon,ij}(\vec{x}), i,j = 1,2,3$, are the components of an arbitrary distributional spatial metric. Note that we use the same labels $t$ and $\vec{x} = (x^1, x^2, x^3)$ for coordinates in the asymptotic past and future only for simplicity; they are obviously defined on non-intersecting regions of the spacetime.) In each of these asymptotic regions the distributional field $(\Phi_{\varepsilon})_{\varepsilon}$ can be written as $(\Phi_{\varepsilon})_{\varepsilon} = (\tilde{\Phi}_{\varepsilon}/f_{\varepsilon,J})_{\varepsilon}$, where $(\tilde{\Phi}_{\varepsilon})_{\varepsilon}$ satisfies

$$ -(\frac{\partial^2}{\partial t^2} \tilde{\Phi})_{\varepsilon} = -(\Delta_{\varepsilon,J} \tilde{\Phi}_{\varepsilon})_{\varepsilon} + (V_{\varepsilon,J} \tilde{\Phi}_{\varepsilon})_{\varepsilon}, \quad (5.2) $$

where $(\Delta_{\varepsilon,in})_{\varepsilon}$ is the flat Laplace operator, $(\Delta_{\varepsilon,out})_{\varepsilon}$ is the Laplace operator associated with the spatial metric $(h_{\varepsilon,ij})_{\varepsilon}$, and the effective potential $V_J$ is given by

$$ \left\{ \begin{aligned} (V_{\varepsilon,J})_{\varepsilon} &= \left( \frac{\Delta_{\varepsilon} f_{\varepsilon,J}}{f_{\varepsilon,J}} \right)_{\varepsilon} + \left( f_{\varepsilon,J}^2 (m^2 + \xi R_{\varepsilon}) \right)_{\varepsilon} = \\ (1 - 6\xi) \left( \frac{\Delta_{\varepsilon} f_{\varepsilon,J}}{f_{\varepsilon,J}} \right)_{\varepsilon} + m^2 (f_{\varepsilon,J}^2)_{\varepsilon} + \xi (K_{\varepsilon,J})_{\varepsilon}, \end{aligned} \right. \quad (5.3) $$

with $(K_{\varepsilon,in})_{\varepsilon} = 0, K_{\varepsilon,out} = K_{\varepsilon,out}(\vec{x})$ the scalar curvature associated with the spatial distributional metric $(h_{\varepsilon,ij})_{\varepsilon}$.

We assume now this condition: (i) the massless ($m = 0$) field with arbitrary coupling $\xi$ in spacetimes which are asymptotically flat in the past and asymptotically static in the future, i.e. $f_{in} = 1$ and $f_{\varepsilon,out} = f_{\varepsilon,out}(\vec{x})$, as those describing the formation of a static BH from matter initially scattered throughout space, and (ii) the massless, conformally coupled field ($m = 0$ and $\xi = 1/6$). With this assumptions for the potential, two different sets of positive-norm distributional modes, $(u_{\varepsilon,\vec{k}}^{(+)})_{\varepsilon}$ and $(v_{\varepsilon,\alpha}^{(+)})_{\varepsilon}$, can be naturally defined by the requirement that they are the solutions of Eq. (5.2) which satisfy the asymptotic conditions:

$$ (u_{\varepsilon,\vec{k}}^{(+)})_{\varepsilon} \underset{\text{past}}{\asymp} (16\pi^3 \omega_{\vec{k}}^{-1/2} e^{-i(\omega_{\vec{k}}t - k\vec{x})} (f_{\varepsilon,in}^{-1})_{\varepsilon} \quad (5.4) $$
and

$$ (v_{\varepsilon,a}^{(+)})_{\varepsilon} \underset{\text{future}}{\propto} (2\bar{\omega}_{\alpha})^{-1/2} e^{-i\bar{\omega}_{\alpha}t} \left( f_{\varepsilon,\text{out}}^{-1} F_{\varepsilon,a}(\vec{x}) \right)_{\varepsilon}, \quad (5.5) $$

where $\vec{k} \in \mathbb{R}^3$, $\omega_{\vec{k}} := \|\vec{k}\|$, $\bar{\omega}_{\alpha} > 0$, and $(F_{\varepsilon,a}(\vec{x}))_{\varepsilon}$ are Colombeau solutions of

