$$ \Xi_{\epsilon}^{+}(r_{\epsilon}, \theta) = \frac{(r_{\epsilon} - r_{E^{-}}(\theta)) \sqrt{(r_{\epsilon} - r_{E^{+}}(\theta))^{2} + \epsilon^{2}}}{\rho_{\epsilon}^{2}(r_{\epsilon}) \Delta_{a, \epsilon} = r_{\epsilon}^{2} - 2mr_{\epsilon} + a^{2}}, \qquad (7.2.6) $$

Where $\rho_{\epsilon}^{2} = \rho_{\epsilon}^{2}(r_{\epsilon}) = r_{\epsilon}^{2} + a^{2} \cos^{2} \theta, \epsilon \in (0, \delta]$, $r_{\epsilon} \ge r_{E^{+}}(\theta) \ge r_{+}$. Thus Colombeau generalized metric (above ergosurface $r = r_{E^{+}}(\theta)$) corresponding to classical Kerr metric (7.2.1) reads

$$ (ds_{\epsilon}^{+2})_{\epsilon} = -\left[ (\Xi_{\epsilon}(r_{\epsilon}, \theta))_{\epsilon} \right] dt^{2} - \left[ \left( \frac{4mr_{\epsilon}a \sin^{2}\theta}{\rho_{\epsilon}^{2}} \right)_{\epsilon} \right] dtd\phi + \left[ \left( \frac{\rho_{\epsilon}^{2}}{\Delta_{a,\epsilon}} \right)_{\epsilon} \right] \left[ (dr_{\epsilon}^{2})_{\epsilon} + (\rho_{\epsilon}^{2})_{\epsilon} d\theta^{2} + \left( r_{\epsilon}^{2} + a^{2} + \frac{2mra^{2}\sin^{2}\theta}{\rho_{\epsilon}^{2}} \right)_{\epsilon} \sin^{2}\theta d\phi^{2} \right]. \qquad (7.2.7) $$

**Remark 7.2.4.** Let $\Xi(\theta)$ be a submanifold given by equation $\phi = \text{const}$, then Colombeau generalized metric (7.2.7) restricted on $\Xi(\theta)$ reads

$$ (ds_{\epsilon}^{+2})_{\epsilon} = -\left[ (\Xi_{\epsilon}(r_{\epsilon}, \theta))_{\epsilon} \right] dt^{2} + \left[ \left( \frac{\rho_{\epsilon}^{2}}{\Delta_{a,\epsilon}} \right)_{\epsilon} \right] \left[ (dr_{\epsilon}^{2})_{\epsilon} + \left[ (\rho_{\epsilon}^{2})_{\epsilon} \right] d\theta^{2} \right]. \qquad (7.2.8) $$

Note that Colombeau generalized metric (7.2.7) nondegenerate on outer ergosurfaces $(r_{\epsilon})_{\epsilon} = r_{E^{+}}(\theta)$, see Fig.7.1.

**Remark 7.2.4.** Note that for small values of the parameter $a$, where we may neglect terms of the order of $a^2$, we get from (7.2.7) effectively the following Colombeau generalized metric

$$ (ds_{\epsilon}^{+2})_{\epsilon} = -\left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\epsilon})_{\epsilon}}\right) dt^{2} + \left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\epsilon})_{\epsilon}}\right)^{-1} \left[ (dr_{\epsilon}^{2})_{\epsilon} \right] + \left[ (r_{\epsilon}^{2})_{\epsilon} \right] (d\theta^{2} + \sin^{2}\theta d\phi^{2}) - \frac{4ma\sin^{2}\theta}{(r_{\epsilon})_{\epsilon}} dtd\phi, \qquad (7.2.9) $$

Where $r_{E^{+}}(\theta) = m + \sqrt{m^2 - a^2 \cos^2 \theta}$.

**Remark 7.2.5.** Note that Colombeau generalized metric (7.2.9) restricted on $\Xi(\theta)$ reads

$$ (ds_{\epsilon}^{+2})_{\epsilon} = -\left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\epsilon})_{\epsilon}}\right) dt^{2} + \left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\epsilon})_{\epsilon}}\right)^{-1} \left[ (dr_{\epsilon}^{2})_{\epsilon} \right] + \left[ (r_{\epsilon}^{2})_{\epsilon} \right] d\theta^{2} \qquad (7.2.10) $$

(I) Let $(\mathbf{R}^{a \ll 1}(r_{\epsilon}, \epsilon))_{\epsilon}$ be Colombeau generalized curvature scalar $(\mathbf{R}(r_{\epsilon}, \epsilon))_{\epsilon}$ corresponding to the metric (7.2.10) with $a \ll 1$. Main singular part sing $[(\mathbf{R}^{a \ll 1}(r_{\epsilon}, \epsilon))_{\epsilon}]$ of the Colombeau generalized curvature scalar $(\mathbf{R}^{a \ll 1}(r_{\epsilon}, \epsilon))_{\epsilon}$ corresponding to the metric (7.2.10) reads

$$ \text{sing}[(\mathbf{R}^{a \ll 1}(r_{\epsilon}, \epsilon))_{\epsilon}] = \tilde{\mathbf{R}} \left( \frac{\epsilon^2}{r_{E_+}(\theta) \left[ (r_{\epsilon} - r_{E_+}(\theta))^2 + \epsilon^2 \right]^{3/2}} \right)_{\epsilon}, \qquad (7.2.11) $$

Where $\text{cl}[(r_{\epsilon})_{\epsilon}] \approx \tilde{\mathbf{R}} r_{E^{+}}(\theta)$, see Appendix Eq. (A.12).

(II) Let $(\mathbf{R}^{\mu\nu(a \ll 1)}(r_{\epsilon}, \epsilon) \mathbf{R}_{\mu\nu}^{(a \ll 1)}(r_{\epsilon}, \epsilon))_{\epsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_{\epsilon}, \epsilon) \mathbf{R}_{\mu\nu}(r_{\epsilon}, \epsilon))_{\epsilon}$ corresponding to the metric (7.2.10) with $a \ll 1$. From Eq. (7.2.10) and Eq. (A.1)-Eq. (A.2) one obtains that main singular part
$$ \Xi_{\varepsilon}^{+}(r_{\varepsilon}, \theta) = \frac{(r_{\varepsilon} - r_{E^{-}}(\theta)) \sqrt{(r_{\varepsilon} - r_{E^{+}}(\theta))^{2} + \varepsilon^{2}}}{\rho_{\varepsilon}^{2}(r_{\varepsilon}) \Delta_{a,\varepsilon} = r_{\varepsilon}^{2} - 2mr_{\varepsilon} + a^{2}}, \qquad (7.2.6) $$

Where $\rho_{\varepsilon}^{2} = \rho_{\varepsilon}^{2}(r_{\varepsilon}) = r_{\varepsilon}^{2} + a^{2} \cos^{2} \theta, \varepsilon \in (0, \delta]$, $r_{\varepsilon} \ge r_{E^{+}}(\theta) \ge r_{+}$. Thus Colombeau generalized metric (above ergosurface $r = r_{E^{+}}(\theta)$) corresponding to classical Kerr metric (7.2.1) reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\left[ (\Xi_{\varepsilon}(r_{\varepsilon}, \theta))_{\varepsilon} \right] dt^{2} - \left[ \left( \frac{4mr_{\varepsilon}a \sin^{2}\theta}{\rho_{\varepsilon}^{2}} \right)_{\varepsilon} \right] dtd\phi + \left[ \left( \frac{\rho_{\varepsilon}^{2}}{\Delta_{a,\varepsilon}} \right)_{\varepsilon} \right] \left[ (dr_{\varepsilon}^{2})_{\varepsilon} + (\rho_{\varepsilon}^{2})_{\varepsilon} \right] d\theta^{2} + \left( r_{\varepsilon}^{2} + a^{2} + \frac{2mra^{2} \sin^{2}\theta}{\rho_{\varepsilon}^{2}} \right)_{\varepsilon} \sin^{2}\theta d\phi^{2}. \qquad (7.2.7) $$

**Remark 7.2.4.** Let $\Xi(\theta)$ be a submanifold given by equation $\phi = \text{const}$, then Colombeau generalized metric (7.2.7) restricted on $\Xi(\theta)$ reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\left[ (\Xi_{\varepsilon}(r_{\varepsilon}, \theta))_{\varepsilon} \right] dt^{2} + \left[ \left( \frac{\rho_{\varepsilon}^{2}}{\Delta_{a,\varepsilon}} \right)_{\varepsilon} \right] \left[ (dr_{\varepsilon}^{2})_{\varepsilon} \right] + \left[ (\rho_{\varepsilon}^{2})_{\varepsilon} \right] d\theta^{2}. \qquad (7.2.8) $$

Note that Colombeau generalized metric (7.2.7) nondegenerate on outer ergosurfaces $(r_{\varepsilon})_{\varepsilon} = r_{E^{+}}(\theta)$, see Pic.7.1.

**Remark 7.2.4.** Note that for small values of the parameter $a$, where we may neglect terms of the order of $a^2$, we get from (7.2.7) effectively the following Colombeau generalized metric

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\varepsilon})_{\varepsilon}}\right) dt^{2} + \left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\varepsilon})_{\varepsilon}}\right)^{-1} \left[ (dr_{\varepsilon}^{2})_{\varepsilon} \right] + \left[ (r_{\varepsilon}^{2})_{\varepsilon} \right] (d\theta^{2} + \sin^{2}\theta d\phi^{2}) - \frac{4ma\sin^{2}\theta}{(r_{\varepsilon})_{\varepsilon}} dtd\phi, \qquad (7.2.9) $$

Where $r_{E^{+}}(\theta) = m + \sqrt{m^2 - a^2 \cos^2 \theta}$.

**Remark 7.2.5.** Note that Colombeau generalized metric (7.2.9) restricted on $\Xi(\theta)$ reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\varepsilon})_{\varepsilon}}\right) dt^{2} + \left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\varepsilon})_{\varepsilon}}\right)^{-1} \left[ (dr_{\varepsilon}^{2})_{\varepsilon} \right] + \left[ (r_{\varepsilon}^{2})_{\varepsilon} \right] d\theta^{2} \qquad (7.2.10) $$

(I) Let $(\mathbf{R}^{a \ll 1}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized curvature scalar $(\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.10) with $a \ll 1$. Main singular part sing $[(\mathbf{R}^{a \ll 1}(r_{\varepsilon}, \varepsilon))_{\varepsilon}]$ of the Colombeau generalized curvature scalar $(\mathbf{R}^{a \ll 1}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.10) reads

$$ \text{sing}[(\mathbf{R}^{a \ll 1}(r_{\varepsilon}, \varepsilon))_{\varepsilon}] = \tilde{\mathbf{R}} \left( \frac{\varepsilon^2}{r_{E_+}(\theta) [(r_{\varepsilon} - r_{E_+}(\theta))^2 + \varepsilon^2]} \right)_{\varepsilon}, \qquad (7.2.11) $$

Where $\text{cl}[(r_{\varepsilon})_{\varepsilon}] \approx \tilde{\mathbf{R}} r_{E^{+}}(\theta)$, see Appendix Eq. (A.12).

