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

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

**Remark 1.7.4.** The element of "area" of the Colombeau generalized hypersurface spanned by three infinitesimal Colombeau generalized displacements is the contravariant antisymmetric Colombeau generalized tensor $(dS_{\epsilon}^{ikl})_{\epsilon}$; the vector dual to it is gotten by multiplying by the tensor $\sqrt{-(g_{\epsilon})_{\epsilon}}e_{iklm}$, so it is equal to

$$ \sqrt{-(g_{\epsilon})_{\epsilon}} (dS_{\epsilon,i})_{\epsilon} = -\frac{1}{6} \sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm} (dS_{\epsilon}^{kim})_{\epsilon}. \qquad (1.7.26) $$

**Remark 1.7.5.** Let $(df_{\epsilon}^{ik})_{\epsilon}$ be the element of two-dimensional Colombeau generalized surface spanned by two infinitesimal Colombeau generalized displacements, the dual Colombeau generalized tensor is defined as

$$ \sqrt{-(g_{\epsilon})_{\epsilon}} (df_{ik,\epsilon}^{*})_{\epsilon} = \frac{1}{2} \sqrt{-(g_{\epsilon})_{\epsilon}} e_{iklm} (df_{\epsilon}^{lm})_{\epsilon}. \qquad (1.7.27) $$

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

## 1.8 Generalized Stokes' theorem

**Remark 1.7.6.** Note that the canonical rules for transforming the various integrals into one another remain the same, since their derivation was formal in character and not related to the tensor properties of the different quantities. Of particular importance is the rule for transforming the integral over a hypersurface into an integral over a four-volume (Gauss' theorem), which is accomplished by the substitution

$$ (dS_{i,\epsilon})_{\epsilon} := [(d\Omega_{\epsilon})_{\epsilon}] \left(\frac{\partial}{\partial x_{\epsilon}^{i}}\right)_{\epsilon}. \qquad (1.7.28) $$

**Remark 1.7.7.** (Stokes' theorem) Note that for the integral of Colombeau generalized vector $(A_{\epsilon}^{i})_{\epsilon}$ we have

$$ (A_{\epsilon}^{i} dS_{i,\epsilon})_{\epsilon} = \left(\frac{\partial A_{\epsilon}^{i}}{\partial x_{\epsilon}^{i}} d\Omega_{\epsilon}\right)_{\epsilon} = \left[\left(\frac{\partial A_{\epsilon}^{i}}{\partial x_{\epsilon}^{i}}\right)_{\epsilon}\right] [(d\Omega_{\epsilon})_{\epsilon}]. \qquad (1.7.29) $$

which is the generalization of Stokes' theorem.

Note that in galilean coordinates the Colombeau generalized differentials $(dA_{i,\epsilon})_{\epsilon}$ of a vector $(A_{i,\epsilon})_{\epsilon}$ form the Colombeau generalized vector, and the derivatives $(\partial A_{i,\epsilon}/\partial x_{\epsilon}^{k})_{\epsilon}$ of the components of a vector with respect to the coordinates form the Colombeau generalized tensor. In Colombeau generalized curvilinear coordinates this is not so; $(dA_{i,\epsilon})_{\epsilon}$ is not a vector, and $(\partial A_{i,\epsilon}/\partial x_{\epsilon}^{k})_{\epsilon}$ is not the Colombeau generalized tensor. This is due to the fact that $(dA_{i,\epsilon})_{\epsilon}$ is the difference of vectors located at different (infinitesimally separated) points of space; at different points in space vectors transform differently, since the coefficients in the transformation formulas (1.7.3), (1.7.4) are Colombeau generalized functions of the generalized coordinates. Thus in order to compare two infinitesimally separated generalized vectors we must subject one of them to a parallel translation to the point where the second is located. Let us consider an arbitrary generalized contravariant vector; if its value at the point $x^{i}$ is $(A_{\epsilon}^{i})_{\epsilon}$, then at the neighboring point $x^{i} + dx^{i}$ it is equal to $(A_{\epsilon}^{i})_{\epsilon} + (dA_{\epsilon}^{i})_{\epsilon} = (A_{\epsilon}^{i} + dA_{\epsilon}^{i})_{\epsilon}$. We subject the vector $(A_{\epsilon}^{i})_{\epsilon}$ to an infinitesimal parallel displacement to the point $x^{i} + dx^{i}$; the change in the vector which results from this we denote
by $(\delta A_{\epsilon}^{i})_{\epsilon}$. Then the difference $(DA_{\epsilon}^{i})_{\epsilon}$ between the two Colombeau generalized vectors which are now located at the same point is

$$ (DA_{\epsilon}^{i})_{\epsilon} = (dA_{\epsilon}^{i})_{\epsilon} - (\delta A_{\epsilon}^{i})_{\epsilon}. \qquad (1.7.30) $$

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

$$ (\delta A_{\epsilon}^{i})_{\epsilon} = -(\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k} dx^{l})_{\epsilon}, \qquad (1.7.31) $$

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

$$ (\Gamma_{i,kl,\epsilon})_{\epsilon} = (g_{im,\epsilon} \Gamma_{km,\epsilon})_{\epsilon}. \qquad (1.7.32) $$

Conversely,

$$ (\Gamma_{kl,\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{im} \Gamma_{m,kl,\epsilon})_{\epsilon}. \qquad (1.7.33) $$

It is also easy to relate the change in the components of a covariant vector under a parallel displacement to the Christoffel symbols. To do this we note that under a parallel displacement, a scalar is unchanged. In particular, the scalar product of two vectors does not change under a parallel displacement. Let $(A_{i,\epsilon})_{\epsilon}$ and $(B_{\epsilon}^{i})_{\epsilon}$ be any covariant and contravariant vectors. Then from $\delta(A_{i,\epsilon}B_{\epsilon}^{i})_{\epsilon} = 0_{\mathbb{R}}$, we have

$$ (B_{\epsilon}^{i} \delta A_{i,\epsilon}) = -(A_{i,\epsilon} \delta B_{\epsilon}^{i})_{\epsilon} = (\Gamma_{kl,\epsilon}^{i} B_{\epsilon}^{k} A_{i,\epsilon} dx^{l})_{\epsilon} \qquad (1.7.34) $$

or, changing the indices,

$$ (B_{\epsilon}^{i} \delta A_{i,\epsilon}) = (\Gamma_{il,\epsilon}^{k} B_{\epsilon}^{i} A_{k,\epsilon} dx^{l})_{\epsilon} \qquad (1.7.35) $$

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

$$ (\delta A_{i,\epsilon})_{\epsilon} = ((\Gamma_{il,\epsilon}^{k} A_{k,\epsilon})_{\epsilon}) dx^{l} \qquad (1.7.36) $$

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

Substituting (1.7.31) and $(dA_{\epsilon}^{i})_{\epsilon} = ((\partial A_{\epsilon}^{i}/\partial x^{l})_{\epsilon}) dx^{l}$ in (1.7.30), we obtain

$$ (DA_{\epsilon}^{i})_{\epsilon} = \left[ \left( \frac{\partial A_{\epsilon}^{i}}{\partial x^{l}} \right)_{\epsilon} + (\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k})_{\epsilon} \right] dx^{l}. \qquad (1.7.37) $$
by $(\delta A_{\epsilon}^{i})_{\epsilon}$. Then the difference $(DA_{\epsilon}^{i})_{\epsilon}$ between the two Colombeau generalized vectors which are now located at the same point is

$$ (DA_{\epsilon}^{i})_{\epsilon} = (dA_{\epsilon}^{i})_{\epsilon} - (\delta A_{\epsilon}^{i})_{\epsilon}. \qquad (1.7.30) $$

