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

To discover the structure of $(W_\varepsilon)_\varepsilon$, let us return to first principles, recalling the Colombeau path-integral quantization procedure such as developed in [34],[35],[36]. 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

$$ (Z_{\varepsilon}[\mathbf{J}_{\varepsilon}])_{\varepsilon} = \left( \int D[\varphi_{\varepsilon}] \exp \left\{ i S_{\mathrm{m}}(\varepsilon) + i \int \mathbf{J}_{\varepsilon}(x) \varphi_{\varepsilon}(x) d^{\mu}x \right\} \right)_{\varepsilon} \qquad (6.19) $$

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

$$ (Z_{\varepsilon}[0])_{\varepsilon} = \left( \int D[\varphi_{\varepsilon}] \exp \left\{ i S_{\mathrm{m}}(\varepsilon) + i \int \mathbf{J}_{\varepsilon}(x) \varphi_{\varepsilon}(x) d^{\mu}x \right\} \right)_{\varepsilon} \Big|_{\mathbf{J}=0} = (\langle 0_{\varepsilon} | 0_{\varepsilon} \rangle)_{\varepsilon} = 1. \qquad (6.20) $$

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

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

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 $(S_m(\varepsilon))_\varepsilon$ in (6.19) as the curved distributional spacetime matter action, and $(\mathbf{J}_\varepsilon(x))_\varepsilon$ as a current density (a scalar density in the case of scalar fields). One can thus set $\mathbf{J}_\varepsilon = 0$ in (6.19) and examine the variation of $(Z_\varepsilon[0])_\varepsilon$:

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

Note that

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

From (6.22) and (6.23) one obtains directly

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

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

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

and

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

Following canonical calculation one obtains

$$ (Z_{\varepsilon}^{\pm}[0])_{\varepsilon} \propto ([\det(-G_{\varepsilon}^{\pm}(x, x'))]^{\frac{1}{2}})_{\varepsilon} \qquad (6.27) $$

where the proportionality constant is metric-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 $\mathfrak{I}$ of generalized vectors $|x\rangle$, normalized by

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

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 $(\mathrm{tr}[\mathfrak{R}_{\varepsilon}])_{\varepsilon}$ of an Colombeau generalized operator $(\mathfrak{R}_{\varepsilon})_{\varepsilon}$ which acts on a linear space $\mathfrak{I}$, is defined by

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

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

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

by Eq.(6.14) one obtains

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

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

$$ \left( \int_{\Lambda}^{\infty} ds (is)^{-1} i \exp[-s \mathcal{F}_{\varepsilon}^{\pm}] \right)_{\varepsilon} = (\mathrm{Ei}(-i \Lambda \mathcal{F}_{\varepsilon}^{\pm}))_{\varepsilon} \qquad (6.34) $$

where $\mathrm{Ei}(x)$ is the exponential integral function.

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

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

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

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

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

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

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 $(W_{\varepsilon}^{\pm})_{\varepsilon}$ using Eq.(6.37) and Eq.(6.31) we get

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

Interchanging the order of integration and taking the limit $x \to x'$ one obtains
$$ (W_{\epsilon}^{\pm})_{\epsilon} = \frac{i}{2} \left( \int_{m^2}^{\infty} dm^2 \int d^n x [-g^{\pm}(x, \epsilon)]^{\frac{1}{2}} G_{\epsilon}^{\pm}(x, x; m^2) \right)_{\epsilon}. \qquad (6.39) $$

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 $(L_{\epsilon;eff}^{\pm}(x))_{\epsilon}$ by

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

whence one get

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

## 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 $-i\epsilon$ 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

$$ \left\{ \begin{aligned} (L_{\epsilon;\mathrm{div}}^{\pm}(x))_{\epsilon} = &-(32\pi^2)^{-1} \left( \lim_{x \to x'} \left[ \left( \Delta_{\pm}^{1/2}(x,x';\epsilon) \right)_{\epsilon} \right] \int_0^{\infty} \frac{ds}{s^3} \exp \left[ -im^2s + \frac{\sigma(x,x')}{2is} \right] \right) \times \\ & \times \left[ a_0^{\pm}(x,x';\epsilon) + isa_1^{\pm}(x,x';\epsilon) + (is)^2 a_2^{\pm}(x,x';\epsilon) \right]_{\epsilon} \end{aligned} \right. \qquad (6.42) $$