(II) Let $(\mathbf{R}^{\mu\nu(a \ll 1)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a \ll 1)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.10) with $a \ll 1$. From Eq. (7.2.10) and Eq. (A.1)-Eq. (A.2) one obtains that main singular part
$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] $$

of the quadratic scalar $(\mathbf{R}^{\mu\nu(a\ll 1)}(r_\varepsilon, \varepsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon$ reads:

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] = \bar{\mathbb{R}} \left( \frac{\varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_\varepsilon - 2m)^2]^3} \right)_\varepsilon . \quad (7.2.12) $$

(III) Let $(\mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)}(r_\varepsilon, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon$ corresponding to the metric (7.2.10) with $a \ll 1$. From Eq.(7.2.10) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] $$

of the Colombeau generalized quadratic scalar

$$ (\mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon))_\varepsilon \text{ reads} $$

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] = \bar{\mathbb{R}} \left( \frac{\varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_\varepsilon - 2m)^2]^3} \right)_\varepsilon . \quad (7.2.13) $$

**Remark 7.2.6.** Note that from Eq.(7.2.11)-Eq.(7.2.13) at outer ergosurfaces $(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)$, (see Pic.7.1) follows that

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{a\ll 1} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] \Big|_{(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)} = \bar{\mathbb{R}} r_{E_+}^{-1}(\theta) (\varepsilon^{-1})_\varepsilon \approx \bar{\mathbb{R}} \infty. \quad (7.2.14) $$

and

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] \Big|_{(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)} = \bar{\mathbb{R}} r_{E_+}^{-4}(\theta) (\varepsilon^{-2})_\varepsilon \approx \bar{\mathbb{R}} \infty, \quad (7.2.15) $$

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\varepsilon, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\varepsilon, \varepsilon) \right)_\varepsilon \right] \Big|_{(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)} = \bar{\mathbb{R}} r_{E_+}^{-4}(\theta) (\varepsilon^{-2})_\varepsilon \approx \bar{\mathbb{R}} \infty. $$

Let $(\mathbf{R}^{a<m}(r_\varepsilon, \varepsilon))_\varepsilon$ be Colombeau generalized curvature scalar $(\mathbf{R}^{a<m}(r_\varepsilon, \varepsilon))_\varepsilon$ corresponding to the metric (7.2.8) with $a < m$. We let now that

$$ \Delta_{\varepsilon} = A_{\varepsilon}(B_{\varepsilon} + C_{\varepsilon}) = A_{\varepsilon} \rho_{\varepsilon}^{2} \Delta_{a,\varepsilon}^{-1}, B_{\varepsilon} r_{\varepsilon}^{2} = \rho_{\varepsilon}^{2}, B_{\varepsilon} + C_{\varepsilon} = \rho_{\varepsilon}^{2} \Delta_{a,\varepsilon}^{-1}, A_{\varepsilon} = \Xi_{\varepsilon}^{+}(r, \theta), D_{\varepsilon} = 0. \quad (7.2.16) $$

From Eq.(7.2.8),Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) we obtain

$$ \begin{aligned} (\mathbf{R}^{a<m}(r_\varepsilon, \varepsilon))_\varepsilon = & \left( \frac{A_\varepsilon}{\Delta_\varepsilon} \left[ \frac{2}{r_\varepsilon} \left( -2\frac{A'_\varepsilon}{A_\varepsilon} - 3\frac{B'_\varepsilon}{B_\varepsilon} + \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} \right) + \frac{2}{r^2} \frac{C_\varepsilon}{B_\varepsilon} - \frac{A''_\varepsilon}{A_\varepsilon} - 2\frac{B''_\varepsilon}{B_\varepsilon} \right. \right. \\ & \left. \left. + \frac{1}{2} \left( \frac{B'_\varepsilon}{B_\varepsilon} \right)^2 - 2\frac{A'_\varepsilon B'_\varepsilon}{A_\varepsilon B_\varepsilon} + \left( \frac{1}{2}\frac{A'_\varepsilon}{A_\varepsilon} + \frac{B'_\varepsilon}{B_\varepsilon} \right) \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} \right] \right)_\varepsilon \\ & \left( \frac{\Delta_{a,\varepsilon}}{\rho_\varepsilon^2} \left[ \frac{2}{r_\varepsilon} \left( -2\frac{A'_\varepsilon}{A_\varepsilon} - 3\frac{B'_\varepsilon}{B_\varepsilon} + \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} \right) + \frac{2}{r^2} \frac{C_\varepsilon}{B_\varepsilon} - \frac{A''_\varepsilon}{A_\varepsilon} - 2\frac{B''_\varepsilon}{B_\varepsilon} \right. \right. \\ & \left. \left. + \frac{1}{2} \left( \frac{B'_\varepsilon}{B_\varepsilon} \right)^2 - 2\frac{A'_\varepsilon B'_\varepsilon}{A_\varepsilon B_\varepsilon} + \left( \frac{1}{2}\frac{A'_\varepsilon}{A_\varepsilon} + \frac{B'_\varepsilon}{B_\varepsilon} \right) \frac{\Delta'_\varepsilon}{\Delta_\varepsilon} \right] \right)_\varepsilon . \end{aligned} \quad (7.2.17) $$

Note that
$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon) \right)_\epsilon \right] $$

of the quadratic scalar $(\mathbf{R}^{\mu\nu(a\ll 1)}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)}(r_\epsilon, \epsilon))_\epsilon$ reads:

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon) \right)_\epsilon \right] = \bar{\mathbb{R}} \left( \frac{\epsilon^4}{4 (r_{E_+}(\theta))^4 [\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_\epsilon . \quad (7.2.12) $$

(III) Let $(\mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)}(r_\epsilon, \epsilon))_\epsilon$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_\epsilon, \epsilon))_\epsilon$ corresponding to the metric (7.2.10) with $a \ll 1$. From Eq.(7.2.10) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon) \right)_\epsilon \right] $$

of the Colombeau generalized quadratic scalar

$$ (\mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon))_\epsilon \text{ reads} $$

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon) \right)_\epsilon \right] = \bar{\mathbb{R}} \left( \frac{\epsilon^4}{4 (r_{E_+}(\theta))^4 [\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_\epsilon . \quad (7.2.13) $$

**Remark 7.2.6.** Note that from Eq.(7.2.11)-Eq.(7.2.13) at outer ergosurfaces $(r_\epsilon)_\epsilon = r_{E_+}(\theta)$, (see Pic.7.1) follows that

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{a\ll 1} (r_\epsilon, \epsilon) \right)_\epsilon \right] \Big|_{(r_\epsilon)_\epsilon = r_{E_+}(\theta)} = \bar{\mathbb{R}} r_{E_+}^{-1}(\theta) (\epsilon^{-1})_\epsilon \approx \bar{\mathbb{R}} \infty. \quad (7.2.14) $$

and

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon) \right)_\epsilon \right] \Big|_{(r_\epsilon)_\epsilon = r_{E_+}(\theta)} = \bar{\mathbb{R}} r_{E_+}^{-4}(\theta) (\epsilon^{-2})_\epsilon \approx \bar{\mathbb{R}} \infty, \quad (7.2.15) $$

$$ \mathrm{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a\ll 1)} (r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a\ll 1)} (r_\epsilon, \epsilon) \right)_\epsilon \right] \Big|_{(r_\epsilon)_\epsilon = r_{E_+}(\theta)} = \bar{\mathbb{R}} r_{E_+}^{-4}(\theta) (\epsilon^{-2})_\epsilon \approx \bar{\mathbb{R}} \infty. $$

Let $(\mathbf{R}^{a<m}(r_\epsilon, \epsilon))_\epsilon$ be Colombeau generalized curvature scalar $(\mathbf{R}^{a<m}(r_\epsilon, \epsilon))_\epsilon$ corresponding to the metric (7.2.8) with $a < m$. We let now that

$$ \Delta_{\epsilon} = A_{\epsilon}(B_{\epsilon} + C_{\epsilon}) = A_{\epsilon}\rho_{\epsilon}^{2}\Delta_{a,\epsilon}^{-1}, B_{\epsilon}r_{\epsilon}^{2} = \rho_{\epsilon}^{2}, B_{\epsilon} + C_{\epsilon} = \rho_{\epsilon}^{2}\Delta_{a,\epsilon}^{-1}, A_{\epsilon} = \Xi_{\epsilon}^{+}(r, \theta), D_{\epsilon} = 0. \quad (7.2.16) $$

From Eq.(7.2.8),Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) we obtain

$$ \begin{aligned} (\mathbf{R}^{a<m}(r_\epsilon, \epsilon))_\epsilon = & \left( \frac{A_\epsilon}{\Delta_\epsilon} \left[ \frac{2}{r_\epsilon} \left( -2\frac{A'_\epsilon}{A_\epsilon} - 3\frac{B'_\epsilon}{B_\epsilon} + \frac{\Delta'_\epsilon}{\Delta_\epsilon} \right) + \frac{2}{r^2} \frac{C_\epsilon}{B_\epsilon} - \frac{A''_\epsilon}{A_\epsilon} - 2\frac{B''_\epsilon}{B_\epsilon} \right. \right. \\ & \left. \left. + \frac{1}{2} \left( \frac{B'_\epsilon}{B_\epsilon} \right)^2 - 2\frac{A'_\epsilon B'_\epsilon}{A_\epsilon B_\epsilon} + \left( \frac{1}{2}\frac{A'_\epsilon}{A_\epsilon} + \frac{B'_\epsilon}{B_\epsilon} \right) \frac{\Delta'_\epsilon}{\Delta_\epsilon} \right] \right)_\epsilon \\ & \left( \frac{\Delta_{a,\epsilon}}{\rho_\epsilon^2} \left[ \frac{2}{r_\epsilon} \left( -2\frac{A'_\epsilon}{A_\epsilon} - 3\frac{B'_\epsilon}{B_\epsilon} + \frac{\Delta'_\epsilon}{\Delta_\epsilon} \right) + \frac{2}{r^2} \frac{C_\epsilon}{B_\epsilon} - \frac{A''_\epsilon}{A_\epsilon} - 2\frac{B''_\epsilon}{B_\epsilon} \right. \right. \\ & \left. \left. + \frac{1}{2} \left( \frac{B'_\epsilon}{B_\epsilon} \right)^2 - 2\frac{A'_\epsilon B'_\epsilon}{A_\epsilon B_\epsilon} + \left( \frac{1}{2}\frac{A'_\epsilon}{A_\epsilon} + \frac{B'_\epsilon}{B_\epsilon} \right) \frac{\Delta'_\epsilon}{\Delta_\epsilon} \right] \right)_\epsilon . \end{aligned} \quad (7.2.17) $$

Note that
$$ \frac{\partial}{\partial r_{\varepsilon}} \sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^{2} + \varepsilon^{2}} = \frac{r_{\varepsilon} - r_{E_{+}}(\theta)}{\sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^{2} + \varepsilon^{2}}} \qquad (7.2.18) $$
$$ \frac{\partial^{2}}{\partial r_{\varepsilon}^{2}} \sqrt{(r_{\varepsilon} - r_{E_{+}})^{2} + \varepsilon^{2}} = \frac{\varepsilon^{2}}{[(r_{\varepsilon} - r_{E_{+}})^{2} + \varepsilon^{2}]^{\frac{3}{2}}}. $$

From Eq.(7.2.17) and Eq.(7.2.18) one obtains that main singular part **sing** $[(\mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon))_{\varepsilon}]$ of the Colombeau generalized curvature scalar $(\mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.8) (mod nonsingular multiplier) reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \bar{\mathbf{R}} \left( \frac{\varepsilon^2}{\left[ (r_{\varepsilon} - r_{E_{+}}(\theta))^2 + \varepsilon^2 \right]^2} \right)_{\varepsilon} . \qquad (7.2.19) $$

**Remark 7.2.7. (I)** Let $(\mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.8) with $a < m$. From Eq.(7.2.8)-Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] $$

of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \bar{\mathbf{R}} \left( \frac{\varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} . \qquad (7.2.20) $$

(II) Let $(\mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.8) with $a < m$. From Eq.(7.2.8), Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] $$

of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \bar{\mathbf{R}} \left( \frac{\varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} . \qquad (7.2.21) $$

## II. Critical Kerr gravitational singularity: a=m.

Note that in contrast with full distributional Kerr spacetime the case of the critical Kerr gravitational singularity considered in this subsection (see Remark 7.2.3) not principal different in comparison with a case of the slow Kerr gravitational singularity considered above. In particular the Eqs.(7.2.19)-(7.2.21) holds with $r_{E_+}(\theta)$ given by Eq.(7.2.22)

$$ r_{E_+}(\theta) = m + \sqrt{m^2 - a^2 \cos^2 \theta} = m(1 + \sin^2 \theta). \qquad (7.2.22) $$

## III. Fast Kerr gravitational singularity: a≠m.

Let $\Xi_{\eta}(\theta)$ be a submanifold given by equations (i) $\phi = \text{const}$ and (ii) $m^2 - a^2 \cos^2 \theta \ge 0$, i.e.
$$ \frac{\partial}{\partial r_{\varepsilon}} \sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^{2} + \varepsilon^{2}} = \frac{r_{\varepsilon} - r_{E_{+}}(\theta)}{\sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^{2} + \varepsilon^{2}}} \qquad (7.2.18) $$
$$ \frac{\partial^{2}}{\partial r_{\varepsilon}^{2}} \sqrt{(r_{\varepsilon} - r_{E_{+}})^{2} + \varepsilon^{2}} = \frac{\varepsilon^{2}}{[(r_{\varepsilon} - r_{E_{+}})^{2} + \varepsilon^{2}]^{\frac{3}{2}}}. $$

From Eq.(7.2.17) and Eq.(7.2.18) one obtains that main singular part **sing** $[(\mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon))_{\varepsilon}]$ of the Colombeau generalized curvature scalar $(\mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.8) (mod nonsingular multiplier) reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \bar{\mathbf{R}} \left( \frac{\varepsilon^2}{\left[ (r_{\varepsilon} - r_{E_+}(\theta))^2 + \varepsilon^2 \right]^2} \right)_\varepsilon . \qquad (7.2.19) $$

**Remark 7.2.7. (I)** Let $(\mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.8) with $a < m$. From Eq.(7.2.8)-Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] $$

of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \bar{\mathbf{R}} \left( \frac{\varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} . \qquad (7.2.20) $$

(II) Let $(\mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.8) with $a < m$. From Eq.(7.2.8), Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] $$

of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \bar{\mathbf{R}} \left( \frac{\varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} . \qquad (7.2.21) $$

## II. Critical Kerr gravitational singularity: $a=m$.

Note that in contrast with full distributional Kerr spacetime the case of the critical Kerr gravitational singularity considered in this subsection (see Remark 7.2.3) not principal different in comparison with a case of the slow Kerr gravitational singularity considered above. In particular the Eqs.(7.2.19)-(7.2.21) holds with $r_{E_+}(\theta)$ given by Eq.(7.2.22)

$$ r_{E_+}(\theta) = m + \sqrt{m^2 - a^2 \cos^2 \theta} = m(1 + \sin^2 \theta). \qquad (7.2.22) $$

## III. Fast Kerr gravitational singularity: $a \ll m$.

Let $\Xi_{\eta}(\theta)$ be a submanifold given by equations (i) $\phi = \text{const}$ and (ii) $m^2 - a^2 \cos^2 \theta \ge 0$, i.e.
$$ \cos^2 \theta \le \frac{m^2}{a^2} = \eta. \qquad (7.2.23) $$

Let $\Theta_\eta$ be a set $\Theta_\eta = \{\theta | \cos^2 \theta \le \eta\}$ and let $\chi(\theta, \eta)$ be the indicator function of the set $\Theta_\eta$, i.e. $\chi(\theta, \eta)$ is the function defined to be identically 1 on $\Theta_\eta$, and is 0 elsewhere.