$$ \begin{gathered} ([-\Delta_{\varepsilon,\text{out}} + V_{\varepsilon,\text{out}}(\vec{x})]F_{\varepsilon,a}(\vec{x}))_{\varepsilon} = \bar{\omega}_{\alpha}^2 (F_{\varepsilon,a}(\vec{x}))_{\varepsilon}, \\ (F_{\varepsilon,a}(\vec{x}))_{\varepsilon}|_{\varepsilon=0} \in C^{\infty}(\mathbb{R}^3) \end{gathered} \quad (5.6) $$

satisfying the normalization

$$ \left( \int_{\Sigma_{\text{out}}} d^3 x \sqrt{\hbar} F_{\varepsilon,a}(\vec{x})^* F_{\varepsilon,\beta}(\vec{x}) \right)_{\varepsilon} = \delta_{\alpha,\beta} \quad (5.7) $$

on a Cauchy surface $\Sigma_{\text{out}}$ in the asymptotic future. Note that each $F_{\varepsilon,a}, \varepsilon \in [0,1]$ can be chosen to be real without loss of generality. There are reasonable situations where the distributional modes $(v_{\varepsilon,a}^{(+)})_{\varepsilon}$, given in Eq. (5.5), together with distributional modes $(v_{\varepsilon,a}^{(-)})_{\varepsilon}$ fail to form a complete set of distributional normal modes. This happens whenever the operator $([-\Delta_{\varepsilon,\text{out}} + V_{\varepsilon,\text{out}}(\vec{x})])_{\varepsilon}$ in Eq. (5.2) happens to possess normalizable i.e., satisfying Eq. (5.7) eigenfunctions with negative eigenvalues, $\bar{\omega}_{\alpha}^2 = -\bar{\Omega}_{\alpha}^2 < 0$. In this case, additional positive-norm modes $(w_{\varepsilon,a}^{(+)})_{\varepsilon}$ with the asymptotic behavior

$$ (w_{\varepsilon,a}^{(+)})_{\varepsilon} \underset{\text{future}}{\propto} (e^{\Omega_{\alpha}t - i\pi/12} + e^{-\Omega_{\alpha}t + i\pi/12}) \left( \frac{F_{\varepsilon,a}(\vec{x})}{\sqrt{2\bar{\Omega}_{\alpha}} f_{\varepsilon,\text{out}}(\vec{x})} \right)_{\varepsilon} \quad (5.8) $$

and their complex conjugates $(w_{\varepsilon,a}^{(-)})_{\varepsilon}$ are necessary in order to expand an arbitrary Colombeau solution of Eq. (5.1). As a direct consequence, at least some of the in-modes $(u_{\varepsilon,\vec{k}}^{(\pm)})_{\varepsilon}$ (typically those with low $\omega_{\vec{k}}$) eventually undergo an exponential growth. This asymptotic divergence is reflected on the unbounded increase of the vacuum fluctuations,

$$ \langle (\Phi_{\varepsilon}^2(\vec{x})) \rangle_{\varepsilon} \underset{\text{future}}{\propto} \frac{\kappa e^{2\bar{\Omega}t}}{2\bar{\Omega}} \left[ \left( \frac{\bar{F}_{\varepsilon}(\vec{x})}{f_{\varepsilon,\text{out}}(\vec{x})} \right)_{\varepsilon} \right]^2 [1 + \mathcal{O}(e^{-\epsilon t})], \quad (5.9) $$

where $\bar{F}(\vec{x})$ is the eigenfunction of Eq. (5.6) associated with the lowest negative eigenvalue allowed, $\bar{\omega}_{\alpha}^2 = -\bar{\Omega}^2$, $\epsilon$ is some positive constant, and $\kappa$ is a dimensionless constant (typically of order unity) whose exact value depends globally on the spacetime structure (since it crucially depends on the projection of each $(u_{\varepsilon,\vec{k}}^{(\pm)})_{\varepsilon}$ on the mode $(w_{\varepsilon,a}^{(\pm)})_{\varepsilon}$ whose $\bar{\omega}_{\alpha}^2 = -\bar{\Omega}^2$; $\kappa$ also depends on the initial state, here assumed to be the vacuum $|0\rangle_{\text{in}}$). As one would expect, these wild quantum fluctuations give an important contribution to the vacuum energy stored in the field. In fact, the expectation value of its distributional energy-momentum tensor, $\langle (T_{\varepsilon,\mu\nu}(\vec{x})) \rangle_{\varepsilon}, \varepsilon \in (0,1]$, in the asymptotic future is found to be dominated by this exponential growth:

$$ \left\{ \begin{aligned} \langle (T_{\varepsilon,00}(\vec{x})) \rangle_{\varepsilon} &\propto \langle (\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle) \rangle_{\varepsilon} \left\{ \frac{(1-4\xi)}{2} \left( \bar{\Omega}^2 + \frac{(D\bar{F}_{\varepsilon})^2}{\bar{F}^2} + m^2 f_{\varepsilon}^2 + \xi K_{\varepsilon} \right)_{\varepsilon} \right. \\ &\left. \qquad + (1-6\xi) \left( \frac{2\xi D^2 f_{\varepsilon}}{f_{\varepsilon}} + \frac{(Df_{\varepsilon})^2}{2f_{\varepsilon}^2} - \frac{D f_{\varepsilon} D^i \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right\} + \mathcal{O}(e^{-\epsilon t}) \end{aligned} \right\}, \quad (5.10) $$
$$
\left\{
\begin{aligned}
& \left\langle \left\langle T_{\varepsilon,0i}(\vec{x}) \right\rangle \right\rangle_{\varepsilon} \underset{\text{future}}{\asymp} \\
& \left\langle \left\langle \left\langle \Phi_{\varepsilon}^{2}(\vec{x}) \right\rangle \right\rangle \right\rangle_{\varepsilon} \left\{ (1-4\xi) \left( \frac{\bar{\Omega}D_{i}\bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}} \right)_{\varepsilon} - (1-6\xi) \left( \frac{\bar{\Omega}D_{i}f_{\varepsilon}}{f_{\varepsilon}} \right)_{\varepsilon} + \mathcal{O}(e^{-\epsilon t}) \right\}, \\
& \\
& \left\langle \left\langle T_{\varepsilon,ij}(\vec{x}) \right\rangle \right\rangle_{\varepsilon} \underset{\text{future}}{\asymp} \\
& \left\langle \left\langle \left\langle \Phi_{\varepsilon}^{2}(\vec{x}) \right\rangle \right\rangle \right\rangle_{\varepsilon} \left\{ (1-2\xi) \left( \frac{D_{i}\bar{F}_{\varepsilon}D_{j}\bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}^{2}} \right)_{\varepsilon} - 2\xi \left( \frac{D_{i}D_{j}\bar{F}_{\varepsilon}}{F_{\varepsilon}} \right)_{\varepsilon} + \xi (\tilde{R}_{\varepsilon,ij})_{\varepsilon} \right. \\
& \qquad \left. + \frac{(1-4\xi)h_{ij}}{2} \left( \bar{\Omega}^{2} - \left( \frac{(D\bar{F}_{\varepsilon})^{2}}{\bar{F}_{\varepsilon}^{2}} - m^{2}f_{\varepsilon}^{2} - \xi K_{\varepsilon} \right)_{\varepsilon} \right) \right. \\
& \qquad \left. + (1-6\xi) \left[ \left( \frac{D_{i}f_{\varepsilon}D_{j}f_{\varepsilon}}{f_{\varepsilon}^{2}} - \frac{D_{i}f_{\varepsilon}D_{j}\bar{F}}{f_{\varepsilon}\bar{F}_{\varepsilon}} - \frac{D_{i}f_{\varepsilon}D_{i}\bar{F}_{\varepsilon}}{f_{\varepsilon}\bar{F}_{\varepsilon}} \right)_{\varepsilon} + \right. \right. \\
& \qquad \left. \left. + (h_{\varepsilon,ij})_{\varepsilon} \left( \frac{2\xi D^{2}f_{\varepsilon}}{f_{\varepsilon}} - \frac{(Df_{\varepsilon})^{2}}{2f_{\varepsilon}^{2}} + \frac{D_{k}f_{\varepsilon}D^{k}\bar{F}_{\varepsilon}}{f_{\varepsilon}\bar{F}_{\varepsilon}} \right)_{\varepsilon} \right] \right) + \mathcal{O}(e^{-\epsilon t}),
\end{aligned}
\right.
\quad (5.11)
$$