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 $(L_{\epsilon;\mathrm{div}}^{\pm}(x))_{\epsilon}$ 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 $\infty \times [\mathrm{geometrical object}]$. This can be done in a variety of ways. For example, in n dimensions, the asymptotic (adiabatic) expansion of $(L_{\epsilon;\mathrm{eff}}^{\pm}(x))_{\epsilon}$ is

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

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

$$ \left\{ \begin{aligned} \left( L_{\varepsilon; \text{eff}}^{\pm}(x) \right)_{\varepsilon} &\propto 2^{-1} (4\pi)^{-n/2} \left( \sum_{j=0}^{\infty} a_j(x; \varepsilon) \int_0^{\infty} \text{id}s (is)^{j-1-n/2} \exp[-im^2 s] \right)_{\varepsilon} = \\ &\quad 2^{-1} (4\pi)^{-n/2} \sum_{j=0}^{\infty} a_j(x; \varepsilon) (m^2)^{n/2-j} \Gamma\left(j - \frac{n}{2}\right), \\ a_j(x; \varepsilon) &= a_j(x, x; \varepsilon). \end{aligned} \right. \quad (6.44) $$

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:

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

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

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

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)

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

Finally one obtains

$$ (L_{\varepsilon; \text{ren}}^{\pm}(x))_{\varepsilon} \propto -\frac{1}{64\pi^2} \left( \int_0^{\infty} \text{id}s \ln(is) \frac{\partial^3}{\partial(is)^3} \left[ \mathcal{F}_{\varepsilon}^{\pm}(x, x; is) e^{-ism^2} \right] \right)_{\varepsilon}. \quad (6.49) $$

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

$$ (S_{\epsilon}[\tilde{g}_{\mu\nu}^{\pm}(x, \epsilon)])_{\epsilon} = (S_{\epsilon}[g_{\mu\nu}^{\pm}(x, \epsilon)])_{\epsilon} + \left( \int d^{n}x \left( \frac{\delta S_{\epsilon}[\tilde{g}_{\mu\nu}^{\pm}(x, \epsilon)]}{\delta \tilde{g}^{\pm \rho \sigma}(x, \epsilon)} \right) \delta \tilde{g}^{\pm \rho \sigma}(x, \epsilon) \right)_{\epsilon} . \quad (6.51) $$

From Eq.(6.51) one obtains

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

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 $j = 2$:

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

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

$$ (W_{\epsilon, \text{div}}^{\pm})_{\epsilon} = 2^{-1}(4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \Gamma\left(2-\frac{n}{2}\right) \left(\int d^n x [-g^{\pm}(x, \epsilon)]^{\frac{1}{2}} a_2(x, \epsilon)\right)_{\epsilon} = 2^{-1}(4\pi)^{-n/2} \left(\frac{m}{\mu}\right)^{n-4} \Gamma\left(2-\frac{n}{2}\right) \left(\int d^n x [-g^{\pm}(x, \epsilon)]^{\frac{1}{2}} [\alpha F_{\epsilon}^{\pm}(x) + \beta G_{\epsilon}^{\pm}(x)]\right)_{\epsilon} + O(n-4) \quad (6.54) $$

where

$$ \begin{aligned} (F_{\epsilon}(x))_{\epsilon} &= (\mathbf{R}^{\pm a\beta\gamma\delta}(x, \epsilon) \mathbf{R}_{a\beta\gamma\delta}^{\pm}(x, \epsilon))_{\epsilon} - 2(\mathbf{R}^{\pm a\beta}(x, \epsilon) \mathbf{R}_{a\beta}^{\pm}(x, \epsilon))_{\epsilon} + \frac{1}{3}(\mathbf{R}^{\pm 2}(x, \epsilon))_{\epsilon}, \\ (G_{\epsilon}^{\pm}(x))_{\epsilon} &= (\mathbf{R}^{\pm a\beta\gamma\delta}(x, \epsilon) \mathbf{R}_{a\beta\gamma\delta}^{\pm}(x, \epsilon))_{\epsilon} \end{aligned} \quad (6.55) $$

and

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

Finally one obtains

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

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

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

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
$$ \langle T_{\mu}^{\mu}(x, \epsilon) \rangle_{\text{ren}} \propto -(2880\pi^2)^{-1} \left[ \left( \left( (r-2m)^2 + \epsilon^2 \right)^{-1} \right)_{\epsilon} + 4(2m)^4 \right]. \quad (6.59) $$