We introduce now the following regularized (above ergosurface $r = r_{E_+}(\theta)$, $\theta \in \Theta_\eta$) quantity

$$ \Xi_{\varepsilon}^{+}(r_{\varepsilon}, \theta, \eta) = \frac{\chi(\theta, \eta)(r_{\varepsilon} - r_{E^{-}}(\theta))\sqrt{(r_{\varepsilon} - r_{E^{+}}(\theta))^{2} + \varepsilon^{2}}}{\rho_{\varepsilon}^{2}(r_{\varepsilon})}, \qquad (7.2.25) $$

where $\rho_{\varepsilon}^2 = \rho_{\varepsilon}^2(r_{\varepsilon}) = r_{\varepsilon}^2 + a^2 \cos^2 \theta$, $\varepsilon \in (0, \delta]$, $r_{\varepsilon} \ge r_{E_{+}}(\theta) > 0$. Thus Colombeau generalized metric (above ergosurface $r = r_{E_+}(\theta)$) corresponding to classical Kerr metric (7.2.1) reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\chi(\theta, \eta) [(\Xi_{\varepsilon}(r_{\varepsilon}, \theta, \eta))_{\varepsilon}] dt^2 - \left[ \left( \frac{4mr_{\varepsilon}a \sin^2 \theta}{\rho_{\varepsilon}^2} \right)_{\varepsilon} \right] dtd\phi + \left( r_{\varepsilon}^2 + a^2 + \frac{2mra^2 \sin^2 \theta}{\rho_{\varepsilon}^2} \right)_{\varepsilon} \sin^2 \theta d\phi^2. \qquad (7.2.26) $$

**Remark 7.2.8.** Note that we will consider the distributional Kerr spacetime not as full distributional BH with Colombeau generalized metric (7.2.7), but only as gravitational singularity located on submanifold $\cup_{\theta \in \Theta_\eta} \Xi_\eta(\theta)$ which coincide with an part of the outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11),Eq.(7.2.19) and Eq.(7.2.20) submanifold $\cup_{\theta \in \Theta_\eta} \Xi_\eta(\theta)$ presented the singular boundary of distributional spacetime with Colombeau generalized metric (7.2.26).

**Remark 7.2.9.** Let $\tilde{\Xi}_\eta(\theta)$ be a submanifold given by equations (i) $\phi = \text{const}$ and (ii) $\cos^2 \theta \le \eta$, then Colombeau generalized metric (7.2.26) restricted on submanifold $\cup_{\theta \in \Theta_\eta} \tilde{\Xi}_\eta(\theta)$ reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\chi(\theta, \eta) [(\Xi_{\varepsilon}(r_{\varepsilon}, \theta, \eta))_{\varepsilon}] dt^2 + \left[ \left( \frac{\rho_{\varepsilon}^2}{\Delta_{a, \varepsilon}} \right)_{\varepsilon} \right] [(dr_{\varepsilon}^2)_{\varepsilon}] + [(\rho_{\varepsilon}^2)_{\varepsilon}] d\theta^2. \qquad (7.2.27) $$

Note that Colombeau generalized metric (7.2.27) nondegenerate on outer ergosurfaces $(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)$, see Pic.7.3. From Eq.(7.2.27) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing $[(R^{a>m}(r_\varepsilon, \varepsilon))_\varepsilon]$ of the Colombeau generalized curvature scalar $(R^{a>m}(r_\varepsilon, \varepsilon))_\varepsilon$ corresponding

<figure><img src="image_5.png" /><figcaption>Fig. 3. Ergosurface, horizon and singularity for slow Kerr black hole.</figcaption></figure>
$$ \cos^2 \theta \le \frac{m^2}{a^2} = \eta. \qquad (7.2.23) $$

Let $\Theta_{\eta}$ be a set $\Theta_{\eta} = \{\theta | \cos^2 \theta \le \eta\}$ and let $\chi(\theta, \eta)$ be the indicator function of the set $\Theta_{\eta}$, i.e. $\chi(\theta, \eta)$ is the function defined to be identically 1 on $\Theta_{\eta}$, and is 0 elsewhere.

We introduce now the following regularized (above ergosurface $r = r_{E+}(\theta)$, $\theta \in \Theta_{\eta}$) quantity

$$ \Xi_{\varepsilon}^{+}(r_{\varepsilon}, \theta, \eta) = \frac{\chi(\theta, \eta)(r_{\varepsilon} - r_{E-}(\theta))\sqrt{(r_{\varepsilon} - r_{E+}(\theta))^2 + \varepsilon^2}}{\rho_{\varepsilon}^2(r_{\varepsilon})}, \qquad (7.2.25) $$

where $\rho_{\varepsilon}^2 = \rho_{\varepsilon}^2(r_{\varepsilon}) = r_{\varepsilon}^2 + a^2 \cos^2 \theta$, $\varepsilon \in (0, \delta]$, $r_{\varepsilon} \ge r_{E+}(\theta) > 0$. Thus Colombeau generalized metric (above ergosurface $r = r_{E+}(\theta)$) corresponding to classical Kerr metric (7.2.1) reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\chi(\theta, \eta) [(\Xi_{\varepsilon}(r_{\varepsilon}, \theta, \eta))_{\varepsilon}] dt^2 - \left[ \left( \frac{4mr_{\varepsilon}a\sin^2\theta}{\rho_{\varepsilon}^2} \right)_{\varepsilon} \right] dtd\phi + \left( r_{\varepsilon}^2 + a^2 + \frac{2mra^2\sin^2\theta}{\rho_{\varepsilon}^2} \right)_{\varepsilon} \sin^2\theta d\phi^2. \qquad (7.2.26) $$

**Remark 7.2.8.** Note that we will consider the distributional Kerr spacetime not as full distributional BH with Colombeau generalized metric (7.2.7), but only as gravitational singularity located on submanifold $\cup_{\theta \in \Theta_{\eta}} \Xi_{\eta}(\theta)$ which coincide with an part of the outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11),Eq.(7.2.19) and Eq.(7.2.20) submanifold $\cup_{\theta \in \Theta_{\eta}} \Xi_{\eta}(\theta)$ presented the singular boundary of distributional spacetime with Colombeau generalized metric (7.2.26).

**Remark 7.2.9.** Let $\tilde{\Xi}_{\eta}(\theta)$ be a submanifold given by equations (i) $\phi = \text{const}$ and (ii) $\cos^2 \theta \le \eta$, then Colombeau generalized metric (7.2.26) restricted on submanifold $\cup_{\theta \in \Theta_{\eta}} \tilde{\Xi}_{\eta}(\theta)$ reads

$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\chi(\theta, \eta) [(\Xi_{\varepsilon}(r_{\varepsilon}, \theta, \eta))_{\varepsilon}] dt^2 + \left[ \left( \frac{\rho_{\varepsilon}^2}{\Delta_{a,\varepsilon}} \right)_{\varepsilon} \right] [(dr_{\varepsilon}^2)_{\varepsilon}] + [(\rho_{\varepsilon}^2)_{\varepsilon}] d\theta^2. \qquad (7.2.27) $$

Note that Colombeau generalized metric (7.2.27) nondegenerate on outer ergosurfaces $(r_{\varepsilon})_{\varepsilon} = r_{E+}(\theta)$, see Pic.7.3. From Eq.(7.2.27) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part sing $[(R^{a>m}(r_{\varepsilon}, \varepsilon))_{\varepsilon}]$ of the Colombeau generalized curvature scalar $(R^{a>m}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding

<figure><img src="image_5.png" /><figcaption>Fig. 3. Ergosurface, horizon and singularity for slow Kerr black hole.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Ergosurface, horizon, and singularity for critical Kerr black hole.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Ergosurface, horizon and singularity for fast Kerr black hole.</figcaption></figure>

to the metric (7.2.27) (mod nonsingular multiplier) reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{a>m} (r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \mathbb{R} \left( \frac{\chi(\theta, \eta) \varepsilon^2}{\left[ (r_{\varepsilon} - r_{E_+}(\theta))^2 + \varepsilon^2 \right]^2} \right)_{\varepsilon} . \quad (7.2.28) $$

**Remark 7.2.10. (I)** Let $(\mathbf{R}^{\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.27) with $a > m$.

From Eq. (7.2.27) and Eq. (A.1)-Eq. (A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a>m)} (r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)} (r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] $$

of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a>m)} (r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)} (r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \mathbb{R} \left( \frac{\chi(\theta, \eta) \varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} . \quad (7.2.29) $$

**(II)** Let $(\mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.27) with $a > m$. From Eq. (7.2.27)
<figure><img src="image_6.png" /><figcaption>Fig. 4. Ergosurface, horizon, and singularity for critical Kerr black hole.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Ergosurface, horizon and singularity for fast Kerr black hole.</figcaption></figure>

to the metric (7.2.27) (mod nonsingular multiplier) reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{a>m} (r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \mathbb{R} \left( \frac{\chi(\theta, \eta) \varepsilon^2}{\left[ (r_{\varepsilon} - r_{E_+}(\theta))^2 + \varepsilon^2 \right]^2} \right)_{\varepsilon} . \qquad (7.2.28) $$

**Remark 7.2.10. (I)** Let $(\mathbf{R}^{\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.27) with $a > m$.

From Eq. (7.2.27) and Eq. (A.1)-Eq. (A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] $$

of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon) \right)_{\varepsilon} \right] = \mathbb{R} \left( \frac{\chi(\theta, \eta) \varepsilon^4}{4 (r_{E_+}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon} . \qquad (7.2.29) $$

**(II)** Let $(\mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}$ corresponding to the metric (7.2.27) with $a > m$. From Eq. (7.2.27)
and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_\epsilon, \epsilon) \right)_\epsilon \right] $$

of the Colombeau generalized quadratic scalar $\left( \mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_\epsilon, \epsilon) \right)_\epsilon$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_\epsilon, \epsilon) \right)_\epsilon \right] = \bar{\mathbb{R}} \left( \frac{\chi(\theta, \eta) \epsilon^4}{4 (r_{E_+}(\theta))^4 [\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_\epsilon . \quad (7.2.30) $$

# 8 Conclusions and Remarks

This paper dealing with an extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1]-[2],[5]-[7] and [14]-[15]. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically.

In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We have shown that a successful approach for dealing with curvature tensor valued distribution is to first impose admissible the nondegeneracy conditions on the metric tensor, and then take its derivatives in the sense of classical distributions in space $S'_{2m}(\mathbb{R}^3)$.

The distributional meaning is then equivalent to the junction condition formalism. Afterwards, through appropriate limiting procedures, it is then possible to obtain well behaved distributional tensors with support on submanifolds of $d \le 3$, as we have shown for the energy-momentum tensors associated with the Schwarzschild spacetimes. The above procedure provides us with what is expected on physical grounds. However, it should be mentioned that the use of new supergeneralized functions (supergeneralized Colombeau algebras $\tilde{G}(\mathbb{R}^3, \Sigma)$), in order to obtain superdistributional curvatures, may renders a more rigorous setting for discussing situations like the ones considered in this paper.

The vacuum energy density of free scalar quantum field $\Phi$ with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations $\langle \Phi^2 \rangle$ has a singular behavior on BHs horizon $r_+$: $\langle \Phi^2(r) \rangle^{-|r-r_+|^{-2}}$. We argue that this vacuum dominance may bear important astrophysical implications.