where $D_i$ is the derivative operator compatible with the distributional metric $(h_{\varepsilon,ij})_\varepsilon$ (so that $\Delta_{out} = D^2$), $(\tilde{R}_{\varepsilon,ij})_\varepsilon$ is the associated distributional Ricci tensor so that $(K_{\varepsilon,out})_\varepsilon = (h_\varepsilon^{ij}\tilde{R}_{\varepsilon,ij})_\varepsilon$, and we have omitted the subscript *out* in $(f_{\varepsilon,out})_\varepsilon$ and $(K_{\varepsilon,out})_\varepsilon$ for simplicity. The Eqs. (5.10-5.12), together with Eq.(5.9), imply that on time scales determined by $\bar{\Omega}^{-1}$, the vacuum fluctuations of the field should overcome any other classical source of energy, therefore taking control over the evolution of the background geometry through the semiclassical Einstein equations (in which $\langle (T_{\varepsilon,\mu\nu}) \rangle_\varepsilon$ is included as a source term for the distributional Einstein tensor). We are then confronted with a startling situation where the quantum fluctuations of a field, whose energy is usually negligible in comparison with classical energy components, are forced by the distributional background spacetime to play a dominant role. We are still left with the task of showing that there exist indeed well-behaved distributional background spacetimes in which the operator $[-\Delta_{\varepsilon,out} + V_{\varepsilon,out}(\vec{x}))]$ possesses negative eigenvalues $\bar{\omega}_a^2 < 0$, condition on which depends Eq(5.9). Experience from quantum mechanics tells us that this typically occurs when $(V_{\varepsilon,out})_\varepsilon$ gets sufficiently negative over a sufficiently large region. It is easy to see from Eq. (5.3) that, except for very special geometries (as the flat one), one can generally find appropriate values of $\xi \in \mathbb{R}$ which make $(V_{\varepsilon,out})_\varepsilon$ as negative as would be necessary in order to guarantee the existence of negative eigenvalues. For distributional BH spacetime using Eq.(5.9)-Eq.(5.12) one obtains

$$
\left\langle \left\langle \Phi_{\varepsilon}^{2}(r) \right\rangle \right\rangle_{\varepsilon} \underset{\text{future}}{\asymp} \frac{\kappa e^{2\bar{\Omega}t}}{2\bar{\Omega}} \left[ \left( \frac{r^{1/2} \bar{F}_{\varepsilon}(r)}{((r-r_{+})^{2} + \varepsilon^{2})^{1/4}} \right)_{\varepsilon} \right]^{2}, \quad r \to r_{+} \qquad (5.13)
$$
$$ \langle (T_{\varepsilon,00}(r)) \rangle_{\varepsilon} \underset{\text{future}}{\times} \left\{ \left\langle \left\langle \Phi_{\varepsilon}^2(r) \right\rangle \right\rangle_{\varepsilon} \left\{ \frac{(1-4\xi)}{2} \left( \bar{\Omega}^2 + \frac{(D\bar{F}_{\varepsilon}(r_+))^2}{\bar{F}^2(r_+)} + m^2((r-r_+)^2 + \varepsilon^2)^{1/2} + \xi K_{\varepsilon} \right) \right\}_{\varepsilon} \right. \\ \left. + (1-6\xi) \left( -\frac{2\xi D^2 f_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} + \frac{(Df_{\varepsilon}(r))^2}{((r-r_+)^2 + \varepsilon^2)^{1/4}} - \frac{D_{if\varepsilon}(r)D^i\bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4}\bar{F}_{\varepsilon}} \right) \right\}_{\varepsilon}, \quad (5.14) \\ r \to r_+, f_{\varepsilon}(r) = ((r-r_+)^2 + \varepsilon^2)^{1/4} $$