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_25.png" /><figcaption>Fig.7.1. 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_26.png" /><figcaption>Fig.7.2. 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 10-11 at 22 GHz.</figcaption></figure>

Color marks the lower limit of observed brightness temperature obtained from visibility amplitudes. Adapted from [54].

The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than $5.3 G\lambda$, 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 (7.1.1) 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 [1]. 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 [1]. 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 accelerated 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

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

**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 $J_z = ma$

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

**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

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

$$ \Xi_{\varepsilon}^{+}(r_{\varepsilon}, \theta) = \frac{(r_{\varepsilon} - r_{E_{-}}(\theta)) \sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^{2} + \varepsilon^{2}}}{\rho_{\varepsilon}^{2}(r_{\varepsilon})}, \quad (7.2.6) \\ \Delta_{a,\varepsilon} = r_{\varepsilon}^{2} - 2mr_{\varepsilon} + a^{2}, $$

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

(7.2.1) reads

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

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

generalized metric (7.2.7) restricted on $\Xi(\theta)$ reads

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

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

**Remark 7.2.4.** Note that for small values of the parameter $a$, where we may neglect terms

of the order of $a^2$, we get from (7.2.7) effectively the following Colombeau generalized metric

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

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

**Remark 7.2.5.** Note that Colombeau generalized metric (7.2.9) restricted on $\Xi(\theta)$ reads
$$ (ds_{\varepsilon}^{+2})_{\varepsilon} = -\left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\varepsilon})_{\varepsilon}}\right) dt^2 + \left(1 - \frac{r_{E_{+}}(\theta)}{(r_{\varepsilon})_{\varepsilon}}\right)^{-1} \left[[(dr_{\varepsilon}^2)_{\varepsilon}] + [(r_{\varepsilon}^2)_{\varepsilon}] d\theta^2\right] \quad (7.2.10) $$

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

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

where $\mathbf{cl}[(r_\varepsilon)_\varepsilon] \approx \tilde{\mathbb{R}} r_{E_+}(\theta)$, see Appendix Eq.(A.12).

(II) Let $(\mathbf{R}^{\mu\nu(a \ll 1)}(r_\varepsilon, \varepsilon)\mathbf{R}_{\mu\nu}^{(a \ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon$ be Colombeau generalized quadratic scalar $(\mathbf{R}^{\mu\nu}(r_\varepsilon, \varepsilon)\mathbf{R}_{\mu\nu}(r_\varepsilon, \varepsilon))_\varepsilon$ corresponding to the metric (7.2.10) with $a \ll 1$. From Eq.(7.2.10)

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

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

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

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

$$ \mathbf{sing}[(\mathbf{R}^{\rho\sigma\mu\nu(a \ll 1)}(r_\varepsilon, \varepsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{(a \ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon] $$
of the Colombeau generalized quadratic scalar $(\mathbf{R}^{\rho\sigma\mu\nu(a \ll 1)}(r_\varepsilon, \varepsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{(a \ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon$ reads

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

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

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

and

$$ \begin{aligned} \mathbf{sing}[(\mathbf{R}^{\mu\nu(a \ll 1)}(r_\varepsilon, \varepsilon)\mathbf{R}_{\mu\nu}^{(a \ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon] \Big|_{(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)} &= \tilde{\mathbb{R}} r_{E_+}^{-4}(\theta)(\varepsilon^{-2})_\varepsilon \approx \tilde{\mathbb{R}} \infty, \\ \mathbf{sing}[(\mathbf{R}^{\rho\sigma\mu\nu(a \ll 1)}(r_\varepsilon, \varepsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{(a \ll 1)}(r_\varepsilon, \varepsilon))_\varepsilon] \Big|_{(r_\varepsilon)_\varepsilon = r_{E_+}(\theta)} &= \tilde{\mathbb{R}} r_{E_+}^{-4}(\theta)(\varepsilon^{-2})_\varepsilon \approx \tilde{\mathbb{R}} \infty. \end{aligned} \quad (7.2.15) $$