# Acknowledgments

To reviewers provided important clarifications.
and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_{\epsilon}, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_{\epsilon}, \epsilon) \right)_{\epsilon} \right] $$

of the Colombeau generalized quadratic scalar $\left( \mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_{\epsilon}, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_{\epsilon}, \epsilon) \right)_{\epsilon}$ reads

$$ \mathbf{sing} \left[ \left( \mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_{\epsilon}, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_{\epsilon}, \epsilon) \right)_{\epsilon} \right] = \bar{\mathbb{R}} \left( \frac{\chi(\theta, \eta) \epsilon^4}{4 (r_{E_+}(\theta))^4 [\epsilon^2 + (r_{\epsilon} - 2m)^2]^3} \right)_{\epsilon} . \quad (7.2.30) $$

# 8 Conclusions and Remarks

This paper dealing with an extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1]-[2],[5]-[7] and [14]-[15]. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically.

In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We have shown that a successful approach for dealing with curvature tensor valued distribution is to first impose admissible the nondegeneracy conditions on the metric tensor, and then take its derivatives in the sense of classical distributions in space $S'_{2m}(\mathbb{R}^3)$.

The distributional meaning is then equivalent to the junction condition formalism. Afterwards, through appropriate limiting procedures, it is then possible to obtain well behaved distributional tensors with support on submanifolds of $d \le 3$, as we have shown for the energy-momentum tensors associated with the Schwarzschild spacetimes. The above procedure provides us with what is expected on physical grounds. However, it should be mentioned that the use of new supergeneralized functions (supergeneralized Colombeau algebras $\tilde{G}(\mathbb{R}^3, \Sigma)$), in order to obtain superdistributional curvatures, may render a more rigorous setting for discussing situations like the ones considered in this paper.

The vacuum energy density of free scalar quantum field $\Phi$ with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations $\langle \Phi^2 \rangle$ has a singular behavior on BHs horizon $r_+$: $\langle \Phi^2(r) \rangle^{-|r-r_+|^{-2}}$. We argue that this vacuum dominance may bear important astrophysical implications.

# Acknowledgments

To reviewers provided important clarifications.
References

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[2] J.F.Colombeau, “Elementary Introduction to New Generalized Functions,” (North Holland, Amsterdam 1985).

[3] T. Müller, F. Grave, Catalogue of Spacetimes, pp.58 arXiv:0904.4184 [gr-qc] https://arxiv.org/abs/0904.4184v2

[4] L. D.Landau E. M. Lifschitz 1975 The Classical Theory of Fields Pergamon Press, Oxford.

[5] J. A. Vickers, J. P. Wilson, “A nonlinear theory of tensor distributions,” gr-qc/9807068

[6] J. A. Vickers, J. P. Wilson, “Invariance of the distributional curvature of the cone under smooth diffeomorphisms,” Class. Quantum Grav. **16**, 579-588 (1999).

[7] J. A. Vickers, “Nonlinear generalised functions in general relativity,” in *Nonlinear Theory of Generalized Functions*, Chapman & Hall/CRC Research Notes in Mathematics **401**, 275-290, eds. M. Grosser, G. Hörmann, M. Kunzinger, M. Oberguggenberger (Chapman & Hall CRC, Boca Raton 1999).

[8] R. Geroch, J. Traschen, “Strings and other distributional sources in general relativity,” Phys. Rev. D **36**, 1017-1031 (1987).

[9] H. Balasin, H. Nachbagauer, “On the distributional nature of the energy-momentum tensor of a black hole or What curves the Schwarzschild geometry?” Class. Quant. Grav. **10**, 2271-2278 (1993).

[10] H. Balasin, H. Nachbagauer, “Distributional energy-momentum tensor of the Kerr-Newman space-time family,” Class. Quant. Grav. **11**, 1453-1461 (1994).

[11] C. J. S. Clarke, J. A. Vickers, J. P. Wilson, “Generalized functions and distributional curvature of cosmic strings,” Class. Quant. Grav. **13**, 2485-2498 (1996).

[12] N. Pantoja, H. Rago, “Energy-momentum tensor valued distributions for the Schwarzschild and Reissner-Nordstrøm geometries,” preprint, gr-qc/9710072 (1997).

[13] N. Pantoja, H. Rago, “Distributional sources in General Relativity: Two point-like examples revisited,” preprint, gr-qc/0009053, (2000).

[14] M. Kunzinger, R. Steinbauer, “Nonlinear distributional geometry,” Acta Appl. Math., (2001).

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[16] M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, “On the foundations of nonlinear generalized functions I, II,” Mem. Am. Math. Soc., **153**, 729, (2001).

[17] M. Grosser, M. Kunzinger, R. Steinbauer, J. Vickers, “A global theory of nonlinear generalized functions,” Adv. Math., to appear (2001).

[18] L. Schwartz, “Sur l’impossibilité de la multiplication des distributions,” C. R. Acad. Sci. Paris **239**, 847-848 (1954).

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[21] G. Debney, R. Kerr, A. Schild, “Solutions of the Einstein and Einstein-Maxwell equations,” J. Math. Phys. **10**, 1842-1854 (1969).

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References

[1] J.F. Colombeau, “New Generalized Functions and Multiplication of Distributions,” (North Holland, Amsterdam, 1984).

[2] J.F.Colombeau, “Elementary Introduction to New Generalized Functions,” (North Holland, Amsterdam 1985).

[3] T. Müller, F. Grave, Catalogue of Spacetimes, pp.58 arXiv:0904.4184 [gr-qc] https://arxiv.org/abs/0904.4184v2

[4] L. D. Landau, E. M. Lifschitz 1975 The Classical Theory of Fields Pergamon Press, Oxford.

[5] J. A. Vickers, J. P. Wilson, “A nonlinear theory of tensor distributions,” gr-qc/9807068

[6] J. A. Vickers, J. P. Wilson, “Invariance of the distributional curvature of the cone under smooth diffeomorphisms,” Class. Quantum Grav. **16**, 579-588 (1999).

[7] J. A. Vickers, “Nonlinear generalised functions in general relativity,” in *Nonlinear Theory of Generalized Functions*, Chapman & Hall/CRC Research Notes in Mathematics **401**, 275-290, eds. M. Grosser, G. Hörmann, M. Kunzinger, M. Oberguggenberger (Chapman & Hall CRC, Boca Raton 1999).

[8] R. Geroch, J. Traschen, “Strings and other distributional sources in general relativity,” Phys. Rev. D **36**, 1017-1031 (1987).

[9] H. Balasin, H. Nachbagauer, “On the distributional nature of the energy-momentum tensor of a black hole or What curves the Schwarzschild geometry?” Class. Quant. Grav. **10**, 2271-2278 (1993).

[10] H. Balasin, H. Nachbagauer, “Distributional energy-momentum tensor of the Kerr-Newman space-time family,” Class. Quant. Grav. **11**, 1453-1461 (1994).

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Appendix A.

Expressions for the Colombeau quantities $(\mathbf{R}(\{\cdot\}, (\cdot)))_\epsilon, (\mathbf{R}^{\mu\nu}(\{\cdot\}, (\cdot))\mathbf{R}_{\mu\nu}(\{\cdot\}, (\cdot)))_\epsilon$ and $(\mathbf{R}^{\rho\sigma\mu\nu}(\{\cdot\}, (\cdot))\mathbf{R}_{\rho\sigma\mu\nu}(\{\cdot\}, (\cdot)))_\epsilon$ in terms of $(A_\epsilon)_{\epsilon}, (B_\epsilon)_{\epsilon}, (C_\epsilon)_{\epsilon}$ and $(D_\epsilon)_{\epsilon}, \epsilon \in (0, 1]$:

Let us introduce now Colombeau generalized metric which has the form

$$ \left\{ \begin{aligned} (ds^2)_\epsilon = & - (A_\epsilon(r)(dx^0)^2)_\epsilon - 2(D_\epsilon(r)dx^0dr)_\epsilon + ((B_\epsilon(r)+C_\epsilon(r))(dr)^2)_\epsilon \\ & + (B_\epsilon(r)r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2])_\epsilon \quad r = \mathbf{cl}[(r_\epsilon)_\epsilon] \in \mathbb{R}. \end{aligned} \right. \quad (A.1) $$

The Colombeau scalars $(\mathbf{R}(\epsilon))_\epsilon, (\mathbf{R}^{\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}(\epsilon))_\epsilon$ and $(R^{\rho\sigma\mu\nu}(\epsilon)R_{\rho\sigma\mu\nu}(\epsilon))_\epsilon$, in terms of Colombeau generalized functions $(A_\epsilon(r))_\epsilon, (B_\epsilon(r))_\epsilon, (C_\epsilon(r))_\epsilon, (D_\epsilon(r))_\epsilon$ is expressed as

$$ \begin{aligned} (\mathbf{R}(\epsilon))_{\epsilon} &= \left( \frac{A_{\epsilon}}{\Delta_{\epsilon}} \left[ \begin{aligned}[t] & \frac{2}{r} \left( -2 \frac{A'_{\epsilon}}{A_{\epsilon}} - 3 \frac{B'_{\epsilon}}{B_{\epsilon}} + \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{2}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{A''_{\epsilon}}{A_{\epsilon}} - 2 \frac{B''_{\epsilon}}{B_{\epsilon}} \\ & + \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 - 2 \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \left( \frac{1}{2} \frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{B'_{\epsilon}}{B_{\epsilon}} \right) \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \end{aligned} \right]_{\epsilon} \right), \\ (\mathbf{R}^{\mu\nu}(\epsilon) \mathbf{R}_{\mu\nu}(\epsilon))_{\epsilon} &= \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{1}{2} \frac{A''_{\epsilon}}{A_{\epsilon}} - \frac{1}{4} \frac{A'_{\epsilon}\Delta'_{\epsilon}}{A_{\epsilon}\Delta_{\epsilon}} + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \frac{1}{r} \frac{A'_{\epsilon}}{A_{\epsilon}} \right) \right)_\epsilon + \\ &+ 2 \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \begin{aligned}[t] & \frac{1}{r} \left( \frac{1}{2} \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} - \frac{A'_{\epsilon}}{A_{\epsilon}} - 2 \frac{B'_{\epsilon}}{B_{\epsilon}} \right) + \frac{1}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} - \\ & - \frac{1}{2} \frac{B''_{\epsilon}}{B_{\epsilon}} + \frac{1}{4} \frac{B'_{\epsilon}\Delta'_{\epsilon}}{B_{\epsilon}\Delta_{\epsilon}} \end{aligned} \right]_{\epsilon} \right) + \\ &\left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \begin{aligned}[t] & \frac{1}{2} \frac{A''_{\epsilon}}{A_{\epsilon}} - \frac{1}{4} \frac{A'_{\epsilon}\Delta'_{\epsilon}}{A_{\epsilon}\Delta_{\epsilon}} + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \frac{B''_{\epsilon}}{B_{\epsilon}} - \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 \\ & - \frac{1}{2} \frac{B'_{\epsilon}\Delta'_{\epsilon}}{B_{\epsilon}\Delta_{\epsilon}} + \frac{1}{r} \left( \frac{A'_{\epsilon}}{A_{\epsilon}} - \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} + 2 \frac{B'_{\epsilon}}{B_{\epsilon}} \right) \end{aligned} \right]_{\epsilon} \right)^2, \\ (\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_{\epsilon} &= \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{A''_{\epsilon}}{A_{\epsilon}} - \frac{1}{2} \frac{A'_{\epsilon}\Delta'_{\epsilon}}{A_{\epsilon}\Delta_{\epsilon}} \right)^2 + 2 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{1}{r} \frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} \right)^2 \right. \\ &+ 4 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \frac{B'_{\epsilon}}{B_{\epsilon}} - \frac{1}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} + \frac{1}{4} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 \right]^2 + \\ &+ 2 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \begin{aligned}[t] & \frac{1}{r} \left( \frac{A'_{\epsilon}}{A_{\epsilon}} + 2 \frac{B'_{\epsilon}}{B_{\epsilon}} - \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \frac{B''_{\epsilon}}{B_{\epsilon}} \\ & - \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 - \frac{1}{2} \frac{B'_{\epsilon}\Delta'_{\epsilon}}{B_{\epsilon}\Delta_{\epsilon}} \end{aligned} \right]^2. \end{aligned} \right) \quad (A.2) $$

Here

$$ (\Delta_{\epsilon})_{\epsilon} = (A_{\epsilon}(r)(B_{\epsilon}(r) + C_{\epsilon}(r)))_{\epsilon} + (D_{\epsilon}^{2}(r))_{\epsilon}. \quad (A.3) $$

Assume that

$$ (\Delta_{\epsilon}(r))_{\epsilon} = 1, (B_{\epsilon}(r))_{\epsilon} = 1, (D_{\epsilon}(r))_{\epsilon} = 0. \quad (A.4) $$

From Eq. (A.2)–Eq. (A.4) one obtains
Appendix A.

