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

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

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

$$ \begin{aligned} K \left( \left[ \mathbf{R}_{\epsilon}^{+} \right]_{2}^{2} \right)_{\epsilon} \Phi(\vec{x}) d^3 x = & K \left( \left[ \mathbf{R}_{\epsilon}^{+} \right]_{3}^{3} \right)_{\epsilon} \Phi(\vec{x}) d^3 x = \\ & \frac{R}{2m} \left( r(h_{\epsilon}^{+})_{\epsilon} + 1 + (h_{\epsilon}^{+})_{\epsilon} \right) \tilde{\Phi}(r) dr = \\ & \frac{R}{2m} \left\{ - \frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} \right\} \tilde{\Phi}(r) dr + \frac{R}{2m} \tilde{\Phi}(r) dr. \end{aligned} \quad (3.6) $$

By replacement $r-2m=u$, from (3.6) we obtain

$$ K \left( \left[ \mathbf{R}_{\epsilon}^{+} \right]_{2}^{2} \right)_{\epsilon} \Phi(x) d^3 x = K \left( \left[ \mathbf{R}_{\epsilon}^{+} \right]_{3}^{3} \right)_{\epsilon} \Phi(x) d^3 x = - \frac{R-2m}{0} \frac{u \tilde{\Phi}(u+2m) du}{(u^2 + \epsilon^2)^{1/2}} + \frac{R-2m}{0} \tilde{\Phi}(u+2m) du. \quad (3.7) $$

By replacement $u=\epsilon\eta$, from (3.7) we obtain the expression

$$ \left\{ \begin{aligned} \mathbf{I}_3^+(\epsilon) = & K \left( \left[ \mathbf{R}_\epsilon^+ \right]_3^3 \right)_\epsilon \Phi(x) d^3 x = \mathbf{I}_2^+(\epsilon) = K \left( \left[ \mathbf{R}_\epsilon^+ \right]_2^2 \right)_\epsilon \Phi(\vec{x}) d^3 x = \\ & -\epsilon \times \left( \frac{R-2m}{0} \frac{\eta \tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} - \frac{R-2m}{0} \tilde{\Phi}(\epsilon\eta + 2m) d\eta \right). \end{aligned} \right. \quad (3.8) $$

From Eq.(3.8) we obtain

$$ \begin{aligned} \mathbf{I}_3^+(\epsilon) = \mathbf{I}_2^+(\epsilon) = & -\epsilon \frac{\tilde{\Phi}(2m)}{0!} \frac{R-2m}{0} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] d\eta - \\ & -\frac{\epsilon^2}{1!} \frac{R-2m}{0} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta = \\ & -\epsilon \tilde{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1} - \left(\frac{R-2m}{\epsilon} - 1\right) \right] - \\ & -\frac{\epsilon^2}{1} \frac{R-2m}{0} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta, \end{aligned} \quad (3.9) $$

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

$$ \left\{ \begin{aligned} \tilde{\Phi}(\epsilon\eta + 2m) &= \sum_{l=0}^{n-1} \frac{\Phi^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{(n)}(\xi), & (3.10) \\ \xi &\triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 \end{aligned} \right. $$

with $\tilde{\Phi}^{(l)}(\xi) \triangleq d^l \tilde{\Phi}/d\xi^l$. Equations (3.9)-(3.10) gives

$$ \left\{ \begin{aligned} & \lim_{\epsilon \to 0} \mathbf{I}_3^+(\epsilon) = \lim_{\epsilon \to 0} \mathbf{I}_2^+(\epsilon) = \\ & \lim_{\epsilon \to 0} \left\{ -\epsilon \tilde{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1} + 1 - \frac{R-2m}{\epsilon} \right] \right\} + \\ & + \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1} \frac{R-2m}{0} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta \right\} = 0 \end{aligned} \right. \quad (3.11) $$
Thus in $S'_{2m}(B_R^+(2m)) \subset S'_{2m}(\mathbb{R}^3) \subset \mathcal{D}'(\mathbb{R}^3)$, where $B^+(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}$ from Eq.(3.11) we obtain

$$ \left\{ \begin{array}{l} w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{3}^{3} = \lim_{\epsilon \to 0} \mathbf{I}_{3}^{+}(\epsilon) = 0, \\ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{2}^{2} = \lim_{\epsilon \to 0} \mathbf{I}_{2}^{+}(\epsilon) = 0. \end{array} \right. \qquad (3.12) $$

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

$$ \left\{ \begin{aligned} & 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} \right) \Phi(x) d^3x = 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} \right) \Phi(x) d^3x = \\ & \qquad \frac{R}{2m} (r^2 (h_{\epsilon}^{+''})_{\epsilon} + 2r (h_{\epsilon}^{+'})_{\epsilon}) \tilde{\Phi}(r) dr = \\ = & \frac{R}{2m} \left\{ - \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \tilde{\Phi}(r) dr. \end{aligned} \right. \qquad (3.13) $$

By replacement $r - 2m = u$, from (3.13) we obtain

$$ \left\{ \begin{aligned} \mathbf{I}_1^+(\epsilon) &= 2_K \left( [\mathbf{R}_\epsilon^+]_1 \right) \Phi(x) d^3x = I_2^+(\epsilon) = 2_K \left( [\mathbf{R}_\epsilon^+]_0 \right) \Phi(x) d^3x \\ &= \frac{R}{2m} (r^2 (h_\epsilon^{+''})_\epsilon + 2r (h_\epsilon^{+'})_\epsilon) \tilde{\Phi}(r) dr = \\ &= \frac{R-2m}{0} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{1/2}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \tilde{\Phi}(u+2m) du. \end{aligned} \right. \qquad (3.14) $$

By replacement $u = \epsilon\eta$, from (3.14) we obtain

$$ \begin{aligned} 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} \right) \Phi(x) d^3x = 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} \right) \Phi(x) d^3x = & \\ = \frac{R}{2m} (r^2 (h_{\epsilon}^{+''})_{\epsilon} + 2r (h_{\epsilon}^{+'})_{\epsilon}) \tilde{\Phi}(r) dr = & \\ = \epsilon_0 \frac{R-2m}{\epsilon} \left\{ -\frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \tilde{\Phi}(\epsilon\eta + 2m) d\eta = & \\ -\epsilon_0 \frac{R-2m}{\epsilon} \frac{\epsilon^2\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - 2m_0 \frac{R-2m}{\epsilon} \frac{\epsilon\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + & \\ \frac{R-2m}{0} \frac{\epsilon^4\eta^3\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} + 2m_0 \frac{R-2m}{\epsilon} \frac{\epsilon^3\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = & \\ \epsilon \left[ -\epsilon_0 \frac{R-2m}{\epsilon} \frac{\eta\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} + \epsilon_0 \frac{R-2m}{\epsilon} \frac{\eta^3\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right] + & \\ 2m \left[ -\epsilon_0 \frac{R-2m}{\epsilon} \frac{\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} + \epsilon_0 \frac{R-2m}{\epsilon} \frac{\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right] = & \end{aligned} \qquad (3.15) $$

From Eq.(3.15) we obtain

$$ \begin{aligned} \mathbf{I}_0^+(\epsilon) = \mathbf{I}_1^+(\epsilon) = 2m \frac{\tilde{\Phi}(2m)}{0! 0} \frac{R-2m}{\epsilon} \left[ -\frac{1}{(\epsilon^2+1)^{1/2}} + \frac{\eta^2}{(\epsilon^2+1)^{3/2}} \right] d\eta + & \\ + \frac{\epsilon}{1! 0} \frac{R-2m}{\epsilon} \tilde{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\epsilon^2+1)^{1/2}} + \frac{\eta^2}{(\epsilon^2+1)^{3/2}} \right] \eta d\eta + & \\ + \frac{\epsilon \tilde{\Phi}(2m)}{0! 0} \frac{R-2m}{\epsilon} \left[ -\frac{1}{(\epsilon^2+1)^{1/2}} + \frac{\eta^2}{(\epsilon^2+1)^{3/2}} \right] d\eta + & \\ + \frac{\epsilon^2}{1! 0} \frac{R-2m}{\epsilon} \tilde{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\epsilon^2+1)^{1/2}} + \frac{\eta^2}{(\epsilon^2+1)^{3/2}} \right] \eta d\eta, & \end{aligned} \qquad (3.16) $$
Thus in $S'_{2m}(B_R^+(2m)) \subset S'_{2m}(\mathbb{R}^3) \subset \mathcal{D}'(\mathbb{R}^3)$, where $B^+(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}$ from Eq.(3.11) we obtain

$$ \left\{ \begin{array}{l} w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{3}^{3} = \lim_{\epsilon \to 0} \mathbf{I}_{3}^{+}(\epsilon) = 0, \\ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{2}^{2} = \lim_{\epsilon \to 0} \mathbf{I}_{2}^{+}(\epsilon) = 0. \end{array} \right. \qquad (3.12) $$

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

$$ \left\{ \begin{aligned} & 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} \right) \Phi(x) \, d^3x = 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} \right) \Phi(x) \, d^3x = \\ & \qquad \frac{R}{2m} (r^2 (h_{\epsilon}^{+''})_{\epsilon} + 2r (h_{\epsilon}^{+'})_{\epsilon}) \tilde{\Phi}(r) \, dr = \\ & = \frac{R}{2m} \left\{ - \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \tilde{\Phi}(r) \, dr. \end{aligned} \right. \qquad (3.13) $$

By replacement $r - 2m = u$, from (3.13) we obtain

$$ \left\{ \begin{aligned} \mathbf{I}_1^+(\epsilon) &= 2_K \left( [\mathbf{R}_\epsilon^+]_1 \right) \Phi(x) \, d^3x = I_2^+(\epsilon) = 2_K \left( [\mathbf{R}_\epsilon^+]_0 \right)_\epsilon \Phi(x) \, d^3x \\ &= \frac{R}{2m} (r^2 (h_\epsilon^{+''})_\epsilon + 2r (h_\epsilon^{+'})_\epsilon) \tilde{\Phi}(r) \, dr = \\ &= \frac{R-2m}{0} \left\{ -\frac{u+2m}{(u^2+\epsilon^2)^{1/2}} + \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \tilde{\Phi}(u+2m) \, du. \end{aligned} \right. \qquad (3.14) $$

By replacement $u = \epsilon\eta$, from (3.14) we obtain

$$ \begin{aligned} & 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} \right) \Phi(x) \, d^3x = 2_K \left( [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} \right) \Phi(x) \, d^3x = \\ & \qquad = \frac{R}{2m} (r^2 (h_{\epsilon}^{+''})_{\epsilon} + 2r (h_{\epsilon}^{+'})_{\epsilon}) \tilde{\Phi}(r) \, dr = \\ & = \epsilon_0 \frac{R-2m}{\epsilon} \left\{ -\frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \tilde{\Phi}(\epsilon\eta + 2m) \, d\eta = \\ & \qquad -\epsilon_0 \frac{R-2m}{\epsilon} \frac{\epsilon^2\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - 2m_0 \frac{R-2m}{\epsilon} \frac{\epsilon\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + \\ & \qquad \frac{R-2m}{0} \frac{\epsilon^4\eta^3\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} + 2m_0 \frac{R-2m}{\epsilon} \frac{\epsilon^3\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = \\ & \qquad \epsilon \left[ -\epsilon_0 \frac{R-2m}{\epsilon} \frac{\eta\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} + \epsilon_0 \frac{R-2m}{\epsilon} \frac{\eta^3\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right] + \\ & \qquad 2m \left[ -\epsilon_0 \frac{R-2m}{\epsilon} \frac{\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} + \epsilon_0 \frac{R-2m}{\epsilon} \frac{\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right]. \end{aligned} \qquad (3.15) $$

From Eq.(3.15) we obtain

$$ \begin{aligned} \mathbf{I}_0^+(\epsilon) = \mathbf{I}_1^+(\epsilon) = 2m \frac{\tilde{\Phi}(2m)}{0! 0} \frac{R-2m}{\epsilon} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \\ + \frac{\epsilon}{1! 0} \frac{R-2m}{\epsilon} \tilde{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta + \\ + \frac{\epsilon \tilde{\Phi}(2m)}{0! 0} \frac{R-2m}{\epsilon} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \\ + \frac{\epsilon^2}{1! 0} \frac{R-2m}{\epsilon} \tilde{\Phi}^{(1)}(\xi) \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta, \end{aligned} \qquad (3.16) $$
Where we have expressed the function $\tilde{\Phi}(\epsilon\eta + 2m)$ as

$$ \left\{ \begin{array}{l} \tilde{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\Phi^{\alpha\beta(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{\alpha\beta(n)}(\xi), \\ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 \end{array} \right. \qquad (3.17) $$

With $\tilde{\Phi}^{(l)}(\xi) \triangleq d^l \tilde{\Phi}/d\xi^l$. Equation (3.17) gives

$$ \begin{aligned} w \lim_{\epsilon \to 0} \mathbf{I}_0^+ (\epsilon) &= w \lim_{\epsilon \to 0} \mathbf{I}_1^+ (\epsilon) = \\ 2m \tilde{\Phi}(2m) \lim_{\epsilon \to 0} \left\{ \begin{array}{l} \frac{R-2m}{\epsilon} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta \\ 2m \tilde{\Phi}(2m) \lim_{s \to \infty} \left[ \begin{array}{l} s \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} - s \frac{d\eta}{(\eta^2+1)^{1/2}} \\ -2m \tilde{\Phi}(2m). \end{array} \right] \end{array} \right\} = \qquad (3.18) \end{aligned} $$

Where use is made of the relation

$$ \lim_{s \to \infty} \left[ \frac{s}{0} \frac{\eta^2 d\eta}{(\eta^2 + 1)^{3/2}} - \frac{s}{0} \frac{d\eta}{(u^2 + 1)^{1/2}} \right] = -1 \qquad (3.19) $$

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

We obtain

$$ w \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} = w \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} = -m \tilde{\Phi}(2m). \qquad (3.20) $$

The supergeneralized Ricci tensor below horizon $[\mathbf{R}_{\epsilon}^{-}]_{\alpha}^{\beta} = [\mathbf{R}_{\epsilon}^{-}]_{\alpha}^{\beta}$

May now be calculated componentwise using the classical formulae

$$ \left\{ \begin{array}{l} \left( [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} \right)_{\epsilon} = \left( [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} \right)_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{-\prime\prime})_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{-\prime})_{\epsilon} \right), \\ \left( [\mathbf{R}_{\epsilon}^{-}]_{2}^{2} \right)_{\epsilon} = \left( [\mathbf{R}_{\epsilon}^{-}]_{3}^{3} \right)_{\epsilon} = \frac{(h_{\epsilon}^{-\prime})_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{-})_{\epsilon}}{r^2}. \end{array} \right. \qquad (3.21) $$

From Eq.(3.21) we obtain

$$ \begin{aligned} h_{\epsilon}^{-}(r) &= -\frac{r-2m}{r} \longmapsto h_{\epsilon}^{-}(r) = \left( \frac{\sqrt{(2m-r)^2 + \epsilon^2}}{r} \right) = -h_{\epsilon}^{+}(r), \quad r < 2m. \\ h_{\epsilon}^{-\prime}(r) &= -h_{\epsilon}^{+\prime}(r) = \frac{r-2m}{r \left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} - \frac{\left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}}{r^2}, \\ r(h_{\epsilon}^{-\prime})_{\epsilon} + 1 + (h_{\epsilon}^{-})_{\epsilon} &= -r(h_{\epsilon}^{+\prime})_{\epsilon} + 1 - (h_{\epsilon}^{+})_{\epsilon} = \frac{r-2m}{\left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} + 1. \qquad (3.22) \\ h_{\epsilon}^{-\prime\prime}(r) &= -h_{\epsilon}^{+\prime\prime}(r) = \frac{r-2m}{r^2 \left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} + \frac{2 \left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}}{r^3}. \\ r^2(h_{\epsilon}^{-\prime\prime})_{\epsilon} + 2r(h_{\epsilon}^{-\prime})_{\epsilon} &= -r^2(h_{\epsilon}^{+\prime\prime})_{\epsilon} - 2r(h_{\epsilon}^{+\prime})_{\epsilon} = \frac{r}{\left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} - \frac{r(r-2m)^2}{\left[ (r-2m)^2 + \epsilon^2 \right]^{3/2}}. \end{aligned} $$
Where we have expressed the function $\tilde{\Phi}(\epsilon\eta + 2m)$ as

$$ \left\{ \begin{array}{l} \tilde{\Phi}(\epsilon\eta + 2m) = \sum_{l=0}^{n-1} \frac{\Phi^{\alpha\beta(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{\alpha\beta(n)}(\xi), \\ \xi \triangleq \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 \end{array} \right. \qquad (3.17) $$

With $\tilde{\Phi}^{(l)}(\xi) \triangleq d^l \tilde{\Phi}/d\xi^l$. Equation (3.17) gives

$$ \begin{aligned} & w \lim_{\epsilon \to 0} \mathbf{I}_0^+ (\epsilon) = w \lim_{\epsilon \to 0} \mathbf{I}_1^+ (\epsilon) = \\ & 2m \tilde{\Phi}(2m) \lim_{\epsilon \to 0} \left\{ \begin{aligned} & \frac{R-2m}{\epsilon} \left[ -\frac{1}{(\eta^2+1)^{1/2}} + \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta \\ & - 2m \tilde{\Phi}(2m) \lim_{s \to \infty} \left[ \begin{aligned} & \frac{s}{\eta^2+1} \frac{\eta^2}{\eta^2+1} \frac{d\eta}{ds} \\ & - \frac{s}{\eta^2+1} \frac{d\eta}{ds} \end{aligned} \right] \end{aligned} \right\} = \\ & = -2m \tilde{\Phi}(2m). \end{aligned} \qquad (3.18) $$

Where use is made of the relation

$$ \lim_{s \to \infty} \left[ \frac{s}{\eta^2 + 1} \frac{d\eta}{ds} - \frac{s}{\eta^2 + 1} \frac{d\eta}{ds} \right] = -1 \qquad (3.19) $$

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

We obtain

$$ w \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} = w \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} = -m \tilde{\Phi}(2m). \qquad (3.20) $$

The supergeneralized Ricci tensor below horizon $[\mathbf{R}_{\epsilon}^{-}]_{\alpha}^{\beta} = [\mathbf{R}_{\epsilon}^{-}]_{\alpha}^{\beta}$

May now be calculated componentwise using the classical formulae

$$ \left\{ \begin{aligned} \left( [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} \right)_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{-\prime\prime})_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{-\prime})_{\epsilon} \right), \\ \left( [\mathbf{R}_{\epsilon}^{-}]_{2}^{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{-}]_{3}^{3} \right)_{\epsilon} = \frac{(h_{\epsilon}^{-\prime})_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{-})_{\epsilon}}{r^2}. \end{aligned} \right. \qquad (3.21) $$

From Eq.(3.21) we obtain

$$ \begin{aligned} h_{\epsilon}^{-}(r) &= -\frac{r-2m}{r} \longmapsto h_{\epsilon}^{-}(r) = \left( \frac{\sqrt{(2m-r)^2 + \epsilon^2}}{r} \right) = -h_{\epsilon}^{+}(r), \quad r < 2m. \\ h_{\epsilon}^{-\prime}(r) &= -h_{\epsilon}^{+\prime}(r) = \frac{r-2m}{r \left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} - \frac{\left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}}{r^2}, \\ r(h_{\epsilon}^{-\prime})_{\epsilon} + 1 + (h_{\epsilon}^{-})_{\epsilon} &= -r(h_{\epsilon}^{+\prime})_{\epsilon} + 1 - (h_{\epsilon}^{+})_{\epsilon} = \frac{r-2m}{\left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} + 1. \end{aligned} \qquad (3.22) \\ h_{\epsilon}^{-\prime\prime}(r) &= -h_{\epsilon}^{+\prime\prime}(r) = \frac{r-2m}{r^2 \left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} + \frac{2 \left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}}{r^3}. \\ r^2(h_{\epsilon}^{-\prime\prime})_{\epsilon} + 2r(h_{\epsilon}^{-\prime})_{\epsilon} &= -r^2(h_{\epsilon}^{+\prime\prime})_{\epsilon} - 2r(h_{\epsilon}^{+\prime})_{\epsilon} = \frac{r}{\left[ (r-2m)^2 + \epsilon^2 \right]^{1/2}} - \frac{r(r-2m)^2}{\left[ (r-2m)^2 + \epsilon^2 \right]^{3/2}}. \end{aligned} $$
Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation $\tilde{\Phi}(r) =_0^\pi \sin\theta d\theta^2_0 d\phi \Phi(x)$ where $\Phi(\vec{x}) \in C^\infty(\mathbb{R}^3)$, $\Phi(x)$ is a function with compact support $K$ such that $K \subseteq B^-(0, 2m) = \{x \in \mathbb{R}^3 | 0 \le \|x\| \le 2m\}$

We get:

$$ K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{2}^{2} \right)_{\epsilon} \Phi(\vec{x}) d^3 x = K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{3}^{3} \right)_{\epsilon} \Phi(\vec{x}) d^3 x = \begin{cases} 2^{2m} (r(h_{\epsilon}^{-})_{\epsilon} + 1 + (h_{\epsilon}^{-})_{\epsilon}) \tilde{\Phi}(r) dr = & (3.23) \\ \frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} \tilde{\Phi}(r) dr + 2^{2m} \tilde{\Phi}(r) dr. & \end{cases} $$