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

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

From Eq.(7.2.8),Eq.(7.2.16) and Eq.(A.1)-Eq.(A.2) we obtain
$$ (\mathbf{R}^{a<m}(r_{\varepsilon}, \varepsilon))_{\varepsilon} = \left( \frac{A_{\varepsilon}}{\Delta_{\varepsilon}} \left[ \frac{2}{r_{\varepsilon}} \left( -2\frac{A'_{\varepsilon}}{A_{\varepsilon}} - 3\frac{B'_{\varepsilon}}{B_{\varepsilon}} + \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} \right) + \frac{2}{r^2} \frac{C_{\varepsilon}}{B_{\varepsilon}} - \frac{A''_{\varepsilon}}{A_{\varepsilon}} - 2\frac{B''_{\varepsilon}}{B_{\varepsilon}} \right. \right. \\ \left. \left. + \frac{1}{2} \left( \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right)^2 - 2\frac{A'_{\varepsilon}B'_{\varepsilon}}{A_{\varepsilon}B_{\varepsilon}} + \left( \frac{1}{2}\frac{A'_{\varepsilon}}{A_{\varepsilon}} + \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right) \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} \right] \right)_{\varepsilon} = \\ \left( \frac{\Delta_{a,\varepsilon}}{\rho_{\varepsilon}^2} \left[ \frac{2}{r_{\varepsilon}} \left( -2\frac{A'_{\varepsilon}}{A_{\varepsilon}} - 3\frac{B'_{\varepsilon}}{B_{\varepsilon}} + \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} \right) + \frac{2}{r^2} \frac{C_{\varepsilon}}{B_{\varepsilon}} - \frac{A''_{\varepsilon}}{A_{\varepsilon}} - 2\frac{B''_{\varepsilon}}{B_{\varepsilon}} \right. \right. \\ \left. \left. + \frac{1}{2} \left( \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right)^2 - 2\frac{A'_{\varepsilon}B'_{\varepsilon}}{A_{\varepsilon}B_{\varepsilon}} + \left( \frac{1}{2}\frac{A'_{\varepsilon}}{A_{\varepsilon}} + \frac{B'_{\varepsilon}}{B_{\varepsilon}} \right) \frac{\Delta'_{\varepsilon}}{\Delta_{\varepsilon}} \right] \right)_{\varepsilon}. \quad (7.2.17) $$

Note that

$$ \begin{aligned} \frac{\partial}{\partial r_{\varepsilon}} \sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^2 + \varepsilon^2} &= \frac{r_{\varepsilon} - r_{E_{+}}(\theta)}{\sqrt{(r_{\varepsilon} - r_{E_{+}}(\theta))^2 + \varepsilon^2}} \\ \frac{\partial^2}{\partial r_{\varepsilon}^2} \sqrt{(r_{\varepsilon} - r_{E_{+}})^2 + \varepsilon^2} &= \frac{\varepsilon^2}{[(r_{\varepsilon} - r_{E_{+}})^2 + \varepsilon^2]^{\frac{3}{2}}}. \end{aligned} \quad (7.2.18) $$

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

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

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

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

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

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

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

$$ \mathbf{sing}[(\mathbf{R}^{\rho\sigma\mu\nu(a<m)}(r_{\varepsilon}, \varepsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{(a<m)}(r_{\varepsilon}, \varepsilon))_{\varepsilon}] = \mathcal{R} \left( \frac{\varepsilon^4}{4(r_{E_{+}}(\theta))^4 [\varepsilon^2 + (r_{\varepsilon} - 2m)^2]^3} \right)_{\varepsilon}. \quad (7.2.21) $$
<figure><img src="image_27.png" /><figcaption>Pic.7. Ergosurface, horizon, and singularity for slow Kerr black hole.</figcaption></figure>

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

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

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

<figure><img src="image_28.png" /><figcaption>Pic.8. Ergosurface, horizon, and singularity for critical Kerr black hole.</figcaption></figure>