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

$$ (\delta A_{\epsilon}^{i})_{\epsilon} = -(\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k} dx^{l})_{\epsilon}, \qquad (1.7.31) $$

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

$$ (\Gamma_{i,kl,\epsilon})_{\epsilon} = (g_{im,\epsilon} \Gamma_{km,\epsilon})_{\epsilon}. \qquad (1.7.32) $$

Conversely,

$$ (\Gamma_{kl,\epsilon}^{i})_{\epsilon} = (g_{\epsilon}^{im} \Gamma_{m,kl,\epsilon})_{\epsilon}. \qquad (1.7.33) $$

It is also easy to relate the change in the components of a covariant vector under a parallel displacement to the Christoffel symbols. To do this we note that under a parallel displacement, a scalar is unchanged. In particular, the scalar product of two vectors does not change under a parallel displacement. Let $(A_{i,\epsilon})_{\epsilon}$ and $(B_{\epsilon}^{i})_{\epsilon}$ be any covariant and contravariant vectors. Then from $\delta(A_{i,\epsilon}B_{\epsilon}^{i})_{\epsilon} = 0_{\mathbb{R}}$, we have

$$ (B_{\epsilon}^{i} \delta A_{i,\epsilon}) = -(A_{i,\epsilon} \delta B_{\epsilon}^{i})_{\epsilon} = (\Gamma_{kl,\epsilon}^{i} B_{\epsilon}^{k} A_{i,\epsilon} dx^{l})_{\epsilon} \qquad (1.7.34) $$

or, changing the indices,

$$ (B_{\epsilon}^{i} \delta A_{i,\epsilon}) = (\Gamma_{il,\epsilon}^{k} B_{\epsilon}^{i} A_{k,\epsilon} dx^{l})_{\epsilon} \qquad (1.7.35) $$

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

$$ (\delta A_{i,\epsilon})_{\epsilon} = ((\Gamma_{il,\epsilon}^{k} A_{k,\epsilon})_{\epsilon}) dx^{l} \qquad (1.7.36) $$

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

Substituting (1.7.31) and $(dA_{\epsilon}^{i})_{\epsilon} = ((\partial A_{\epsilon}^{i}/\partial x^{l})_{\epsilon}) dx^{l}$ in (1.7.30), we obtain

$$ (DA_{\epsilon}^{i})_{\epsilon} = \left[ \left( \frac{\partial A_{\epsilon}^{i}}{\partial x^{l}} \right)_{\epsilon} + (\Gamma_{kl,\epsilon}^{i} A_{\epsilon}^{k})_{\epsilon} \right] dx^{l}. \qquad (1.7.37) $$
## 1.9 The Colombeau Generalized Curvature Tensor

In this subsection we derive the general formula for the change in a vector after parallel displacement around any infinitesimal closed contour $\gamma$. This generalized change $(\Delta A_{k,\epsilon})_{\epsilon} \in \mathbb{R}$ can clearly be written in the form $(\delta A_{k,\epsilon})_{\epsilon}$, where the Colombeau integral is taken over the given regular contour $\gamma$. Substituting in place of $(\delta A_{k,\epsilon})_{\epsilon}$ the expression (1.7.36), we get

$$ (\Delta A_{k,\epsilon})_{\epsilon} = (\gamma \Gamma_{kl,\epsilon}^{i}(x) A_{i}(x) dx^{l})_{\epsilon} \in \tilde{\mathbb{R}}, \qquad (1.8.1) $$

where for any $i, k, l = 0, 1, 2, 3$: $(\Gamma_{kl,\epsilon}^{i}(x))_{\epsilon} \in \mathcal{G}(\mathbb{R}^4)$, $x = (x^0, x^1, x^2, x^3)$, $A_i(x) \in \mathcal{D}(G)$, $G \subseteq \mathbb{R}^4$. Note that the vector $A_i$ which appears in the integrand obviously changes as we move along the contour $\gamma$.

**Definition 1.8.1.** We will say that generalized change $(\Delta A_{k,\epsilon})_{\epsilon}$ exists in the sense of the Schwartz distributions if for any $A_i(x) \in \mathcal{D}(G)$ the limit: $\lim_{\epsilon \to 0} \Delta A_{k,\epsilon}$ exists. Of course in this case obviously $\mathrm{cl}[(\Gamma_{kl,\epsilon}^{i}(x))_{\epsilon}] \in \mathcal{D}'(G)$ and $\mathrm{cl}[(\Delta A_{k,\epsilon})_{\epsilon}] \in \mathbb{R}$.

For the further transformation of this Colombeau integral, we must note the following. The values of the vector $A_i$ at points inside the contour are not unique; they depend on the path along which we approach the particular point. However, as we shall see from the result obtained below, this non-uniqueness is related to terms of second order. We may therefore, with the first-order accuracy which is sufficient for the transformation, regard the components of the vector $(A_{i,\epsilon})_{\epsilon}$ at points inside the infinitesimal contour $\gamma$ as being uniquely determined by their values on the contour itself by the formulas

$$ (\delta A_i(x))_{\epsilon} = (\Gamma_{il,\epsilon}^n(x) A_{n,\epsilon}(x) dx^l)_{\epsilon}, \qquad (1.8.2) $$

i.e., by the derivatives

$$ \frac{\partial A_{i, \epsilon}(x)}{\partial x^l} = (\Gamma_{il, \epsilon}^n(x) A_{n, \epsilon}(x))_{\epsilon}. \qquad (1.8.3) $$

Now applying generalized Stokes' theorem (see Theorem 1.7.1) to the integral (1.8.1) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f_{\epsilon}^{im})_{\epsilon}$, we get:

$$ \begin{aligned} (\Delta A_{k,\epsilon})_{\epsilon} &= \frac{1}{2} \left[ \left( \frac{\partial (\Gamma_{km,\epsilon}^{i}(x) A_i(x))}{\partial x^l} \right)_{\epsilon} - \left( \frac{\partial (\Gamma_{kl,\epsilon}^{i}(x) A_i(x))}{\partial x^m} \right)_{\epsilon} \right] (\Delta f_{\epsilon}^{im})_{\epsilon} = \\ &= \frac{1}{2} \left[ A_i(x) \left( \frac{\partial (\Gamma_{km,\epsilon}^{i}(x))}{\partial x^l} \right)_{\epsilon} - A_i(x) \left( \frac{\partial (\Gamma_{kl,\epsilon}^{i}(x))}{\partial x^m} \right)_{\epsilon} + \right. \\ & \qquad \left. \left( \frac{\partial A_i(x)}{\partial x^l} \right)_{\epsilon} (\Gamma_{km,\epsilon}^{i}(x))_{\epsilon} - \left( \frac{\partial}{\partial x^m} \right)_{\epsilon} (\Gamma_{kl,\epsilon}^{i}(x))_{\epsilon} \right] (\Delta f_{\epsilon}^{im})_{\epsilon}. \end{aligned} \qquad (1.8.4) $$

**Definition 1.8.2.** Colombeau generalized $k$-form $(3c9_{\epsilon})_{\epsilon}$ on a differentiable manifold $M$ is a smooth section of the bundle of alternating Colombeau generalized $k$-tensors on $M$.