By replacement $r - 2m = u$, from Eq. (3.23) we obtain

$$ \left\{ \begin{aligned} K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{2}^{2} \right)_{\epsilon} \Phi(x) d^3 x = & K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{3}^{3} \right)_{\epsilon} \Phi(x) d^3 x = \\ & \frac{u \tilde{\Phi}(u+2m) du}{(u^2 + \epsilon^2)^{1/2}} + \frac{u \tilde{\Phi}(u+2m) du}{(u^2 + \epsilon^2)^{1/2}}. \end{aligned} \right. \qquad (3.24) $$

By replacement $u = \epsilon\eta$, from (3.23) we obtain

$$ \mathbf{I}_3^{-}(\epsilon) =_K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_3^3 \right)_{\epsilon} \Phi(x) d^3 x = \mathbf{I}_2^{-}(\epsilon) =_K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_2^2 \right)_{\epsilon} \Phi(\vec{x}) d^3 x = \epsilon \times \left( \frac{0}{-\frac{2m}{\epsilon}} \frac{\eta \tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \frac{0}{-\frac{2m}{\epsilon}} \tilde{\Phi}(\epsilon\eta + 2m)d\eta \right), \qquad (3.25) $$

Which is calculated to give

$$ \begin{aligned} \mathbf{I}_3^{-}(\epsilon) = \mathbf{I}_2^{-}(\epsilon) ={}& \epsilon \frac{\tilde{\Phi}(2m)^0}{0!} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] d\eta + \\ & + \frac{\epsilon^2 0}{1! - \frac{2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta = \\ & + \epsilon \tilde{\Phi}(2m) \left[ 1 - \sqrt{\left(\frac{2m}{\epsilon}\right)^2 + 1 + \frac{2m}{\epsilon}} \right] + \\ & + \frac{\epsilon^2 0}{1 - \frac{2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta, \end{aligned} \qquad (3.26) $$

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

$$ \left\{ \begin{aligned} \tilde{\Phi}(\epsilon\eta + 2m) = & \sum_{l=0}^{n-1} \frac{\Phi^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{(n)}(\xi), & (3.27) \\ \xi \triangleq & \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 & \end{aligned} \right. $$

With $\tilde{\Phi}^{(l)} \triangleq d^l \tilde{\Phi}/dr^l$. Equation (3.27) gives
Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation $\tilde{\Phi}(r) =_0^\pi \sin\theta d\theta^2_0 d\phi \Phi(x)$ where $\Phi(\vec{x}) \in C^\infty(\mathbb{R}^3)$, $\Phi(x)$ is a function with compact support $K$ such that $K \subseteq B^-(0, 2m) = \{x \in \mathbb{R}^3 | 0 \le \|x\| \le 2m\}$

We get:

$$ K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{2}^{2} \right)_{\epsilon} \Phi(\vec{x}) d^3 x = K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{3}^{3} \right)_{\epsilon} \Phi(\vec{x}) d^3 x = \begin{cases} 2^{2m} (r(h_{\epsilon}^{-})_{\epsilon} + 1 + (h_{\epsilon}^{-})_{\epsilon}) \tilde{\Phi}(r) dr = & \epsilon > 0 \\ \frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} \tilde{\Phi}(r) dr + 2^{2m} \tilde{\Phi}(r) dr = & \epsilon = 0 \end{cases} \quad (3.23) $$

By replacement $r - 2m = u$, from Eq. (3.23) we obtain

$$ \left\{ \begin{aligned} K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{2}^{2} \right)_{\epsilon} \Phi(x) d^3 x = & K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_{3}^{3} \right)_{\epsilon} \Phi(x) d^3 x = \\ & \frac{u \tilde{\Phi}(u+2m) du}{(u^2 + \epsilon^2)^{1/2}} + \frac{u \tilde{\Phi}(u+2m) du}{(u^2 + \epsilon^2)^{1/2}} \end{aligned} \right. \quad (3.24) $$

By replacement $u = \epsilon\eta$, from (3.23) we obtain

$$ \mathbf{I}_3^{-}(\epsilon) =_K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_3^3 \right)_{\epsilon} \Phi(x) d^3 x = \mathbf{I}_2^{-}(\epsilon) =_K \left( \left[ \mathbf{R}_{\epsilon}^{-} \right]_2^2 \right)_{\epsilon} \Phi(\vec{x}) d^3 x = \\ \epsilon \times \left( \frac{0}{-\frac{2m}{\epsilon}} \frac{\eta \tilde{\Phi}(\epsilon\eta + 2m)d\eta}{(\eta^2 + 1)^{1/2}} + \frac{0}{-\frac{2m}{\epsilon}} \tilde{\Phi}(\epsilon\eta + 2m)d\eta \right), \quad (3.25) $$

Which is calculated to give

$$ \begin{aligned} \mathbf{I}_3^{-}(\epsilon) = \mathbf{I}_2^{-}(\epsilon) ={}& \epsilon \frac{\tilde{\Phi}(2m)^0}{0!} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] d\eta + \\ & + \frac{\epsilon^2 0}{1! - \frac{2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta = \\ & + \epsilon \tilde{\Phi}(2m) \left[ 1 - \sqrt{\left(\frac{2m}{\epsilon}\right)^2 + 1 + \frac{2m}{\epsilon}} \right] + \\ & + \frac{\epsilon^2 0}{1 - \frac{2m}{\epsilon}} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta, \end{aligned} \quad (3.26) $$

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

$$ \left\{ \begin{aligned} \tilde{\Phi}(\epsilon\eta + 2m) = & \sum_{l=0}^{n-1} \frac{\Phi^{(l)}(2m)}{l!} (\epsilon\eta)^l + \frac{1}{n!} (\epsilon\eta)^n \Phi^{(n)}(\xi), & (3.27) \\ \xi \triangleq & \theta\epsilon\eta + 2m, \quad 1 > \theta > 0, \quad n=1 & \end{aligned} \right. $$

With $\tilde{\Phi}^{(l)} \triangleq d^l \tilde{\Phi}/dr^l$. Equation (3.27) gives
$$
\left\{
\begin{aligned}
& \lim_{\epsilon \to 0} \mathbf{I}_3^-(\epsilon) = \lim_{\epsilon \to 0} \mathbf{I}_2^-(\epsilon) = \\
& \qquad \lim_{\epsilon \to 0} \left\{ \epsilon \tilde{\Phi}(2m) \left[ 1 - \sqrt{\left(\frac{2m}{\epsilon}\right)^2 + 1 + \frac{2m}{\epsilon}} \right] \right\} + \\
& \qquad + \lim_{\epsilon \to 0} \left\{ \frac{\epsilon^2}{2} \frac{0}{-2m} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta \right\} = 0.
\end{aligned}
\right.
\qquad (3.28)
$$

Thus in $S'_{2m}(B_R^-(2m)) \subset S'_{2m}(\mathbb{R}^3)$, where $B^-(0, 2m) = \{x \in \mathbb{R}^3 | 0 \le \|x\| \le 2m\}$ from Eq.(3.28) we obtain

$$
\left\{
\begin{aligned}
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{3}^{3} &= \lim_{\epsilon \to 0} \mathbf{I}_{3}^{-}(\epsilon) = 0. \\
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{2}^{2} &= \lim_{\epsilon \to 0} \mathbf{I}_{2}^{-}(\epsilon) = 0.
\end{aligned}
\right.
\qquad (3.29)
$$

For $([\mathbf{R}_\epsilon^-]_1^1)_\epsilon, ([\mathbf{R}_\epsilon^-]_0^0)_\epsilon$ we get:

$$
\begin{align}
2K \left( [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} \right) \Phi(x) d^3 x &= 2K \left( [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} \right) \Phi(x) d^3 x = \nonumber \\
&= 2^{2m} \left\{ \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \tilde{\Phi}(r) dr. \tag{3.30}
\end{align}
$$

By replacement $r - 2m = u$, from (3.30) we obtain

$$
\begin{align}
I_1^+(\epsilon) &= 2 \left( [\mathbf{R}_\epsilon^-]_1^1 \right) \Phi(x) \, d^3x = I_2^+(\epsilon) = 2 \left( [\mathbf{R}_\epsilon^-]_0^0 \right) \Phi(x) \, d^3x \nonumber \\
&= {}_0^{2m} \left( r^2 (h_\epsilon^-)^\prime \right)_\epsilon + 2r (h_\epsilon^-)^\prime \right)_\epsilon \tilde{\Phi}(r) dr = \nonumber \\
&= {}_{-2m}^0 \left\{ \frac{u+2m}{(u^2+\epsilon^2)^{1/2}} - \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \tilde{\Phi}(u+2m) du. \tag{3.31}
\end{align}
$$

By replacement $u = \epsilon\eta$, from (3.31) we obtain

$$
\begin{align}
2K \left( [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} \right) \Phi(x) d^3 x &= 2K \left( [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} \right) \Phi(x) d^3 x = \nonumber \\
&= \epsilon^0 {}_{-2m}^{0} \left\{ \frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \tilde{\Phi}(\epsilon\eta + 2m) d\eta = \nonumber \\
&\quad - \epsilon^0 {}_{-2m}^{0} \frac{\epsilon^2\eta\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} + 2m^0 {}_{-2m}^{0} \frac{\epsilon\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - \nonumber \\
&\quad - \epsilon^0 {}_{-2m}^{0} \frac{\epsilon^4\eta^3\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} - 2m^0 {}_{-2m}^{0} \frac{\epsilon^3\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = \nonumber \\
&= \epsilon^0 {}_{-2m}^{0} \frac{\eta\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} - \epsilon^0 {}_{-2m}^{0} \frac{\eta^3\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} + \nonumber \\
&\quad + 2m \left[ \epsilon^0 {}_{-2m}^{0} \frac{\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{1/2}} - \epsilon^0 {}_{-2m}^{0} \frac{\eta^2\tilde{\Phi}(\epsilon\eta + 2m) d\eta}{(\eta^2 + 1)^{3/2}} \right]. \tag{3.32}
\end{align}
$$

which is calculated to give
$$
\left\{
\begin{aligned}
& \lim_{\epsilon \to 0} \mathbf{I}_3^-(\epsilon) = \lim_{\epsilon \to 0} \mathbf{I}_2^-(\epsilon) = \\
& \qquad \lim_{\epsilon \to 0} \left\{ \epsilon \tilde{\Phi}(2m) \left[ 1 - \sqrt{\left(\frac{2m}{\epsilon}\right)^2 + 1 + \frac{2m}{\epsilon}} \right] \right\} + \\
& \qquad + \lim_{\epsilon \to 0} \left\{ \frac{\epsilon^2}{2} \frac{0}{-2m} \left[ \frac{\eta}{(\eta^2 + 1)^{1/2}} + 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta \right\} = 0.
\end{aligned}
\right.
\tag{3.28}
$$

Thus in $S'_{2m}(B_R^-(2m)) \subset S'_{2m}(\mathbb{R}^3)$, where $B^-(0, 2m) = \{x \in \mathbb{R}^3 | 0 \le \|x\| \le 2m\}$ from Eq.(3.28)
we obtain

$$
\left\{
\begin{aligned}
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{3}^{3} &= \lim_{\epsilon \to 0} \mathbf{I}_{3}^{-}(\epsilon) = 0. \\
w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{2}^{2} &= \lim_{\epsilon \to 0} \mathbf{I}_{2}^{-}(\epsilon) = 0.
\end{aligned}
\right.
\qquad (3.29)
$$

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

$$
\begin{align}
2K \left( [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} \right) \Phi(x) \, d^3 x &= 2K \left( [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} \right) \Phi(x) \, d^3 x = \nonumber \\
&= 2^{2m} \left\{ \frac{r}{[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{r(r-2m)^2}{[(r-2m)^2 + \epsilon^2]^{3/2}} \right\} \tilde{\Phi}(r) dr. \tag{3.30}
\end{align}
$$

By replacement $r - 2m = u$, from (3.30) we obtain

$$
\begin{align}
I_1^+(\epsilon) &= 2 \left( [\mathbf{R}_\epsilon^-]_1^1 \right) \Phi(x) \, d^3x = I_2^+(\epsilon) = 2 \left( [\mathbf{R}_\epsilon^-]_0^0 \right) \Phi(x) \, d^3x \nonumber \\
&= \int_0^{2m} (r^2 (h_\epsilon^-)')_\epsilon + 2r (h_\epsilon^-)'_\epsilon) \tilde{\Phi}(r) dr = \nonumber \\
&= \int_{-2m}^0 \left\{ \frac{u+2m}{(u^2+\epsilon^2)^{1/2}} - \frac{u^2(u+2m)}{(u^2+\epsilon^2)^{3/2}} \right\} \tilde{\Phi}(u+2m) du. \tag{3.31}
\end{align}
$$

By replacement $u = \epsilon\eta$, from (3.31) we obtain

$$
\begin{align}
& 2K \left( [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} \right) \Phi(x) \, d^3x = 2K \left( [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} \right) \Phi(x) \, d^3x = \nonumber \\
& \qquad \int_{-\frac{2m}{\epsilon}}^{0} (r^2 (h_{\epsilon}^{-})')_{\epsilon} + 2r (h_{\epsilon}^{-})'_{\epsilon}) \tilde{\Phi}(r) dr = \nonumber \\
&= \epsilon^{0} \int_{-\frac{2m}{\epsilon}}^{0} \left\{ \frac{\epsilon\eta + 2m}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - \frac{\epsilon^2\eta^2(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} \right\} \tilde{\Phi}(\epsilon\eta + 2m) d\eta = \nonumber \\
& \qquad - \frac{\epsilon^2\eta\tilde{\Phi}(\epsilon\eta + 2m)}{\epsilon^2\eta^2 + \epsilon^2} + 2m^{0} \frac{\epsilon\tilde{\Phi}(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{1/2}} - \nonumber \\
& \qquad - \frac{\epsilon^4\eta^3\tilde{\Phi}(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} - 2m^{0} \frac{\epsilon^3\eta^2\tilde{\Phi}(\epsilon\eta + 2m)}{(\epsilon^2\eta^2 + \epsilon^2)^{3/2}} = \nonumber \\
& \qquad \epsilon^{0} \frac{\eta\tilde{\Phi}(\epsilon\eta + 2m)}{(\epsilon^2 + 1)^{1/2}} - \frac{\epsilon^0 \eta^3\tilde{\Phi}(\epsilon\eta + 2m)}{(\epsilon^2 + 1)^{3/2}} + \nonumber \\
& \qquad + 2m \left[ \frac{0 \eta\tilde{\Phi}(\epsilon\eta + 2m)}{-\frac{2m}{\epsilon}(\epsilon^2 + 1)^{1/2}} - \frac{0 \eta^2\tilde{\Phi}(\epsilon\eta + 2m)}{-\frac{2m}{\epsilon}(\epsilon^2 + 1)^{3/2}} \right]. \tag{3.32}
\end{align}
$$

which is calculated to give
$$ \begin{aligned} \mathbf{I}_0^-(\epsilon) = \mathbf{I}_1^-(\epsilon) = 2m \frac{\tilde{\Phi}(2m)}{0!} \epsilon^{\frac{2m}{\epsilon}} \epsilon^0 \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \right. \\ \left. + \frac{\epsilon^{\frac{2m}{\epsilon}}}{1!} \tilde{\Phi}^{(1)}(\xi) \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta + O(\epsilon^2). \end{aligned} \quad (3.33) $$

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

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

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

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

where use is made of the relation

$$ \lim_{s \to \infty} \left[ \frac{0}{-s} \frac{d\eta}{(u^2+1)^{1/2}} - \frac{0}{-s} \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} \right] = 1. \quad (3.36) $$

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

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

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

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

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

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

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

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

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

$$ \begin{aligned} (\mathbf{R}^{\pm\mu\nu}(\epsilon) \mathbf{R}_{\mu\nu}^{\pm}(\epsilon)) = & \\ \left( \left( \frac{1}{2} (h_{\epsilon}^{\pm})'' + \frac{1}{r} (h_{\epsilon}^{\pm})' \right)^2 \right)_{\epsilon} + 2 \left[ \left( -\frac{h_{\epsilon}^{\pm}}{r} + \frac{1}{r^2} \right)^2 \right]_{\epsilon} + \left( \left[ \frac{1}{2} (h_{\epsilon}^{\pm})'' + \frac{(h_{\epsilon}^{\pm})'}{r} \right]^2 \right)_{\epsilon} \asymp & \\ \asymp & \frac{1}{4 [(r-2m)^2 + \epsilon^2]} + \frac{1}{(2m)^4}, & \end{aligned} \quad (3.41) $$
$$ 
\begin{aligned}
\mathbf{I}_0^-(\epsilon) = \mathbf{I}_1^-(\epsilon) = 2m \frac{\tilde{\Phi}(2m)}{0!} \epsilon^{\frac{2m}{\epsilon}} \epsilon^0 \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] d\eta + \right. \\
\left. + \frac{\epsilon^{\frac{2m}{\epsilon}}}{1!} \tilde{\Phi}^{(1)}(\xi) \left[ \frac{1}{(\eta^2+1)^{1/2}} - \frac{\eta^2}{(\eta^2+1)^{3/2}} \right] \eta d\eta + O(\epsilon^2).
\end{aligned} 
\quad (3.33) $$

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

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

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

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

where use is made of the relation

$$ \lim_{s \to \infty} \left[ \frac{0}{-s} \frac{d\eta}{(u^2+1)^{1/2}} - \frac{0}{-s} \frac{\eta^2 d\eta}{(\eta^2+1)^{3/2}} \right] = 1. \quad (3.36) $$

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

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

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

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

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

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

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

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

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

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

$$ (3.40) \quad (\mathbf{R}^{\rho\sigma\mu\nu} (\epsilon) \mathbf{R}_{\rho\sigma\mu\nu} (\epsilon))_{\epsilon} = \left( \left( (h_{\epsilon}^{\pm})'' \right)^2 + 2 \left( \frac{h_{\epsilon}^{\pm}}{r} \right)'^2 + 4 \frac{1}{r^4} + 2 \left( \frac{h_{\epsilon}^{\pm}}{r} \right)'^2 \right)_{\epsilon} \asymp \frac{1}{[(r-2m)^2 + \epsilon^2]} + \frac{4}{(2m)^4}. \quad (3.42) $$

## 3.2 Examples of Distributional Geometries. Calculation of the Distributi-onal Quadratic Scalars by using Nonsmooth Regula- rization via Horizon

Let us consider again the Schwarzschild metric (3.1)

$$ \left\{ \begin{array}{l} ds^2 = h(r)dt^2 - h(r)^{-1}dr^2 + r^2d\Omega^2, \\ h(r) = -1 + \frac{2m}{r} = -\frac{r-2m}{r}, \\ h^{-1}(r) = -\frac{r-2m}{r}. \end{array} \right. \qquad (3.43) $$

We rewrite now the Schwarzschild metric (3.43) above Horizon ($r \ge 2m$) in the form

$$ \left\{ \begin{array}{l} ds^{+2} = -A^{+}(r)dt^2 + (A^{+}(r))^{-1}(r)dr^2 + r^2d\Omega^2, \\ A^{+}(r) = \frac{r-2m}{r}, \\ (A^{+}(r))^{-1} = \frac{r}{r-2m}. \end{array} \right. \qquad (3.44) $$

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

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

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

$$ \left\{ \begin{array}{l} (\tilde{ds}_{\epsilon}^{+2})_{\epsilon} = (-A_{\epsilon}^{+}(r)dt^2 + (A_{\epsilon}^{+}(r))^{-1}dr^2)_{\epsilon} + r^2d\Omega^2, \\ (A_{\epsilon}^{+}(r))_{\epsilon} = \left( \frac{\sqrt{r^2 + \epsilon^2 - 2m}}{r} \right)_{\epsilon}, \\ ((A_{\epsilon}^{+}(r))^{-1})_{\epsilon} = \left( \frac{r}{\sqrt{r^2 + \epsilon^2 - 2m}} \right)_{\epsilon}. \end{array} \right. \qquad (3.46) $$

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

$$ \left\{ \begin{array}{l} ds^{-2} = A^{-}(r)dt^2 - (A^{-}(r))^{-1}dr^2 + r^2d\Omega^2, \\ A^{-}(r) = \frac{2m-r}{r}, (A^{-}(r))^{-1} = \frac{r}{2m-r}. \end{array} \right. \qquad (3.47) $$

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

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

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

$$ (3.40) \quad (\mathbf{R}^{\rho\sigma\mu\nu} (\epsilon) \mathbf{R}_{\rho\sigma\mu\nu} (\epsilon))_{\epsilon} = \left( \left( (h_{\epsilon}^{\pm})'' \right)^2 + 2 \left( \frac{h_{\epsilon}^{\pm}}{r} \right)'^2 + 4 \frac{1}{r^4} + 2 \left( \frac{h_{\epsilon}^{\pm}}{r} \right)'^2 \right)_{\epsilon} \asymp \frac{1}{[(r-2m)^2 + \epsilon^2]} + \frac{4}{(2m)^4}. \quad (3.42) $$

## 3.2 Examples of Distributional Geometries. Calculation of the Distributi-onal Quadratic Scalars by using Nonsmooth Regula- rization via Horizon