$$ \left\{ \begin{array}{l} \left\langle (T_{\varepsilon,0i}(r)) \right\rangle_{\varepsilon} \underset{\text{future}}{\times} \left\{ \left\langle \left\langle \Phi_{\varepsilon}^2(r) \right\rangle \right\rangle_{\varepsilon} \left\{ (1-4\xi) \left( \frac{\bar{\Omega}D_i\bar{F}_{\varepsilon}(r_+)}{\bar{F}_{\varepsilon}(r_+)} \right)_{\varepsilon} - (1-6\xi) \left( \frac{\bar{\Omega}D_{if\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} \right)_{\varepsilon} \right\} \right\} \\ r \to r_+, f_{\varepsilon}(r) = ((r-r_+)^2 + \varepsilon^2)^{1/4} \end{array} \right. \quad (5.15) $$

$$ \langle (T_{\varepsilon,ij}(r)) \rangle_{\varepsilon} \underset{\text{future}}{\times} \left\{ \left\langle \left\langle \Phi_{\varepsilon}^2(r) \right\rangle \right\rangle_{\varepsilon} \left\{ (1-2\xi) \left( \frac{D_i \bar{F}_{\varepsilon} D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}^2} \right)_{\varepsilon} - 2\xi \left( \frac{D_i D_j \bar{F}_{\varepsilon}}{F_{\varepsilon}} \right)_{\varepsilon} + \xi (\tilde{R}_{\varepsilon,ij})_{\varepsilon} \right. \right. \\ + \frac{(1-4\xi)h_{ij}}{2} \left( \bar{\Omega}^2 - \left( \frac{(D\bar{F}_{\varepsilon}(r_+))^2}{\bar{F}_{\varepsilon}^2(r_+)} - m^2((r-r_+)^2 + \varepsilon^2)^{1/2} - \xi K_{\varepsilon} \right)_{\varepsilon} \right) \\ + (1-6\xi) \left[ \left( \frac{D_{if\varepsilon} D_j f_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/2}} - \frac{D_{if\varepsilon} D_j \bar{F}}{((r-r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} - \frac{D_{if\varepsilon} D_i \bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/2} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right. \\ \left. + (h_{\varepsilon,ij})_{\varepsilon} \left( \frac{2\xi D^2 f_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4}} - \frac{(Df_{\varepsilon})^2}{2((r-r_+)^2 + \varepsilon^2)^{1/2}} + \frac{D_k f_{\varepsilon} D^k \bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right] \right\} \quad (5.16) \\ r \to r_+ \end{array} \right. $$

**Remark 5.1.** Note that in spite of the unbounded growth at $r \to r_+$ in Eq.(5.13)-Eq.(5.16), $\langle (T_{\varepsilon,\mu\nu}) \rangle_{\varepsilon}$ is covariantly conserved: $(\nabla_{\mu} \langle T_{\varepsilon,\nu}^{\mu} \rangle)_{\varepsilon} = 0$. In the static case $(f_{\varepsilon,out})_{\varepsilon} = (f_{\varepsilon,out}(\vec{x}))_{\varepsilon}$, for instance for distributional BH geometry, this implies that the total vacuum energy is kept constant, although it continuously flows from spatial regions where its density is negative to spatial regions where it is positive.

**Remark 5.2.** Note that the singular behavior at $r \to r_+$ appearing in Eq.(5.13)-Eq.(5.16) leads only to asymptotic divergences, i.e. all the quantities remain finite everywhere except horizon.

# 6. Distributional SAdS BH spacetime-induced vacuum dominance
## 6.1. Adiabatic expansion of Green functions

Using equation of motion Eq.(5.2) one can obtain corresponding distributional generalization of the canonical Green functions equations. In particular for the distributional propagator

$$iG_{\varepsilon}^{\pm}(x,x') = \langle 0 | T(\varphi_{\varepsilon}^{\pm}(x)\varphi_{\varepsilon}^{\pm}(x')) | 0 \rangle, \varepsilon \in (0,1] \quad (6.1)$$

one obtains directly

$$([\Box_{\varepsilon,x} + m^2 + \xi \mathbf{R}^{\pm}(x, \varepsilon)] G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} = -[-g^{\pm}(x, \varepsilon)]^{-1/2} \delta^n(x - x'). \quad (6.2)$$