Expressions for the Colombeau quantities $(\mathbf{R}(\{\cdot\}, (\cdot)))_\epsilon, (\mathbf{R}^{\mu\nu}(\{\cdot\}, (\cdot))\mathbf{R}_{\mu\nu}(\{\cdot\}, (\cdot)))_\epsilon$ and $(\mathbf{R}^{\rho\sigma\mu\nu}(\{\cdot\}, (\cdot))\mathbf{R}_{\rho\sigma\mu\nu}(\{\cdot\}, (\cdot)))_\epsilon$ in terms of $(A_\epsilon)_{\epsilon}, (B_\epsilon)_{\epsilon}, (C_\epsilon)_{\epsilon}$ and $(D_\epsilon)_{\epsilon}, \epsilon \in (0, 1]$:

Let us introduce now Colombeau generalized metric which has the form

$$ \left\{ \begin{aligned} (ds^2)_\epsilon = & - (A_\epsilon(r)(dx^0)^2)_\epsilon - 2(D_\epsilon(r)dx^0dr)_\epsilon + ((B_\epsilon(r)+C_\epsilon(r))(dr)^2)_\epsilon \\ & + (B_\epsilon(r)r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2])_\epsilon \quad r = \mathbf{cl}[(r_\epsilon)_\epsilon] \in \mathbb{R}. \end{aligned} \right. \qquad (A.1) $$

The Colombeau scalars $(\mathbf{R}(\epsilon))_\epsilon, (\mathbf{R}^{\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}(\epsilon))_\epsilon$ and $(R^{\rho\sigma\mu\nu}(\epsilon)R_{\rho\sigma\mu\nu}(\epsilon))_\epsilon$, in terms of Colombeau generalized functions $(A_\epsilon(r))_\epsilon, (B_\epsilon(r))_\epsilon, (C_\epsilon(r))_\epsilon, (D_\epsilon(r))_\epsilon$ is expressed as

$$ \begin{aligned} (\mathbf{R}(\epsilon))_{\epsilon} &= \left( \frac{A_{\epsilon}}{\Delta_{\epsilon}} \left[ \frac{2}{r} \left( -2 \frac{A'_{\epsilon}}{A_{\epsilon}} - 3 \frac{B'_{\epsilon}}{B_{\epsilon}} + \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{2}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{A''_{\epsilon}}{A_{\epsilon}} - 2 \frac{B''_{\epsilon}}{B_{\epsilon}} \right. \right. \\ & \left. \left. + \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 - 2 \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \left( \frac{1}{2} \frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{B'_{\epsilon}}{B_{\epsilon}} \right) \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right] \right)_{\epsilon}, \\ (\mathbf{R}^{\mu\nu}(\epsilon) \mathbf{R}_{\mu\nu}(\epsilon))_{\epsilon} &= \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{1}{2} \frac{A''_{\epsilon}}{A_{\epsilon}} - \frac{1}{4} \frac{A'_{\epsilon}\Delta'_{\epsilon}}{A_{\epsilon}\Delta_{\epsilon}} + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \frac{1}{r} \frac{A'_{\epsilon}}{A_{\epsilon}} \right) \right)_{\epsilon} + \\ & + 2 \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \left( \frac{1}{2} \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} - \frac{A'_{\epsilon}}{A_{\epsilon}} - 2 \frac{B'_{\epsilon}}{B_{\epsilon}} \right) + \frac{1}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} - \right. \right. \\ & \left. \left. - \frac{1}{2} \frac{B''_{\epsilon}}{B_{\epsilon}} + \frac{1}{4} \frac{B'_{\epsilon}\Delta'_{\epsilon}}{B_{\epsilon}\Delta_{\epsilon}} \right] \right)_{\epsilon} + \\ & \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{2} \frac{A''_{\epsilon}}{A_{\epsilon}} - \frac{1}{4} \frac{A'_{\epsilon}\Delta'_{\epsilon}}{A_{\epsilon}\Delta_{\epsilon}} + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \frac{B''_{\epsilon}}{B_{\epsilon}} - \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 \right. \right. \\ & \left. \left. - \frac{1}{2} \frac{B'_{\epsilon}\Delta'_{\epsilon}}{B_{\epsilon}\Delta_{\epsilon}} + \frac{1}{r} \left( \frac{A'_{\epsilon}}{A_{\epsilon}} - \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} + 2 \frac{B'_{\epsilon}}{B_{\epsilon}} \right) \right] \right)_{\epsilon}, \\ (\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_{\epsilon} &= \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{A''_{\epsilon}}{A_{\epsilon}} - \frac{1}{2} \frac{A'_{\epsilon}\Delta'_{\epsilon}}{A_{\epsilon}\Delta_{\epsilon}} \right)^2 + 2 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{1}{r} \frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} \right)^2 \right. \\ & + 4 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \frac{B'_{\epsilon}}{B_{\epsilon}} - \frac{1}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} + \frac{1}{4} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 \right]^2 + \\ & + 2 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \left( \frac{A'_{\epsilon}}{A_{\epsilon}} + 2 \frac{B'_{\epsilon}}{B_{\epsilon}} - \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{1}{2} \frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \frac{B''_{\epsilon}}{B_{\epsilon}} - \right. \\ & \left. \left. \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 - \frac{1}{2} \frac{B'_{\epsilon}\Delta'_{\epsilon}}{B_{\epsilon}\Delta_{\epsilon}} \right] \right)_{\epsilon}. \end{aligned} \qquad (A.2) $$

Here

$$ (\Delta_{\epsilon})_{\epsilon} = (A_{\epsilon}(r)(B_{\epsilon}(r) + C_{\epsilon}(r)))_{\epsilon} + (D_{\epsilon}^{2}(r))_{\epsilon}. \qquad (A.3) $$

Assume that

$$ (\Delta_{\epsilon}(r))_{\epsilon} = 1, (B_{\epsilon}(r))_{\epsilon} = 1, (D_{\epsilon}(r))_{\epsilon} = 0. \qquad (A.4) $$

From Eq.(A.2)-Eq.(A.4) one obtains
$$ (\mathbf{R}(\epsilon))_{\epsilon} = \left(-\frac{4A'_{\epsilon}}{r} + \frac{2A_{\epsilon}C_{\epsilon}}{r^2} - A''_{\epsilon}\right)_{\epsilon}, \\ (\mathbf{R}^{\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}(\epsilon))_{\epsilon} = \left(\left(\frac{1}{2}A''_{\epsilon} + \frac{1}{r}A'_{\epsilon}\right)^2\right)_{\epsilon} + 2\left(\left[-\frac{A'_{\epsilon}}{r} + \frac{A_{\epsilon}C_{\epsilon}}{r^2}\right]^2\right)_{\epsilon} + \left(\left[\frac{1}{2}A''_{\epsilon} + \frac{1}{r}A'_{\epsilon}\right]^2\right)_{\epsilon}, \quad (A.5) \\ (\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_{\epsilon} = \left((A''_{\epsilon})^2 + 2\left(\frac{A'_{\epsilon}}{r}\right)^2 + 4\frac{(A_{\epsilon}C_{\epsilon})^2}{r^4} + 2\left(\frac{A'_{\epsilon}}{r}\right)^2\right)_{\epsilon}. $$

We choose now

$$ B_{\epsilon}(r_{\epsilon}) = 1, C_{\epsilon}(r_{\epsilon}) = -1 + A_{\epsilon}^{-1}(r_{\epsilon}), D_{\epsilon}(r_{\epsilon}) = 0, \quad (A.6) $$

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

$$ (ds_{\epsilon}^{2})_{\epsilon} = -(A_{\epsilon}(r_{\epsilon}) dt^{2})_{\epsilon} + (A_{\epsilon}^{-1}(r_{\epsilon}) dr_{\epsilon}^{2})_{\epsilon} + r_{\epsilon}^{2} d\Omega^{2}, \quad (A.7) $$

Where $A_{\epsilon}(r)$

$$ A_{\epsilon}(r_{\epsilon}) = -r_{\epsilon}^{-1}\sqrt{(r_{\epsilon} - 2m)^{2} + \epsilon^{2}}, \quad r_{\epsilon} \geq 2m. \quad (A.8) $$

By differentiation we obtain

$$ \begin{gathered} \Delta_{\epsilon} = A_{\epsilon}(B_{\epsilon} + C_{\epsilon}) = 1, \Delta'_{\epsilon} = 0, \\ A'_{\epsilon}(r_{\epsilon}) = \frac{-2m(r_{\epsilon} - 2m)}{r_{\epsilon}^{2}\sqrt{(r_{\epsilon} - 2m)^{2} + \epsilon^{2}}}, \\ A''_{\epsilon}(r) = \frac{2m(-16m^{3} + 24m^{2}r_{\epsilon} - 12mr_{\epsilon}^{2} - 4m\epsilon^{2} + 2r_{\epsilon}^{3} + r_{\epsilon}\epsilon^{2})}{r_{\epsilon}^{3}\left[(r_{\epsilon} - 2m)^{2} + \epsilon^{2}\right]^{3/2}} = \frac{4m(r_{\epsilon} - 2m)^{3} + (r_{\epsilon} - 4m)\epsilon^{2}}{r_{\epsilon}^{3}\left[(r_{\epsilon} - 2m)^{2} + \epsilon^{2}\right]^{3/2}}. \end{gathered} \quad (A.9) $$

From Eqs.(A.2)-(A.5) and Eq.(A.9) we obtain

$$ \begin{aligned} (\mathbf{R}(r, \epsilon))_{\epsilon} = & \left( \frac{A_{\epsilon}}{\Delta_{\epsilon}} \left[ \frac{2}{r_{\epsilon}} \left( -2\frac{A'_{\epsilon}}{A_{\epsilon}} - 3\frac{B'_{\epsilon}}{B_{\epsilon}} + \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{2}{r_{\epsilon}^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{A'_{\epsilon}}{A_{\epsilon}} - 2\frac{B''_{\epsilon}}{B_{\epsilon}} \right. \right. \\ & \left. \left. + \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 - 2\frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \left( \frac{1}{2}\frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{B'_{\epsilon}}{B_{\epsilon}} \right) \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right] \right)_{\epsilon} = \\ & \left( A_{\epsilon} \left[ \frac{2}{r_{\epsilon}} \left( -2\frac{A'_{\epsilon}}{A_{\epsilon}} - \frac{2A_{\epsilon}(1-A_{\epsilon}^{-1})}{r_{\epsilon}^2} - \frac{A''_{\epsilon}}{A_{\epsilon}} \right) \right] \right)_{\epsilon} = \\ & \left( -\frac{4A'_{\epsilon}}{r_{\epsilon}} - \frac{2A_{\epsilon}}{r_{\epsilon}^2} + \frac{2}{r_{\epsilon}^2} - A''_{\epsilon} \right)_{\epsilon} = \\ & \left( \frac{8m(r_{\epsilon}-2m)}{r_{\epsilon}^3 \sqrt{(r_{\epsilon}-2m)^2 + \epsilon^2}} \right) + 2r_{\epsilon}^{-3} \left( \sqrt{(r_{\epsilon}-2m)^2 + \epsilon^2} \right)_{\epsilon} - \frac{2}{(r_{\epsilon})_{\epsilon}} - \\ & \left( \frac{2m(-16m^3 + 24m^2r_{\epsilon} - 12mr_{\epsilon}^2 - 4m\epsilon^2 + 2r_{\epsilon}^3 + r_{\epsilon}\epsilon^2)}{r_{\epsilon}^3 [(r_{\epsilon}-2m)^2 + \epsilon^2]^{3/2}} \right)_{\epsilon}. \end{aligned} \quad (A.10) $$

Finally from Eq.(A.10) one obtains the following expression for the distributional Colombeau scalar
$$ (\mathbf{R}(\epsilon))_{\epsilon} = \left(-\frac{4A'_{\epsilon}}{r} + \frac{2A_{\epsilon}C_{\epsilon}}{r^2} - A''_{\epsilon}\right)_{\epsilon}, \\ (\mathbf{R}^{\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}(\epsilon))_{\epsilon} = \left(\left(\frac{1}{2}A''_{\epsilon} + \frac{1}{r}A'_{\epsilon}\right)^2\right)_{\epsilon} + 2\left(\left[-\frac{A'_{\epsilon}}{r} + \frac{A_{\epsilon}C_{\epsilon}}{r^2}\right]^2\right)_{\epsilon} + \left(\left[\frac{1}{2}A''_{\epsilon} + \frac{1}{r}A'_{\epsilon}\right]^2\right)_{\epsilon}, \quad (A.5) \\ (\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_{\epsilon} = \left((A''_{\epsilon})^2 + 2\left(\frac{A'_{\epsilon}}{r}\right)^2 + 4\frac{(A_{\epsilon}C_{\epsilon})^2}{r^4} + 2\left(\frac{A'_{\epsilon}}{r}\right)^2\right)_{\epsilon}. $$