Let us consider again the Schwarzschild metric (3.1)

$$ \left\{ \begin{array}{l} ds^2 = h(r)dt^2 - h(r)^{-1}dr^2 + r^2d\Omega^2, \\ h(r) = -1 + \frac{2m}{r} = -\frac{r-2m}{r}, \\ h^{-1}(r) = -\frac{r-2m}{r}. \end{array} \right. \qquad (3.43) $$

We rewrite now the Schwarzschild metric (3.43) above Horizon ($r \ge 2m$) in the form

$$ \left\{ \begin{array}{l} ds^{+2} = -A^{+}(r)dt^2 + (A^{+}(r))^{-1}(r)dr^2 + r^2d\Omega^2, \\ A^{+}(r) = \frac{r-2m}{r}, \\ (A^{+}(r))^{-1} = \frac{r}{r-2m}. \end{array} \right. \qquad (3.44) $$

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

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

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

$$ \left\{ \begin{array}{l} (\tilde{ds}_{\epsilon}^{+2})_{\epsilon} = (-A_{\epsilon}^{+}(r)dt^2 + (A_{\epsilon}^{+}(r))^{-1}dr^2)_{\epsilon} + r^2d\Omega^2, \\ (A_{\epsilon}^{+}(r))_{\epsilon} = \left( \frac{\sqrt{r^2 + \epsilon^2 - 2m}}{r} \right)_{\epsilon}, \\ ((A_{\epsilon}^{+}(r))^{-1})_{\epsilon} = \left( \frac{r}{\sqrt{r^2 + \epsilon^2 - 2m}} \right)_{\epsilon}. \end{array} \right. \qquad (3.46) $$

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

$$ \left\{ \begin{array}{l} ds^{-2} = A^{-}(r)dt^2 - (A^{-}(r))^{-1}dr^2 + r^2d\Omega^2, \\ A^{-}(r) = \frac{2m-r}{r}, (A^{-}(r))^{-1} = \frac{r}{2m-r}. \end{array} \right. \qquad (3.47) $$

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

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

Thus below Horizon ($r < 2m$) the corresponding distributional metric $(\tilde{ds}_{\epsilon}^{-2})_{\epsilon}$ takes the form
$$ \left\{ \begin{array}{l} (d\tilde{s}_{\epsilon}^{-2})_{\epsilon} = (A_{\epsilon}^{-}(r)dt^2 - (A_{\epsilon}^{-}(r))^{-1}dr^2)_{\epsilon} + r^2 d\Omega^2, \\ (A_{\epsilon}^{-}(r))_{\epsilon} = \left(\frac{2m - \sqrt{r^2 + \epsilon^2}}{r}\right)_{\epsilon}, \quad ((A_{\epsilon}^{-}(r))^{-1})_{\epsilon} = \left(\frac{r}{2m - \sqrt{r^2 + \epsilon^2}}\right)_{\epsilon}. \end{array} \right. \qquad (3.49) $$

From Eq.(3.46) one obtains

$$ \left\{ \begin{aligned} (A_{\epsilon}^{+})' &= \left(-\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r}\right)' = -\frac{1}{\sqrt{r^2 + \epsilon^2}} + \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^2} \\ (A_{\epsilon}^{+})'' &= \frac{r}{(r^2 + \epsilon^2)^{3/2}} - 2\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^3} + \frac{1}{r\sqrt{r^2 + \epsilon^2}} \end{aligned} \right. \qquad (3.50) $$

From Eq.(3.46) using Eq.(A.5) one obtains

$$ \begin{aligned} (\mathbf{R}(\epsilon))_{\epsilon} = & \left(-\frac{4A'_{\epsilon}}{r} + \frac{2A_{\epsilon}C_{\epsilon}}{r^2} - A''_{\epsilon}\right)_{\epsilon} = \\ & -\frac{4}{r}\left(-\frac{1}{\sqrt{r^2 + \epsilon^2}} + \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^2}\right)_{\epsilon} + \frac{2}{r^2} - \\ & -\left(\frac{r}{(r^2 + \epsilon^2)^{3/2}} - 2\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^3} + \frac{1}{r\sqrt{r^2 + \epsilon^2}}\right)_{\epsilon}. \end{aligned} \qquad (3.51) $$

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

$$ (\mathbf{R}(\epsilon))_{\epsilon} = \frac{6}{(2m)^2} - \frac{1}{(2m)^2} + \frac{1}{(2m)^3} = \frac{1}{(2m)^3} + \frac{5}{(2m)^2} \qquad (3.52) $$

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

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

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

Where

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

**Remark 3.3.** Notethat the coordinates $x^0/c$ and $r$ are time and space coordinates, respectively, only if

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

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

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

The metric (3.53)-(3.55) takes the form
$$ \left\{ \begin{array}{l} (d\tilde{s}_{\epsilon}^{-2})_{\epsilon} = (A_{\epsilon}^{-}(r)dt^2 - (A_{\epsilon}^{-}(r))^{-1}dr^2)_{\epsilon} + r^2 d\Omega^2, \\ (A_{\epsilon}^{-}(r))_{\epsilon} = \left(\frac{2m - \sqrt{r^2 + \epsilon^2}}{r}\right)_{\epsilon}, \quad ((A_{\epsilon}^{-}(r))^{-1})_{\epsilon} = \left(\frac{r}{2m - \sqrt{r^2 + \epsilon^2}}\right)_{\epsilon}. \end{array} \right. \qquad (3.49) $$

From Eq.(3.46) one obtains

$$ \left\{ \begin{aligned} (A_{\epsilon}^{+})' &= \left(-\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r}\right)' = -\frac{1}{\sqrt{r^2 + \epsilon^2}} + \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^2} \\ (A_{\epsilon}^{+})'' &= \frac{r}{(r^2 + \epsilon^2)^{3/2}} - 2\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^3} + \frac{1}{r\sqrt{r^2 + \epsilon^2}} \end{aligned} \right. \qquad (3.50) $$

From Eq.(3.46) using Eq.(A.5) one obtains

$$ \begin{aligned} (\mathbf{R}(\epsilon))_{\epsilon} = & \left(-\frac{4A'_{\epsilon}}{r} + \frac{2A_{\epsilon}C_{\epsilon}}{r^2} - A''_{\epsilon}\right)_{\epsilon} = \\ & -\frac{4}{r}\left(-\frac{1}{\sqrt{r^2 + \epsilon^2}} + \frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^2}\right)_{\epsilon} + \frac{2}{r^2} - \\ & -\left(\frac{r}{(r^2 + \epsilon^2)^{3/2}} - 2\frac{\sqrt{r^2 + \epsilon^2} - 2m}{r^3} + \frac{1}{r\sqrt{r^2 + \epsilon^2}}\right)_{\epsilon}. \end{aligned} \qquad (3.51) $$

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

$$ (\mathbf{R}(\epsilon))_{\epsilon} = \frac{6}{(2m)^2} - \frac{1}{(2m)^2} + \frac{1}{(2m)^3} = \frac{1}{(2m)^3} + \frac{5}{(2m)^2} \qquad (3.52) $$

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

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

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

Where

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

**Remark 3.3.** Notethat the coordinates $x^0/c$ and $r$ are time and space coordinates, respectively, only if

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

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

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

The metric (3.53)-(3.55) takes the form
$$ds^2 = g_{\mu\nu} dx^\mu dx^\nu \qquad (3.57)$$

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

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

From Eq.(3.54) one obtains

$$\begin{aligned} A(r) &= \Omega^2 \left( \frac{\rho(r)}{\rho(r) - |a|} \right), \quad B = \frac{(\rho(r) - |a|)^2}{\rho^2(r)}, \\ C(r) &= \left( \frac{\rho(r) - |a|}{\rho(r)} \right) - \frac{(\rho(r) - |a|)^2}{\rho^2(r)} - \left( \frac{\rho(r)}{\rho(r) - |a|} \right) (f'(r))^2, \quad (3.59) \\ D(r) &= \Omega \left( \frac{\rho(r)}{\rho(r) - |a|} \right) f'(r), f'(r) \triangleq df(r)/dr. \end{aligned}$$

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

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

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

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

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

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

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

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

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

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

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

$$\begin{aligned} g_{00}^{\pm}(\epsilon) &= -A_{\epsilon}^{\pm}(r), \quad g_{0\alpha}^{\pm}(\epsilon) = -D_{\epsilon}^{\pm}(r) \frac{x^{\alpha}}{r}, \\ g_{\alpha\beta}^{\pm}(\epsilon) &= B_{\epsilon}^{\pm}(r) \delta^{\alpha\beta} + C_{\epsilon}^{\pm}(r) \frac{x^{\alpha} x^{\beta}}{r^2}. \end{aligned} \qquad (3.65)$$
$$ds^2 = g_{\mu\nu} dx^\mu dx^\nu \qquad (3.57)$$

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

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

From Eq.(3.54) one obtains

$$A(r) = \Omega^2 \left( \frac{\rho(r)}{\rho(r) - |a|} \right), \quad B = \frac{(\rho(r) - |a|)^2}{\rho^2(r)}, \\ C(r) = \left( \frac{\rho(r) - |a|}{\rho(r)} \right) - \frac{(\rho(r) - |a|)^2}{\rho^2(r)} - \left( \frac{\rho(r)}{\rho(r) - |a|} \right) (f'(r))^2, \qquad (3.59) \\ D(r) = \Omega \left( \frac{\rho(r)}{\rho(r) - |a|} \right) f'(r), f'(r) \triangleq \frac{df(r)}{dr}.$$

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

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

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

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

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

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

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

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

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

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

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

$$g_{00}^{\pm}(\epsilon) = -A_{\epsilon}^{\pm}(r), \quad g_{0\alpha}^{\pm}(\epsilon) = -D_{\epsilon}^{\pm}(r) \frac{x^{\alpha}}{r}, \\ g_{\alpha\beta}^{\pm}(\epsilon) = B_{\epsilon}^{\pm}(r) \delta^{\alpha\beta} + C_{\epsilon}^{\pm}(r) \frac{x^{\alpha} x^{\beta}}{r^2}. \qquad (3.65)$$
Using now Eq. A2 one obtains that the Colombeau curvature scalars $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon}$ in terms of Colombeau generalized functions $(A_{\epsilon}^{\pm}(r))_{\epsilon}, (B_{\epsilon}^{\pm}(r))_{\epsilon}, (C_{\epsilon}^{\pm}(r))_{\epsilon}, (D_{\epsilon}^{\pm}(r))_{\epsilon}$ is expressed as

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

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

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

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

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

$$ \left\{ \pm \left( \frac{(r^2 + \epsilon^2)^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^4} \left\{ \left[ \frac{5}{2} \left[ \frac{3a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} \right]^2 - \frac{2a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] + 2 \left[ \frac{3a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} + \frac{a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \right\} \right)_{\epsilon} = \right. \quad (3.68) $$

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

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

$$ \begin{aligned} (\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon} = & \left( \frac{12a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^6} \left[ 1 + \frac{a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] \right)^2 \mp \\ & \mp \frac{4a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^5} \left[ 1 + \frac{a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \left[ \frac{2\epsilon^2}{r^2(r^2 + \epsilon^2)} + \frac{9\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} \right] + \\ & \frac{a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^4} \left[ \frac{4\epsilon^4}{r^4(r^2 + \epsilon^2)} + \frac{81\epsilon^4}{(r^2 + \epsilon^2)^3} \right]. \end{aligned} \quad (3.69) $$

**Remark 3.8.** Note that (i) on horizon $r = a$ Colombeau scalars $(\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon}$ well defined and becomes to infinite large Colombeau generalized numbers.
Using now Eq. A2 one obtains that the Colombeau curvature scalars $(\mathbf{R}^{\pm}(\epsilon))_{\epsilon}$ in terms of Colombeau generalized functions $(A_{\epsilon}^{\pm}(r))_{\epsilon}, (B_{\epsilon}^{\pm}(r))_{\epsilon}, (C_{\epsilon}^{\pm}(r))_{\epsilon}, (D_{\epsilon}^{\pm}(r))_{\epsilon}$ is expressed as

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

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

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

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

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

$$ \left\{ \pm \left( \frac{(r^2 + \epsilon^2)^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^4} \left\{ \left[ \frac{5}{2} \left[ \frac{3a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} \right]^2 - \frac{2a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right] + 2 \left[ \frac{3a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} + \frac{a\epsilon^2}{r^2(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \right\} \right)_{\epsilon} = \right. \quad (3.68) $$

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

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

$$ (\mathbf{R}^{\pm\rho\sigma\mu\nu}(\epsilon) \mathbf{R}_{\rho\sigma\mu\nu}^{\pm}(\epsilon))_{\epsilon} = \left( \frac{12a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^6} \left[ 1 + \frac{a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \mp \frac{4a^2}{(\sqrt{r^2 + \epsilon^2} - |a|)^5} \left[ 1 + \frac{a\epsilon^2}{(r^2 + \epsilon^2)^{\frac{3}{2}}} \right]^2 \left[ \frac{2\epsilon^2}{r^2(r^2 + \epsilon^2)} + \frac{9\epsilon^2}{(r^2 + \epsilon^2)^{\frac{5}{2}}} \right] + \frac{4\epsilon^4}{(\sqrt{r^2 + \epsilon^2} - |a|)^4} \left[ \frac{4\epsilon^4}{r^4(r^2 + \epsilon^2)} + \frac{81\epsilon^4}{(r^2 + \epsilon^2)^3} \right] \right) \quad (3.69) $$

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

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

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

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

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

and

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

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

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

# 4 Quantum Scalar Field in Curved Distributional Spacetime

## 4.1 Canonical Quantization in Curved Distributional Spacetime

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

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

The corresponding equation of motion is

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

Here

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

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

$$ (S_{\epsilon})_{\epsilon} = (dx^0 L_{\epsilon})_{\epsilon}, (L_{\epsilon})_{\epsilon} = (d^{D-1}x_{\epsilon})_{\epsilon}. \qquad (4.4) $$

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

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

Where $\underline{x}$ labels a point on a surface of constant $x^0$, the $x^0$ argument of $(\varphi_{\epsilon})_{\epsilon}$ is suppressed, $n^\mu$ is the
defined and becomes to infinite large Colombeau generalized numbers, (ii) for $r \neq a$ Colombeau scalars finite

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

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

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

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

and

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

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

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

# 4 Quantum Scalar Field in Curved Distributional Spacetime

## 4.1 Canonical Quantization in Curved Distributional Spacetime

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

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

The corresponding equation of motion is

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

Here

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

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

$$ (S_{\epsilon})_{\epsilon} = (dx^0 L_{\epsilon})_{\epsilon}, (L_{\epsilon})_{\epsilon} = (d^{D-1}x_{\epsilon})_{\epsilon}. \qquad (4.4) $$

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

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

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

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

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

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

Where

$$ (j_{\epsilon}^{\mu}(\varphi_{\epsilon}, \phi_{\epsilon}))_{\epsilon} = (i/\hbar) \left( |g_{\epsilon}|^{1/2} g_{\epsilon}^{\mu\nu} (\bar{\varphi}_{\epsilon} \partial_{\nu}\phi_{\epsilon} - \varphi_{\epsilon} \partial_{\nu}\bar{\phi}_{\epsilon}) \right)_{\epsilon} \quad (4.8) $$

This bracket is called the generalized Klein-Gordon inner product, and $(\langle\phi_\epsilon, \phi_\epsilon\rangle)_\epsilon$ the generalized Klein Gordon norm of $(\varphi_\epsilon)_\epsilon$. The generalized current density $(j_\epsilon^\mu(\varphi_\epsilon, \phi_\epsilon))_\epsilon$ is divergenceless,i.e. $(\partial_\mu j_\epsilon^\mu(\varphi_\epsilon, \phi_\epsilon))_\epsilon = 0$ when the Colombeau generalized functions $(\varphi_\epsilon)_\epsilon$ and $(\phi_\epsilon)_\epsilon$ satisfy the KG equation (4.2), hence the value of the integral in (4.7) is independent of the spacelike surface 3a3 over which it is evaluated, provided the functions vanish at spatial infinity. The generalized KG inner product satisfies the relations

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

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

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

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

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

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

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

Note that from Eq.(4.11) follows

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

Note that if $(\phi_\epsilon)_\epsilon$ is a positive norm solution with unit norm hf with, then $(a(\varphi_\epsilon))_\epsilon$ and $a^\dagger(\varphi_\epsilon)$ satisfy the commutation relation $([a^\dagger(\varphi_\epsilon), a(\phi_\epsilon)])_\epsilon = 1$. Suppose now that $|3a8\rangle$ is a normalized quantum state satisfying $(a(\phi_\epsilon)|3a8\rangle)_\epsilon = 1$, then for each $n$, the state $|n, 3a8\rangle = ((1/\sqrt{n!})(a(\phi_\epsilon))^n|3a8\rangle)_\epsilon$ is a normalized eigenstate of the number operator $(N[(\phi_\epsilon)])_\epsilon = (a^\dagger(\varphi_\epsilon)a(\varphi_\epsilon))_\epsilon$ with eigenvalue $n$. The span of all these states defines a Fock space of the distributional $(\varphi_\epsilon)_\epsilon$-wavepacket “n-particle excitations” above the state $|3a8\rangle$. If we want to construct the full Hilbert space of the field theory in curved distributional spacetime, how can we proceed? We should find a decomposition of the space of complex Colombeau solutions to the wave equation (4.2) S into a direct sum of a positive norm subspace $S_p$ and its complex conjugate $\overline{S_p}$, such that all brackets between solutions from the two subspaces vanish. That is, we must find a direct sum decomposition:

$$ S = S_p \oplus \overline{S_p} \quad (4.14) $$
unit normal to the surface, and $(|h_{\varepsilon}|)_{\varepsilon}$ is the determinant of the induced spatial metric $(h_{ij}(\varepsilon))_{\varepsilon}$. To quantize, the Colombeau generalized field $(\varphi_{\varepsilon})_{\varepsilon}$ and its conjugate momentum $(\pi_{\varepsilon}(\underline{x}))_{\varepsilon}$ are now promoted to hermitian operators and required to satisfy the canonical commutation relation,

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

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

$$ (\langle\varphi_{\varepsilon}, \phi_{\varepsilon}\rangle)_{\varepsilon} = (\Sigma d\Sigma_{\mu} j_{\varepsilon}^{\mu})_{\varepsilon}, \varepsilon \in (0, 1]. \quad (4.7) $$

Where

$$ (j_{\varepsilon}^{\mu}(\varphi_{\varepsilon}, \phi_{\varepsilon}))_{\varepsilon} = (i/\hbar) \left( |g_{\varepsilon}|^{1/2} g_{\varepsilon}^{\mu\nu} (\bar{\varphi}_{\varepsilon} \partial_{\nu}\phi_{\varepsilon} - \varphi_{\varepsilon} \partial_{\nu}\bar{\phi}_{\varepsilon}) \right)_{\varepsilon}. \quad (4.8) $$

This bracket is called the generalized Klein-Gordon inner product, and $(\langle\phi_{\varepsilon}, \phi_{\varepsilon}\rangle)_{\varepsilon}$ the generalized Klein Gordon norm of $(\varphi_{\varepsilon})_{\varepsilon}$. The generalized current density $(j_{\varepsilon}^{\mu}(\varphi_{\varepsilon}, \phi_{\varepsilon}))_{\varepsilon}$ is divergenceless,i.e. $(\partial_{\mu}j_{\varepsilon}^{\mu}(\varphi_{\varepsilon}, \phi_{\varepsilon}))_{\varepsilon} = 0$ when the Colombeau generalized functions $(\varphi_{\varepsilon})_{\varepsilon}$ and $(\phi_{\varepsilon})_{\varepsilon}$ satisfy the KG equation (4.2), hence the value of the integral in (4.7) is independent of the spacelike surface 3a3 over which it is evaluated, provided the functions vanish at spatial infinity. The generalized KG inner product satisfies the relations

$$ (\langle\bar{\varphi}_{\varepsilon}, \phi_{\varepsilon}\rangle)_{\varepsilon} = -(\langle\bar{\phi}_{\varepsilon}, \bar{\phi}_{\varepsilon}\rangle)_{\varepsilon} = (\langle\phi_{\varepsilon}, \varphi_{\varepsilon}\rangle)_{\varepsilon}, \varepsilon \in (0, 1]. \quad (4.9) $$

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

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

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

$$ (a^{\dagger}(\phi_{\varepsilon}))_{\varepsilon} = -(a(\bar{\phi}_{\varepsilon}))_{\varepsilon}. \quad (4.11) $$