Special interest attaches to the short distance behaviour of the Green functions, such as $(G_{\varepsilon}^{\pm}(x,x'))_{\varepsilon}$ in the limit $\|x - x'\| \to 0$ with a fixed $\varepsilon \in (0,1]$. We obtain now an adiabatic expansion of $(G_{\varepsilon}^{\pm}(x,x'))_{\varepsilon}$. Introducing Riemann normal coordinates $y^{\mu}$ for the point $x$, with origin at the point $x'$ we have expanding

$$(g_{\mu\nu}^{\pm}(x, \varepsilon))_{\varepsilon} = \eta_{\mu\nu} + \frac{1}{3} [(\mathbf{R}_{\mu\nu\beta}^{\pm}(\varepsilon))_{\varepsilon}] y^{\alpha} y^{\beta} - \frac{1}{6} [(\mathbf{R}_{\mu\nu\beta;\gamma}^{\pm}(\varepsilon))_{\varepsilon}] y^{\alpha} y^{\beta} y^{\gamma} + \\ + \left[ \frac{1}{20} (\mathbf{R}_{\mu\nu\beta;\gamma\delta}^{\pm}(\varepsilon))_{\varepsilon} + \frac{2}{45} [(\mathbf{R}_{a\mu\beta\lambda}^{\pm}(\varepsilon))_{\varepsilon}] (\mathbf{R}_{\gamma\nu\delta}^{\pm\lambda}(\varepsilon))_{\varepsilon} \right] y^{\alpha} y^{\beta} y^{\gamma} y^{\delta} + \dots \quad (6.3)$$

where $\eta_{\mu\nu}$ is the Minkowski metric tensor, and the coefficients are all evaluated at $y = 0$. Defining now

$$(\mathcal{L}_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} = \left[ \left( (-g_{\mu\nu}^{\pm}(x, \varepsilon))^{1/4} \right)_{\varepsilon} \right] (G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} \quad (6.4)$$

and its Colombeau-Fourier transform by

$$(\mathcal{L}_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} = (2\pi)^{-n} \left( \int d^n k e^{-iky} \mathcal{L}_{\varepsilon}^{\pm}(k) \right)_{\varepsilon} \quad (6.5)$$

where $ky = \eta^{a\beta}k_{a}y_{\beta}$, one can work in a sort of localized momentum space. Expanding (6.2) in normal coordinates and converting to $k$-space, $(\mathcal{L}_{\varepsilon}^{\pm}(k))_{\varepsilon}$ can readily be solved by iteration to any adiabatic order. The result to adiabatic order four (i.e., four derivatives of the metric) is

$$(\mathcal{L}_{\varepsilon}^{\pm}(k))_{\varepsilon} = (k^2 - m^2)^{-1} - \left(\frac{1}{6} - \xi\right)(k^2 - m^2)^{-2}(\mathbf{R}^{\pm}(\varepsilon))_{\varepsilon} + \\ + \frac{i}{2}\left(\frac{1}{6} - \xi\right)\partial^{\alpha}(k^2 - m^2)^{-2}(\mathbf{R}_{;\alpha}^{\pm}(\varepsilon))_{\varepsilon} - \\ - \frac{1}{3}\left[(a_{\alpha}^{\pm\beta}(\varepsilon))_{\varepsilon}\right]\partial^{\alpha}\partial^{\beta}(k^2 - m^2)^{-2} + \left[\left(\frac{1}{6} - \xi\right)^2(\mathbf{R}^{\pm 2}(\varepsilon))_{\varepsilon} + \frac{2}{3}(a_{\lambda}^{\pm\gamma}(\varepsilon))_{\varepsilon}\right](k^2 - m^2)^{-3} \quad (6.6)$$