We choose now

$$ B_{\epsilon}(r_{\epsilon}) = 1, C_{\epsilon}(r_{\epsilon}) = -1 + A_{\epsilon}^{-1}(r_{\epsilon}), D_{\epsilon}(r_{\epsilon}) = 0, \quad (A.6) $$

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

$$ (ds_{\epsilon}^{2})_{\epsilon} = -(A_{\epsilon}(r_{\epsilon}) dt^{2})_{\epsilon} + (A_{\epsilon}^{-1}(r_{\epsilon}) dr_{\epsilon}^{2})_{\epsilon} + r_{\epsilon}^{2} d\Omega^{2}, \quad (A.7) $$

Where $A_{\epsilon}(r)$

$$ A_{\epsilon}(r_{\epsilon}) = -r_{\epsilon}^{-1}\sqrt{(r_{\epsilon} - 2m)^{2} + \epsilon^{2}}, \quad r_{\epsilon} \geq 2m. \quad (A.8) $$

By differentiation we obtain

$$ \begin{gathered} \Delta_{\epsilon} = A_{\epsilon}(B_{\epsilon} + C_{\epsilon}) = 1, \Delta'_{\epsilon} = 0, \\ A'_{\epsilon}(r_{\epsilon}) = \frac{-2m(r_{\epsilon} - 2m)}{r_{\epsilon}^{2}\sqrt{(r_{\epsilon} - 2m)^{2} + \epsilon^{2}}}, \\ A''_{\epsilon}(r) = \frac{2m(-16m^{3} + 24m^{2}r_{\epsilon} - 12mr_{\epsilon}^{2} - 4m\epsilon^{2} + 2r_{\epsilon}^{3} + r_{\epsilon}\epsilon^{2})}{r_{\epsilon}^{3}\left[(r_{\epsilon} - 2m)^{2} + \epsilon^{2}\right]^{3/2}} = \frac{4m(r_{\epsilon} - 2m)^{3} + (r_{\epsilon} - 4m)\epsilon^{2}}{r_{\epsilon}^{3}\left[(r_{\epsilon} - 2m)^{2} + \epsilon^{2}\right]^{3/2}}. \end{gathered} \quad (A.9) $$

From Eqs.(A.2)-(A.5) and Eq.(A.9) we obtain

$$ \begin{aligned} (\mathbf{R}(r, \epsilon))_{\epsilon} = & \left( \frac{A_{\epsilon}}{\Delta_{\epsilon}} \left[ \frac{2}{r_{\epsilon}} \left( -2\frac{A'_{\epsilon}}{A_{\epsilon}} - 3\frac{B'_{\epsilon}}{B_{\epsilon}} + \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{2}{r_{\epsilon}^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{A'_{\epsilon}}{A_{\epsilon}} - 2\frac{B''_{\epsilon}}{B_{\epsilon}} \right. \right. \\ & \left. + \frac{1}{2} \left( \frac{B'_{\epsilon}}{B_{\epsilon}} \right)^2 - 2\frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \left( \frac{1}{2}\frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{B'_{\epsilon}}{B_{\epsilon}} \right) \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right] \right)_{\epsilon} = \\ & \left( A_{\epsilon} \left[ \frac{2}{r_{\epsilon}} \left( -2\frac{A'_{\epsilon}}{A_{\epsilon}} - \frac{2A_{\epsilon}(1-A_{\epsilon}^{-1})}{r_{\epsilon}^2} - \frac{A''_{\epsilon}}{A_{\epsilon}} \right) \right] \right)_{\epsilon} = \\ & \left( -\frac{4A'_{\epsilon}}{r_{\epsilon}} - \frac{2A_{\epsilon}}{r_{\epsilon}^2} + \frac{2}{r_{\epsilon}^2} - A''_{\epsilon} \right)_{\epsilon} = \\ & \left( \frac{8m(r_{\epsilon} - 2m)}{r_{\epsilon}^3 \sqrt{(r_{\epsilon} - 2m)^2 + \epsilon^2}} \right) + 2r_{\epsilon}^{-3} \left( \sqrt{(r_{\epsilon} - 2m)^2 + \epsilon^2} \right)_{\epsilon} - \frac{2}{(r_{\epsilon})_{\epsilon}} - \\ & \left( \frac{2m(-16m^3 + 24m^2r_{\epsilon} - 12mr_{\epsilon}^2 - 4m\epsilon^2 + 2r_{\epsilon}^3 + r_{\epsilon}\epsilon^2)}{r_{\epsilon}^3 [(r_{\epsilon} - 2m)^2 + \epsilon^2]^{3/2}} \right)_{\epsilon}. \end{aligned} \quad (A.10) $$

Finally from Eq.(A.10) one obtains the following expression for the distributional Colombeau scalar
$$ (\mathbf{R}(r, \varepsilon))_{\varepsilon} $$

$$ (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} = \left( \frac{8m(r_{\varepsilon} - 2m)}{r_{\varepsilon}^3 \sqrt{(r - 2m)^2 + \varepsilon^2}} \right)_{\varepsilon} + 2 \left[ (r_{\varepsilon}^{-3})_{\varepsilon} \right] \left( \sqrt{(r_{\varepsilon} - 2m)^2 + \varepsilon^2} \right)_{\varepsilon} - \frac{2}{(r_{\varepsilon}^2)_{\varepsilon}} - \left( \frac{4m(r_{\varepsilon} - 2m)^3 + (r_{\varepsilon} - 4m)\varepsilon^2}{r_{\varepsilon}^3 [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} \right)_{\varepsilon}. \quad (A.11) $$

**Remark A.1.** Note that from Eq.(A.11) follows that: if st $((r_{\varepsilon})_{\varepsilon}) \neq 0$, i.e. $(r_{\varepsilon})_{\varepsilon} \not\approx_{\tilde{\mathbb{R}}} 2m$ then $(r_{\varepsilon})_{\varepsilon} \not\approx_{\tilde{\mathbb{R}}} 2m \Rightarrow (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} \sim (\varepsilon^2)_{\varepsilon} \approx_{\tilde{\mathbb{R}}} 0$.

We assume now that $cl [(r_{\varepsilon})_{\varepsilon}] \approx_{\tilde{\mathbb{R}}} 2m$ and therefore from Eq.(A.11) we obtain

$$ (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} \approx_{\tilde{\mathbb{R}}} \left( \frac{4m^2 \varepsilon^2}{8m^3 [(r_{\varepsilon}-2m)^2 + \varepsilon^2]^{3/2}} \right)_{\varepsilon}. \quad (A.12) $$

**Remark A.2.** Note that from Eq.(A.12) at horizon $r = \tilde{\mathbb{R}}$ $2m$ follows that:

$$ (\mathbf{R}(r, \varepsilon))_{\varepsilon} = \left( \frac{4m^2 \varepsilon^2}{8m^3 [\varepsilon^2]^{3/2}} \right)_{\varepsilon} = (4m)^{-1} (\varepsilon^{-1})_{\varepsilon} \approx_{\tilde{\mathbb{R}}} \infty. \quad (A.13) $$

**Remark A.3.** Note that from Eq.(A.11) follows that:

$$ w\text{-lim}_{\varepsilon \to 0} \mathbf{R}(r, \varepsilon)^{-} \delta(r - 2m). \quad (A.14) $$

**Remark A.4.** Let $[(r_{\varepsilon} - 2m)_{\varepsilon}] \approx_{\tilde{\mathbb{R}}} 0$, then from Eq.(A.13) we obtain

$$ [(\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon}] \approx_{\tilde{\mathbb{R}}} \left[ \left( \frac{\varepsilon^2}{2m [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} \right)_{\varepsilon} \right]. \quad (A.15) $$

From Eqs.(A.2) and Eq.(A.9) we obtain

$$ \begin{aligned} (\mathbf{R}^{\mu\nu} (r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu} (r_{\varepsilon}, \varepsilon))_{\varepsilon} = &+2 \left( \left[ \frac{1}{r_{\varepsilon}} A'_{\varepsilon} + \frac{-A_{\varepsilon} + 1}{r_{\varepsilon}^2} \right]^2 \right)_{\varepsilon} + 2 \left( \left[ \frac{1}{2} A''_{\varepsilon} + \frac{1}{r_{\varepsilon}} A'_{\varepsilon} \right]^2 \right)_{\varepsilon} = \\ & 2 \left( \left[ \left( \frac{1}{r_{\varepsilon}^3} \sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2} + \frac{1}{r_{\varepsilon}^2} \right)_{\varepsilon} - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_{\varepsilon} + \\ & 2 \left( \left[ \frac{4m(r_{\varepsilon} - 2m)^3 + (r_{\varepsilon} - 4m)\varepsilon^2}{r_{\varepsilon}^3 [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_{\varepsilon}. \end{aligned} \quad (A.16) $$

**Remark A.5.** Note that from Eq.(A.16) follows that: if st $((r_{\varepsilon})_{\varepsilon}) \neq 0$, i.e. $(r_{\varepsilon})_{\varepsilon} \not\approx_{\tilde{\mathbb{R}}} 2m$ then

$$ \begin{aligned} r \not\approx_{\tilde{\mathbb{R}}} 2m \Rightarrow & (\mathbf{R}^{\mu\nu} (r, \varepsilon) \mathbf{R}_{\mu\nu} (r, \varepsilon))_{\varepsilon} \approx_{\tilde{\mathbb{R}}} K(r), \\ & K(r) = 12 \frac{r_s^2}{r^6}, r_s = 2m. \end{aligned} \quad (A.17) $$
$$ (\mathbf{R}(r, \varepsilon))_{\varepsilon} $$

$$ (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} = \left( \frac{8m(r_{\varepsilon} - 2m)}{r_{\varepsilon}^3 \sqrt{(r - 2m)^2 + \varepsilon^2}} \right)_{\varepsilon} + 2 \left[ (r_{\varepsilon}^{-3})_{\varepsilon} \right] \left( \sqrt{(r_{\varepsilon} - 2m)^2 + \varepsilon^2} \right)_{\varepsilon} - \frac{2}{(r_{\varepsilon}^2)_{\varepsilon}} - \left( \frac{4m(r_{\varepsilon} - 2m)^3 + (r_{\varepsilon} - 4m)\varepsilon^2}{r_{\varepsilon}^3 [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} \right)_{\varepsilon}. \quad (A.11) $$

**Remark A.1.** Note that from Eq.(A.11) follows that: if $\mathrm{st}((r_{\varepsilon})_{\varepsilon}) \neq 0$, i.e. $(r_{\varepsilon})_{\varepsilon} \not\approx_{\tilde{\mathbb{R}}} 2m$ then $(r_{\varepsilon})_{\varepsilon} \not\approx_{\tilde{\mathbb{R}}} 2m \Rightarrow (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} \sim (\varepsilon^2)_{\varepsilon} \approx_{\tilde{\mathbb{R}}} 0$.

We assume now that $\mathrm{cl}[(r_{\varepsilon})_{\varepsilon}] \approx_{\tilde{\mathbb{R}}} 2m$ and therefore from Eq.(A.11) we obtain

$$ (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} \approx_{\tilde{\mathbb{R}}} \left( \frac{4m^2 \varepsilon^2}{8m^3 [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} \right)_{\varepsilon}. \quad (A.12) $$

**Remark A.2.** Note that from Eq.(A.12) at horizon $r = \tilde{r} = 2m$ follows that:

$$ (\mathbf{R}(r, \varepsilon))_{\varepsilon} = \left( \frac{4m^2 \varepsilon^2}{8m^3 [\varepsilon^2]^{3/2}} \right)_{\varepsilon} = (4m)^{-1} (\varepsilon^{-1})_{\varepsilon} \approx_{\tilde{\mathbb{R}}} \infty. \quad (A.13) $$

**Remark A.3.** Note that from Eq.(A.11) follows that:

$$ w\text{-lim}_{\varepsilon \to 0} \mathbf{R}(r, \varepsilon)^{-} \delta(r - 2m). \quad (A.14) $$

**Remark A.4.** Let $[(r_{\varepsilon} - 2m)_{\varepsilon}] \approx_{\tilde{\mathbb{R}}} 0$, then from Eq.(A.13) we obtain

$$ [(\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon}] \approx_{\tilde{\mathbb{R}}} \left[ \left( \frac{\varepsilon^2}{2m [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} \right)_{\varepsilon} \right]. \quad (A.15) $$