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

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

Note that from Eq.(4.11) follows

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

Note that if $(\phi_{\varepsilon})_{\varepsilon}$ is a positive norm solution with unit norm hf with, then $(a(\varphi_{\varepsilon}))_{\varepsilon}$ and $a^{\dagger}(\varphi_{\varepsilon})$ satisfy the commutation relation $([a^{\dagger}(\varphi_{\varepsilon}), a(\phi_{\varepsilon})])_{\varepsilon} = 1$. Suppose now that $|3a8\rangle$ is a normalized quantum state satisfying $(a(\phi_{\varepsilon})|3a8\rangle)_{\varepsilon} = 1$, then for each $n$, the state $|n, 3a8\rangle = ((1/\sqrt{n!})(a(\phi_{\varepsilon}))^{n}|3a8\rangle)_{\varepsilon}$ is a normalized eigenstate of the number operator $(N[(\phi_{\varepsilon})])_{\varepsilon} = (a^{\dagger}(\varphi_{\varepsilon})a(\varphi_{\varepsilon}))_{\varepsilon}$ with eigenvalue $n$. The span of all these states defines a Fock space of the distributional $(\varphi_{\varepsilon})_{\varepsilon}$-wavepacket “n-particle excitations” above the state $|3a8\rangle$. If we want to construct the full Hilbert space of the field theory in curved distributional spacetime, how can we proceed? We should find a decomposition of the space of complex Colombeau solutions to the wave equation (4.2) S into a direct sum of a positive norm subspace $S_p$ and its complex conjugate $\overline{S_p}$, such that all brackets between solutions from the two subspaces vanish. That is, we must find a direct sum decomposition:

$$ S = S_p \oplus \overline{S_p} \quad (4.14) $$
Such that

$$ (\langle\langle\varphi_{\varepsilon}, \varphi_{\varepsilon}\rangle\rangle)_{\varepsilon} > 0, \forall (\varphi_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\mathbf{P}} \qquad (4.15) $$

and

$$ (\langle\langle\varphi_{\varepsilon}, \phi_{\varepsilon}\rangle\rangle)_{\varepsilon} > 0, \forall (\varphi_{\varepsilon})_{\varepsilon}, (\phi_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\mathbf{P}}. \qquad (4.16) $$

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

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

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

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

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

## 4.2 Defining distributional outgoing modes

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

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

Where

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

Where $d\Omega_{d-2}^2$ is the metric of the $(d-2)$-sphere, and the AdS curvature radius squared $L^2$ is related to the cosmological constant by $L^2 = -(d-2)(d-1)/239b$. The parameter $r_0$ is proportional to the mass $M$ of the spacetime: $M = (d-2)A_{d-2}r_0^{d-3}/163c0$, where $A_{d-2} = 23c0^{(d-1)/2}/393[(d-1)/2]$. The distributional Schwarzschild geometry corresponds to $L \to \infty$. The corresponding equation of motion (4.2) for massless case are

$$ \begin{aligned} (\nabla_{\mu} \nabla^{\mu} \varphi_{\varepsilon})_{\varepsilon} &= \frac{(d-2)\gamma}{4(d-1)} (R_{\varepsilon})_{\varepsilon}, \\ (G_{\varepsilon, \mu\nu})_{\varepsilon} + \Lambda (g_{\varepsilon, \mu\nu})_{\varepsilon} &= 8\pi G (T_{\varepsilon, \mu\nu})_{\varepsilon}, \\ (T_{\varepsilon, \mu\nu})_{\varepsilon} &= \delta(x). \end{aligned} \qquad (4.21) $$

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

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

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

$$ (\langle\langle\varphi_{\varepsilon}, \varphi_{\varepsilon}\rangle\rangle)_{\varepsilon} > 0, \forall (\varphi_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\mathbf{P}} \qquad (4.15) $$

and

$$ (\langle\langle\varphi_{\varepsilon}, \phi_{\varepsilon}\rangle\rangle)_{\varepsilon} > 0, \forall (\varphi_{\varepsilon})_{\varepsilon}, (\phi_{\varepsilon})_{\varepsilon} \in \mathbf{S}_{\mathbf{P}}. \qquad (4.16) $$

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

$$ (\lvert\lvert a(\varphi_{\varepsilon}), a(\phi_{\varepsilon}) \rvert\rvert)_{\varepsilon} = (\lvert\lvert a^{\dagger}(\varphi_{\varepsilon}), a^{\dagger}(\phi_{\varepsilon}) \rvert\rvert)_{\varepsilon} = 0. \qquad (4.17) $$

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

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

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

## 4.2 Defining distributional outgoing modes

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

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

Where

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

Where $d\Omega_{d-2}^2$ is the metric of the $(d-2)$-sphere, and the AdS curvature radius squared $L^2$ is related to the cosmological constant by $L^2 = -(d-2)(d-1)/239b$. The parameter $r_0$ is proportional to the mass $M$ of the spacetime: $M = (d-2)A_{d-2}r_0^{d-3}/163c0$, where $A_{d-2} = 23c0^{(d-1)/2}/393[(d-1)/2]$. The distributional Schwarzschild geometry corresponds to $L \to \infty$. The corresponding equation of motion (4.2) for massless case are

$$ \begin{array}{l} (\nabla_{\mu} \nabla^{\mu} \varphi_{\varepsilon})_{\varepsilon} = \frac{(d-2)\gamma}{4(d-1)} (R_{\varepsilon})_{\varepsilon}, \\ (G_{\varepsilon,\mu\nu})_{\varepsilon} + \Lambda (g_{\varepsilon,\mu\nu})_{\varepsilon} = 8\pi G (T_{\varepsilon,\mu\nu})_{\varepsilon}, \\ (T_{\varepsilon,\mu\nu})_{\varepsilon} \sim \delta(x). \end{array} \qquad (4.21) $$

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

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

Where $Y_{lm}(\theta)$ denotes the $d$-dimensional scalar spherical harmonics, satisfying
$$22062126_{d-2}Y_{lm}(\theta) = -l(l+d-3)Y_{lm}(\theta), \qquad (4.23)$$

Where $22062126_{d-2}$ the Laplace-Beltrami operator.Substituting the decomposition (4.22) into Eq. (6) one get a radial wave equation

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

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

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

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

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

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

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

For most spacetimes of interest the potential $(V_{\varepsilon}(r_{\varepsilon}^{*}(r)))_{\varepsilon} = 0$ as $r = r_{+}$, i.e.

($|r_{\varepsilon}^{*}(r)|$)$_{\varepsilon} = +\infty$, and in this limit solutions to the wave equation (4.26) behave as

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

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

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

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

$$(r_{\varepsilon}^{*}(r))_{\varepsilon} = (f_{\varepsilon}^{-1}(r) dr)_{\varepsilon}^{-} [(f_{\varepsilon}'(r_{+}))_{\varepsilon}]^{-1} (\ln(|r - r_{+}| + \varepsilon))_{\varepsilon} \text{ as } r^{-}r_{+}, \qquad (4.29)$$

Where $(f_{\varepsilon}'(r_{+}))_{\varepsilon} > 0$. Therefore near the horizon, outgoing modes behave as

$$\left\{ \begin{array}{l} (\exp[-i\omega(t - r_{\varepsilon}^{*}(r))])_{\varepsilon} = \{(\exp[-i\omega v_{\varepsilon}^{*}(t, r)])_{\varepsilon}\} (\exp[-2i\omega r_{\varepsilon}^{*}(r)])_{\varepsilon} = \\ = \{(\exp[-i\omega v_{\varepsilon}^{*}(t, r)])_{\varepsilon}\} \{([|r - r_{+}| + \varepsilon]^{2i\omega/f_{\varepsilon}'(r_{+})})_{\varepsilon}\}, \end{array} \right. \qquad (4.30)$$

Where $(v_{\varepsilon}^{*}(t, r))_{\varepsilon} = t + (r_{\varepsilon}^{*}(r))_{\varepsilon}$. Now Eq. (4.30) shows that outgoing modes is Colombeau generalized function of class $G(\mathbb{R})$.
$$22062126_{d-2}Y_{lm}(\theta) = -l(l+d-3)Y_{lm}(\theta), \qquad (4.23)$$

Where $22062126_{d-2}$ the Laplace-Beltrami operator.Substituting the decomposition (4.22) into Eq. (6) one get a radial wave equation

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

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

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

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

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

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

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

For most spacetimes of interest the potential $(V_{\varepsilon}(r_{\varepsilon}^{*}(r)))_{\varepsilon} = 0$ as $r = r_{+}$, i.e.

($|r_{\varepsilon}^{*}(r)|$)$_{\varepsilon} = +\infty$, and in this limit solutions to the wave equation (4.26) behave as

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

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

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

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

$$(r_{\varepsilon}^{*}(r))_{\varepsilon} = (f_{\varepsilon}^{-1}(r) dr)_{\varepsilon}^{-} [(f_{\varepsilon}'(r_{+}))_{\varepsilon}]^{-1} (\ln(|r - r_{+}| + \varepsilon))_{\varepsilon} \text{ as } r^{-}r_{+}, \qquad (4.29)$$

Where $(f_{\varepsilon}'(r_{+}))_{\varepsilon} > 0$. Therefore near the horizon, outgoing modes behave as

$$\left\{ \begin{array}{l} (\exp[-i\omega(t - r_{\varepsilon}^{*}(r))])_{\varepsilon} = \{(\exp[-i\omega v_{\varepsilon}^{*}(t, r)])_{\varepsilon}\} (\exp[-2i\omega r_{\varepsilon}^{*}(r)])_{\varepsilon} = \\ = \{(\exp[-i\omega v_{\varepsilon}^{*}(t, r)])_{\varepsilon}\} \{([|r - r_{+}| + \varepsilon]^{2i\omega/f_{\varepsilon}'(r_{+})})_{\varepsilon}\}, \end{array} \right. \qquad (4.30)$$

Where $(v_{\varepsilon}^{*}(t, r))_{\varepsilon} = t + (r_{\varepsilon}^{*}(r))_{\varepsilon}$. Now Eq. (4.30) shows that outgoing modes is Colombeau generalized function of class $G(\mathbb{R})$.
# 5 Energy-Momentum Tensor Calculation by using Colombeau Distributional Modes

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

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

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

$$ -\left(\frac{\partial^2 \tilde{\Phi}}{\partial t^2}\right)_{\tilde{\epsilon}} = -(\Delta_{\tilde{\epsilon},J} \tilde{\Phi}_{\tilde{\epsilon}})_{\tilde{\epsilon}} + (V_{\tilde{\epsilon},J} \tilde{\Phi}_{\tilde{\epsilon}})_{\tilde{\epsilon}}, \qquad (5.2) $$

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

$$ \left\{ \begin{array}{l} (V_{\tilde{\epsilon},J})_{\tilde{\epsilon}} = \left(\frac{\Delta_{\tilde{\epsilon},J} f_{\tilde{\epsilon},J}}{f_{\tilde{\epsilon},J}}\right)_{\tilde{\epsilon}} + (f_{\tilde{\epsilon},J}^2(m^2 + \xi R_{\tilde{\epsilon}}))_{\tilde{\epsilon}} = \\ (1 - 6\xi) \left(\frac{\Delta_{\tilde{\epsilon},J} f_{\tilde{\epsilon},J}}{f_{\tilde{\epsilon},J}}\right)_{\tilde{\epsilon}} + m^2 (f_{\tilde{\epsilon},J}^2)_{\tilde{\epsilon}} + \xi (K_{\tilde{\epsilon},J})_{\tilde{\epsilon}}, \end{array} \right. \qquad (5.3) $$

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

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

$$ (u_{\tilde{\epsilon},\vec{k}}^{(+)})_{\tilde{\epsilon}} \asymp (16\pi^3 \omega_{\vec{k}}^{-1/2}) e^{-i(\omega_{\vec{k}}t - \vec{k} \cdot \vec{x})} (f_{\tilde{\epsilon},in}^{-1})_{\tilde{\epsilon}} \qquad (5.4) $$

and

$$ (v_{\tilde{\epsilon},\alpha}^{(+)})_{\tilde{\epsilon}} \asymp (2\varpi_{\alpha})^{-1/2} e^{-i\varpi_{\alpha}t} (f_{\tilde{\epsilon},out}^{-1} F_{\tilde{\epsilon},\alpha}(\vec{x}))_{\tilde{\epsilon}}, \qquad (5.5) $$

Where $\vec{k} \in \mathbb{R}^3$, $\omega_{\vec{k}} := \|\vec{k}\|$, $\varpi_{\alpha} > 0$, and $(F_{\tilde{\epsilon},\alpha}(\vec{x}))_{\tilde{\epsilon}}$ are Colombeau solutions of

$$ \left( \begin{array}{l} \left( [-\Delta_{\tilde{\epsilon},out} + V_{\tilde{\epsilon},out}(\vec{x})] F_{\tilde{\epsilon},\alpha}(\vec{x}) \right)_{\tilde{\epsilon}} = \varpi_{\alpha}^2 (F_{\tilde{\epsilon},\alpha}(\vec{x}))_{\tilde{\epsilon}}, \\ (F_{\tilde{\epsilon},\alpha}(\vec{x}))_{\tilde{\epsilon}} \Big|_{\tilde{\epsilon}=0} \in C^{\infty}(\mathbb{R}^3) \end{array} \right. \qquad (5.6) $$
# 5 Energy-Momentum Tensor Calculation by using Colombeau Distributional Modes

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

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

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

$$ - \left( \frac{\partial^2 \tilde{\Phi}}{\partial t^2} \right)_{\epsilon} = - \left( \Delta_{\epsilon,J} \tilde{\Phi}_{\epsilon} \right)_{\epsilon} + \left( V_{\epsilon,J} \tilde{\Phi}_{\epsilon} \right)_{\epsilon}, \qquad (5.2) $$

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

$$ \left\{ \begin{array}{l} (V_{\epsilon,J})_{\epsilon} = \left( \frac{\Delta_{\epsilon} J f_{\epsilon,J}}{f_{\epsilon,J}} \right)_{\epsilon} + (f_{\epsilon,J}^2 (m^2 + \xi R_{\epsilon}))_{\epsilon} = \\ (1 - 6\xi) \left( \frac{\Delta_{\epsilon} J f_{\epsilon,J}}{f_{\epsilon,J}} \right)_{\epsilon} + m^2 (f_{\epsilon,J}^2)_{\epsilon} + \xi (K_{\epsilon,J})_{\epsilon}, \end{array} \right. \qquad (5.3) $$

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

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

$$ (u_{\epsilon,\vec{k}}^{(+)})_{\epsilon} \asymp (16\pi^3 \omega_{\vec{k}})_{\epsilon}^{-1/2} e^{-i(\omega_{\vec{k}}t - \vec{k} \cdot \vec{x})} (f_{\epsilon,in}^{-1})_{\epsilon} \qquad (5.4) $$

and

$$ (v_{\epsilon,\alpha}^{(+)})_{\epsilon} \asymp (2\varpi_{\alpha})_{\epsilon}^{-1/2} e^{-i\varpi_{\alpha}t} (f_{\epsilon,out}^{-1} F_{\epsilon,\alpha}(\vec{x}))_{\epsilon}, \qquad (5.5) $$

Where $\vec{k} \in \mathbb{R}^3$, $\omega_{\vec{k}} := \|\vec{k}\|$, $\varpi_{\alpha} > 0$, and $(F_{\epsilon,\alpha}(\vec{x}))_{\epsilon}$ are Colombeau solutions of

$$ \left( \begin{array}{l} \left( [-\Delta_{\epsilon,out} + V_{\epsilon,out}(\vec{x})] F_{\epsilon,\alpha}(\vec{x}) \right)_{\epsilon} = \varpi_{\alpha}^2 (F_{\epsilon,\alpha}(\vec{x}))_{\epsilon}, \\ (F_{\epsilon,\alpha}(\vec{x}))_{\epsilon} \Big|_{\epsilon=0} \in C^{\infty}(\mathbb{R}^3) \end{array} \right. \qquad (5.6) $$
Satisfying the normalization

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

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

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

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

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

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

$$ \left\{ \langle\langle T_{\varepsilon,00}(\vec{x}) \rangle\rangle_{\text{future}} \asymp \left\{ \langle\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle\rangle_{\varepsilon} \right\} \left\{ \frac{(1-4\xi)}{2} \left( \bar{\Omega}^2 + \frac{(D\bar{F}_{\varepsilon})^2}{\bar{F}^2} + m^2 f_{\varepsilon}^2 + \xi K_{\varepsilon} \right) \right. \right. \qquad (5.10) \\ \left. \left. + (1-6\xi) \left( \frac{2\xi D^2 f_{\varepsilon}}{f_{\varepsilon}} + \frac{(D f_{\varepsilon})^2}{2f_{\varepsilon}^2} - \frac{D_i f_{\varepsilon} D^i \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right) + \mathcal{O}(e^{-\epsilon t}) \right\} \right\} $$

$$ \left\{ \langle\langle T_{\varepsilon,0i}(\vec{x}) \rangle\rangle_{\text{future}} \asymp \left\{ \langle\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle\rangle_{\varepsilon} \right\} \left\{ (1-4\xi) \left( \frac{\bar{\Omega}D_i \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}} \right)_{\varepsilon} - (1-6\xi) \left( \frac{\bar{\Omega}D_i f_{\varepsilon}}{f_{\varepsilon}} \right)_{\varepsilon} + \mathcal{O}(e^{-\epsilon t}) \right\}, \right. \qquad (5.11) $$
Satisfying the normalization

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

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

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

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

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

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

$$ \left\{ \begin{aligned} & \langle\langle T_{\varepsilon,00}(\vec{x}) \rangle\rangle_{\text{future}} \asymp \{\langle\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle\rangle_{\varepsilon}\} \left\{ \frac{(1-4\xi)}{2} \left( \bar{\Omega}^2 + \frac{(D\bar{F}_{\varepsilon})^2}{\bar{F}^2} + m^2 f_{\varepsilon}^2 + \xi K_{\varepsilon} \right) \right. \\ & \qquad \left. + (1-6\xi) \left( \frac{2\xi D^2 f_{\varepsilon}}{f_{\varepsilon}} + \frac{(D f_{\varepsilon})^2}{2f_{\varepsilon}^2} - \frac{D_i f_{\varepsilon} D^i \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right) + \mathcal{O}(e^{-\epsilon t}) \right\}, \end{aligned} \right. \qquad (5.10) $$

$$ \left\{ \begin{aligned} & \langle\langle T_{\varepsilon,0i}(\vec{x}) \rangle\rangle_{\text{future}} \underset{\text{future}}{\asymp} \{\langle\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle\rangle_{\varepsilon}\} \left\{ (1-4\xi) \left( \frac{\bar{\Omega} D_i \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}} \right)_{\varepsilon} - (1-6\xi) \left( \frac{\bar{\Omega} D_i f_{\varepsilon}}{f_{\varepsilon}} \right)_{\varepsilon} + \mathcal{O}(e^{-\epsilon t}) \right\}, \end{aligned} \right. \qquad (5.11) $$
$$ \begin{aligned} & \langle (T_{\varepsilon,ij}(\vec{x})) \rangle_{\text{future}} \underset{\varepsilon}{\asymp} \{ (\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle)_{\varepsilon} \} \{ (1-2\xi) \left( \frac{D_i \bar{F}_{\varepsilon} D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}^2} \right) - 2\xi \left( \frac{D_i D_j \bar{F}_{\varepsilon}}{F_{\varepsilon}} \right) + \xi (\tilde{R}_{\varepsilon,ij})_{\varepsilon} \\ & \qquad + \frac{(1-4\xi)h_{ij}}{2} \left( \bar{\Omega}^2 - \left( \frac{(D\bar{F}_{\varepsilon})^2}{\bar{F}_{\varepsilon}^2} - m^2 f_{\varepsilon}^2 - \xi K_{\varepsilon} \right) \right) \\ & \qquad + (1-6\xi) \left[ \left( \frac{D_i f_{\varepsilon} D_j f_{\varepsilon}}{f_{\varepsilon}^2} - \frac{D_i f D_j \bar{F}}{f_{\varepsilon} \bar{F}_{\varepsilon}} - \frac{D_j f_{\varepsilon} D_i \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right)_{\varepsilon} + \right. \\ & \qquad \qquad \left. + (h_{\varepsilon,ij})_{\varepsilon} \left( \frac{2\xi D^2 f_{\varepsilon}}{f_{\varepsilon}} - \frac{(Df_{\varepsilon})^2}{2f_{\varepsilon}^2} + \frac{D_k f_{\varepsilon} D^k \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right) \right] + \mathcal{O}(e^{-\epsilon t}), \end{aligned} \quad (5.12) $$