## III. Fast Kerr gravitational singularity: $a > m$.
<figure><img src="image_29.png" /><figcaption>Pic.9. Ergosurface, horizon, and singularity for fast Kerr black hole.</figcaption></figure>

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

$$ \cos^2\theta \le \frac{m^2}{a^2} = \eta. \qquad (7.2.23) $$

Let $\Theta_{\eta}$ be a set $\Theta_{\eta} = \{\theta|\cos^2\theta \le \eta\}$ and let $\chi(\theta, \eta)$ be the indicator function of the set $\Theta_{\eta}$, i.e. $\chi(\theta, \eta)$ is the function defined to be identically 1 on $\Theta_{\eta}$, and is 0 elsewhere. We introduce now the following regularized (above ergosurface $r = r_{E_{+}}(\theta)$, $\theta \in \Theta_{\eta}$) quantity

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

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

(7.2.1) reads

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

**Remark 7.2.8.** 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 $\cup_{\theta \in \Theta_{\eta}} \Xi_{\eta}(\theta)$ which coincide with an part of the outer ergosurface of classical Kerr spacetime. In accordance with Eq.(7.2.11),Eq.(7.2.19) and

Eq.(7.2.20) submanifold $\cup_{\theta \in \Theta_{\eta}} \Xi_{\eta}(\theta)$ presented the singular boundary of distributional spacetime with Colombeau generalized metric (7.2.26).

**Remark 7.2.9.** Let $\tilde{\Xi}_{\eta}(\theta)$ be a submanifold given by equations (i) $\phi = \text{const}$ and (ii) $\cos^2\theta \le \eta$, then Colombeau generalized metric (7.2.26) restricted on submanifold $\cup_{\theta \in \Theta_{\eta}} \tilde{\Xi}_{\eta}(\theta)$ reads
$$ (ds_{\epsilon}^{+2})_{\epsilon} = -\chi(\theta, \eta)[(\Xi_{\epsilon}(r_{\epsilon}, \theta, \eta))_{\epsilon}]dt^{2} + \left[ \left( \frac{\rho_{\epsilon}^{2}}{\Delta_{a,\epsilon}} \right)_{\epsilon} \right] [(dr_{\epsilon}^{2})_{\epsilon}] + [(\rho_{\epsilon}^{2})_{\epsilon}]d\theta^{2}. \quad (7.2.27) $$

Note that Colombeau generalized metric (7.2.27) nondegenerate on outer ergosurfaces

$(r_{\epsilon})_{\epsilon} = r_{E_{+}}(\theta)$, see Pic.7.3.

From Eq.(7.2.27) and and Eq.(A.1)-Eq.(A.2) one obtains that main singular part $sing[(R^{a>m}(r_{\epsilon}, \epsilon))_{\epsilon}]$ of the Colombeau generalized curvature scalar $(R^{a>m}(r_{\epsilon}, \epsilon))_{\epsilon}$ corresponding to the metric (7.2.27) (mod nonsingular multiplier) reads

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

**Remark 7.2.10. (I)** Let $(\mathbf{R}^{\mu\nu(a>m)}(r_{\epsilon}, \epsilon)\mathbf{R}_{\mu\nu}^{(a>m)}(r_{\epsilon}, \epsilon))_{\epsilon}$ be Colombeau generalized quadratic

scalar $(\mathbf{R}^{\mu\nu}(r_{\epsilon}, \epsilon)\mathbf{R}_{\mu\nu}(r_{\epsilon}, \epsilon))_{\epsilon}$ corresponding to the metric (7.2.27) with $a > m$. From Eq.(7.2.27) and Eq.(A.1)-Eq.(A.2) one obtains that main singular part

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

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

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

(II) Let $(\mathbf{R}^{\rho\sigma\mu\nu(a>m)}(r_{\epsilon}, \epsilon)\mathbf{R}_{\rho\sigma\mu\nu}^{(a>m)}(r_{\epsilon}, \epsilon))_{\epsilon}$ be Colombeau generalized quadratic

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

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

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

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

## 8. Conclusions and remarks.

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

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

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

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

# 9. Acknowledgments

To reviewers provided important clarifications.