Equivalently, $(3c9_{\epsilon})_{\epsilon}$ associates to each $x \in M$ an alternating Colombeau generalized $k$-tensor $(3c9_{x,\epsilon})_{\epsilon}$, in such a way that in any chart for $M$, the coefficients $(3c9_{i_1...i_k,\epsilon})_{\epsilon}$ are Colombeau generalized functions.

**Theorem 1.8.1.** (Generalized Stokes' Theorem) Let $(3c9_{\epsilon})_{\epsilon}$ be Colombeau generalized differential form. Then the Colombeau integral of a differential form $(3c9_{\epsilon})_{\epsilon}$ over the boundary of some orientable manifold $\Sigma \subset M$ is equal to the integral of its exterior Colombeau derivative $(d3c9_{\epsilon})_{\epsilon}$ over the whole of $\Sigma$, i.e.,
## 1.9 The Colombeau Generalized Curvature Tensor

In this subsection we derive the general formula for the change in a vector after parallel displacement around any infinitesimal closed contour $\gamma$. This generalized change $(\Delta A_{k,\varepsilon})_{\varepsilon} \in \mathbb{R}$ can clearly be written in the form $(\delta A_{k,\varepsilon})_{\varepsilon}$, where the Colombeau integral is taken over the given regular contour $\gamma$. Substituting in place of $(\delta A_{k,\varepsilon})_{\varepsilon}$ the expression (1.7.36), we get

$$ (\Delta A_{k,\varepsilon})_{\varepsilon} = (\gamma \Gamma_{kl,\varepsilon}^{i}(x) A_{i}(x) dx^{l})_{\varepsilon} \in \tilde{\mathbb{R}}, \qquad (1.8.1) $$

where for any $i, k, l = 0, 1, 2, 3$: $(\Gamma_{kl,\varepsilon}^{i}(x))_{\varepsilon} \in \mathcal{G}(\mathbb{R}^4)$, $x = (x^0, x^1, x^2, x^3)$, $A_i(x) \in \mathcal{D}(G)$, $G \subseteq \mathbb{R}^4$. Note that the vector $A_i$ which appears in the integrand obviously changes as we move along the contour $\gamma$.

**Definition 1.8.1.** We will say that generalized change $(\Delta A_{k,\varepsilon})_{\varepsilon}$ exists in the sense of the Schwartz distributions if for any $A_i(x) \in \mathcal{D}(G)$ the limit: $\lim_{\varepsilon \to 0} \Delta A_{k,\varepsilon}$ exists. Of course in this case obviously $\mathrm{cl}[(\Gamma_{kl,\varepsilon}^{i}(x))_{\varepsilon}] \in \mathcal{D}'(G)$ and $\mathrm{cl}[(\Delta A_{k,\varepsilon})_{\varepsilon}] \in \mathbb{R}$.

For the further transformation of this Colombeau integral, we must note the following. The values of the vector $A_i$ at points inside the contour are not unique; they depend on the path along which we approach the particular point. However, as we shall see from the result obtained below, this non-uniqueness is related to terms of second order. We may therefore, with the first-order accuracy which is sufficient for the transformation, regard the components of the vector $(A_{i,\varepsilon})_{\varepsilon}$ at points inside the infinitesimal contour $\gamma$ as being uniquely determined by their values on the contour itself by the formulas

$$ (\delta A_i(x))_{\varepsilon} = (\Gamma_{il,\varepsilon}^n(x) A_{n,\varepsilon}(x) dx^l)_{\varepsilon}, \qquad (1.8.2) $$

i.e., by the derivatives

$$ \frac{\partial A_{i,\varepsilon}(x)}{\partial x^l} = (\Gamma_{il,\varepsilon}^n(x) A_{n,\varepsilon}(x))_{\varepsilon}. \qquad (1.8.3) $$

Now applying generalized Stokes' theorem (see Theorem 1.7.1) to the integral (1.8.1) and considering that the area enclosed by the contour has the infinitesimal value $(\Delta f_{\varepsilon}^{im})_{\varepsilon}$, we get:

$$ \begin{aligned} (\Delta A_{k,\varepsilon})_{\varepsilon} &= \frac{1}{2} \left[ \left( \frac{\partial (\Gamma_{km,\varepsilon}^{i}(x) A_i(x))}{\partial x^l} \right)_{\varepsilon} - \left( \frac{\partial (\Gamma_{kl,\varepsilon}^{i}(x) A_i(x))}{\partial x^m} \right)_{\varepsilon} \right] (\Delta f_{\varepsilon}^{im})_{\varepsilon} = \\ &= \frac{1}{2} \left[ A_i(x) \left( \frac{\partial (\Gamma_{km,\varepsilon}^{i}(x))}{\partial x^l} \right)_{\varepsilon} - A_i(x) \left( \frac{\partial (\Gamma_{kl,\varepsilon}^{i}(x))}{\partial x^m} \right)_{\varepsilon} + \right. \\ & \qquad \left. \left( \frac{\partial A_i(x)}{\partial x^l} \right)_{\varepsilon} (\Gamma_{km,\varepsilon}^{i}(x))_{\varepsilon} - \left( \frac{\partial}{\partial x^m} \right)_{\varepsilon} (\Gamma_{kl,\varepsilon}^{i}(x))_{\varepsilon} \right] (\Delta f_{\varepsilon}^{im})_{\varepsilon}. \end{aligned} \qquad (1.8.4) $$

**Definition 1.8.2.** Colombeau generalized $k$-form $(3c9_{\varepsilon})_{\varepsilon}$ on a differentiable manifold $M$ is a smooth section of the bundle of alternating Colombeau generalized $k$-tensors on $M$.

Equivalently, $(3c9_{\varepsilon})_{\varepsilon}$ associates to each $x \in M$ an alternating Colombeau generalized $k$-tensor $(3c9_{x,\varepsilon})_{\varepsilon}$, in such a way that in any chart for $M$, the coefficients $(3c9_{i_1...i_k,\varepsilon})_{\varepsilon}$ are Colombeau generalized functions.

**Theorem 1.8.1.** (Generalized Stokes' Theorem) Let $(3c9_{\varepsilon})_{\varepsilon}$ be Colombeau generalized differential form. Then the Colombeau integral of a differential form $(3c9_{\varepsilon})_{\varepsilon}$ over the boundary of some orientable manifold $\Sigma \subset M$ is equal to the integral of its exterior Colombeau derivative $(d3c9_{\varepsilon})_{\varepsilon}$ over the whole of $\Sigma$, i.e.,
$$ \partial\Sigma (3c9_\epsilon)_\epsilon = (\partial\Sigma 3c9_\epsilon)_\epsilon = (\Sigma d3c9_\epsilon)_\epsilon = \Sigma (d3c9_\epsilon)_\epsilon. \qquad (1.8.5) $$

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

**Example 1.8.1.** For example, for the integral of Colombeau generalized vector $(A_{i,\epsilon}(x))_\epsilon$

we have

$$ (\Gamma A_{i,\epsilon} dx^i)_\epsilon = \left( \Sigma df^{ki} \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_\epsilon = \frac{1}{2} \left( \left[ (df^k)_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} - \frac{\partial A_{i,\epsilon}}{\partial x^k} \right) \right)_\epsilon = \\ \frac{1}{2} \left[ (df^k)_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} - \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_\epsilon = \frac{1}{2} \left[ (df^k)_{\epsilon} \right] \left[ \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} \right)_\epsilon - \left( \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_\epsilon \right], \qquad (1.8.6) $$

where $\Gamma = \partial\Sigma$ and $(df^k)_{\epsilon} = (dx^i_\epsilon dx^k_\epsilon)_{\epsilon} - (dx^k_\epsilon dx^i_\epsilon)_{\epsilon}$ is the infinitesimal element of surface which is given by the antisymmetric tensor of second rank $(df^k)_{\epsilon}$.