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

$$ \begin{aligned} & \langle (\Phi_{\varepsilon}^2(r)) \rangle_{\text{future}} \underset{\varepsilon}{\asymp} \frac{\kappa e^{2\bar{\Omega}t}}{2\bar{\Omega}} \left[ \left( \frac{r^{1/2} \bar{F}_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} \right)_{\varepsilon} \right]^2, \quad r \to r_+ \quad (5.13) \\ & \{ (\langle \Phi_{\varepsilon}^2(r) \rangle)_{\varepsilon} \} \left\{ \begin{aligned} & \frac{(1-4\xi)}{2} \left( \bar{\Omega}^2 + \frac{(D\bar{F}_{\varepsilon}(r_+))^2}{\bar{F}^2(r_+)} + m^2 ((r-r_+)^2 + \varepsilon^2)^{1/2} + \xi K_{\varepsilon} \right)_{\varepsilon} \\ & + (1-6\xi) \left( \frac{2\xi D^2 f_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} + \frac{(Df_{\varepsilon}(r))^2}{((r-r_+)^2 + \varepsilon^2)^{1/4}} - \frac{D_i f_{\varepsilon}(r) D^i \bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \end{aligned} \right\}, \quad (5.14) \\ & \left\{ \begin{aligned} & \langle (T_{\varepsilon,0i}(r)) \rangle_{\text{future}} \underset{\varepsilon}{\asymp} \{ (\langle \Phi_{\varepsilon}^2(r) \rangle)_{\varepsilon} \} \left\{ (1-4\xi) \left( \frac{\bar{\Omega} D_i \bar{F}_{\varepsilon}(r_+)}{\bar{F}_{\varepsilon}(r_+)} \right)_{\varepsilon} - (1-6\xi) \left( \frac{\bar{\Omega} D_i f_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} \right)_{\varepsilon} \right\}, \quad (5.15) \\ & r \to r_+, f_{\varepsilon}(r) = ((r-r_+)^2 + \varepsilon^2)^{1/4} \end{aligned} \right. \end{aligned} $$
$$ \begin{aligned} & \langle (T_{\varepsilon,ij}(\vec{x})) \rangle_{\text{future}} \underset{\varepsilon}{\asymp} \\ & \{ (\langle \Phi_{\varepsilon}^2(\vec{x}) \rangle)_{\varepsilon} \} \{ (1-2\xi) \left( \frac{D_i \bar{F}_{\varepsilon} D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}^2} \right) - 2\xi \left( \frac{D_i D_j \bar{F}_{\varepsilon}}{F_{\varepsilon}} \right) + \xi (\tilde{R}_{\varepsilon,ij})_{\varepsilon} \\ & \qquad + \frac{(1-4\xi)h_{ij}}{2} \left( \bar{\Omega}^2 - \left( \frac{(D\bar{F}_{\varepsilon})^2}{\bar{F}_{\varepsilon}^2} - m^2 f_{\varepsilon}^2 - \xi K_{\varepsilon} \right) \right) \\ & \qquad + (1-6\xi) \left[ \left( \frac{D_i f_{\varepsilon} D_j f_{\varepsilon}}{f_{\varepsilon}^2} - \frac{D_i f D_j \bar{F}}{f_{\varepsilon} \bar{F}_{\varepsilon}} - \frac{D_j f_{\varepsilon} D_i \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right)_{\varepsilon} + \right. \\ & \qquad \left. + (h_{\varepsilon,ij})_{\varepsilon} \left( \frac{2\xi D^2 f_{\varepsilon}}{f_{\varepsilon}} - \frac{(Df_{\varepsilon})^2}{2f_{\varepsilon}^2} + \frac{D_k f_{\varepsilon} D^k \bar{F}_{\varepsilon}}{f_{\varepsilon} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right] + \mathcal{O}(e^{-\epsilon t}), \end{aligned} \quad (5.12) $$

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

$$ \begin{aligned} & \langle (\Phi_{\varepsilon}^2(r)) \rangle_{\text{future}} \underset{\varepsilon}{\asymp} \frac{\kappa e^{2\bar{\Omega}t}}{2\bar{\Omega}} \left[ \left( \frac{r^{1/2} \bar{F}_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} \right)_{\varepsilon} \right]^2, \quad r \to r_+ \quad (5.13) \\ & \{ (\langle \Phi_{\varepsilon}^2(r) \rangle)_{\varepsilon} \} \left\{ \frac{(1-4\xi)}{2} \left( \bar{\Omega}^2 + \frac{(D\bar{F}_{\varepsilon}(r_+))^2}{\bar{F}^2(r_+)} + m^2 \left( (r-r_+)^2 + \varepsilon^2 \right)^{1/2} + \xi K_{\varepsilon} \right)_{\varepsilon} \right. \\ & \qquad \left. + (1-6\xi) \left( -\frac{2\xi D^2 f_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} + \frac{(Df_{\varepsilon}(r))^2}{((r-r_+)^2 + \varepsilon^2)^{1/4}} - \frac{D_i f_{\varepsilon}(r) D^i \bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right\}, \quad (5.14) \\ & \left\{ \begin{aligned} & \langle (T_{\varepsilon,0i}(r)) \rangle_{\text{future}} \underset{\varepsilon}{\asymp} \\ & \{ (\langle \Phi_{\varepsilon}^2(r) \rangle)_{\varepsilon} \} \left\{ (1-4\xi) \left( \frac{\bar{\Omega} D_i \bar{F}_{\varepsilon}(r_+)}{\bar{F}_{\varepsilon}(r_+)} \right)_{\varepsilon} - (1-6\xi) \left( \frac{\bar{\Omega} D_i f_{\varepsilon}(r)}{((r-r_+)^2 + \varepsilon^2)^{1/4}} \right)_{\varepsilon} \right\}, \quad (5.15) \\ & r \to r_+, f_{\varepsilon}(r) = ((r-r_+)^2 + \varepsilon^2)^{1/4} \end{aligned} \right. \end{aligned} $$
$$ ((T_{\varepsilon,ij}(r)))_{\text{future}} \underset{\text{future}}{\asymp} \\ \{\langle\langle\Phi_{\varepsilon}^2(r)\rangle\rangle\}_{\varepsilon} \left\{ \begin{aligned} & (1-2\xi) \left( \frac{D_i \bar{F}_{\varepsilon} D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}^2} \right)_{\varepsilon} - 2\xi \left( \frac{D_i D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}} \right)_{\varepsilon} + \xi (\bar{R}_{\varepsilon,ij})_{\varepsilon} \\ & + \frac{(1-4\xi)h_{ij}}{2} \left( \bar{\Omega}^2 - \left( \frac{(D \bar{F}_{\varepsilon} (r_+))^2}{\bar{F}_{\varepsilon}^2 (r_+)} - m^2 ((r-r_+)^2 + \varepsilon^2)^{1/2} - \xi K_{\varepsilon} \right)_{\varepsilon} \right) \\ & + (1-6\xi) \left[ \frac{D_i f_{\varepsilon} D_j f_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/2}} - \frac{D_i f D_j \bar{F}}{((r-r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} - \frac{D_j f_{\varepsilon} D_i \bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/2} \bar{F}_{\varepsilon}} \right]_{\varepsilon} + \\ & + (h_{\varepsilon,ij})_{\varepsilon} \left( \frac{2\xi D^2 f_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4}} - \frac{(D f_{\varepsilon})^2}{2 ((r-r_+)^2 + \varepsilon^2)^{1/2}} + \frac{D_k f_{\varepsilon} D^k \bar{F}_{\varepsilon}}{((r-r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \end{aligned} \right]_{\varepsilon} \quad (5.16) $$

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

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

# 6 Distributional SAdS BH Spacetime-induced Vacuum Dominance

## 6.1 Adiabatic Expansion of Green Functions

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

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

One obtains directly

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

Special interest attaches to the short distance behaviour of the Green functions, such as $(G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon}$ in the limit $\|x - x'\| \to 0$ with a fixed $\varepsilon \in (0, 1]$. We obtain now an adiabatic expansion of $(G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon}$. Introducing Riemann normal coordinates $y^{\mu}$ for the point $x$, with origin at the point $x_5f_3$ we have expanding

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

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

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

and its Colombeau-Fourier transform by
$$ \begin{aligned} & ((T_{\varepsilon,ij}(r)})_{\text{future}} \underset{\text{future}}{\asymp} \\ & \{\langle\langle\Phi_{\varepsilon}^2(r)\rangle\rangle\}_{\varepsilon} \left\{ \left(1 - 2\varepsilon\right) \left(\frac{D_i \bar{F}_{\varepsilon} D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}^2}\right)_{\varepsilon} - 2\varepsilon \left(\frac{D_i D_j \bar{F}_{\varepsilon}}{\bar{F}_{\varepsilon}}\right)_{\varepsilon} + \xi \left(\bar{R}_{\varepsilon,ij}\right)_{\varepsilon} \right. \\ & \left. + \frac{(1 - 4\varepsilon)h_{ij}}{2} \left(\bar{\Omega}^2 - \left(\frac{D\bar{F}_{\varepsilon}(r_+)}{\bar{F}_{\varepsilon}^2(r_+)}\right)^2 - m^2 \left((r - r_+)^2 + \varepsilon^2\right)^{1/2} - \xi K_{\varepsilon}\right) \right\} \\ & + (1 - 6\varepsilon) \left[ \left( \frac{D_i f_{\varepsilon} D_j f_{\varepsilon}}{((r - r_+)^2 + \varepsilon^2)^{1/2}} - \frac{D_i f D_j \bar{F}}{((r - r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} - \frac{D_j f_{\varepsilon} D_i \bar{F}_{\varepsilon}}{((r - r_+)^2 + \varepsilon^2)^{1/2} \bar{F}_{\varepsilon}} \right)_{\varepsilon} + (h_{\varepsilon,ij})_{\varepsilon} \left( \frac{2\xi D^2 f_{\varepsilon}}{((r - r_+)^2 + \varepsilon^2)^{1/4}} - \frac{(D f_{\varepsilon})^2}{2 \left((r - r_+)^2 + \varepsilon^2\right)^{1/2}} + \frac{D_k f_{\varepsilon} D^k \bar{F}_{\varepsilon}}{((r - r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right]_{\varepsilon} \\ & \qquad + (h_{\varepsilon,ij})_{\varepsilon} \left. \left( \frac{2\xi D^2 f_{\varepsilon}}{((r - r_+)^2 + \varepsilon^2)^{1/4}} - \frac{(D f_{\varepsilon})^2}{2 \left((r - r_+)^2 + \varepsilon^2\right)^{1/2}} + \frac{D_k f_{\varepsilon} D^k \bar{F}_{\varepsilon}}{((r - r_+)^2 + \varepsilon^2)^{1/4} \bar{F}_{\varepsilon}} \right)_{\varepsilon} \right]_{\varepsilon} \end{aligned} \quad (5.16) $$

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

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

# 6 Distributional SAdS BH Spacetime-induced Vacuum Dominance

## 6.1 Adiabatic Expansion of Green Functions

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

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

One obtains directly

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

Special interest attaches to the short distance behaviour of the Green functions, such as $(G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon}$ in the limit $\|x - x'\| \to 0$ with a fixed $\varepsilon \in (0, 1]$. We obtain now an adiabatic expansion of $(G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon}$. Introducing Riemann normal coordinates $y^{\mu}$ for the point $x$, with origin at the point $x_5f_3$ we have expanding

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

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

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

and its Colombeau-Fourier transform by
$$ (\frac{\pm}{\epsilon}(x, x'))_{\epsilon} = (2\pi)^{-n} (d^n k e_{\epsilon}^{-iky \pm} (k))_{\epsilon} \qquad (6.5) $$

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

$$ \begin{aligned} (\frac{\pm}{\epsilon}(k))_{\epsilon} = & (k^2 - m^2)^{-1} - (\frac{1}{6} - \xi) (k^2 - m^2)^{-2} (\mathbf{R}^{\pm}(\epsilon))_{\epsilon} + \\ & + \frac{i}{2} (\frac{1}{6} - \xi) \partial^{\alpha} (k^2 - m^2)^{-2} (\mathbf{R}_{;\alpha}^{\pm}(\epsilon))_{\epsilon} - \\ & - \frac{1}{3} \left[ (a_{\alpha\beta}^{\pm}(\epsilon))_{\epsilon} \right] \partial^{\alpha} \partial^{\beta} (k^2 - m^2)^{-2} + \left[ \left(\frac{1}{6} - \xi\right)^2 (\mathbf{R}^{\pm 2}(\epsilon))_{\epsilon} + \frac{2}{3} (a_{\lambda}^{\pm\lambda}(\epsilon))_{\epsilon} \right] (k^2 - m^2)^{-3} \end{aligned} \qquad (6.6) $$

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

$$ \begin{aligned} (a_{\alpha\beta}^{\pm}(\epsilon))_{\epsilon} \asymp & \left(\frac{1}{2} - \xi\right) (\mathbf{R}_{;\alpha\beta}^{\pm}(\epsilon))_{\epsilon} + \frac{1}{120} (\mathbf{R}_{;\alpha\beta}^{\pm}(\epsilon))_{\epsilon} - \frac{1}{140} (\mathbf{R}_{\alpha\beta;\lambda}^{\pm}(\epsilon))_{\epsilon} - \\ & - \frac{1}{30} \left[ (\mathbf{R}_{\alpha}^{\pm\lambda}(\epsilon))_{\epsilon} \right] (\mathbf{R}_{\lambda\beta}^{\pm}(\epsilon))_{\epsilon} + \frac{1}{60} \left[ (\mathbf{R}_{\alpha}^{\pm\kappa\lambda}(\epsilon))_{\epsilon} \right] (\mathbf{R}_{\kappa\lambda}^{\pm}(\epsilon))_{\epsilon} + \\ & + \frac{1}{60} \left[ (\mathbf{R}_{\lambda}^{\pm\mu\kappa}(\epsilon))_{\epsilon} \right] (\mathbf{R}_{\lambda\mu\kappa\beta}^{\pm}(\epsilon))_{\epsilon}, \end{aligned} \qquad (6.7) $$

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

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

where

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

and, to adiabatic order 4,

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

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

itbpF2.386in1.7694in0inFigure

Pic.3. The contour in the complex $k^0$ plane $C$ to be used in the evaluation of the integral giving $^+$. The cross indicates the pole at

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

If one uses the canonical integral representation

<figure><img src="image_2.png" /></figure>
$$ (\frac{\pm}{\epsilon}(x, x'))_{\epsilon} = (2\pi)^{-n} (d^n k e_{\epsilon}^{-iky \pm} (k))_{\epsilon} \qquad (6.5) $$

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

$$ \begin{aligned} (\frac{\pm}{\epsilon}(k))_{\epsilon} = & (k^2 - m^2)^{-1} - (\frac{1}{6} - \xi) (k^2 - m^2)^{-2} (\mathbf{R}^{\pm}(\epsilon))_{\epsilon} + \\ & + \frac{i}{2} (\frac{1}{6} - \xi) \partial^{\alpha} (k^2 - m^2)^{-2} (\mathbf{R}_{;\alpha}^{\pm}(\epsilon))_{\epsilon} - \\ & - \frac{1}{3} \left[ (a_{\alpha\beta}^{\pm}(\epsilon))_{\epsilon} \right] \partial^{\alpha} \partial^{\beta} (k^2 - m^2)^{-2} + \left[ \left(\frac{1}{6} - \xi\right)^2 (\mathbf{R}^{\pm 2}(\epsilon))_{\epsilon} + \frac{2}{3} (a_{\lambda}^{\pm\lambda}(\epsilon))_{\epsilon} \right] (k^2 - m^2)^{-3} \end{aligned} \qquad (6.6) $$

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

$$ \begin{aligned} (a_{\alpha\beta}^{\pm}(\epsilon))_{\epsilon} \asymp & \left(\frac{1}{2} - \xi\right) (\mathbf{R}_{;\alpha\beta}^{\pm}(\epsilon))_{\epsilon} + \frac{1}{120} (\mathbf{R}_{;\alpha\beta}^{\pm}(\epsilon))_{\epsilon} - \frac{1}{140} (\mathbf{R}_{\alpha\beta;\lambda}^{\pm}(\epsilon))_{\epsilon} - \\ & - \frac{1}{30} \left[ (\mathbf{R}_{\alpha}^{\pm\lambda}(\epsilon))_{\epsilon} \right] (\mathbf{R}_{\lambda\beta}^{\pm}(\epsilon))_{\epsilon} + \frac{1}{60} \left[ (\mathbf{R}_{\alpha}^{\pm\kappa\lambda}(\epsilon))_{\epsilon} \right] (\mathbf{R}_{\kappa\lambda}^{\pm}(\epsilon))_{\epsilon} + \\ & + \frac{1}{60} \left[ (\mathbf{R}_{\lambda}^{\pm\mu\kappa}(\epsilon))_{\epsilon} \right] (\mathbf{R}_{\lambda\mu\kappa\beta}(\epsilon))_{\epsilon}, \end{aligned} \qquad (6.7) $$

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

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

where

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

and, to adiabatic order 4,

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

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

itbpF2.386in1.7694in0inFigure

Pic.3. The contour in the complex $k^0$ plane $C$ to be used in the evaluation of the integral giving $^+$. The cross indicates the pole at

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

If one uses the canonical integral representation

<figure><img src="image_2.png" /></figure>
$$(k^2 - m^2 + i\epsilon)^{-1} = -i_0^\infty ds e^{is(k^2 - m^2 + i\epsilon)} \qquad (6.11)$$

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

$$ (\epsilon^\pm(x, x'))_\epsilon = -i (4\pi)^{-n/2} \left( \int_0^\infty ids(is)^{-n/2} \exp\left[-im^2s + \frac{\sigma(x, x')}{2is}\right]_\epsilon^\pm (x, x'; is) \right)_\epsilon \quad (6.12) \\ \sigma(x, x') = \frac{1}{2} y_\alpha y^\alpha. $$

The function $\sigma(x, x')$ which is one-half of the square of the proper distance between $x$ and $x'$, while the function $(\epsilon(x, x'; is))_\epsilon$ has the following asymptotic adiabatic expansion

$$(\epsilon^\pm(x, x'; is))_\epsilon \approx (a_0^\pm(x, x'; \epsilon))_\epsilon + is(a_1^\pm(x, x'; \epsilon))_\epsilon + (is)^2(a_2^\pm(x, x'; \epsilon))_\epsilon + \dots \quad (6.13)$$

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

$$ \left\{ -i (4\pi)^{-n/2} \left( \left[ (\Delta_{\pm}^{1/2}(x, x'; \epsilon))_0^\infty ids(is)^{-n/2} \exp\left[-im^2s + \frac{\sigma(x, x')}{2is}\right]_\epsilon (x, x'; is) \right)_\epsilon \right) \right. \quad (6.14) $$

Where $(\Delta_\pm(x, x'; \epsilon))_\epsilon$ is the distributional Van Vleck determinant

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

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

$$(\epsilon^\pm(x, x'; is))_\epsilon \approx \sum_{j=0}^\infty (is)^j (a_2^\pm(x, x'; \epsilon))_\epsilon \quad (6.16)$$

With $(a_0^\pm(x, x'; \epsilon))_\epsilon = 1$, the other $(a_j^\pm(x, x'; \epsilon))_\epsilon$ being given by canonical recursion relations which enable their adiabatic expansions to be obtained. The expansions (6.13) and (6.16) are, however, only asymptotic approximations in the limit of large adiabatic parameter $T$.

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

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

In which, strictly, a small imaginary part $i\epsilon$ should be subtracted from $\sigma$. Since we have not imposed global boundary conditions on the distributional Green function Colombeau solution of (6.2), the expansion (6.17) does not determine the particular vacuum state in (6.1). In particular, the "$i\epsilon$" in the expansion of $(G_\epsilon^\pm(x, x'))_\epsilon$ only ensures that (6.17) represents the expectation value, in some set of states, of a time-ordered product of fields. Under some circumstances the use of "$i\epsilon$" in the exact representation (6.14) may give additional information concerning the global nature of the states
$$ (k^2 - m^2 + i\epsilon)^{-1} = -i_0^\infty ds e^{is(k^2 - m^2 + i\epsilon)} \qquad (6.11) $$

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

$$ (\epsilon^\pm(x, x'))_\epsilon = -i(4\pi)^{-n/2} \left( \int_0^\infty ids(is)^{-n/2} \exp\left[-im^2s + \frac{\sigma(x, x')}{2is}\right]_\epsilon^\pm (x, x'; is) \right)_\epsilon \quad (6.12) \\ \sigma(x, x') = \frac{1}{2}y_\alpha y^\alpha. $$

The function $\sigma(x, x')$ which is one-half of the square of the proper distance between $x$ and $x'$, while the function $(\epsilon(x, x'; is))_\epsilon$ has the following asymptotic adiabatic expansion

$$ (\epsilon^\pm(x, x'; is))_\epsilon \approx (a_0^\pm(x, x'; \epsilon))_\epsilon + is(a_1^\pm(x, x'; \epsilon))_\epsilon + (is)^2(a_2^\pm(x, x'; \epsilon))_\epsilon + \dots \quad (6.13) $$

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

$$ \left\{ -i (4\pi)^{-n/2} \left( \left[ (\Delta_{\pm}^{1/2}(x, x'; \epsilon))_0^\infty ids(is)^{-n/2} \exp\left[-im^2s + \frac{\sigma(x, x')}{2is}\right]_\epsilon (x, x'; is) \right)_\epsilon \right) \right. \quad (6.14) $$

Where $(\Delta_\pm(x, x'; \epsilon))_\epsilon$ is the distributional Van Vleck determinant

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

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

$$ (\epsilon^\pm(x, x'; is))_\epsilon \approx \sum_{j=0}^\infty (is)^j (a_2^\pm(x, x'; \epsilon))_\epsilon \quad (6.16) $$

With $(a_0^\pm(x, x'; \epsilon))_\epsilon = 1$, the other $(a_j^\pm(x, x'; \epsilon))_\epsilon$ being given by canonical recursion relations which enable their adiabatic expansions to be obtained. The expansions (6.13) and (6.16) are, however, only asymptotic approximations in the limit of large adiabatic parameter $T$.