where $\partial_{\alpha} = \partial/\partial k^{\alpha}$,

$$(a_{\alpha\beta}^{\pm}(\varepsilon))_{\varepsilon} \propto \left(\frac{1}{2} - \xi\right)(\mathbf{R}_{;\alpha\beta}^{\pm}(\varepsilon))_{\varepsilon} + \frac{1}{120}(\mathbf{R}_{;\alpha\beta}^{\pm}(\varepsilon))_{\varepsilon} - \frac{1}{140}(\mathbf{R}_{\alpha\beta;\lambda}^{\pm\lambda}(\varepsilon))_{\varepsilon} - \\ - \frac{1}{30}\left[(\mathbf{R}_{\alpha}^{\pm\lambda}(\varepsilon))_{\varepsilon}\right](\mathbf{R}_{\lambda\beta}^{\pm}(\varepsilon))_{\varepsilon} + \frac{1}{60}\left[(\mathbf{R}_{\alpha}^{\pm\lambda\lambda}(\varepsilon))_{\varepsilon}\right](\mathbf{R}_{\kappa\lambda}^{\pm}(\varepsilon))_{\varepsilon} + \\ + \frac{1}{60}\left[(\mathbf{R}_{\lambda}^{\pm\mu\kappa}(\varepsilon))_{\varepsilon}\right](\mathbf{R}_{\lambda\mu\kappa\beta}(\varepsilon))_{\varepsilon}, \quad (6.7)$$

and we are using the symbol $\propto$ to indicate that this is an asymptotic expansion. One ensures that Eq.(6.5) represents a time-ordered product by performing the $k^0$ integral
along the appropriate contour in Pic.3. This is equivalent to replacing $m^2$ by $m^2 - i\epsilon$.
Similarly, the adiabatic expansions of other Green functions can be obtained by using
the other contours in Pic.3.Substituting Eq.(6.6) into Eq.(6.5) gives

$$ \left\{ \left( \int d^n k e^{-iky} \left[ a_0^{\pm}(x,x';\epsilon) + a_1^{\pm}(x,x';\epsilon) \left(-\frac{\partial}{\partial m^2}\right) + a_2^{\pm}(x,x';\epsilon) \left(\frac{\partial}{\partial m^2}\right)^2 \right] (k^2 - m^2)^{-1} \right)_{\epsilon} \right\} (2\pi)^{-n} = \tag{6.8} $$

where

$$ (a_0^{\pm}(x,x';\epsilon))_{\epsilon} = 1 \qquad (6.9) $$

and, to adiabatic order 4,

$$ \left\{ \begin{array}{l} (a_1^{\pm}(x,x';\epsilon))_{\epsilon} = \\ \left(\frac{1}{6} - \xi\right)(\mathbf{R}^{\pm}(\epsilon))_{\epsilon} - \frac{i}{2}\left(\frac{1}{6} - \xi\right)[(\mathbf{R}_{;a}^{\pm}(\epsilon))_{\epsilon}]y^{\alpha} - \frac{1}{3}\left[(a_{\alpha\beta}^{\pm}(\epsilon))_{\epsilon}\right]y^{\alpha}y^{\beta} \quad (6.10) \\ \\ (a_2^{\pm}(x,x';\epsilon))_{\epsilon} = \frac{1}{2}\left(\frac{1}{6} - \xi\right)(\mathbf{R}^{\pm 2}(\epsilon))_{\epsilon} + \frac{1}{3}(a_{\lambda}^{\pm\lambda}(\epsilon))_{\epsilon} \end{array} \right. $$

with all geometric quantities on the right-hand side of Eq.(6.10) evaluated at $x'$.

<figure><img src="image_24.png" /><figcaption>Fig.4. The contour in the complex $k^0$ plane $\mathbb{C}$ to be used in the evaluation of the integral giving $\mathcal{L}^+$. The cross indicates the pole at</figcaption></figure>

$$ k^0 = (|\mathbf{k}|^2 + m^2)^{1/2}. $$

If one uses the canonical integral representation

$$ (k^2 - m^2 + i\epsilon)^{-1} = -i \int_0^\infty ds e^{is(k^2 - m^2 + i\epsilon)} \qquad (6.11) $$

in Eq.(6.8), then the $d^n k$ integration may be interchanged with the $ds$ integration, and
performed explicitly to yield (dropping the $i\epsilon$)

$$ \begin{aligned} (\mathcal{L}_{\epsilon}^{\pm}(x,x'))_{\epsilon} &= -i(4\pi)^{-n/2} \left( \int_0^{\infty} ids(is)^{-n/2} \exp\left[-im^2s + \frac{\sigma(x,x')}{2is}\right] \mathcal{F}_{\epsilon}^{\pm}(x,x';is) \right)_{\epsilon} \quad (6.12) \\ \sigma(x,x') &= \frac{1}{2}y_a y^{\alpha}. \end{aligned} $$
The function $\sigma(x, x')$ which is one-half of the square of the proper distance between $x$ and $x'$, while the function $(\mathcal{F}_{\varepsilon}(x, x'; is))_{\varepsilon}$ has the following asymptotic adiabatic expansion

$$ (\mathcal{F}_{\varepsilon}^{\pm}(x, x'; is))_{\varepsilon} \propto (a_{0}^{\pm}(x, x'; \varepsilon))_{\varepsilon} + is(a_{1}^{\pm}(x, x'; \varepsilon))_{\varepsilon} + (is)^{2}(a_{2}^{\pm}(x, x'; \varepsilon))_{\varepsilon} + \dots \quad (6.13) $$