Substituting the values of the derivatives (1.4.3) into Eq.(1.4.4), we get

$$ (\Delta A_{k,\epsilon})_{\epsilon} = \frac{1}{2} (R_{klm,\epsilon}^{i}(x) A_{i,\epsilon}(x) \Delta f_{\epsilon}^{im})_{\epsilon}, \qquad (1.8.7) $$

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

$$ (R_{klm,\epsilon}^{i}(x))_{\epsilon} = \left( \frac{\partial (\Gamma_{km,\epsilon}^{i}(x))}{\partial x^l} \right)_{\epsilon} - \left( \frac{\partial (\Gamma_{kl,\epsilon}^{i}(x))}{\partial x^m} \right)_{\epsilon} + \\ (\Gamma_{ni,\epsilon}^{i}(x) \Gamma_{km,\epsilon}^{n}(x))_{\epsilon} - (\Gamma_{nm,\epsilon}^{i}(x) \Gamma_{kl,\epsilon}^{n}(x))_{\epsilon}. \qquad (1.8.8) $$

**Definition 1.8.3.** The tensor field $(R_{kim,\epsilon}^{i}(x))_{\epsilon}$ is called the distributional curvature tensor or the distributional Riemann tensor.

**Remark 1.8.1.** Note that in general case for any $i, k, l = 0, 1, 2, 3$: $cl [(R_{klm,\epsilon}^{i}(x))_{\epsilon}] \in \mathcal{G}(\mathbb{R}^4)$.

**Definition 1.8.4.** We will say that the distributional Riemann tensor $(R_{klm,\epsilon}^{i}(x))_{\epsilon}$ exists in the sense of the Schwartz distributions if for any $i, k, l = 0, 1, 2, 3$ and for any $A_i(x) \in \mathcal{D}(\mathcal{G})$ the limit:

$$ \lim_{\epsilon \to 0} G R_{klm,\epsilon}^{i}(x) A_{i}(x) d^{4}x \qquad (1.8.9) \quad \text{exists.} $$

**Definition 1.8.5.** We will say that the distributional Riemann tensor $(R_{klm,\epsilon}^{i}(x))_{\epsilon}$ exists in the classical sense at point $\tilde{x} \in \mathbb{R}$ if there exists standard part of point value of Colombeau generalized function $(R_{klm,\epsilon}^{i}(x))_{\epsilon}$ at point $\tilde{x} \in \mathbb{R}$, i.e. $\text{st}(cl [(R_{klm,\epsilon}^{i}(\tilde{x}))_{\epsilon}]) \in \mathbb{R}$.

From the expression (1.8.8) it follows directly that $\forall x \in \mathbb{R}$ the distributional curvature tensor is antisymmetric in the indices $l$ and $m$:

$$ (R_{klm,\epsilon}^{i}(x))_{\epsilon} = -(R_{kml,\epsilon}^{i}(x))_{\epsilon} \qquad (1.8.10) $$

and therefore $\forall z = (x_\epsilon)_{\epsilon} \in \tilde{\mathbb{R}}$ the following identity holds

$$ (R_{klm,\epsilon}^{i}(x_{\epsilon}))_{\epsilon} = -(R_{kml,\epsilon}^{i}(x_{\epsilon}))_{\epsilon}. \qquad (1.8.11) $$

## 1.10 Generalized Einstein's field equations

The action functional for the gravitational field reads

$$ (R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon}. \qquad (1.9.1) $$
$$ \partial\Sigma (3c9_\epsilon)_\epsilon = (\partial\Sigma 3c9_\epsilon)_\epsilon = (\Sigma d3c9_\epsilon)_\epsilon = \Sigma (d3c9_\epsilon)_\epsilon. \qquad (1.8.5) $$

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

**Example 1.8.1.** For example, for the integral of Colombeau generalized vector $(A_{i,\epsilon}(x))_\epsilon$

we have

$$ (\Gamma A_{i,\epsilon} dx^i)_\epsilon = \left(\Sigma df^{ki} \frac{\partial A_{i,\epsilon}}{\partial x^k}\right)_\epsilon = \frac{1}{2} \left( \left[ (df^k)_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} - \frac{\partial A_{i,\epsilon}}{\partial x^k} \right) \right)_\epsilon = \\ \frac{1}{2} \left[ (df^k)_{\epsilon} \right] \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} - \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_\epsilon = \frac{1}{2} \left[ (df^k)_{\epsilon} \right] \left[ \left( \frac{\partial A_{k,\epsilon}}{\partial x^i} \right)_\epsilon - \left( \frac{\partial A_{i,\epsilon}}{\partial x^k} \right)_\epsilon \right], \qquad (1.8.6) $$

where $\Gamma = \partial\Sigma$ and $(df^k)_{\epsilon} = (dx^i_\epsilon dx^k_\epsilon)_{\epsilon} - (dx^k_\epsilon dx^i_\epsilon)_{\epsilon}$ is the infinitesimal element of surface which is given by the antisymmetric tensor of second rank $(df^k)_{\epsilon}$.

Substituting the values of the derivatives (1.4.3) into Eq.(1.4.4), we get

$$ (\Delta A_{k,\epsilon})_{\epsilon} = \frac{1}{2} (R_{klm,\epsilon}^{i}(x) A_{i,\epsilon}(x) \Delta f_{\epsilon}^{im})_{\epsilon}, \qquad (1.8.7) $$

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

$$ (R_{klm,\epsilon}^{i}(x))_{\epsilon} = \left( \frac{\partial (\Gamma_{km,\epsilon}^{i}(x))}{\partial x^l} \right)_{\epsilon} - \left( \frac{\partial (\Gamma_{kl,\epsilon}^{i}(x))}{\partial x^m} \right)_{\epsilon} + (\Gamma_{ni,\epsilon}^{i}(x) \Gamma_{km,\epsilon}^{n}(x))_{\epsilon} - (\Gamma_{nm,\epsilon}^{i}(x) \Gamma_{kl,\epsilon}^{n}(x))_{\epsilon}. \qquad (1.8.8) $$

**Definition 1.8.3.** The tensor field $(R_{kim,\epsilon}^{i}(x))_{\epsilon}$ is called the distributional curvature tensor or the distributional Riemann tensor.

**Remark 1.8.1.** Note that in general case for any $i, k, l = 0, 1, 2, 3$: $cl[(R_{klm,\epsilon}^{i}(x))_{\epsilon}] \in \mathcal{G}(\mathbb{R}^4)$.

**Definition 1.8.4.** We will say that the distributional Riemann tensor $(R_{klm,\epsilon}^{i}(x))_{\epsilon}$ exists in the sense of the Schwartz distributions if for any $i, k, l = 0, 1, 2, 3$ and for any $A_i(x) \in \mathcal{D}(\mathcal{G})$ the limit:

$$ \lim_{\epsilon \to 0} G R_{klm,\epsilon}^{i}(x) A_{i}(x) d^{4}x \qquad (1.8.9) \quad \text{exists.} $$

**Definition 1.8.5.** We will say that the distributional Riemann tensor $(R_{klm,\epsilon}^{i}(x))_{\epsilon}$ exists in the classical sense at point $\tilde{x} \in \mathbb{R}$ if there exists standard part of point value of Colombeau generalized function $(R_{klm,\epsilon}^{i}(x))_{\epsilon}$ at point $\tilde{x} \in \mathbb{R}$, i.e. $\text{st}(cl[(R_{klm,\epsilon}^{i}(\tilde{x}))_{\epsilon}]) \in \mathbb{R}$.