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

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

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

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

$$ \left( \frac{2}{(-g(\epsilon))^\frac{1}{2}} \frac{\delta W_\epsilon}{\delta g^{\mu\nu}(\epsilon)} \right)_\epsilon = (\langle \mathbf{T}_{\mu\nu}(\epsilon) \rangle)_\epsilon \qquad (6.18) $$

To discover the structure of $(W_\epsilon)_\epsilon$, let us return to first principles, recalling the Colombeau path-integral quantization procedure such as developed in [1]. Our notation will imply a treatment for the scalar field, but the formal manipulations are identical for fields of higher spins. Note that the generating functional

$$ (Z_\epsilon [\mathbf{J}_\epsilon])_\epsilon = (D[\varphi_\epsilon] \exp \{iS_m(\epsilon) + i\mathbf{J}_\epsilon(x) \varphi_\epsilon(x) d^m x\})_\epsilon \qquad (6.19) $$

was interpreted physically as the vacuum persistence amplitude $(\langle \mathbf{out}_\epsilon, 0 | \mathbf{in}_\epsilon \rangle)_\epsilon$. The presence of the external distributional current density $(\mathbf{J}_\epsilon)_\epsilon$ can cause the initial vacuum state $(|0, \mathbf{in}_\epsilon\rangle)_\epsilon$ to be unstable, i.e., it can bring about the production of particles. In flat space, in the limit $(\mathbf{J}_\epsilon)_\epsilon = 0$, no particles are produced, and one have the normalization condition

$$ (Z_\epsilon [0])_\epsilon = (D[\varphi_\epsilon] \exp \{iS_m(\epsilon) + i\mathbf{J}_\epsilon(x) \varphi_\epsilon(x) d^m x\})_\epsilon \Big|_{\mathbf{J}=0} = (\langle 0_\epsilon | 0_\epsilon \rangle)_\epsilon = 1. \qquad (6.20) $$

However, when distributional spacetime is curved, we have seen that, in general,

$$ (|0, \mathbf{out}_\epsilon\rangle)_\epsilon \neq (|0, \mathbf{in}_\epsilon\rangle)_\epsilon, \qquad (6.21) $$

even in the absence of source currents J. Hence (6.19) will no longer apply. Path-integral quantization still works in curved distributional spacetime; one simply treats $(S_m(\epsilon))_\epsilon$ in (6.19) as the curved distributional spacetime matter action, and $(\mathbf{J}_\epsilon(x))_\epsilon$ as a current density (a scalar density in the case of scalar fields). One can thus set $\mathbf{J}_\epsilon = 0$ in (6.19) and examine the variation of $(Z_\epsilon[0])_\epsilon$:

$$ (\delta Z_\epsilon[0])_\epsilon = iD[\varphi_\epsilon] \delta S_m(\epsilon) \exp [iS_m(\varphi_\epsilon; \epsilon)] = i (\langle \mathbf{out}_\epsilon, 0 | \delta S_m(\epsilon) | 0, \mathbf{in}_\epsilon \rangle)_\epsilon. \qquad (6.22) $$

Note that

$$ \left( \frac{2}{(-g(\epsilon))^\frac{1}{2}} \frac{\delta S_m(\epsilon)}{\delta g^{\mu\nu}(\epsilon)} \right)_\epsilon = (\mathbf{T}_{\mu\nu}(\epsilon))_\epsilon. \qquad (6.23) $$

From (6.22) and (6.23) one obtains directly

$$ \left( \frac{2}{(-g(\epsilon))^\frac{1}{2}} \frac{\delta Z_\epsilon[0]}{\delta g^{\mu\nu}(\epsilon)} \right)_\epsilon = i (\langle \mathbf{out}_\epsilon, 0 | \mathbf{T}_{\mu\nu}(\epsilon) | 0, \mathbf{in}_\epsilon \rangle)_\epsilon \qquad (6.24) $$

Noting that the matter action $S_m(\epsilon)$ appears exponentiated in (6.19), one obtains directly

$$ Z_\epsilon[0] = (\exp(iW_\epsilon))_\epsilon \qquad (6.25) $$

and

$$ (\exp(W_\epsilon))_\epsilon = -i (\ln \langle \mathbf{out}_\epsilon, 0 | \mathbf{in}_\epsilon \rangle)_\epsilon. \qquad (6.26) $$

Following canonical calculation one obtains
## 6.2 Effective Action for the Quantum Matter Fields in Curved Distributional Spcetime

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

$$ \left( \frac{2}{-(g(\epsilon))^{\frac{1}{2}}} \frac{\delta W_{\epsilon}}{\delta g^{\mu\nu}(\epsilon)} \right)_{\epsilon} = (\langle \mathbf{T}_{\mu\nu}(\epsilon) \rangle)_{\epsilon} \qquad (6.18) $$

To discover the structure of $(W_\epsilon)_\epsilon$, let us return to first principles, recalling the Colombeau path-integral quantization procedure such as developed in [1]. Our notation will imply a treatment for the scalar field, but the formal manipulations are identical for fields of higher spins. Note that the generating functional

$$ (Z_{\epsilon}[\mathbf{J}_{\epsilon}])_{\epsilon} = (D[\varphi_{\epsilon}] \exp\{iS_{\mathbf{m}}(\epsilon) + i\mathbf{J}_{\epsilon}(x)\varphi_{\epsilon}(x) d^{m}x\})_{\epsilon} \qquad (6.19) $$

was interpreted physically as the vacuum persistence amplitude $(\langle \mathbf{out}_\epsilon, 0 | \mathbf{in}_\epsilon \rangle)_\epsilon$. The presence of the external distributional current density $(\mathbf{J}_\epsilon)_\epsilon$ can cause the initial vacuum state $(|0, \mathbf{in}_\epsilon\rangle)_\epsilon$ to be unstable, i.e., it can bring about the production of particles. In flat space, in the limit $(\mathbf{J}_\epsilon)_\epsilon = 0$, no particles are produced, and one have the normalization condition

$$ (Z_{\epsilon}[0])_{\epsilon} = (D[\varphi_{\epsilon}] \exp\{iS_{\mathbf{m}}(\epsilon) + i\mathbf{J}_{\epsilon}(x)\varphi_{\epsilon}(x) d^{m}x\})_{\epsilon}|_{\mathbf{J}=0} = (\langle 0_{\epsilon}|0_{\epsilon} \rangle)_{\epsilon} = 1. \qquad (6.20) $$

However, when distributional spacetime is curved, we have seen that, in general,

$$ (|0, \mathbf{out}_\epsilon\rangle)_{\epsilon} \neq (|0, \mathbf{in}_\epsilon\rangle)_{\epsilon}, \qquad (6.21) $$

even in the absence of source currents J. Hence (6.19) will no longer apply. Path-integral quantization still works in curved distributional spacetime; one simply treats $(S_{\mathbf{m}}(\epsilon))_\epsilon$ in (6.19) as the curved distributional spacetime matter action, and $(\mathbf{J}_\epsilon(x))_\epsilon$ as a current density (a scalar density in the case of scalar fields). One can thus set $\mathbf{J}_\epsilon = 0$ in (6.19) and examine the variation of $(Z_\epsilon[0])_\epsilon$:

$$ (\delta Z_{\epsilon}[0])_{\epsilon} = iD[\varphi_{\epsilon}] \delta S_{\mathbf{m}}(\epsilon) \exp[iS_{\mathbf{m}}(\varphi_{\epsilon}; \epsilon)] = i (\langle \mathbf{out}_{\epsilon}, 0 | \delta S_{\mathbf{m}}(\epsilon) | 0, \mathbf{in}_{\epsilon} \rangle)_{\epsilon}. \qquad (6.22) $$

Note that

$$ \left( \frac{2}{-(g(\epsilon))^{\frac{1}{2}}} \frac{\delta S_{\mathbf{m}}(\epsilon)}{\delta g^{\mu\nu}(\epsilon)} \right)_{\epsilon} = (\mathbf{T}_{\mu\nu}(\epsilon))_{\epsilon}. \qquad (6.23) $$

From (6.22) and (6.23) one obtains directly

$$ \left( \frac{2}{-(g(\epsilon))^{\frac{1}{2}}} \frac{\delta Z_{\epsilon}[0]}{\delta g^{\mu\nu}(\epsilon)} \right)_{\epsilon} = i (\langle \mathbf{out}_{\epsilon}, 0 | \mathbf{T}_{\mu\nu}(\epsilon) | 0, \mathbf{in}_{\epsilon} \rangle)_{\epsilon} \qquad (6.24) $$

Noting that the matter action $S_{\mathbf{m}}(\epsilon)$ appears exponentiated in (6.19), one obtains directly

$$ Z_{\epsilon}[0] = (\exp(iW_{\epsilon}))_{\epsilon} \qquad (6.25) $$

and

$$ (\exp(W_{\epsilon}))_{\epsilon} = -i (\ln \langle \mathbf{out}_{\epsilon}, 0 | \mathbf{in}_{\epsilon} \rangle)_{\epsilon}. \qquad (6.26) $$

Following canonical calculation one obtains
$$ (Z_{\varepsilon}^{\pm}[0])_{\varepsilon} \propto \left( \left[ \det(-G_{\varepsilon}^{\pm}(x, x')) \right]^{\frac{1}{2}} \right)_{\varepsilon} \qquad (6.27) $$

Where the proportionality constant is $\propto$ -independent and can be ignored. Thus we obtain

$$ (W_{\varepsilon}^{\pm})_{\varepsilon} = -i (\ln Z_{\varepsilon}^{\pm}[0])_{\varepsilon} = -\frac{i}{2} (\mathrm{tr} [\ln (-\hat{G}_{\varepsilon}^{\pm})])_{\varepsilon}. \qquad (6.28) $$

In (6.28) $(\hat{G}_{\varepsilon}^{\pm})_{\varepsilon}$ is to be interpreted as an Colombeau generalized operator which acts on a linear space $\mathfrak{S}$ of generalized vectors $|x\rangle$, normalized by

$$ \langle\langle x | x' \rangle\rangle_{\varepsilon} = \delta(x - x') \left( [-g^{\pm}(x, \varepsilon)]^{-\frac{1}{2}} \right)_{\varepsilon} \qquad (6.29) $$

in such a way that

$$ (G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} = \langle x | \hat{G}_{\varepsilon}^{\pm} | x' \rangle_{\varepsilon}. \qquad (6.30) $$

**Remark 6.1.** Note that the trace tr[] of an Colombeau generalized operator $(\Re_{\varepsilon})_{\varepsilon}$ which acts on a linear space $\mathfrak{S}$, is defined by

$$ (\mathrm{tr}[\Re_{\varepsilon}])_{\varepsilon} = \left( d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} \Re_{xx;\varepsilon} \right)_{\varepsilon} = \left( d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} \langle x | \Re_{xx;\varepsilon} | x' \rangle \right)_{\varepsilon}. \qquad (6.31) $$

Writing now the Colombeau generalized operator $(\hat{G}_{\varepsilon}^{\pm})_{\varepsilon}$ as

$$ (\hat{G}_{\varepsilon}^{\pm})_{\varepsilon} = - (\varepsilon^{-1})_{\varepsilon} = -i (\int_{0}^{\infty} ds \exp [-s_{\varepsilon}^{\pm}])_{\varepsilon}, \qquad (6.32) $$

By Eq. (6.14) one obtains

$$ \left\{ \begin{array}{l} \langle\langle x | \exp [-s_{\varepsilon}^{\pm}] | x' \rangle\rangle = \\ i (4\pi)^{-n/2} \left[ \left( \Delta_{\pm}^{1/2} (x, x'; \varepsilon) \right)_{\varepsilon} \right] \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right]^{\pm} (x, x'; is) (is)^{-n/2} \end{array} \right. \qquad (6.33) $$

Now, assuming $(\varepsilon)_{\varepsilon}$ to have a small negative imaginary part, we obtain

$$ (\int_{\Lambda}^{\infty} ds (is)^{-1} i \exp [-s_{\varepsilon}^{\pm}])_{\varepsilon} = (Ei(-i\Lambda_{\varepsilon}^{\pm}))_{\varepsilon} \qquad (6.34) $$

Where $Ei(x)$ is the exponential integral function.

**Remark 6.2.** Note that for $x \to 0$

$$ Ei(x) = \gamma + \ln(-x) + O(x) \qquad (6.35) $$

Where $\gamma$ is the Euler's constant. Substituting now (6.35) into (6.34) and letting $\Lambda \to 0$ we obtain

$$ (\ln(-\hat{G}_{\varepsilon}^{\pm}))_{\varepsilon} = -(\ln(\varepsilon))_{\varepsilon} = (\int_{0}^{\infty} ds \exp[-s_{\varepsilon}^{\pm}] (is)^{-1})_{\varepsilon}, \qquad (6.36) $$

Which is correct up to the addition of a metric-independent infinite large Colombeau constant $\Omega \in \tilde{\mathbb{R}}$ that can be ignored in what follows. Thus, in the generalized De Witt-Schwinger representation (6.33) or (6.14) we have

$$ \langle x | \ln(-\hat{G}_{\varepsilon}^{\pm}) | x' \rangle_{\varepsilon} = (\int_{m^2}^{\infty} G_{\varepsilon}^{\pm}(x, x'; m^2) dm^2)_{\varepsilon}, \qquad (6.37) $$

Where the integral with respect to $m^2$ brings down the extra power of $(is)^{-1}$ that appears in
$$ (Z_{\varepsilon}^{\pm}[0])_{\varepsilon} \propto \left( \left[ \det(-G_{\varepsilon}^{\pm}(x, x')) \right]^{\frac{1}{2}} \right)_{\varepsilon} \qquad (6.27) $$

Where the proportionality constant is ignored. Thus we obtain

$$ (W_{\varepsilon}^{\pm})_{\varepsilon} = -i (\ln Z_{\varepsilon}^{\pm}[0])_{\varepsilon} = -\frac{i}{2} (\mathrm{tr} [\ln (-\hat{G}_{\varepsilon}^{\pm})])_{\varepsilon}. \qquad (6.28) $$

In (6.28) $(\hat{G}_{\varepsilon}^{\pm})_{\varepsilon}$ is to be interpreted as an Colombeau generalized operator which acts on a linear space $\Re$ of generalized vectors $|x\rangle$, normalized by

$$ \langle\langle x | x' \rangle\rangle_{\varepsilon} = \delta(x - x') \left( [-g^{\pm}(x, \varepsilon)]^{-\frac{1}{2}} \right)_{\varepsilon} \qquad (6.29) $$

in such a way that

$$ (G_{\varepsilon}^{\pm}(x, x'))_{\varepsilon} = \langle x | \hat{G}_{\varepsilon}^{\pm} | x' \rangle_{\varepsilon}. \qquad (6.30) $$

**Remark 6.1.** Note that the trace tr[] of an Colombeau generalized operator $(\Re_{\varepsilon})_{\varepsilon}$ which acts on a linear space $\Re$, is defined by

$$ (\mathrm{tr}[\Re_{\varepsilon}])_{\varepsilon} = \left( d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} \Re_{xx;\varepsilon} \right)_{\varepsilon} = \left( d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} \langle x | \Re_{xx;\varepsilon} | x' \rangle \right)_{\varepsilon}. \qquad (6.31) $$

Writing now the Colombeau generalized operator $(\hat{G}_{\varepsilon}^{\pm})_{\varepsilon}$ as

$$ (\hat{G}_{\varepsilon}^{\pm})_{\varepsilon} = - (\varepsilon^{-1})_{\varepsilon} = -i (\int_{0}^{\infty} ds \exp [-s_{\varepsilon}^{\pm}])_{\varepsilon}, \qquad (6.32) $$

By Eq. (6.14) one obtains

$$ \left\{ \begin{array}{l} \langle\langle x | \exp [-s_{\varepsilon}^{\pm}] | x' \rangle\rangle = \\ i (4\pi)^{-n/2} \left[ \left( \Delta_{\pm}^{1/2} (x, x'; \varepsilon) \right)_{\varepsilon} \right] \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right]^{\pm} (x, x'; is) (is)^{-n/2} \end{array} \right. \qquad (6.33) $$

Now, assuming $(\varepsilon)_{\varepsilon}$ to have a small negative imaginary part, we obtain

$$ (\int_{\Lambda} ds (is)^{-1} i \exp [-s_{\varepsilon}^{\pm}])_{\varepsilon} = (Ei(-i\Lambda_{\varepsilon}^{\pm}))_{\varepsilon} \qquad (6.34) $$

Where $Ei(x)$ is the exponential integral function.

**Remark 6.2.** Note that for $x \to 0$

$$ Ei(x) = \gamma + \ln(-x) + O(x) \qquad (6.35) $$

Where $\gamma$ is the Euler's constant. Substituting now (6.35) into (6.34) and letting $\Lambda \to 0$ we obtain

$$ (\ln(-\hat{G}_{\varepsilon}^{\pm}))_{\varepsilon} = -(\ln(\varepsilon))_{\varepsilon} = (\int_{0}^{\infty} ds \exp[-s_{\varepsilon}^{\pm}] (is)^{-1})_{\varepsilon}, \qquad (6.36) $$

Which is correct up to the addition of a metric-independent infinite large Colombeau constant $\Omega \in \tilde{\mathbb{R}}$ that can be ignored in what follows. Thus, in the generalized De Witt-Schwinger representation (6.33) or (6.14) we have

$$ \langle\langle x | \ln(-\hat{G}_{\varepsilon}^{\pm}) | x' \rangle\rangle_{\varepsilon} = (\int_{m^2}^{\infty} G_{\varepsilon}^{\pm}(x, x'; m^2) dm^2)_{\varepsilon}, \qquad (6.37) $$

Where the integral with respect to $m^2$ brings down the extra power of $(is)^{-1}$ that appears in
Eq.(6.36). Returning now to the expression (6.28) for $(W_{\epsilon}^{\pm})_{\epsilon}$ using Eq.(6.37) and Eq.(6.31) we get

$$ (W_{\epsilon}^{\pm})_{\epsilon} = \frac{i}{2} \left[ \left( d^n x \left[-g^{\pm}(x, \epsilon)\right]^{\frac{1}{2}} \right) \right]_{\epsilon} \left( \lim_{x \to x'm^2} \frac{G_{\epsilon}^{\pm}(x, x'; m^2)}{m^2} dm^2 \right)_{\epsilon} \quad (6.38) $$

Interchanging the order of integration and taking the limit $x \to x'$ one obtains

$$ (W_{\epsilon}^{\pm})_{\epsilon} = \frac{i}{2} \left( \int_{m^2}^{\infty} dm^2 d^n x \left[-g^{\pm}(x, \epsilon)\right]^{\frac{1}{2}} G_{\epsilon}^{\pm}(x, x; m^2) \right)_{\epsilon}. \quad (6.39) $$

Colombeau quantity $(W_{\epsilon}^{\pm})_{\epsilon}$ is colled as the one-loop effective action. In the case of fermion effective actions, there would be a remaining trace over spinorial indices. From Eq.(6.39) we may define an effective Lagrangian density $(L_{\epsilon;\text{eff}}^{\pm}(x))_{\epsilon}$ by

$$ (W_{\epsilon}^{\pm})_{\epsilon} = \left( d^n x \left[-g^{\pm}(x, \epsilon)\right]^{\frac{1}{2}} L_{\epsilon;\text{eff}}^{\pm}(x) \right)_{\epsilon} \quad (6.40) $$

Whence one get

$$ (L_{\epsilon}^{\pm}(x))_{\epsilon} = \left[-g^{\pm}(x, \epsilon)\right]_{\epsilon}^{\frac{1}{2}} \epsilon^{\pm}(x) = \frac{i}{2} \left( \lim_{x \to x'm^2} \int dm^2 G_{\epsilon}^{\pm}(x, x'; m^2) \right)_{\epsilon}. \quad (6.41) $$

## 6.3 Stress-tensor Renormalization

Note that $(L_{\epsilon}^{\pm}(x))_{\epsilon}$ diverges at the lower end of the s integral because the $\sigma/2s$ damping factor in the exponent vanishes in the limit $x \to x'$. (Convergence at the upper end is guaranteed by the $-ie$ that is implicitly added to $m^2$ in the De Witt-Schwinger representation of $(L_{\epsilon}^{\pm}(x))_{\epsilon}$. In four dimensions, the potentially divergent terms in the DeWitt-Schwinger expansion of $(L_{\epsilon}^{\pm}(x))_{\epsilon}$ are

$$ \left( L_{\epsilon;\text{div}}^{\pm}(x) \right)_{\epsilon} = - (32\pi^2)^{-1} \left( \lim_{x \to x'} \left[ \left( \Delta_{\pm}^{1/2}(x, x'; \epsilon) \right)_{\epsilon} \right]_{0}^{\infty} \frac{ds}{s^3} \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right] \right) \times \left[ a_0^{\pm}(x, x'; \epsilon) + i sa_1^{\pm}(x, x'; \epsilon) + (is)^2 a_2^{\pm}(x, x'; \epsilon) \right]_{\epsilon} \quad (6.42) $$

Where the coefficients $a_0^{\pm}$, $a_1^{\pm}$ and $a_2^{\pm}$ are given by Eq.(6.9)-Eq.(6.10). The remaining terms in this asymptotic expansion, involving $a_3^{\pm}$ and higher, are finite in the limit $x \to x'$.