From Eqs.(A.2) and Eq.(A.9) we obtain

$$ \begin{aligned} (\mathbf{R}^{\mu\nu} (r_{\varepsilon}, \varepsilon) \mathbf{R}_{\mu\nu} (r_{\varepsilon}, \varepsilon))_{\varepsilon} = &+2 \left( \left[ \frac{1}{r_{\varepsilon}} A'_{\varepsilon} + \frac{-A_{\varepsilon} + 1}{r_{\varepsilon}^2} \right]^2 \right)_{\varepsilon} + 2 \left( \left[ \frac{1}{2} A''_{\varepsilon} + \frac{1}{r_{\varepsilon}} A'_{\varepsilon} \right]^2 \right)_{\varepsilon} = \\ & 2 \left( \left[ \left( \frac{1}{r_{\varepsilon}^3} \sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2} + \frac{1}{r_{\varepsilon}^2} \right)_{\varepsilon} - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_{\varepsilon} + \\ & 2 \left( \left[ \frac{4m(r_{\varepsilon} - 2m)^3 + (r_{\varepsilon} - 4m)\varepsilon^2}{r_{\varepsilon}^3 [(r_{\varepsilon} - 2m)^2 + \varepsilon^2]^{3/2}} - 2 \frac{m}{r_{\varepsilon}^3} \frac{r_{\varepsilon} - 2m}{\sqrt{\varepsilon^2 + (r_{\varepsilon} - 2m)^2}} \right]^2 \right)_{\varepsilon}. \end{aligned} \quad (A.16) $$

**Remark A.5.** Note that from Eq.(A.16) follows that: if $\mathrm{st}((r_{\varepsilon})_{\varepsilon}) \neq 0$, i.e. $(r_{\varepsilon})_{\varepsilon} \not\approx_{\tilde{\mathbb{R}}} 2m$ then

$$ \begin{aligned} r \not\approx_{\tilde{\mathbb{R}}} 2m \Rightarrow & (\mathbf{R}^{\mu\nu} (r, \varepsilon) \mathbf{R}_{\mu\nu} (r, \varepsilon))_{\varepsilon} \approx_{\tilde{\mathbb{R}}} K(r), \\ & K(r) = 12 \frac{r_s^2}{r^6}, r_s = 2m. \end{aligned} \quad (A.17) $$
We assume now that $(r_\epsilon)_\epsilon \approx_{\mathbb{R}} 2m$ and therefore from Eq.(A.16) we obtain

$$ (\mathbf{R}^{\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}(r_\epsilon, \epsilon))_\epsilon \approx_{\mathbb{R}} K(r_s) + \left( \frac{\epsilon^4}{4m^4 [\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_\epsilon \quad (A.18) $$

**Remark A.6.** Note that from Eq.(A.18) at horizon $r = \tilde{r} \approx 2m$ follows that:

$$ (\mathbf{R}^{\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}(r_\epsilon, \epsilon))_\epsilon = \left( \frac{1}{4m^4\epsilon^2} \right)_\epsilon \approx_{\mathbb{R}} \infty, \quad (A.19) $$

**Remark C.6.** Let $[(r_\epsilon - 2m)_\epsilon] \approx_{\mathbb{R}} 0$, then from Eq.(A1.3) and Eq.(C.12) we obtain

$$ [(\mathbf{R}^{\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}(r_\epsilon, \epsilon))_\epsilon] \approx_{\mathbb{R}} K(r_s) + \left[ \left( \frac{\epsilon^4}{4m^4 (\epsilon^2 + (r_\epsilon - 2m)^2)^3} \right)_\epsilon \right] \quad (A.20) $$

From Eqs.(A.2) and Eq.(C.3) we obtain

$$ \begin{aligned} & (\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_\epsilon = \\ & \left( (A''_\epsilon)^2 + 2 \left( \frac{A'_\epsilon}{r} \right)^2 \right)_\epsilon + 4 \left( \left[ \frac{1}{r^2} (1 - A_\epsilon) \right]^2 \right)_\epsilon + 2 \left( \left[ \frac{A'_\epsilon}{r} \right]^2 \right)_\epsilon = \\ & \left( A''_\epsilon + 4 \frac{A'^2_\epsilon}{r^2} \right)_\epsilon + 4 \left( \frac{1}{r^4} (1 - A_\epsilon)^2 \right)_\epsilon = \\ & \left( \left[ \frac{4m(r-2m)^3 + (r-4m)\epsilon^2}{r^3 [(r-2m)^2 + \epsilon^2]^{3/2}} \right]^2 \right)_\epsilon - \frac{8m^2(r-2m)^2}{r^6 [(r-2m)^2 + \epsilon^2]} + \\ & \frac{4}{r^4} \left( 1 + r^{-1} \sqrt{(r-2m)^2 + \epsilon^2} \right)^2. \end{aligned} \quad (A.21) $$

**Remark C.7.** Note that from Eq.(C.15) follows that:

$$ r \not\approx_{\mathbb{R}} 2m \implies (\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_\epsilon \approx_{\mathbb{R}} K(r), \quad (A.22) $$

see Definition 1.5.2.(i).

We assume now that $(r_\epsilon)_\epsilon \approx_{\mathbb{R}} 2m$ and therefore from Eq.(C.10) we obtain

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_\epsilon, \epsilon))_\epsilon \approx_{\mathbb{R}} K(r_s) + \left( \frac{\epsilon^4}{4m^4 [\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_\epsilon \quad (A.23) $$

**Remark C.8.** Let $[(r_\epsilon - 2m)_\epsilon] \approx_{\mathbb{R}} 0$, then from Eq.(A1.3) and Eq.(C.12) we obtain

$$ [(\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_\epsilon] = K(r_s) + \left( \frac{\epsilon^4}{4m^4 [\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_\epsilon \quad (A.24) $$

**Remark C.9.** Note that from Eq.(C.15) at horizon $r = 2m$ follows that:

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_\epsilon \approx_{\mathbb{R}} \infty, \quad (A.25) $$

see Definition 1.5.2.(ii).

**Remark A.2.6.** We assume now there exist an fundamental generalized length $(l_\epsilon)_\epsilon$
We assume now that $(r_\epsilon)_\epsilon \approx_{\mathbb{R}} 2m$ and therefore from Eq.(A.16) we obtain

$$ (\mathbf{R}^{\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}(r_\epsilon, \epsilon))_{\epsilon} \approx_{\mathbb{R}} K(r_s) + \left( \frac{\epsilon^4}{4m^4[\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_{\epsilon} \quad (A.18) $$

**Remark A.6.** Note that from Eq.(A.18) at horizon $r = \tilde{r} \approx 2m$ follows that:

$$ (\mathbf{R}^{\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}(r_\epsilon, \epsilon))_{\epsilon} = \left( \frac{1}{4m^4\epsilon^2} \right)_{\epsilon} \approx_{\mathbb{R}} \infty, \quad (A.19) $$

**Remark C.6.** Let $[(r_\epsilon - 2m)_\epsilon] \approx_{\mathbb{R}} 0$, then from Eq.(A1.3) and Eq.(C.12) we obtain

$$ [(\mathbf{R}^{\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\mu\nu}(r_\epsilon, \epsilon))_{\epsilon}] \approx_{\mathbb{R}} K(r_s) + \left[ \left( \frac{\epsilon^4}{4m^4(\epsilon^2 + (r_\epsilon - 2m)^2)^3} \right)_{\epsilon} \right] \quad (A.20) $$

From Eqs.(A.2) and Eq.(C.3) we obtain

$$ \begin{aligned} & (\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_{\epsilon} = \\ & \qquad \left( (A''_{\epsilon})^2 + 2 \left( \frac{A'_{\epsilon}}{r} \right)^2 \right)_{\epsilon} + 4 \left( \left[ \frac{1}{r^2} (1 - A_{\epsilon}) \right]^2 \right)_{\epsilon} + 2 \left( \left[ \frac{A'_{\epsilon}}{r} \right]^2 \right)_{\epsilon} = \\ & \qquad \left( A''_{\epsilon} + 4 \frac{A'_{\epsilon}^2}{r^2} \right)_{\epsilon} + 4 \left( \frac{1}{r^4} (1 - A_{\epsilon})^2 \right)_{\epsilon} = \\ & \qquad \left( \left[ \frac{4m(r-2m)^3 + (r-4m)\epsilon^2}{r^3 [(r-2m)^2 + \epsilon^2]^{3/2}} \right]^2 \right)_{\epsilon} - \frac{8m^2(r-2m)^2}{r^6 [(r-2m)^2 + \epsilon^2]} + \\ & \qquad \frac{4}{r^4} \left( 1 + r^{-1} \sqrt{(r-2m)^2 + \epsilon^2} \right)^2. \end{aligned} \quad (A.21) $$

**Remark C.7.** Note that from Eq.(C.15) follows that:

$$ r \not\approx_{\mathbb{R}} 2m \implies (\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_{\epsilon} \approx_{\mathbb{R}} K(r), \quad (A.22) $$

see Definition 1.5.2.(i).

We assume now that $(r_\epsilon)_\epsilon \approx_{\mathbb{R}} 2m$ and therefore from Eq.(C.10) we obtain

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(r_\epsilon, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r_\epsilon, \epsilon))_{\epsilon} \approx_{\mathbb{R}} K(r_s) + \left( \frac{\epsilon^4}{4m^4[\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_{\epsilon} \quad (A.23) $$

**Remark C.8.** Let $[(r_\epsilon - 2m)_\epsilon] \approx_{\mathbb{R}} 0$, then from Eq.(A1.3) and Eq.(C.12) we obtain

$$ [(\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_{\epsilon}] = K(r_s) + \left( \frac{\epsilon^4}{4m^4[\epsilon^2 + (r_\epsilon - 2m)^2]^3} \right)_{\epsilon} \quad (A.24) $$

**Remark C.9.** Note that from Eq.(C.15) at horizon $r = 2m$ follows that:

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(r, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu}(r, \epsilon))_{\epsilon} \approx_{\mathbb{R}} \infty, \quad (A.25) $$

see Definition 1.5.2.(ii).

**Remark A.2.6.** We assume now there exist an fundamental generalized length $(l_\epsilon)_\epsilon$
$$ (l_{\varepsilon})_{\varepsilon \in (0, \eta]} = a(\varepsilon)_{\varepsilon \in (0, \eta]}, \eta \ll 1, \\ (l_{\varepsilon})_{\varepsilon \in (\eta, 1]} = a, \qquad (A.26) $$

such that $|(r_{\varepsilon} - \bar{p})_{\varepsilon}| \ge (l_{\varepsilon})_{\varepsilon} = a(\varepsilon)_{\varepsilon}$. It meant there exist a thickness $th_{hor} = (l_{\varepsilon})_{\varepsilon}$ of BH horizon. We introduce a norm $||th_{hor}||$ of a thickness $th_{hor}$ by formula

$$ ||th_{hor}|| = \sup_{\varepsilon \in (0, \eta]} |l_{\varepsilon}| = \eta, \qquad (A.27) $$

By using (A.20) we get the estimate

$$ (\mathbf{R}^{\rho\sigma\mu\nu} (r_{\varepsilon}, \varepsilon) \mathbf{R}_{\rho\sigma\mu\nu} (r_{\varepsilon}, \varepsilon))_{\varepsilon} \approx \mathbb{R} K(r_s) + \left( \frac{\varepsilon^4}{4m^4 \left[ (r_{\varepsilon} - 2m)^2 + \varepsilon^2 \right]^3} \right)_{\varepsilon \in (0, \eta]} = \\ K(r_s) + \left( \frac{1}{4m^4 \left[ (r_{\varepsilon} - 2m)^2 + \varepsilon^2 \right]} \right)_{\varepsilon \in (0, \eta]} \times \left( \frac{\varepsilon^2}{[(r_{\varepsilon} - 2m)^2 + \varepsilon^2]} \right)_{\varepsilon \in (0, \eta]} \times \\ \times \left( \frac{\varepsilon^2}{[(r_{\varepsilon} - 2m)^2 + \varepsilon^2]} \right)_{\varepsilon \in (0, \eta]} \le \\ K(r_s) + \frac{1}{4m^4 [a^2 + 1]^2} \left( \frac{1}{[(r_{\varepsilon} - 2m)^2 + \varepsilon^2]} \right)_{\varepsilon \in (0, \eta]} \le \\ K(r_s) + \frac{1}{4m^4 [a^2 + 1]^2} \left( \frac{1}{(r - 2m)^2} \right)_{r-2m \in (0, \eta]}. \qquad (A.28) $$

## Biography of author(s)

<figure><img src="image_8.png" /></figure>

### Jaykov Foukzon

Currently works at the Center for Mathematical Sciences, Technion - Israel Institute of Technology, Haifa, Israel.

Jaykov Foukzon (born October 7, 1954). Professor, Doctor Sc. (Phys. and Math.). Nationality: Soviet Union, Education: Moscow State University (1975) Doctoral adviser: Lev Pontryagin. Soviet and Russian mathematician and physicist. Institutions: Russian Space Research Institute, Institute of Control Sciences of Russian Academy of Sciences, Central Aerohydrodynamic Institute (TsAGI) Russia.