# Appendix A.

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

Let us introduce now Colombeau generalized metric which has the form

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

The Colombeau scalars $(\mathbf{R}(\epsilon))_\epsilon$, $(\mathbf{R}^{\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}(\epsilon))_\epsilon$ and $(\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_\epsilon$, in terms of Colombeau generalized functions $(A_\epsilon(r))_\epsilon$, $(B_\epsilon(r))_\epsilon$, $(C_\epsilon(r))_\epsilon$, $(D_\epsilon(r))_\epsilon$ is expressed as
$$ (\mathbf{R}(\epsilon))_{\epsilon} = \left( \frac{A_{\epsilon}}{\Delta_{\epsilon}} \left[ \frac{2}{r} \left( -2\frac{A'_{\epsilon}}{A_{\epsilon}} - 3\frac{B'_{\epsilon}}{B_{\epsilon}} + \frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right) + \frac{2}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{A''_{\epsilon}}{A_{\epsilon}} - 2\frac{B''_{\epsilon}}{B_{\epsilon}} \right. \right. \\ \left. \left. + \frac{1}{2}\left(\frac{B'_{\epsilon}}{B_{\epsilon}}\right)^2 - 2\frac{A'_{\epsilon}B'_{\epsilon}}{A_{\epsilon}B_{\epsilon}} + \left(\frac{1}{2}\frac{A'_{\epsilon}}{A_{\epsilon}} + \frac{B'_{\epsilon}}{B_{\epsilon}}\right)\frac{\Delta'_{\epsilon}}{\Delta_{\epsilon}} \right] \right)_{\epsilon}, $$

$$ (\mathbf{R}^{\mu\nu}(\epsilon)\mathbf{R}_{\mu\nu}(\epsilon))_{\epsilon} = \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{1}{2} \frac{A_{\epsilon}''}{A_{\epsilon}} - \frac{1}{4} \frac{A_{\epsilon}' \Delta_{\epsilon}'}{A_{\epsilon} \Delta_{\epsilon}} + \frac{1}{2} \frac{A_{\epsilon}' B_{\epsilon}'}{A_{\epsilon} B_{\epsilon}} + \frac{1}{r} \frac{A_{\epsilon}'}{A_{\epsilon}} \right)^2 \right)_{\epsilon} + \\ + 2 \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \left( \frac{1}{2} \frac{\Delta_{\epsilon}'}{\Delta_{\epsilon}} - \frac{A_{\epsilon}'}{A_{\epsilon}} - 2 \frac{B_{\epsilon}'}{B_{\epsilon}} \right) + \frac{1}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} - \frac{1}{2} \frac{A_{\epsilon}' B_{\epsilon}'}{A_{\epsilon} B_{\epsilon}} - \right. \right. \\ \left. \left. - \frac{1}{2} \frac{B_{\epsilon}''}{B_{\epsilon}} + \frac{1}{4} \frac{B_{\epsilon}' \Delta_{\epsilon}'}{B_{\epsilon} \Delta_{\epsilon}} \right]^2 \right)_{\epsilon} + \\ \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{2} \frac{A_{\epsilon}''}{A_{\epsilon}} - \frac{1}{4} \frac{A_{\epsilon}' \Delta_{\epsilon}'}{A_{\epsilon} \Delta_{\epsilon}} + \frac{1}{2} \frac{A_{\epsilon}' B_{\epsilon}'}{A_{\epsilon} B_{\epsilon}} + \frac{B_{\epsilon}''}{B_{\epsilon}} - \frac{1}{2} \left( \frac{B_{\epsilon}'}{B_{\epsilon}} \right)^2 \right. \right. \\ \left. \left. - \frac{1}{2} \frac{B_{\epsilon}' \Delta_{\epsilon}'}{B_{\epsilon} \Delta_{\epsilon}} + \frac{1}{r} \left( \frac{A_{\epsilon}'}{A_{\epsilon}} - \frac{\Delta_{\epsilon}'}{\Delta_{\epsilon}} + 2 \frac{B_{\epsilon}'}{B_{\epsilon}} \right) \right]^2 \right)_{\epsilon}, \quad (A.2) $$