Using Eq.(6.4), equation (6.12) gives a representation of $(G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon}$:

$$ \left\{ \begin{array}{l} (G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} = \\ -i(4\pi)^{-n/2} \left( \left[ (\Delta_{\pm}^{1/2}(x, x'; \varepsilon))_{\varepsilon} \right]_{0}^{\infty} \int ds (is)^{-n/2} \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right] \mathcal{F}_{\varepsilon}(x, x'; is) \right)_{\varepsilon} \end{array} \right. \quad (6.14) $$

where $(\Delta_{\pm}(x, x'; \varepsilon))_{\varepsilon}$ is the distributional Van Vleck determinant

$$ (\Delta_{\pm}(x, x'; \varepsilon))_{\varepsilon} = -\det[\partial_{\mu}\partial_{\nu}\sigma(x, x')] (g^{\pm}(x, \varepsilon)g^{\pm}(x', \varepsilon)]^{-1/2})_{\varepsilon} \quad (6.15) $$

In the normal coordinates about $x'$ that we are currently using, $(\Delta_{\pm}(x, x'; \varepsilon))_{\varepsilon}$ reduces to $([g^{\pm}(x, \varepsilon)]^{-1/2})_{\varepsilon}$. The full asymptotic expansion of $(\mathcal{F}_{\varepsilon}^{\pm}(x, x'; is))_{\varepsilon}$ to all adiabatic orders are

$$ (\mathcal{F}_{\varepsilon}^{\pm}(x, x'; is))_{\varepsilon} \propto \sum_{j=0}^{\infty} (is)^j (a_{2}^{\pm}(x, x'; \varepsilon))_{\varepsilon} \quad (6.16) $$

with $(a_{0}^{\pm}(x, x'; \varepsilon))_{\varepsilon} = 1$, the other $(a_{j}^{\pm}(x, x'; \varepsilon))_{\varepsilon}$ being given by canonical recursion relations which enable their adiabatic expansions to be obtained. The expansions (6.13) and (6.16) are, however, only asymptotic approximations in the limit of large adiabatic parameter $T$.

If (6.16) is substituted into (6.14) the integral can be performed to give the adiabatic expansion of the Feynman propagator in coordinate space:

$$ (G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} \propto -(4\pi i)^{-n/2} \left( \Delta_{\pm}^{1/2}(x, x'; \varepsilon) \sum_{j=0}^{\infty} a_{j}^{\pm}(x, x'; \varepsilon) \left(-\frac{\partial}{\partial m^2}\right)^j \times \left[ \left(-\frac{2m^2}{\sigma}\right)^{\frac{n-2}{4}} H_{(n-2)/2}^{(2)}\left((2m^2\sigma)^{\frac{1}{2}}\right) \right]_{\varepsilon} \right) \quad (6.17) $$

which, strictly, a small imaginary part $i\epsilon$ should be subtracted from $\sigma$. Since we have not imposed global boundary conditions on the distributional Green function Colombeau solution of (6.2), the expansion (6.17) does not determine the particular vacuum state in (6.1). In particular, the "$i\epsilon$" in the expansion of $(G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon}$ only ensures that (6.17) represents the expectation value, in some set of states, of a time-ordered product of fields. Under some circumstances the use of "$i\epsilon$" in the exact representation (6.14) may give additional information concerning the global nature of the states

## 6.2. Effective action for the quantum matter fields in curved distributional spacetime

As in classical case one can obtain Colombeau generalized quantity $(W_{\varepsilon})_{\varepsilon}$, called the effective action for the quantum matter fields in curved distributional spacetime, which, when functionally differentiated, yields