Let us determine now the precise form of the geometrical $(L_{\epsilon;\text{div}}^{\pm}(x))_{\epsilon}$ terms, to compare them with the conventional gravitational Lagrangian that appears in (2.38). This is a delicate matter because (6.48) is, of course, infinite. What we require is to display the divergent terms in the form $\infty \times$ [geometrical object]. This can be done in a variety of ways. For example, in n dimensions, the asymptotic (adiabatic) expansion of $(L_{\epsilon;\text{eff}}^{\pm}(x))_{\epsilon}$ is

$$ \left( L_{\epsilon;\text{eff}}^{\pm}(x) \right)_{\epsilon} \approx 2^{-1} (4\pi)^{-n/2} \left( \lim_{x \to x'} \left[ \left( \Delta_{\pm}^{1/2}(x, x'; \epsilon) \right)_{\epsilon} \right]_{j=0}^{\infty} a_j(x, x'; \epsilon) \times \right. \\ \left. \times \int_0^{\infty} ids(is)^{j-1-n/2} \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right] \right)_{\epsilon} \quad (6.43) $$

of which the first $n/2 + 1$ terms are divergent as $\sigma \to 0$. If n is treated as a variable which can be
Eq.(6.36). Returning now to the expression (6.28) for $(W_{\epsilon}^{\pm})_{\epsilon}$ using Eq.(6.37) and Eq.(6.31) we get

$$ (W_{\epsilon}^{\pm})_{\epsilon} = \frac{i}{2} \left[ \left( d^n x \left[-g^{\pm}(x, \epsilon)\right]^{\frac{1}{2}} \right) \right]_{\epsilon} \left( \lim_{x \to x'm^2} \frac{G_{\epsilon}^{\pm}(x, x'; m^2)}{m^2} dm^2 \right)_{\epsilon} \quad (6.38) $$

Interchanging the order of integration and taking the limit $x \to x'$ one obtains

$$ (W_{\epsilon}^{\pm})_{\epsilon} = \frac{i}{2} \left( \int_{m^2}^{\infty} dm^2 d^n x \left[-g^{\pm}(x, \epsilon)\right]^{\frac{1}{2}} G_{\epsilon}^{\pm}(x, x; m^2) \right)_{\epsilon}. \quad (6.39) $$

Colombeau quantity $(W_{\epsilon}^{\pm})_{\epsilon}$ is colled as the one-loop effective action. In the case of fermion effective actions, there would be a remaining trace over spinorial indices. From Eq.(6.39) we may define an effective Lagrangian density $(L_{\epsilon;\text{eff}}^{\pm}(x))_{\epsilon}$ by

$$ (W_{\epsilon}^{\pm})_{\epsilon} = \left( d^n x \left[-g^{\pm}(x, \epsilon)\right]^{\frac{1}{2}} L_{\epsilon;\text{eff}}^{\pm}(x) \right)_{\epsilon} \quad (6.40) $$

Whence one get

$$ (L_{\epsilon}^{\pm}(x))_{\epsilon} = \left[-g^{\pm}(x, \epsilon)\right]_{\epsilon}^{\frac{1}{2}} \epsilon^{\pm}(x) = \frac{i}{2} \left( \lim_{x \to x'm^2} \int dm^2 G_{\epsilon}^{\pm}(x, x'; m^2) \right)_{\epsilon}. \quad (6.41) $$

## 6.3 Stress-tensor Renormalization

Note that $(L_{\epsilon}^{\pm}(x))_{\epsilon}$ diverges at the lower end of the s integral because the $\sigma/2s$ damping factor in the exponent vanishes in the limit $x \to x'$. (Convergence at the upper end is guaranteed by the $-ie$ that is implicitly added to $m^2$ in the De Witt-Schwinger representation of $(L_{\epsilon}^{\pm}(x))_{\epsilon}$. In four dimensions, the potentially divergent terms in the DeWitt-Schwinger expansion of $(L_{\epsilon}^{\pm}(x))_{\epsilon}$ are

$$ \left( L_{\epsilon; \text{div}}^{\pm}(x) \right)_{\epsilon} = - (32\pi^2)^{-1} \left( \lim_{x \to x'} \left[ \left( \Delta_{\pm}^{1/2}(x, x'; \epsilon) \right)_{\epsilon} \right]_{0}^{\infty} \frac{ds}{s^3} \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right] \right) \times \left[ a_0^{\pm}(x, x'; \epsilon) + i sa_1^{\pm}(x, x'; \epsilon) + (is)^2 a_2^{\pm}(x, x'; \epsilon) \right]_{\epsilon} \quad (6.42) $$

Where the coefficients $a_0^{\pm}$, $a_1^{\pm}$ and $a_2^{\pm}$ are given by Eq.(6.9)-Eq.(6.10). The remaining terms in this asymptotic expansion, involving $a_3^{\pm}$ and higher, are finite in the limit $x \to x'$.

Let us determine now the precise form of the geometrical $(L_{\epsilon;\text{div}}^{\pm}(x))_{\epsilon}$ terms, to compare them with the conventional gravitational Lagrangian that appears in (2.38). This is a delicate matter because (6.48) is, of course, infinite. What we require is to display the divergent terms in the form $\infty \times$ [geometrical object]. This can be done in a variety of ways. For example, in n dimensions, the asymptotic (adiabatic) expansion of $(L_{\epsilon;\text{eff}}^{\pm}(x))_{\epsilon}$ is

$$ \left( L_{\epsilon; \text{eff}}^{\pm}(x) \right)_{\epsilon} \approx 2^{-1} (4\pi)^{-n/2} \left( \lim_{x \to x'} \left[ \left( \Delta_{\pm}^{1/2}(x, x'; \epsilon) \right)_{\epsilon} \right]_{j=0}^{\infty} a_j(x, x'; \epsilon) \times \times_0^{\infty} ids(is)^{j-1-n/2} \exp \left[ -im^2 s + \frac{\sigma(x, x')}{2is} \right] \right)_{\epsilon} \quad (6.43) $$

of which the first $n/2 + 1$ terms are divergent as $\sigma \to 0$. If n is treated as a variable which can be
analytically continued throughout the complex plane, then we may take the $x \to x'$ limit

$$ \left\{ \begin{array}{l} \left( L_{\varepsilon; \text{eff}}^{\pm} (x) \right)_{\varepsilon} \propto 2^{-1} (4\pi)^{-n/2} \left( \begin{array}{l} \displaystyle \sum_{j=0}^{\infty} a_j (x; \varepsilon) \sum_{0}^{\infty} ids(is) \sum_{j=0}^{j-1-n/2} (is)^{j-1-n/2} \exp[-im^2 s] \\ \displaystyle - 2^{-1} (4\pi)^{-n/2} \sum_{j=0}^{\infty} a_j (x; \varepsilon) (m^2)^{n/2-j} \Gamma\left(j - \frac{n}{2}\right), \end{array} \right)_{\varepsilon} = \\ a_j (x; \varepsilon) = a_j (x, x; \varepsilon). \end{array} \right. \quad (6.44) $$

From Eq. (6.44) follows we shall wish to retain the units of $L_{\varepsilon; \text{eff}}^{\pm}(x)$ as $(\text{length})^{-4}$, even when $n \neq 4$. It is therefore necessary to introduce an arbitrary mass scale $\mu$ and to rewrite Eq. (6.44) as

$$ \left( L_{\varepsilon; \text{eff}}^{\pm} (x) \right)_{\varepsilon} \propto 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \left( \sum_{j=0}^{\infty} a_j (x; \varepsilon) (m^2)^{4-2j} \Gamma\left(j - \frac{n}{2}\right) \right)_{\varepsilon}. \quad (6.45) $$

If $n \to 4$, the first three terms of Eq. (6.45) diverge because of poles in the $\Gamma$-functions:

$$ \left\{ \begin{array}{l} \Gamma\left(-\frac{n}{4}\right) = \frac{4}{n(n-2)}\left(\frac{2}{4-n} - \gamma\right) + O(n-4), \\ \Gamma\left(1-\frac{n}{2}\right) = \frac{4}{(2-n)}\left(\frac{2}{4-n} - \gamma\right) + O(n-4), \\ \Gamma\left(2-\frac{n}{2}\right) = \frac{2}{4-n} - \gamma + O(n-4). \end{array} \right. \quad (6.46) $$

Denoting these first three terms by $(L_{\varepsilon; \text{div}}^{\pm}(x))_{\varepsilon}$, we have

$$ (4\pi)^{-n/2} \left\{ \frac{1}{n-4} + \frac{1}{2} \left[ \gamma + \ln \left( \frac{m^2}{\mu^2} \right) \right] \right\} \left( L_{\varepsilon; \text{div}}^{\pm}(x) \right)_{\varepsilon} = \left( \left[ \frac{4m^4 a_0(x; \varepsilon)}{n(n-2)} - \frac{2m^2 a_1(x; \varepsilon)}{n-2} + a_2(x; \varepsilon) \right] \right)_{\varepsilon}. \quad (6.47) $$

The functions $a_0(x; \varepsilon)$, $a_1(x; \varepsilon)$ and $a_2(x; \varepsilon)$ are given by taking the coincidence limits of (6.9)-(6.10)

$$ \left\{ \begin{array}{l} (a_0^{\pm}(x; \varepsilon))_{\varepsilon} = 1, \\ (a_1^{\pm}(x; \varepsilon))_{\varepsilon} = \left(\frac{1}{6} - \xi\right) (\mathbf{R}^{\pm}(\varepsilon))_{\varepsilon}, \\ (a_2^{\pm}(x; \varepsilon))_{\varepsilon} = \frac{1}{180} \left(\mathbf{R}_{\alpha\beta\gamma\delta}^{\pm}(x, \varepsilon) \mathbf{R}^{\pm\alpha\beta\gamma\delta}(x, \varepsilon)\right)_{\varepsilon} - \frac{1}{180} \left(\mathbf{R}^{\pm\alpha\beta}(x, \varepsilon) \mathbf{R}_{\alpha\beta}^{\pm}(x, \varepsilon)\right)_{\varepsilon} - \\ -\frac{1}{6} \left(\frac{1}{5} - \xi\right) \left(\Box \mathbf{R}^{\pm}(x, \varepsilon)\right)_{\varepsilon} + \frac{1}{2} \left(\frac{1}{6} - \xi\right) \left(\mathbf{R}^{\pm 2}(x, \varepsilon)\right)_{\varepsilon}. \end{array} \right. \quad (6.48) $$

Finally one obtains

$$ (L_{\varepsilon; \text{ren}}^{\pm}(x))_{\varepsilon} \propto -\frac{1}{64\pi^2} \left( \sum_{0}^{\infty} ids \ln(is) \frac{\partial^3}{\partial(is)^3} \left[ \frac{\pm}{\varepsilon} (x, x; is) e^{-ism^2} \right] \right)_{\varepsilon}. \quad (6.49) $$

Special interest attaches to field theories in distributional spacetime in which the classical action $(S_{\varepsilon})_{\varepsilon}$ is invariant under distributional conformal transformations, i.e.,

$$ (g_{\mu\nu}(x, \varepsilon))_{\varepsilon} \to (\Omega_{\varepsilon}^2(x) g_{\mu\nu}(x, \varepsilon))_{\varepsilon} \triangleq (\bar{g}_{\mu\nu}^{\pm}(x, \varepsilon))_{\varepsilon}. \quad (6.50) $$

From the definitions one has

$$ (S_{\varepsilon} [\bar{g}_{\mu\nu}^{\pm}(x, \varepsilon)])_{\varepsilon} = (S_{\varepsilon} [g_{\mu\nu}^{\pm}(x, \varepsilon)])_{\varepsilon} + \left( d^n x \left( \frac{\delta S_{\varepsilon}}{\delta \bar{g}^{\pm \rho \sigma}(x, \varepsilon)} \right) \delta \bar{g}^{\pm \rho \sigma}(x, \varepsilon) \right)_{\varepsilon}. \quad (6.51) $$
analytically continued throughout the complex plane, then we may take the $x \to x'$ limit

$$ \left\{ \begin{array}{l} \left( L_{\varepsilon;\text{eff}}^{\pm}(x) \right)_{\varepsilon} \asymp 2^{-1} (4\pi)^{-n/2} \left( \begin{array}{l} \displaystyle \sum_{j=0}^{\infty} a_j(x;\varepsilon) \sum_{0}^{\infty} ids(is) \sum_{j=0}^{j-1-n/2} (is)^{j-1-n/2} \exp[-im^2s] \\ \displaystyle - 2^{-1} (4\pi)^{-n/2} \sum_{j=0}^{\infty} a_j(x;\varepsilon) (m^2)^{n/2-j} \Gamma\left(j-\frac{n}{2}\right), \end{array} \right)_{\varepsilon} = \\ a_j(x;\varepsilon) = a_j(x,x;\varepsilon). \end{array} \right. \qquad (6.44) $$

From Eq. (6.44) follows we shall wish to retain the units of $L_{\varepsilon;\text{eff}}^{\pm}(x)$ as $(\text{length})^{-4}$, even when $n \neq 4$. It is therefore necessary to introduce an arbitrary mass scale $\mu$ and to rewrite Eq. (6.44) as

$$ \left( L_{\varepsilon;\text{eff}}^{\pm}(x) \right)_{\varepsilon} \asymp 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \left( \sum_{j=0}^{\infty} a_j(x;\varepsilon) (m^2)^{4-2j} \Gamma\left(j-\frac{n}{2}\right) \right)_{\varepsilon}. \qquad (6.45) $$

If $n \to 4$, the first three terms of Eq. (6.45) diverge because of poles in the $\Gamma$-functions:

$$ \left\{ \begin{array}{l} \Gamma\left(-\frac{n}{4}\right) = \frac{4}{n(n-2)}\left(\frac{2}{4-n} - \gamma\right) + O(n-4), \\ \Gamma\left(1-\frac{n}{2}\right) = \frac{4}{(2-n)}\left(\frac{2}{4-n} - \gamma\right) + O(n-4), \\ \Gamma\left(2-\frac{n}{2}\right) = \frac{2}{4-n} - \gamma + O(n-4). \end{array} \right. \qquad (6.46) $$

Denoting these first three terms by $(L_{\varepsilon;\text{div}}^{\pm}(x))_{\varepsilon}$, we have

$$ (4\pi)^{-n/2} \left\{ \frac{1}{n-4} + \frac{1}{2} \left[ \gamma + \ln \left( \frac{m^2}{\mu^2} \right) \right] \right\} \left( L_{\varepsilon;\text{div}}^{\pm}(x) \right)_{\varepsilon} = \left( \left[ \frac{4m^4 a_0(x;\varepsilon)}{n(n-2)} - \frac{2m^2 a_1(x;\varepsilon)}{n-2} + a_2(x;\varepsilon) \right] \right)_{\varepsilon}. \qquad (6.47) $$

The functions $a_0(x;\varepsilon)$, $a_1(x;\varepsilon)$ and $a_2(x;\varepsilon)$ are given by taking the coincidence limits of (6.9)-(6.10)

$$ \left\{ \begin{array}{l} (a_0^{\pm}(x;\varepsilon))_{\varepsilon} = 1, \\ (a_1^{\pm}(x;\varepsilon))_{\varepsilon} = \left(\frac{1}{6} - \xi\right) (\mathbf{R}^{\pm}(\varepsilon))_{\varepsilon}, \\ (a_2^{\pm}(x;\varepsilon))_{\varepsilon} = \frac{1}{180} \left(\mathbf{R}_{\alpha\beta\gamma\delta}^{\pm}(x,\varepsilon) \mathbf{R}^{\pm\alpha\beta\gamma\delta}(x,\varepsilon)\right)_{\varepsilon} - \frac{1}{180} \left(\mathbf{R}^{\pm\alpha\beta}(x,\varepsilon) \mathbf{R}_{\alpha\beta}^{\pm}(x,\varepsilon)\right)_{\varepsilon} - \\ -\frac{1}{6} \left(\frac{1}{5} - \xi\right) \left(\Box \mathbf{R}^{\pm}(x,\varepsilon)\right)_{\varepsilon} + \frac{1}{2} \left(\frac{1}{6} - \xi\right) \left(\mathbf{R}^{\pm 2}(x,\varepsilon)\right)_{\varepsilon}. \end{array} \right. \qquad (6.48) $$

Finally one obtains

$$ (L_{\varepsilon;\text{ren}}^{\pm}(x))_{\varepsilon} \asymp -\frac{1}{64\pi^2} \left( \sum_{0}^{\infty} ids \ln(is) \frac{\partial^3}{\partial(is)^3} \left[ \frac{\pm}{\varepsilon} (x, x; is) e^{-ism^2} \right] \right)_{\varepsilon}. \qquad (6.49) $$

Special interest attaches to field theories in distributional spacetime in which the classical action $(S_{\varepsilon})_{\varepsilon}$ is invariant under distributional conformal transformations, i.e.,

$$ (g_{\mu\nu}(x,\varepsilon))_{\varepsilon} \to (\Omega_{\varepsilon}^2(x) g_{\mu\nu}(x,\varepsilon))_{\varepsilon} \triangleq (\bar{g}_{\mu\nu}^{\pm}(x,\varepsilon))_{\varepsilon}. \qquad (6.50) $$

From the definitions one has

$$ (S_{\varepsilon} [\bar{g}_{\mu\nu}^{\pm}(x, \varepsilon)])_{\varepsilon} = (S_{\varepsilon} [g_{\mu\nu}^{\pm}(x, \varepsilon)])_{\varepsilon} + \left( d^n x \left( \frac{\delta S_{\varepsilon}}{\delta \bar{g}^{\pm \rho \sigma}(x, \varepsilon)} \right) \delta \bar{g}^{\pm \rho \sigma}(x, \varepsilon) \right)_{\varepsilon}. \qquad (6.51) $$
From Eq. (6.51) one obtains

$$ T_{\rho}^{\pm} \rho[g_{\mu\nu}^{\pm}(x, \varepsilon), \varepsilon] = - \left. \left( \frac{\Omega_{\varepsilon}^{2}(x)}{[-g(x, \varepsilon)]^{\frac{1}{2}}} \frac{\delta S_{\varepsilon}[g_{\mu\nu}^{\pm}(x, \varepsilon)]}{\delta \Omega_{\varepsilon}(x)} \right) \right|_{\Omega_{\varepsilon}=1}, \qquad (6.52) $$

and it is clear that if the classical action is invariant under the conformal transformations (6.50), then the classical stress-tensor is traceless. Because conformal transformations are essentially a rescaling of lengths at each spacetime point $x$, the presence of a mass and hence a fixed length scale in the theory will always break the conformal invariance. Therefore we are led to the massless limit of the regularization and renormalization procedures used in the previous section. Although all the higher order ($j > 2$) terms in the DeWitt-Schwinger expansion of the effective Lagrangian (6.45) are infrared divergent at $n = 4$ as $m \to 0$, we can still use this expansion to yield the ultraviolet divergent terms arising from $j = 0, 1$, and $2$ in the four-dimensional case. We may put $m = 0$ immediately in the $j = 0$ and $1$ terms in the expansion, because they are of positive power for $n \sim 4$. These terms therefore vanish. The only nonvanishing potentially ultraviolet divergent term is therefore $j = 2$:

$$ 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} a_2(x, \varepsilon) \Gamma\left(2 - \frac{n}{2}\right), \qquad (6.53) $$

Which must be handled carefully. Substituting for $a_2(x)$ with $\xi = \xi(n)$ from (6.48), and rearranging terms, we may write the divergent term in the effective action arising from (6.53) as follows:

$$ \begin{aligned} \langle W_{\varepsilon, \mathrm{div}}^{\pm} \rangle_{\varepsilon} = & 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \Gamma\left(2 - \frac{n}{2}\right) \left(d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} a_2(x, \varepsilon)\right)_{\varepsilon} = \\ & 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \Gamma\left(2 - \frac{n}{2}\right) \left(d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} \left[\alpha \Gamma^{\pm}_{\varepsilon}(x) + \beta G^{\pm}_{\varepsilon}(x)\right]\right)_{\varepsilon} + O(n-4) \end{aligned} \qquad (6.54) $$

Where

$$ \begin{aligned} \langle \Gamma_{\varepsilon}(x) \rangle_{\varepsilon} = & \left( \mathbf{R}^{\pm \alpha \beta \gamma \delta}(x, \varepsilon) \mathbf{R}_{\alpha \beta \gamma \delta}^{\pm}(x, \varepsilon) \right)_{\varepsilon} - 2 \left( \mathbf{R}^{\pm \alpha \beta}(x, \varepsilon) \mathbf{R}_{\alpha \beta}^{\pm}(x, \varepsilon) \right)_{\varepsilon} + \frac{1}{3} \left( \mathbf{R}^{\pm 2}(x, \varepsilon) \right)_{\varepsilon}, \\ \langle G_{\varepsilon}^{\pm}(x) \rangle_{\varepsilon} = & \left( \mathbf{R}^{\pm \alpha \beta \gamma \delta}(x, \varepsilon) \mathbf{R}_{\alpha \beta \gamma \delta}^{\pm}(x, \varepsilon) \right)_{\varepsilon} \end{aligned} \qquad (6.55) $$

and

$$ \alpha = \frac{1}{120}, \quad \beta = -\frac{1}{360}. \qquad (6.56) $$

Finally one obtains:

$$ \begin{aligned} \langle T_{\mu}^{\alpha}(x, \varepsilon) \rangle_{\mathrm{ren}} = & - (1/2880\pi^2) \left[ \alpha \left( \Gamma_{\varepsilon}(x) - \frac{2}{3} \Box \mathbf{R}^{\pm}(x, \varepsilon) \right)_{\varepsilon} + \beta \left( G_{\varepsilon}^{\pm}(x) \right)_{\varepsilon} \right] = \\ & - (1/2880\pi^2) \left[ \left( \mathbf{R}_{\alpha\beta\gamma\delta}^{\pm}(x, \varepsilon) \mathbf{R}^{\pm\alpha\beta\gamma\delta}(x, \varepsilon) \right)_{\varepsilon} - \left( \mathbf{R}_{\alpha\beta}^{\pm}(x, \varepsilon) \mathbf{R}^{\pm\alpha\beta}(x, \varepsilon) \right)_{\varepsilon} - \Box \mathbf{R}^{\pm}(x, \varepsilon) \right]. \end{aligned} \qquad (6.57) $$