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_{\epsilon} = \left( \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{A_{\epsilon}''}{A_{\epsilon}} - \frac{1}{2} \frac{A_{\epsilon}' \Delta_{\epsilon}'}{A_{\epsilon} \Delta_{\epsilon}} \right)^2 + 2 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left( \frac{1}{r} \frac{A_{\epsilon}'}{A_{\epsilon}} + \frac{1}{2} \frac{A_{\epsilon}' B_{\epsilon}'}{A_{\epsilon} B_{\epsilon}} \right)^2 \right. \\ + 4 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \frac{B_{\epsilon}'}{B_{\epsilon}} - \frac{1}{r^2} \frac{A_{\epsilon}C_{\epsilon} + D_{\epsilon}^2}{A_{\epsilon}B_{\epsilon}} + \frac{1}{4} \left( \frac{B_{\epsilon}'}{B_{\epsilon}} \right)^2 \right]^2 + \\ + 2 \frac{A_{\epsilon}^2}{\Delta_{\epsilon}^2} \left[ \frac{1}{r} \left( \frac{A_{\epsilon}'}{A_{\epsilon}} + 2 \frac{B_{\epsilon}'}{B_{\epsilon}} - \frac{\Delta_{\epsilon}'}{\Delta_{\epsilon}} \right) + \frac{1}{2} \frac{A_{\epsilon}' B_{\epsilon}'}{A_{\epsilon} B_{\epsilon}} + \frac{B_{\epsilon}''}{B_{\epsilon}} \right. \\ \left. - \frac{1}{2} \left( \frac{B_{\epsilon}'}{B_{\epsilon}} \right)^2 - \frac{1}{2} \frac{B_{\epsilon}' \Delta_{\epsilon}'}{B_{\epsilon} \Delta_{\epsilon}} \right]^2 \right)_{\epsilon}. $$

Here

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

Assume that

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

From Eq.(A.2)-Eq.(A.4) one obtains

$$ (\mathbf{R}(\epsilon))_{\epsilon} = \left( -\frac{4A'_{\epsilon}}{r} + \frac{2A_{\epsilon}C_{\epsilon}}{r^2} - A''_{\epsilon} \right)_{\epsilon}, $$

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

$$ (\mathbf{R}^{\rho\sigma\mu\nu}(\epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(\epsilon))_{\epsilon} = \left( (A_{\epsilon}^{\prime\prime})^2 + 4\left(\frac{A_{\epsilon}^{\prime}}{r}\right)^2 + 4\frac{(A_{\epsilon}C_{\epsilon})^2}{r^4} \right)_{\epsilon}. $$

We choose now

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

and rewrite Colombeau generalized object corresponding to Schwarzschild metric above horizon in the following form
$$ (ds_{\varepsilon}^2)_{\varepsilon} = -(A_{\varepsilon}(r_{\varepsilon})dt^2)_{\varepsilon} + (A_{\varepsilon}^{-1}(r_{\varepsilon})dr_{\varepsilon}^2)_{\varepsilon} + r_{\varepsilon}^2 d\Omega^2, \qquad (A.7) $$

where $A_{\varepsilon}(r)$

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

By differentiation we obtain

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

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

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

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

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

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

We assume now that $\mathrm{cl}[(r_{\varepsilon})_{\varepsilon}] \approx_{\mathbb{R}} 2m$ and therefore from Eq.(A.11) we obtain
$$ (\mathbf{R}(r_{\varepsilon}, \varepsilon))_{\varepsilon} \approx_{\mathbb{R}} \left( \frac{4m^2 \varepsilon^2}{8m^3 \left[ (r_{\varepsilon} - 2m)^2 + \varepsilon^2 \right]^{3/2}} \right)_{\varepsilon}. \quad (A.12) $$

**Remark A.2.** Note that from Eq.(A.12) at horizon $r = \tilde{r}_2 2m$ follows that:

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

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

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

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

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

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

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

**Remark A.5.** Note that from Eq.(A.16) follows that: if $\text{st}((r_{\varepsilon})_{\varepsilon}) \neq 0$, i.e. $(r_{\varepsilon})_{\varepsilon} \approx_{\mathbb{R}} 2m$ then

$$ \begin{aligned} r \approx_{\mathbb{R}} 2m \Rightarrow & (\mathbf{R}^{\mu\nu}(r, \varepsilon)\mathbf{R}_{\mu\nu}(r, \varepsilon))_{\varepsilon} \approx_{\mathbb{R}} K(r), \\ K(r) = 12 \frac{r_s^2}{r^6}, & r_s = 2m. \end{aligned} \quad (A.17) $$