From the expression (1.8.8) it follows directly that $\forall x \in \mathbb{R}$ the distributional curvature tensor is antisymmetric in the indices $l$ and $m$:

$$ (R_{klm,\epsilon}^{i}(x))_{\epsilon} = -(R_{kml,\epsilon}^{i}(x))_{\epsilon} \qquad (1.8.10) $$

and therefore $\forall z = (x_\epsilon)_{\epsilon} \in \tilde{\mathbb{R}}$ the following identity holds

$$ (R_{klm,\epsilon}^{i}(x_{\epsilon}))_{\epsilon} = -(R_{kml,\epsilon}^{i}(x_{\epsilon}))_{\epsilon}. \qquad (1.8.11) $$

## 1.10 Generalized Einstein's field equations

The action functional for the gravitational field reads

$$ (R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon}. \qquad (1.9.1) $$
The invariant Colombeau integral (1.9.1) can be transformed by means of Gauss’ theorem to the integral of an expression not containing the second derivatives. Thus Colombeau integral (2.8.2) can be presented in the following form

$$ (R_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = (G_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} + \left( \frac{\partial (\sqrt{-g_{\varepsilon}} w_{\varepsilon}^{i})}{\partial x^{i}} \right)_{\varepsilon} d\Omega, \qquad (1.9.2) $$

where $(G_{\varepsilon})_{\varepsilon}$ contains only the tensor $(g_{ik,\varepsilon})_{\varepsilon}$ and its first derivatives, and the integrand of the second integral has the form of a divergence of a certain quantity $(w_{\varepsilon}^{i})_{\varepsilon}$. According to Gauss’ theorem, this second integral can be transformed into an integral over a hypersurface surrounding the four-volume over which the integration is carried out in the other two integrals. When we vary the action, the variation of the second term on the right vanishes, since in the principle of least action, the variations of the field at the limits of the region of integration are zero. Consequently, we may write

$$ \delta (R_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = (\delta R_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = (\delta G_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon}. \qquad (1.9.3) $$

The left side is Colombeau scalar; therefore the expression on the right is also Colombeau scalar (the quantity $(G_{\varepsilon})_{\varepsilon}$ itself is, of course, not Colombeau scalar). The quantity $(G_{\varepsilon})_{\varepsilon}$ satisfies the condition imposed above, since it contains only the $(g_{ik,\varepsilon})_{\varepsilon}$ and its Colombeau derivatives. Thus finally we obtain

$$ \delta \mathbf{S} [(g_{\varepsilon})_{\varepsilon}] = -\frac{c^3}{16\pi\kappa} (\delta G_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = -\frac{c^3}{16\pi k} (\delta R_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon}. \qquad (1.9.4) $$

The constant $\kappa$ is called the gravitational constant. The dimensions of $\kappa$ follow from (1.9.4). Its numerical value is $\kappa = 6.67 \times 10^{-8} \text{sm}^3 \times \text{gr}^{-1} \times \text{sec}^{-2}$.

We now proceed to the derivation of the equations of the gravitational field. These equations are obtained from the principle of least action $\delta((S_{m,\varepsilon})_{\varepsilon} + (S_{g_{\varepsilon}})_{\varepsilon}) = 0_{\tilde{\mathbb{R}}}$, where $(S_{m,\varepsilon})_{\varepsilon}$ and $(S_{g_{\varepsilon}})_{\varepsilon}$ are the distributional actions of the gravitational field and matter respectively. We now subject the gravitational Colombeau metric field, that is, the quantities $g_{ik}$, to variation. Calculating the variation $\delta (S_{g_{\varepsilon}})_{\varepsilon}$, we get

$$ \begin{aligned} \delta (R_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = (\delta R_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = (\delta g_{\varepsilon}^{ik} R_{ik,\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} = & \\ \{ (R_{ik,\varepsilon} \sqrt{-g_{\varepsilon}} \delta g_{\varepsilon}^{ik} d\Omega)_{\varepsilon} + (R_{ik,\varepsilon} g_{\varepsilon}^{ik} \delta \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} + (g_{\varepsilon}^{ik} \sqrt{-g_{\varepsilon}} \delta R_{ik,\varepsilon} d\Omega)_{\varepsilon} \} & (1.9.5) \\ \{ (R_{ik,\varepsilon} \sqrt{-g_{\varepsilon}} \delta g_{\varepsilon}^{ik} )_{\varepsilon} + (R_{ik,\varepsilon} g_{\varepsilon}^{ik} \delta \sqrt{-g_{\varepsilon}} )_{\varepsilon} + (g_{\varepsilon}^{ik} \sqrt{-g_{\varepsilon}} \delta R_{ik,\varepsilon} )_{\varepsilon} \} d\Omega. & \end{aligned} $$

Thus, the variation $\mathbf{S} [(g_{\varepsilon})_{\varepsilon}]$ is equal to

$$ \mathbf{S} [(g_{\varepsilon})_{\varepsilon}] = -\frac{c^3}{16\pi\kappa} \left\{ \left\{ R_{ik,\varepsilon} - \frac{1}{2} g_{ik,\varepsilon} R_{\varepsilon} \right\} \sqrt{-g_{\varepsilon}} \delta g_{\varepsilon}^{ik} d\Omega \right\}_{\varepsilon}. \qquad (1.9.10) $$

**Remark 1.9.1.** We note that if we had started from the expression

$$ \delta \mathbf{S}_g [(g_{\varepsilon})_{\varepsilon}] = -\frac{c^3}{16\pi\kappa} (\delta G_{\varepsilon} \sqrt{-g_{\varepsilon}} d\Omega)_{\varepsilon} \qquad (1.9.11) $$

for the action of the field, then we get

$$ \delta \mathbf{S} [(g_{\varepsilon})_{\varepsilon}] = -\frac{c^3}{16\pi\kappa} \delta (g_{\varepsilon}^{ik})_{\varepsilon} d\Omega \left\{ \left( \frac{\partial \{G_{\varepsilon} \sqrt{-g_{\varepsilon}}\}}{\partial g_{\varepsilon}^{ik}} \right)_{\varepsilon} - \left( \frac{\partial}{\partial x^l} \frac{\partial \{G_{\varepsilon} \sqrt{-g_{\varepsilon}}\}}{\partial \frac{\partial g_{\varepsilon}^{ik}}{\partial x^l}} \right)_{\varepsilon} \right\}. \qquad (1.9.12) $$
The invariant Colombeau integral (1.9.1) can be transformed by means of Gauss’ theorem to the integral of an expression not containing the second derivatives. Thus Colombeau integral (2.8.2) can be presented in the following form

$$ (R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = (G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} + \left( \frac{\partial (\sqrt{-g_{\epsilon}} w_{\epsilon}^{i})}{\partial x^{i}} \right)_{\epsilon} d\Omega, \qquad (1.9.2) $$

where $(G_{\epsilon})_{\epsilon}$ contains only the tensor $(g_{ik,\epsilon})_{\epsilon}$ and its first derivatives, and the integrand of the second integral has the form of a divergence of a certain quantity $(w_{\epsilon}^{i})_{\epsilon}$. According to Gauss’ theorem, this second integral can be transformed into an integral over a hypersurface surrounding the four-volume over which the integration is carried out in the other two integrals. When we vary the action, the variation of the second term on the right vanishes, since in the principle of least action, the variations of the field at the limits of the region of integration are zero. Consequently, we may write

$$ \delta (R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = (\delta R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = (\delta G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon}. \qquad (1.9.3) $$