Note that from Eq. (3.42) for $r \to 2m$ follows that:

$$ \langle (\mathbf{R}^{\rho\sigma\mu\nu}(\varepsilon))_{\varepsilon} (\mathbf{R}_{\rho\sigma\mu\nu}(\varepsilon))_{\varepsilon} \rangle \approx \left( \left[ (r-2m)^2 + \varepsilon^2 \right]^{-1} \right)_{\varepsilon} + 4(2m)^4. \qquad (6.58) $$

Thus for the case of the distributional Schwarzchild spacetime given by the distributional metric (3.40) using Eq. (6.57) and Eq. (6.58) for $r \to 2m$ one obtains:

$$ \langle T_{\mu}^{\mu}(x, \varepsilon) \rangle_{\mathrm{ren}} \approx - (2880\pi^2)^{-1} \left[ \left( \left[ (r-2m)^2 + \varepsilon^2 \right]^{-1} \right)_{\varepsilon} + 4(2m)^4 \right]. \qquad (6.59) $$
From Eq.(6.51) one obtains

$$ T_{\rho}^{\pm} \left[ g_{\mu\nu}^{\pm}(x, \varepsilon), \varepsilon \right] = - \left. \left( \frac{\Omega_{\varepsilon}^{2}(x)}{[-g(x, \varepsilon)]^{\frac{1}{2}}} \frac{\delta S_{\varepsilon}[g_{\mu\nu}^{\pm}(x, \varepsilon)]}{\delta \Omega_{\varepsilon}(x)} \right) \right|_{\Omega_{\varepsilon}=1}, \qquad (6.52) $$

and it is clear that if the classical action is invariant under the conformal transformations (6.50), then the classical stress-tensor is traceless. Because conformal transformations are essentially a rescaling of lengths at each spacetime point $x$, the presence of a mass and hence a fixed length scale in the theory will always break the conformal invariance. Therefore we are led to the massless limit of the regularization and renormalization procedures used in the previous section. Although all the higher order ($j > 2$) terms in the DeWitt-Schwinger expansion of the effective Lagrangian (6.45) are infrared divergent at $n = 4$ as $m \to 0$, we can still use this expansion to yield the ultraviolet divergent terms arising from $j = 0, 1$, and $2$ in the four-dimensional case. We may put $m = 0$ immediately in the $j = 0$ and $1$ terms in the expansion, because they are of positive power for $n \sim 4$. These terms therefore vanish. The only nonvanishing potentially ultraviolet divergent term is therefore $j = 2$:

$$ 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} a_2(x, \varepsilon) \Gamma\left(2-\frac{n}{2}\right), \qquad (6.53) $$

Which must be handled carefully. Substituting for $a_2(x)$ with $\xi = \xi(n)$ from (6.48), and rearranging terms, we may write the divergent term in the effective action arising from (6.53) as follows:

$$ \begin{aligned} \left( W_{\varepsilon, \mathrm{div}}^{\pm} \right)_{\varepsilon} = & 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \Gamma\left(2-\frac{n}{2}\right) \left( d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} a_2(x, \varepsilon) \right)_{\varepsilon} = \\ & 2^{-1} (4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \Gamma\left(2-\frac{n}{2}\right) \left( d^n x \left[-g^{\pm}(x, \varepsilon)\right]^{\frac{1}{2}} \left[\alpha \Gamma^{\pm}_{\varepsilon}(x) + \beta G^{\pm}_{\varepsilon}(x)\right] \right)_{\varepsilon} + O(n-4) \end{aligned} \qquad (6.54) $$

Where

$$ \begin{aligned} (F_{\varepsilon}(x))_{\varepsilon} = & \left( \mathbf{R}^{\pm \alpha \beta \gamma \delta}(x, \varepsilon) \mathbf{R}_{\alpha \beta \gamma \delta}^{\pm}(x, \varepsilon) \right)_{\varepsilon} - 2 \left( \mathbf{R}^{\pm \alpha \beta}(x, \varepsilon) \mathbf{R}_{\alpha \beta}^{\pm}(x, \varepsilon) \right)_{\varepsilon} + \frac{1}{3} \left( \mathbf{R}^{\pm 2}(x, \varepsilon) \right)_{\varepsilon}, \\ (G_{\varepsilon}^{\pm}(x))_{\varepsilon} = & \left( \mathbf{R}^{\pm \alpha \beta \gamma \delta}(x, \varepsilon) \mathbf{R}_{\alpha \beta \gamma \delta}^{\pm}(x, \varepsilon) \right)_{\varepsilon} \end{aligned} \qquad (6.55) $$

and

$$ \alpha = \frac{1}{120}, \quad \beta = -\frac{1}{360}. \qquad (6.56) $$

Finally one obtains:

$$ \begin{aligned} \langle T_{\mu}^{\alpha}(x, \varepsilon) \rangle_{\mathrm{ren}} = & - (1/2880\pi^2) \left[ \alpha (F_{\varepsilon}(x) - \frac{2}{3} \Box \mathbf{R}^{\pm}(x, \varepsilon))_{\varepsilon} + \beta (G_{\varepsilon}^{\pm}(x))_{\varepsilon} \right] = \\ & - (1/2880\pi^2) \left[ \left( \mathbf{R}_{\alpha\beta\gamma\delta}^{\pm}(x, \varepsilon) \mathbf{R}^{\pm\alpha\beta\gamma\delta}(x, \varepsilon) \right)_{\varepsilon} - \left( \mathbf{R}_{\alpha\beta}^{\pm}(x, \varepsilon) \mathbf{R}^{\pm\alpha\beta}(x, \varepsilon) \right)_{\varepsilon} - \Box \mathbf{R}^{\pm}(x, \varepsilon) \right]. \end{aligned} \qquad (6.57) $$

Note that from Eq.(3.42) for $r \to 2m$ follows that:

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(\varepsilon) \mathbf{R}_{\rho\sigma\mu\nu}(\varepsilon))_{\varepsilon} \approx \left( \left[ (r-2m)^2 + \varepsilon^2 \right]^{-1} \right)_{\varepsilon} + 4(2m)^4. \qquad (6.58) $$

Thus for the case of the distributional Schwarzchild spacetime given by the distributional metric (3.40) using Eq.(6.57) and Eq.(6.58) for $r \to 2m$ one obtains:

$$ \langle T_{\mu}^{\mu}(x, \varepsilon) \rangle_{\mathrm{ren}} \approx -(2880\pi^2)^{-1} \left[ \left( \left[ (r-2m)^2 + \varepsilon^2 \right]^{-1} \right)_{\varepsilon} + 4(2m)^4 \right]. \qquad (6.59) $$
This result in a good agreement with Eq.(5.14)-Eq.(5.16). .

# 7 Novel explanation of the Active Galactic Nuclei. The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on BHs horizon.

## 7.1 The Current Paradigm for Active Galactic Nuclei. High Energy Emission from Galactic Jets

Accretion of gas onto the supermassive Kerr black holes lurking at the center of active galactic nuclei (AGN) gives rise to powerful relativistic jets.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Jet from Black Hole in a Galaxy Pictor A The active galaxy Pictor A lies nearly 500 million light-years from Earth and contains a supermassive black hole at its centre. This is a composite radio and X-ray image.</figcaption></figure>

We remind that in the standard theory of an accretion disk around a black hole it is assumed that there is no coupling between the disk and the central black hole [51]. However, in the presence of a magnetic field, a magnetic coupling between the disk and the black hole could exist and play an important role in the balance and transportation of energy and angular momentum. In the absence of the magnetic coupling, the energy source for the radiation of the disk is the gravitational energy of the disk (i.e., the gravitational binding energy between the disk and the black hole). But, if the magnetic coupling exists and the black hole is rotating, the rotational energy of the black hole provides an additional energy source for the radiation of the disk. With the magnetic coupling, the black hole exerts a torque on the disk, which transfers energy and angular momentum between the black hole and the disk. If the black hole rotates faster than the disk, energy and angular momentum are extracted from the black hole and transfered to the disk. The energy deposited into the disk is eventually radiated away by the disk, which will increase the efficiency of the disk and make the disk brighter than usual. If the black hole rotates slower than the disk, energy and angular momentum are transfered from the disk to the black hole, which will lower the efficiency of the disk and make the disk dimmer than usual. Therefore, the magnetic coupling between the
This result in a good agreement with Eq.(5.14)-Eq.(5.16). .

# 7 Novel explanation of the Active Galactic Nuclei. The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on BHs horizon.

## 7.1 The Current Paradigm for Active Galactic Nuclei. High Energy Emission from Galactic Jets

Accretion of gas onto the supermassive Kerr black holes lurking at the center of active galactic nuclei (AGN) gives rise to powerful relativistic jets.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Jet from Black Hole in a Galaxy Pictor A The active galaxy Pictor A lies nearly 500 million light-years from Earth and contains a supermassive black hole at its centre. This is a composite radio and X-ray image.</figcaption></figure>

We remind that in the standard theory of an accretion disk around a black hole it is assumed that there is no coupling between the disk and the central black hole [51]. However, in the presence of a magnetic field, a magnetic coupling between the disk and the black hole could exist and play an important role in the balance and transportation of energy and angular momentum. In the absence of the magnetic coupling, the energy source for the radiation of the disk is the gravitational energy of the disk (i.e., the gravitational binding energy between the disk and the black hole). But, if the magnetic coupling exists and the black hole is rotating, the rotational energy of the black hole provides an additional energy source for the radiation of the disk. With the magnetic coupling, the black hole exerts a torque on the disk, which transfers energy and angular momentum between the black hole and the disk. If the black hole rotates faster than the disk, energy and angular momentum are extracted from the black hole and transfered to the disk. The energy deposited into the disk is eventually radiated away by the disk, which will increase the efficiency of the disk and make the disk brighter than usual. If the black hole rotates slower than the disk, energy and angular momentum are transfered from the disk to the black hole, which will lower the efficiency of the disk and make the disk dimmer than usual. Therefore, the magnetic coupling between the
black hole and the disk has important effects on the radiation properties of the disk [52]-[53].

The current paradigm for AGN is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed $10^{11}$ to $10^{12}$ K [55]. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Fourier coverage (uv-coverage) of the fringe fitted data (i.e.,reliable fringe detections) of the Radio Astron observations of BL Lac on 2013 November 10-11at 22 GHz. Color marks the lower limit of observed brightness temperature obtained from visibility amplitudes. Adapted from [54].</figcaption></figure>

The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than $5.3G\lambda$, must indeed exceed $2 \times 10^{13}$ K and can reach as high as $\sim 3 \times 10^{14}$ K [55]. As follows from Fig. 7.2, these visibilities correspond to the structural scales of 30-40 $\mu$as oriented along position angles of 25°-30°. These values are indeed close to the width of the inner jet and the normal to its direction. The observed, $T_{b,obs}$, and intrinsic, $T_{b,int}$, brightness temperatures are related by

$$T_{b,obs} = \delta(1+z)^{-1} T_{b,int} \qquad (7.1.1)$$

With $\delta = 7.2$. The estimated by (4.1.1) a lower limit of the intrinsic brightness temperature in the core component of our Radio Astron observations of $T_{b,int} > 2.910^{12}$ K. It is commonly considered that inverse Compton losses limit the intrinsic brightness temperature for incoherent synchrotron sources, such as AGN, to about $10^{12}$ K [53]. In case of a strong flare, the "Compton catastrophe" is calculated to take about one day to drive the brightness temperature below $10^{12}$ K [53]. The estimated lower limit for the intrinsic brightness temperature of the core in the Radio Astron image of $T_{b,int} > 2.910^{12}$ K is therefore more than an order of magnitude larger than the equipartition brightness temperature limit established in [55] and at least several times larger than the limit established by inverse Compton cooling.

**Remark 4.1.1.** Note that if the estimate of the maximum brightness temperature given in [53], is closer to actual values, it would imply $T_{b,int} \times 10^{13}$ K. This is difficult to reconcile with current incoherent synchrotron emission models from relativistic electrons, requiring alternative models such as emission from relativistic protons.

**Remark 4.1.2.** However the proton, as we know, is 1836 times heavier than an electron and absolutely huge energy is required to accelerate it to sublight speed.
black hole and the disk has important effects on the radiation properties of the disk [52]-[53].

The current paradigm for AGN is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed $10^{11}$ to $10^{12}$ K [55]. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Fourier coverage (uv-coverage) of the fringe fitted data (i.e.,reliable fringe detections) of the Radio Astron observations of BL Lac on 2013 November 10-11at 22 GHz. Color marks the lower limit of observed brightness temperature obtained from visibility amplitudes. Adapted from [54].</figcaption></figure>

The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than $5.3G\lambda$, must indeed exceed $2 \times 10^{13}$ K and can reach as high as $\sim 3 \times 10^{14}$ K [55]. As follows from Fig. 7.2, these visibilities correspond to the structural scales of 30-40 $\mu$as oriented along position angles of 25°-30°. These values are indeed close to the width of the inner jet and the normal to its direction. The observed, $T_{b,obs}$, and intrinsic, $T_{b,int}$, brightness temperatures are related by

$$T_{b,obs} = \delta(1+z)^{-1} T_{b,int} \qquad (7.1.1)$$

With $\delta = 7.2$. The estimated by (4.1.1) a lower limit of the intrinsic brightness temperature in the core component of our Radio Astron observations of $T_{b,int} > 2.910^{12}$ K. It is commonly considered that inverse Compton losses limit the intrinsic brightness temperature for incoherent synchrotron sources, such as AGN, to about $10^{12}$ K [53]. In case of a strong flare, the "Compton catastrophe" is calculated to take about one day to drive the brightness temperature below $10^{12}$ K [53]. The estimated lower limit for the intrinsic brightness temperature of the core in the Radio Astron image of $T_{b,int} > 2.910^{12}$ K is therefore more than an order of magnitude larger than the equipartition brightness temperature limit established in [55] and at least several times larger than the limit established by inverse Compton cooling.

**Remark 4.1.1.** Note that if the estimate of the maximum brightness temperature given in [53], is closer to actual values, it would imply $T_{b,int} \times 10^{13}$ K. This is difficult to reconcile with current incoherent synchrotron emission models from relativistic electrons, requiring alternative models such as emission from relativistic protons.

**Remark 4.1.2.** However the proton, as we know, is 1836 times heavier than an electron and absolutely huge energy is required to accelerate it to sublight speed.
**Remark 4.1.3.** These alternative models such as emission from relativistic protons can be supported by semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes presented by rotating gravitational singularities.

## 7.2 The Colombeau distributional Kerr spacetime in Boyer-Lindquist form.

The classical Kerr metric in Boyer-Lindquist form reads

$$ds^2 = -\Xi(r, \theta) dt^2 - \frac{4mra \sin^2 \theta}{\rho^2} dtd\phi + \frac{\rho^2}{\Delta_a} dr^2 + \rho^2 d\theta^2 + \left( r^2 + a^2 + \frac{2mra^2 \sin^2 \theta}{\rho^2} \right) \sin^2 \theta d\phi^2, \qquad (7.2.1)$$

Where

$$\begin{gathered} \rho^2 = r^2 + a^2 \cos^2 \theta, \quad \Delta_a = r^2 - 2mr + a^2, \\ \Xi(r, \theta) = 1 - \frac{2mr}{\rho^2} = \frac{r^2 - 2mr + a^2 \cos^2 \theta}{\rho^2}. \end{gathered} \qquad (7.2.2)$$

**Remark 7.2.1.** Note that For small values of the parameter $a$, where we may neglect terms of the order of $a^2$, we get from (7.2.1) the Lense-Thirring metric with $J_z = ma$

$$ds^2 = - \left(1 - \frac{2m}{r}\right) dt^2 + \left(1 - \frac{2m}{r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2) - \frac{4ma \sin^2 \theta}{r} dtd\phi. \quad (7.2.3)$$

### I. Slow Kerr gravitational singularity: $a < m$.

Note that

$$\Xi(r, \theta) = \frac{r^2 - 2mr + a^2 \cos^2 \theta}{\rho^2} = \frac{(r - r_{E+}(\theta))(r - r_{E-}(\theta))}{\rho^2}, \qquad (7.2.4)$$

Where $r_{E\pm}(\theta) = m \pm \sqrt{m^2 - a^2 \cos^2 \theta}$ and $\Delta_a = r^2 - 2mr + a^2 = (r - r_+)(r - r_-)$, where $r_\pm = m \pm \sqrt{m^2 - a^2}$.

**Remark 7.2.2.** Let $\Xi(\theta)$ be a submanifold given by equation $\phi = \text{const}$, then metric (7.2.1) restricted on submanifold $\cup_{\theta} \Xi(\theta)$ reads

$$ds^2 = -\Xi(r, \theta) dt^2 + \frac{\rho^2}{\Delta_a} dr^2 + \rho^2 d\theta^2. \qquad (7.2.5)$$

Note that: (i) the metric (7.2.5) is degenerate on outer ergosurfaces $r = r_{E+}(\theta)$ and inner ergosurfaces $r - r_{E-}(\theta)$, (ii) the metric (7.2.5) is singular on horizon $r = r_+$, (iii) the metric (7.2.5) is singular on submanifold given by equation $r = r_-$.

**Remark 7.2.3.** Note that we will be 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 $\cup_{\theta} \Xi(\theta)$ which coincide with outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11), Eq.(7.2.19) and Eq.(7.2.20) submanifold $\cup_{\theta} \Xi(\theta)$ presented the singular boundary of distributional spacetime presented by Colombeau generalized metric (7.2.7).

We introduce now the following regularized (above ergosurface $r = r_{E+}(\theta)$) quantity
**Remark 4.1.3.** These alternative models such as emission from relativistic protons can be supported by semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes presented by rotating gravitational singularities.

## 7.2 The Colombeau distributional Kerr spacetime in Boyer-Lindquist form.

The classical Kerr metric in Boyer-Lindquist form reads

$$ds^2 = -\Xi(r, \theta) dt^2 - \frac{4mra \sin^2 \theta}{\rho^2} dtd\phi + \frac{\rho^2}{\Delta_a} dr^2 + \rho^2 d\theta^2 + \left( r^2 + a^2 + \frac{2mra^2 \sin^2 \theta}{\rho^2} \right) \sin^2 \theta d\phi^2, \qquad (7.2.1)$$

Where

$$\begin{gathered} \rho^2 = r^2 + a^2 \cos^2 \theta, \quad \Delta_a = r^2 - 2mr + a^2, \\ \Xi(r, \theta) = 1 - \frac{2mr}{\rho^2} = \frac{r^2 - 2mr + a^2 \cos^2 \theta}{\rho^2}. \end{gathered} \qquad (7.2.2)$$

**Remark 7.2.1.** Note that For small values of the parameter $a$, where we may neglect terms of the order of $a^2$, we get from (7.2.1) the Lense-Thirring metric with $J_z = ma$

$$ds^2 = - \left(1 - \frac{2m}{r}\right) dt^2 + \left(1 - \frac{2m}{r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2) - \frac{4ma \sin^2 \theta}{r} dtd\phi. \quad (7.2.3)$$

### I. Slow Kerr gravitational singularity: $a < m$.

Note that

$$\Xi(r, \theta) = \frac{r^2 - 2mr + a^2 \cos^2 \theta}{\rho^2} = \frac{(r - r_{E+}(\theta))(r - r_{E-}(\theta))}{\rho^2}, \qquad (7.2.4)$$

Where $r_{E\pm}(\theta) = m \pm \sqrt{m^2 - a^2 \cos^2 \theta}$ and $\Delta_a = r^2 - 2mr + a^2 = (r - r_+)(r - r_-)$, where $r_\pm = m \pm \sqrt{m^2 - a^2}$.

**Remark 7.2.2.** Let $\Xi(\theta)$ be a submanifold given by equation $\phi = \text{const}$, then metric (7.2.1) restricted on submanifold $\cup_{\theta} \Xi(\theta)$ reads

$$ds^2 = -\Xi(r, \theta) dt^2 + \frac{\rho^2}{\Delta_a} dr^2 + \rho^2 d\theta^2. \qquad (7.2.5)$$

Note that: (i) the metric (7.2.5) is degenerate on outer ergosurfaces $r = r_{E+}(\theta)$ and inner ergosurfaces $r - r_{E-}(\theta)$, (ii) the metric (7.2.5) is singular on horizon $r = r_+$, (iii) the metric (7.2.5) is singular on submanifold given by equation $r = r_-$.

**Remark 7.2.3.** Note that we will be 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 $\cup_{\theta} \Xi(\theta)$ which coincide with outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11), Eq.(7.2.19) and Eq.(7.2.20) submanifold $\cup_{\theta} \Xi(\theta)$ presented the singular boundary of distributional spacetime presented by Colombeau generalized metric (7.2.7).

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