We assume now that $(r_{\varepsilon})_{\varepsilon} \approx_{\mathbb{R}} 2m$ and therefore from Eq.(A.16) we obtain

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

**Remark A.6.** Note that from Eq.(A.18) at horizon $r = \tilde{r}_2 2m$ follows that:

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

**Remark A.7.** Let $[(r_{\varepsilon} - 2m)_{\varepsilon}] \approx_{\mathbb{R}} 0$, then from Eq.(A13) and Eq.(A.12) we obtain
$$ [(\mathbf{R}^{\mu\nu}(r_{\varepsilon}, \varepsilon)\mathbf{R}_{\mu\nu}(r_{\varepsilon}, \varepsilon))_{\varepsilon}] \approx_{\mathbb{R}} K(r_s) + \left[ \left( \frac{\varepsilon^4}{4m^4(\varepsilon^2 + (r_{\varepsilon} - 2m)^2)^3} \right)_{\varepsilon} \right] \quad (A.20) $$

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

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

**Remark A.8.** Note that from Eq.(C.15) follows that:

$$ r \approx_{\mathbb{R}} 2m \Rightarrow (\mathbf{R}^{\rho\sigma\mu\nu}(r, \varepsilon)\mathbf{R}_{\rho\sigma\mu\nu}(r, \varepsilon))_{\varepsilon} \approx_{\mathbb{R}} K(r), \quad (A.22) $$

see Definition 1.5.2.(i).

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

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

**Remark A.9.** Let $[(r_{\varepsilon} - 2m)_{\varepsilon}] \approx_{\mathbb{R}} 0$, then from Eq.(A.13) and Eq.(A.12) we obtain

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

**Remark A.10.** Note that from Eq.(A.15) at horizon $r = 2m$ follows that:

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

see Definition 1.5.2.(ii).

**Remark A.11.** We assume now there exist an fundamental generalized length $(l_{\varepsilon})_{\varepsilon}$

$$ \begin{aligned} (l_{\varepsilon})_{\varepsilon \in (0, \eta]} &= a(\varepsilon)_{\varepsilon \in (0, \eta]}, \quad \eta \ll 1, \\ (l_{\varepsilon})_{\varepsilon \in (\eta, 1]} &= a, \end{aligned} \quad (A.26) $$

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

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

By using (A.20) we get the estimate
$$ (\mathbf{R}^{\rho\sigma\mu\nu}(r_{\epsilon}, \epsilon)\mathbf{R}_{\rho\sigma\mu\nu}(r_{\epsilon}, \epsilon))_{\epsilon} \approx_{\mathbb{R}} K(r_s) + \left( \frac{\epsilon^4}{4m^4[(r_{\epsilon}-2m)^2 + \epsilon^2]^3} \right)_{\epsilon \in (0, \eta]} = \\ K(r_s) + \left( \frac{1}{4m^4[(r_{\epsilon}-2m)^2 + \epsilon^2]^2} \right)_{\epsilon \in (0, \eta]} \times \left( \frac{\epsilon^2}{[(r_{\epsilon}-2m)^2 + \epsilon^2]^3} \right)_{\epsilon \in (0, \eta]} \times \\ \times \left( \frac{\epsilon^2}{[(r_{\epsilon}-2m)^2 + \epsilon^2]^2} \right)_{\epsilon \in (0, \eta]} \leq \\ K(r_s) + \frac{1}{4m^4[a^2+1]^2} \left( \frac{1}{[(r_{\epsilon}-2m)^2 + \epsilon^2]^2} \right)_{\epsilon \in (0, \eta]} \leq \\ K(r_s) + \frac{1}{4m^4[a^2+1]^2} \left( \frac{1}{(r-2m)^2} \right)_{r-2m \in (0, \eta]} \leq \quad (A.28) $$

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