The left side is Colombeau scalar; therefore the expression on the right is also Colombeau scalar (the quantity $(G_{\epsilon})_{\epsilon}$ itself is, of course, not Colombeau scalar). The quantity $(G_{\epsilon})_{\epsilon}$ satisfies the condition imposed above, since it contains only the $(g_{ik,\epsilon})_{\epsilon}$ and its Colombeau derivatives. Thus finally we obtain

$$ \delta \mathbf{S} [(g_{\epsilon})_{\epsilon}] = - \frac{c^3}{16\pi\kappa} (\delta G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = - \frac{c^3}{16\pi k} (\delta R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon}. \qquad (1.9.4) $$

The constant $\kappa$ is called the gravitational constant. The dimensions of $\kappa$ follow from (1.9.4). Its numerical value is $\kappa = 6.67 \times 10^{-8} \text{sm}^3 \times \text{gr}^{-1} \times \text{sec}^{-2}$.

We now proceed to the derivation of the equations of the gravitational field. These equations are obtained from the principle of least action $\delta((S_{m,\epsilon})_{\epsilon} + (S_{g_{\epsilon}})_{\epsilon}) = 0_{\tilde{\mathbb{R}}}$, where $(S_{m,\epsilon})_{\epsilon}$ and $(S_{g_{\epsilon}})_{\epsilon}$ are the distributional actions of the gravitational field and matter respectively. We now subject the gravitational Colombeau metric field, that is, the quantities $g_{ik}$, to variation. Calculating the variation $\delta (S_{g_{\epsilon}})_{\epsilon}$, we get

$$ \begin{aligned} \delta (R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = (\delta R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = (\delta g_{\epsilon}^{ik} R_{ik,\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} = & \\ \{ (R_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega)_{\epsilon} + (R_{ik,\epsilon} g_{\epsilon}^{ik} \delta \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} + (g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} d\Omega)_{\epsilon} \} & (1.9.5) \\ \{ (R_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} )_{\epsilon} + (R_{ik,\epsilon} g_{\epsilon}^{ik} \delta \sqrt{-g_{\epsilon}} )_{\epsilon} + (g_{\epsilon}^{ik} \sqrt{-g_{\epsilon}} \delta R_{ik,\epsilon} )_{\epsilon} \} d\Omega. & \end{aligned} $$

Thus, the variation $\mathbf{S}[(g_{\epsilon})_{\epsilon}]$ is equal to

$$ \mathbf{S}[(g_{\epsilon})_{\epsilon}] = -\frac{c^3}{16\pi\kappa} \left\{ \left\{ R_{ik,\epsilon} - \frac{1}{2} g_{ik,\epsilon} R_{\epsilon} \right\} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega \right\}_{\epsilon}. \qquad (1.9.10) $$

**Remark 1.9.1.** We note that if we had started from the expression

$$ \delta \mathbf{S}_g [(g_{\epsilon})_{\epsilon}] = - \frac{c^3}{16\pi\kappa} (\delta G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega)_{\epsilon} \qquad (1.9.11) $$

for the action of the field, then we get

$$ \delta \mathbf{S} [(g_{\epsilon})_{\epsilon}] = -\frac{c^3}{16\pi\kappa} \delta (g_{\epsilon}^{ik})_{\epsilon} d\Omega \left\{ \left( \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial g_{\epsilon}^{ik}} \right)_{\epsilon} - \left( \frac{\partial}{\partial x^l} \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial \frac{\partial g_{\epsilon}^{ik}}{\partial x^l}} \right)_{\epsilon} \right\}. \qquad (1.9.12) $$
Comparing Eq.(1.9.12) with Eq.(1.9.10), we get

$$ (R_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} R_{\epsilon})_{\epsilon} = \left\{ \left( \frac{1}{\sqrt{-g_{\epsilon}}} \right)_{\epsilon} \right\} \left\{ \left( \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial g_{\epsilon}^{ik}} \right)_{\epsilon} - \left( \frac{\partial}{\partial x^l} \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial \frac{\partial g_{\epsilon}^{ik}}{\partial x^l}} \right)_{\epsilon} \right\}. \quad (1.9.13) $$

For the variation of the action of the matter we can write

$$ (\delta \mathbf{S}_{m,\epsilon})_{\epsilon} = \frac{1}{2c} (T_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega)_{\epsilon}, \quad (1.9.14) $$

where $(T_{ik,\epsilon})_{\epsilon} \in \mathcal{G}(\mathbb{R}^4)$ is the generalized energy-momentum tensor of the matter fields.

Thus, from the principle of least action

$$ \delta \{ \mathbf{S} [(g_{\epsilon})_{\epsilon}] + (\mathbf{S}_{m,\epsilon})_{\epsilon} \} = 0_{\mathbb{R}} \quad (1.9.15) $$

one obtains

$$ -\frac{c^3}{16\pi\kappa} \left( \left\{ R_{ik,\epsilon} - \frac{1}{2}g_{ik,\epsilon}R_{\epsilon} - \frac{8\pi\kappa}{c^4}T_{ik,\epsilon} \right\} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega \right)_{\epsilon} \quad (1.9.16) $$

From Eq.(1.9.16), since of the arbitrariness of the $(\delta g_{\epsilon}^{ik})_{\epsilon} \in \mathcal{G}(\mathbb{R}^4)$ finally we get

$$ (R_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} R_{\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} (T_{ik,\epsilon})_{\epsilon} \quad (1.9.17) $$

or, in mixed components,

$$ (R_{i,\epsilon}^k)_{\epsilon} - \frac{1}{2} \delta_i^k (R_{\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} (T_{i,\epsilon}^k)_{\epsilon}. \quad (1.9.18) $$

They are called the generalized Einstein equations.

Contracting (1.9.18) on the indices i and k, we get

$$ (R_{\epsilon})_{\epsilon} = -\frac{8\pi\kappa}{c^4} (T_{i,\epsilon}^i)_{\epsilon} = -\frac{8\pi\kappa}{c^4} (T_{\epsilon})_{\epsilon}. \quad (1.9.19) $$

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

$$ (R_{ik,\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} \left\{ (T_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} T_{\epsilon})_{\epsilon} \right\}. \quad (1.9.20) $$

Note that the generalized Einstein equations of the gravitational field are nonlinear Colombeau equations.
Comparing Eq.(1.9.12) with Eq.(1.9.10), we get

$$ (R_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} R_{\epsilon})_{\epsilon} = \left\{ \left( \frac{1}{\sqrt{-g_{\epsilon}}} \right)_{\epsilon} \right\} \left\{ \left( \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial g_{\epsilon}^{ik}} \right)_{\epsilon} - \left( \frac{\partial}{\partial x^l} \frac{\partial \{G_{\epsilon} \sqrt{-g_{\epsilon}}\}}{\partial \frac{\partial g_{\epsilon}^{ik}}{\partial x^l}} \right)_{\epsilon} \right\}. \quad (1.9.13) $$

For the variation of the action of the matter we can write

$$ (\delta \mathbf{S}_{m,\epsilon})_{\epsilon} = \frac{1}{2c} (T_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega)_{\epsilon}, \quad (1.9.14) $$

where $(T_{ik,\epsilon})_{\epsilon} \in \mathcal{G}(\mathbb{R}^4)$ is the generalized energy-momentum tensor of the matter fields.