**Skills and Activities:** Optimal Control, Nonlinear Dynamics, Quantum Chaos, Control Theory, Advanced Control Theory, Systems Dynamics, Theory of Reference Frames (TR), Waves, Non-Standard Model, Standard Model, Calculations, Quantum Field Theory, Number Theory, Electromagnetic Fields, Partial Differential Equations, Actuarial Mathematics, Geometry, Computational Physics, Quantum Mechanics, Theoretical Physics, Foundations of Quantum Mechanics, Mathematical Physics, First-order Logic, Special and General Relativity, General Relativity, Theoretical Particle Physics, Fundamental Physics, Probability Theory, Gravitational Physics, Field Theory, Theoretical High Energy Physics, High Energy Physics Theory, Particle Physics, Foundations of Physics, Supersymmetry,
$$ (l_{\epsilon})_{\epsilon \in (0, \eta]} = a(\epsilon)_{\epsilon \in (0, \eta]}, \eta \ll 1, \\ (l_{\epsilon})_{\epsilon \in (\eta, 1]} = a, \qquad (A.26) $$

such that $|(r_{\epsilon} - \bar{p})_{\epsilon}| \ge (l_{\epsilon})_{\epsilon} = a(\epsilon)_{\epsilon}$. It meant there exist a thickness $th_{hor} = (l_{\epsilon})_{\epsilon}$ of BH horizon. We introduce a norm $\|th_{hor}\|$ of a thickness $th_{hor}$ by formula

$$ \|th_{hor}\| = \sup_{\epsilon \in (0, \eta]} |l_{\epsilon}| = \eta, \qquad (A.27) $$

By using (A.20) we get the estimate

$$ (\mathbf{R}^{\rho\sigma\mu\nu} (r_{\epsilon}, \epsilon) \mathbf{R}_{\rho\sigma\mu\nu} (r_{\epsilon}, \epsilon))_{\epsilon} \approx \mathbb{R} K(r_s) + \left( \frac{\epsilon^4}{4m^4 [(r_{\epsilon}-2m)^2 + \epsilon^2]^3} \right)_{\epsilon \in (0, \eta]} = \\ K(r_s) + \left( \frac{1}{4m^4 [(r_{\epsilon}-2m)^2 + \epsilon^2]} \right)_{\epsilon \in (0, \eta]} \times \left( \frac{\epsilon^2}{[(r_{\epsilon}-2m)^2 + \epsilon^2]} \right)_{\epsilon \in (0, \eta]} \times \\ \times \left( \frac{\epsilon^2}{[(r_{\epsilon}-2m)^2 + \epsilon^2]} \right)_{\epsilon \in (0, \eta]} \le \\ K(r_s) + \frac{1}{4m^4 [a^2+1]^2} \left( \frac{1}{[(r_{\epsilon}-2m)^2 + \epsilon^2]} \right)_{\epsilon \in (0, \eta]} \le \\ K(r_s) + \frac{1}{4m^4 [a^2+1]^2} \left( \frac{1}{(r-2m)^2} \right)_{r-2m \in (0, \eta)}. \qquad (A.28) $$

## Biography of author(s)

<figure><img src="image_8.png" /></figure>

### Jaykov Foukzon

Currently works at the Center for Mathematical Sciences, Technion - Israel Institute of Technology, Haifa, Israel.

Jaykov Foukzon (born October 7, 1954). Professor, Doctor Sc. (Phys. and Math.). Nationality: Soviet Union, Education: Moscow State University (1975) Doctoral adviser: Lev Pontryagin. Soviet and Russian mathematician and physicist. Institutions: Russian Space Research Institute, Institute of Control Sciences of Russian Academy of Sciences, Central Aerohydrodynamic Institute (TsAGI) Russia.

**Skills and Activities:** Optimal Control, Nonlinear Dynamics, Quantum Chaos, Control Theory, Advanced Control Theory, Systems Dynamics, Theory of Reference Frames (TR), Waves, Non-Standard Model, Standard Model, Calculations, Quantum Field Theory, Number Theory, Electromagnetic Fields, Partial Differential Equations, Actuarial Mathematics, Geometry, Computational Physics, Quantum Mechanics, Theoretical Physics, Foundations of Quantum Mechanics, Mathematical Physics, First-order Logic, Special and General Relativity, General Relativity, Theoretical Particle Physics, Fundamental Physics, Probability Theory, Gravitational Physics, Field Theory, Theoretical High Energy Physics, High Energy Physics Theory, Particle Physics, Foundations of Physics, Supersymmetry,
Classical Mechanics, Quantum Cosmology, Conformal Field Theory, Algebraic Quantum Field Theory, Renormalization-group Theory, Quantum Gravity, Electrodynamics, Set Theory, Logic, IT Systems, Algebra, Analysis, Basic, Numbers, Deterministic Quantum Physics, Mathematical Analysis, Differential Geometry.

**Books:** Jaykov Foukzon, Elena R. Men'kova, Alexander A. Potapov. EPRB Paradox Resolution. Bell inequalities revisited. 03/2019; LAMBERT. ISBN: 978-613-9-45511-9 Jaykov Foukzon, Alexander Potapov, Elena Men'kova. Schrdingers Cat Paradox Resolution. A New Quantum Mechanical Formalism Based on the Probability Representation of Continuous Observables. 12/2017; LAMBERT. ISBN: 978-3-659-93543-5 S. A. Podosenov, A. A. Potapov, J. Foukzon, E. R. Menkova. Fields, Fractals, Singularities and Quantum Control. 10/2016; in press. S. A. Podosenov, A. A. Potapov, J. Foukzon, E. R. Menkova. Nonholonomic, Fractal and Bound Structures in Relativistic Continua, Electrodynamics, Quantum Mechanics and Cosmology. In three books. Publishing house URSS, Moscow, LENAND, 03/2016. ISBN 978-5-9710-2456-9 Jaykov Foukzon, Alexander Potapov, Elena Menkova. Large deviations principles of Non-Freidlin-Wentzell type. 07/2015; LAMBERT. ISBN: 978-3-659-66379-6

<figure><img src="image_9.png" /><figcaption>Alexander Alexeevich Potapov</figcaption></figure>

Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia.

Alexander Alexeevich Potapov (born May 4, 1951). Prof., Doctor Sc. (Phys. and Math.). Head of laboratory of IRE RAS (Kotelnikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia), Academician of the Academy of Engineering Sciences named after A.M. Prokhorov, Academician of RANS, Chief Editor of the International Journal Nonlinear World, Honorable Radio Operator of the Russian Federation. Potapov is the author of the basic researches in the field of fractal approaches to radio physics, radio engineering, radar, electrodynamics, electronics, aerial engineering and control problems and broad spectrum of adjacent scientific and technology paths. Potapov is the initiator of the pioneer researches and development works in the field of fractal theory application in radio physics, scaling effects and fractional operators in the radio systems in Russia. Potapov is the founder of the world well-known Russian scientific school of the fractal methods. He is a president of the joint Chinese-Russian Laboratory of the information technologies and fractal signal processing. Potapov is the Emeritus Professor of the Jinan University (China). He is the academician Prokhorov Prizewinner. Potapov is the author of more than 700 scientific publications, including 20 monographs and 1 textbook.

**Scientific area:** radio physics, dispersion and diffraction of electromagnetic waves, radio location, recognition and signal processing and images, fractal antennas, deterministic chaos, modern topology, fractal analysis, fractional operators, scaling effects.
Classical Mechanics, Quantum Cosmology, Conformal Field Theory, Algebraic Quantum Field Theory, Renormalization-group Theory, Quantum Gravity, Electrodynamics, Set Theory, Logic, IT Systems, Algebra, Analysis, Basic, Numbers, Deterministic Quantum Physics, Mathematical Analysis, Differential Geometry.

**Books:** Jaykov Foukzon, Elena R. Men'kova, Alexander A. Potapov. EPRB Paradox Resolution. Bell inequalities revisited. 03/2019; LAMBERT. ISBN: 978-613-9-45511-9 Jaykov Foukzon, Alexander Potapov, Elena Men'kova. Schrdingers Cat Paradox Resolution. A New Quantum Mechanical Formalism Based on the Probability Representation of Continuous Observables. 12/2017; LAMBERT. ISBN: 978-3-659-93543-5 S. A. Podosenov, A. A. Potapov, J. Foukzon, E. R. Menkova. Fields, Fractals, Singularities and Quantum Control. 10/2016; in press. S. A. Podosenov, A. A. Potapov, J. Foukzon, E. R. Menkova. Nonholonomic, Fractal and Bound Structures in Relativistic Continua, Electrodynamics, Quantum Mechanics and Cosmology. In three books. Publishing house URSS, Moscow, LENAND, 03/2016. ISBN 978-5-9710-2456-9 Jaykov Foukzon, Alexander Potapov, Elena Menkova. Large deviations principles of Non-Freidlin-Wentzell type. 07/2015; LAMBERT. ISBN: 978-3-659-66379-6

<figure><img src="image_9.png" /><figcaption>Alexander Alexeevich Potapov</figcaption></figure>

Kotel'nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009, Russia.

Alexander Alexeevich Potapov (born May 4, 1951). Prof., Doctor Sc. (Phys. and Math.). Head of laboratory of IRE RAS (Kotelnikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia), Academician of the Academy of Engineering Sciences named after A.M. Prokhorov, Academician of RANS, Chief Editor of the International Journal Nonlinear World, Honorable Radio Operator of the Russian Federation. Potapov is the author of the basic researches in the field of fractal approaches to radio physics, radio engineering, radar, electrodynamics, electronics, aerial engineering and control problems and broad spectrum of adjacent scientific and technology paths. Potapov is the initiator of the pioneer researches and development works in the field of fractal theory application in radio physics, scaling effects and fractional operators in the radio systems in Russia. Potapov is the founder of the world well-known Russian scientific school of the fractal methods. He is a president of the joint Chinese-Russian Laboratory of the information technologies and fractal signal processing. Potapov is the Emeritus Professor of the Jinan University (China). He is the academician Prokhorov Prizewinner. Potapov is the author of more than 700 scientific publications, including 20 monographs and 1 textbook.

**Scientific area:** radio physics, dispersion and diffraction of electromagnetic waves, radio location, recognition and signal processing and images, fractal antennas, deterministic chaos, modern topology, fractal analysis, fractional operators, scaling effects.
<figure><img src="image_10.png" /></figure>

## Menkova Elena Romanovna

All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia.

Menkova Elena Romanovna (born July 27, 1955) Ph. D. Senior researcher of All-Russian Research Institute for Optical and Physical Measurements, VNIIOFI. Laboratory of Generation and Measurement of Electromagnetic Pulse Parameters. Education: The Moscow Mendeleev Institute of Chemical Technology (1978). The Moscow Institute of Electronic Mechanical Engineering. Automated Management Systems, qualification improvement faculty (1987).

Managed the literature searches and physical interpretation of models. Scientific area: radiation and measurement of pulse electromagnetic fields and their interaction with the conducting bodies. The research interest focuses on the works connected with problems of relativistic mechanics and electrodynamics, noninertial frames of reference in special and general relativity.

© Copyright 2019 The Author(s), Licensee Book Publisher International, This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## DISCLAIMER

This chapter is an extended version of the article published by the same authors in the following journal with CC BY license. British Journal of Mathematics & Computer Science, 13(6): 1-54, 2016.

## Reviewers Information

1. Jos Antonio de Freitas Pacheco, University of Nice-Sophia Antipolis, France.

2. Francisco Bulnes, Department in Mathematics and Engineering, Tecnolgico de Estudios Superiores de Chalco, Mexico.
<figure><img src="image_10.png" /></figure>

## Menkova Elena Romanovna

All-Russian Research Institute for Optical and Physical Measurements, Moscow, 119361, Russia.

Menkova Elena Romanovna (born July 27, 1955) Ph. D. Senior researcher of All-Russian Research Institute for Optical and Physical Measurements, VNIIOFI. Laboratory of Generation and Measurement of Electromagnetic Pulse Parameters. Education: The Moscow Mendeleev Institute of Chemical Technology (1978). The Moscow Institute of Electronic Mechanical Engineering. Automated Management Systems, qualification improvement faculty (1987).

Managed the literature searches and physical interpretation of models. Scientific area: radiation and measurement of pulse electromagnetic fields and their interaction with the conducting bodies. The research interest focuses on the works connected with problems of relativistic mechanics and electrodynamics, noninertial frames of reference in special and general relativity.

© Copyright 2019 The Author(s), Licensee Book Publisher International, This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## DISCLAIMER

This chapter is an extended version of the article published by the same authors in the following journal with CC BY license. British Journal of Mathematics & Computer Science, 13(6): 1-54, 2016.

## Reviewers Information

1. Jos Antonio de Freitas Pacheco, University of Nice-Sophia Antipolis, France.

2. Francisco Bulnes, Department in Mathematics and Engineering, Tecnolgico de Estudios Superiores de Chalco, Mexico.