Thus, from the principle of least action

$$ \delta \{ \mathbf{S} [(g_{\epsilon})_{\epsilon}] + (\mathbf{S}_{m,\epsilon})_{\epsilon} \} = 0_{\mathbb{R}} \quad (1.9.15) $$

one obtains

$$ -\frac{c^3}{16\pi\kappa} \left( \left\{ R_{ik,\epsilon} - \frac{1}{2}g_{ik,\epsilon}R_{\epsilon} - \frac{8\pi\kappa}{c^4}T_{ik,\epsilon} \right\} \sqrt{-g_{\epsilon}} \delta g_{\epsilon}^{ik} d\Omega \right)_{\epsilon} \quad (1.9.16) $$

From Eq.(1.9.16), since of the arbitrariness of the $(\delta g_{\epsilon}^{ik})_{\epsilon} \in \mathcal{G}(\mathbb{R}^4)$ finally we get

$$ (R_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} R_{\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} (T_{ik,\epsilon})_{\epsilon} \quad (1.9.17) $$

or, in mixed components,

$$ (R_{i,\epsilon}^k)_{\epsilon} - \frac{1}{2} \delta_i^k (R_{\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} (T_{i,\epsilon}^k)_{\epsilon}. \quad (1.9.18) $$

They are called the generalized Einstein equations.

Contracting (1.9.18) on the indices i and k, we get

$$ (R_{\epsilon})_{\epsilon} = -\frac{8\pi\kappa}{c^4} (T_{i,\epsilon}^i)_{\epsilon} = -\frac{8\pi\kappa}{c^4} (T_{\epsilon})_{\epsilon}. \quad (1.9.19) $$

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

$$ (R_{ik,\epsilon})_{\epsilon} = \frac{8\pi\kappa}{c^4} \left\{ (T_{ik,\epsilon})_{\epsilon} - \frac{1}{2} (g_{ik,\epsilon} T_{\epsilon})_{\epsilon} \right\}. \quad (1.9.20) $$

Note that the generalized Einstein equations of the gravitational field are nonlinear Colombeau equations.
# 1.11 The breakdown of canonical formalism of Riemann geometry for the singular solutions of the Einstein field equations

## 1.11.1 The densitized Einstein field equations revisited

The densitized Einstein field equations originally considered in A. Einstein and N. Rosen paper [32], see also [46].

**Remark 1.10.1.** Note that if some components of the Riemann curvature tensor $R_{klm}^i(\hat{x})$ become infinite at point $\hat{x}^0$ one obtain the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood $\Omega$ of the point $\hat{x}^0 \in \Omega$, i.e. in such neighborhood $\Omega$ Riemann curvature tensor $R_{klm}^i(\hat{x})$ will be changed by formula (1.10.7) see remark 1.10.2.

**Remark 1.10.2.** Let $\Gamma$ be infinitesimal closed contour and let $\Sigma_{\Gamma}$ be the corresponding surface spanning by $\Gamma$, see Pic.1. We assume now that: (i) christoffel symbol $\Gamma_{kl}^i(\hat{x})$ become infinite at singular point $\hat{x}^0$ by formulae

$$ \left\{ \begin{array}{l} \Gamma_{kl}^i(\hat{x}) \approx \Xi_{kl}(\hat{x}) (x_i - x_i^0)^{-\delta}, \delta \ge 1 \\ \Xi_{kl}(\hat{x}) \in C^{\infty}(\Sigma_{\Gamma}) \end{array} \right. \qquad (1.10.1) $$

and (ii) $\hat{x}^0 \in \Sigma_{\Gamma}$. Let us derive now to similarly canonical calculation [3]-[4] the general formula for the regularized change $\widetilde{\Delta A}_k$ in a vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$. This regularized change $\widetilde{\Delta A}_k$ can clearly be written in the form

$$ \widetilde{\Delta A}_k = \Gamma \Phi (\hat{x} - \hat{x}^0) \delta A_k, \qquad (1.10.2) $$

where $\Phi(\hat{x} - \hat{x}^0) = \int_{i=0}^4 (x_i - x_i^0)^{2\delta}, \delta \ge 1$ and where the integral is taken over the given contour $\Gamma$. Substituting in place of $\delta A_k$ the canonical expression $\delta A_k = \Gamma_{kl}^i(\hat{x}) A_k dx^l$ (see [4], Eq. (85.5)) we obtain

$$ \widetilde{\Delta A}_k = \Gamma \Phi (\hat{x} - \hat{x}^0) \delta A_k = \Gamma \Phi (\hat{x} - \hat{x}^0) \Gamma_{kl}^i(\hat{x}) A_k dx^l, \qquad (1.10.3) $$

where

$$ \frac{\partial A_i}{\partial x^l} = \Gamma_{kl}^i(\hat{x}) A_k. \qquad (1.10.4) $$

Pic.1. Infinitesimal closed contour $\Gamma$ and corresponding singular surface $\Sigma_{\Gamma} \ni \hat{x}^0$ spanning by $\Gamma$. Due to the degeneracy of the metric (1.10.12)

at point $r=0$,
the Levi-Civita connection $\Gamma_{kj}^{+l}(\{\}) = \frac{1}{2}[g^{lm}(\{\})] [(g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})]
is not available on $\mathbb{R}_+^3 \cup \{0\}$ in canonical sense
but only in an distributional sense.

5b5
Pic.2. Infinitesimal closed contour $\Gamma$ with a singularity at point $\hat{x}^0$ on Horizon and corresponding singular surface $\Sigma_{\Gamma} \ni \hat{x}^0$ spanning by $\Gamma$.

Due to the degeneracy of the metric (1.10.12) at $r = 2m$, the Levi-Civita connection $\Gamma_{kj}^{+l}(\{\}) = \frac{1}{2}[g^{lm}(\{\})] [(g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})]$ is not available on horizon in canonical sense but only in an distributional sense.

Now applying Stokes' theorem (see [4], Eq. (6.19)) to the integral (1.3) and considering that the area
## 1.11 The breakdown of canonical formalism of Riemann geometry for the singular solutions of the Einstein field equations

### 1.11.1 The densitized Einstein field equations revisited

The densitized Einstein field equations originally considered in A. Einstein and N. Rosen paper [32], see also [46].

**Remark 1.10.1.** Note that if some components of the Riemann curvature tensor $R_{klm}^i(\hat{x})$ become infinite at point $\hat{x}^0$ one obtain the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood $\Omega$ of the point $\hat{x}^0 \in \Omega$, i.e. in such neighborhood $\Omega$ Riemann curvature tensor $R_{klm}^i(\hat{x})$ will be changed by formula (1.10.7) see remark 1.10.2.

**Remark 1.10.2.** Let $\Gamma$ be infinitesimal closed contour and let $\Sigma_{\Gamma}$ be the corresponding surface spanning by $\Gamma$, see Pic.1. We assume now that: (i) christoffel symbol $\Gamma_{kl}^i(\hat{x})$ become infinite at singular point $\hat{x}^0$ by formulae

$$ \left\{ \begin{array}{l} \Gamma_{kl}^i(\hat{x}) \approx \Xi_{kl}(\hat{x}) (x_i - x_i^0)^{-\delta}, \delta \ge 1 \\ \Xi_{kl}(\hat{x}) \in C^{\infty}(\Sigma_{\Gamma}) \end{array} \right. \qquad (1.10.1) $$

and (ii) $\hat{x}^0 \in \Sigma_{\Gamma}$. Let us derive now to similarly canonical calculation [3]-[4] the general formula for the regularized change $\widetilde{\Delta A}_k$ in a vector $A_i(\hat{x})$ after parallel displacement around infinitesimal closed contour $\Gamma$. This regularized change $\widetilde{\Delta A}_k$ can clearly be written in the form

$$ \widetilde{\Delta A}_k = \Gamma \Phi (\hat{x} - \hat{x}^0) \delta A_k, \qquad (1.10.2) $$

where $\Phi(\hat{x} - \hat{x}^0) = \delta_{i=0}^4 (x_i - x_i^0)^{2\delta}$, $\delta \ge 1$ and where the integral is taken over the given contour $\Gamma$. Substituting in place of $\delta A_k$ the canonical expression $\delta A_k = \Gamma_{kl}^i(\hat{x}) A_k dx^l$ (see [4], Eq. (85.5)) we obtain

$$ \widetilde{\Delta A}_k = \Gamma \Phi (\hat{x} - \hat{x}^0) \delta A_k = \Gamma \Phi (\hat{x} - \hat{x}^0) \Gamma_{kl}^i(\hat{x}) A_k dx^l, \qquad (1.10.3) $$

where

$$ \frac{\partial A_i}{\partial x^l} = \Gamma_{kl}^i(\hat{x}) A_k. \qquad (1.10.4) $$

Pic.1. Infinitesimal closed contour $\Gamma$ and corresponding singular surface $\Sigma_{\Gamma} \ni \hat{x}^0$ spanning by $\Gamma$. Due to the degeneracy of the metric (1.10.12)

at point $r=0$, the Levi-Civita connection $\Gamma_{kj}^{+l}(\{\}) = \frac{1}{2}[g^{lm}(\{\})] [(g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})]$ is not available on $\mathbb{R}_+^3 \cup \{0\}$ in canonical sense but only in an distributional sense.

5b5

Pic.2. Infinitesimal closed contour $\Gamma$ with a singularity at point $\hat{x}^0$ on Horizon and corresponding singular surface $\Sigma_{\Gamma} \ni \hat{x}^0$ spanning by $\Gamma$.

Due to the degeneracy of the metric (1.10.12) at $r=2m$, the Levi-Civita connection $\Gamma_{kj}^{+l}(\{\}) = \frac{1}{2}[g^{lm}(\{\})] [(g_{mk,j}(\{\}) + g_{mj,k}(\{\}) - g_{kj,m}(\{\})]$ is not available on horizon in canonical sense but only in an distributional sense.

Now applying Stokes' theorem (see [4], Eq. (6.19)) to the integral (1.3) and considering that the area
enclosed by the contour has the infinitesimal value $\Delta f^{lm}$, we get

$$ \begin{aligned} \widetilde{\Delta A_k} &= \Gamma^i \Phi (\hat{x} - \hat{x}^0) \Gamma^i_{kl} (\hat{x}) A_k dx^l = \\ &= \frac{1}{2} \Sigma_{\Gamma} \left[ \frac{\partial (\Gamma^i_{km} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^l} - \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^m} \right] df^{lm} \approx \\ &\approx \left[ \frac{\partial (\Gamma^i_{km} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^l} - \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^m} \right] \frac{\Delta f^{lm}}{2} = \\ &\quad \left[ \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{km} (\hat{x} - \hat{x}^0) A_i)}{\partial x^l} + (\Gamma^i_{km} (\hat{x}) A_i) \frac{\partial \Phi (\hat{x} - \hat{x}^0)}{\partial x^l} - \right. \\ &\quad \left. - \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i)}{\partial x^m} - (\Gamma^i_{kl} (\hat{x}) A_i) \frac{\partial \Phi (\hat{x} - \hat{x}^0)}{\partial x^m} \right] \frac{\Delta f^{lm}}{2} = \\ &\quad \left[ \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{km} (\hat{x}) A_i)}{\partial x^l} - \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i)}{\partial x^m} - \right. \\ &\quad \left. A_i (\hat{x}) \Phi (\hat{x} - \hat{x}^0) \frac{2\delta \Gamma^i_{km} (\hat{x})}{x_l - x_l^0} - A_i (\hat{x}) \Phi (\hat{x} - \hat{x}^0) \frac{2\delta \Gamma^i_{kl} (\hat{x})}{x_m - x_m^0} \right] \frac{\Delta f^{lm}}{2} \end{aligned} \quad (1.10.5) $$

Substituting the values of the derivatives (1.4) into Eq.(1.5), we get finally:

$$ \widetilde{\Delta A_k} = \tilde{R}^i_{klm} \frac{A_i(\hat{x}) \Phi(\hat{x} - \hat{x}^0) \Delta f^{lm}}{2}, \quad (1.10.6) $$

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

$$ \tilde{R}^i_{klm} = R^i_{klm} + 2\delta \left[ \frac{\Gamma^i_{km}(\hat{x})}{x_l - x_l^0} - \frac{\Gamma^i_{kl}(\hat{x})}{x_m - x_m^0} \right]. \quad (1.10.7) $$

Here $R^i_{klm}$ is the classical Riemann curvature tensor. That $\tilde{R}^i_{klm}$ is a tensor is clear from the fact that in (1.10.6) the left side is a vector—the difference $\widetilde{\Delta A}_k$ between the values of vectors at one and the same point. Note that a similar results was obtained by many authors [5]-[17] by using Colombeau nonlinear generalized functions [1]-[2].

**Definition1.10.1.** The tensor $\tilde{R}^i_{klm}$ is called the generalized curvature tensor or the generalized Riemann tensor.

**Definition1.10.2.** The generalized Ricci curvature tensor $\tilde{R}_{km}$ is defined as

$$ \tilde{R}_{km} = \tilde{R}^i_{kim}. \quad (1.10.8) $$

**Definition1.10.3.** The generalized Ricci scalar $\tilde{R}$ is defined as

$$ \tilde{R} = g^{km} \tilde{R}_{km}. \quad (1.10.9) $$

**Definition1.10.3.** The generalized Einstein tensor $\tilde{G}_{km}$ is defined as

$$ \tilde{G}_{km} = \tilde{R}_{km} - \frac{1}{2} g_{km} \tilde{R}. \quad (1.10.10) $$

**Remark 1.10.3. (I)** Note that the Schwarzschild spacetime is well defined only for $r > 2m$

The boundary of the manifold $\{r > 2m\}$ in $\mathbb{R}^3 \times \mathbb{R}$ is the submanifold $\{r = 2m\}$ of

$\mathbb{R}^3 \times \mathbb{R}$, diffeomorphic to a product $S^2 \times \mathbb{R}$. This submanifold is colled the event horizon, or simply the horizon [33],[34].
enclosed by the contour has the infinitesimal value $\Delta f^{lm}$, we get

$$ 
\begin{aligned}
\widetilde{\Delta A_k} &= \Gamma^i \Phi (\hat{x} - \hat{x}^0) \Gamma^i_{kl} (\hat{x}) A_k dx^l = \\
&= \frac{1}{2} \Sigma_{\Gamma} \left[ \frac{\partial (\Gamma^i_{km} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^l} - \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^m} \right] df^{lm} \approx \\
&\approx \left[ \frac{\partial (\Gamma^i_{km} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^l} - \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i \Phi (\hat{x} - \hat{x}^0))}{\partial x^m} \right] \frac{\Delta f^{lm}}{2} = \\
&\quad \left[ \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{km} (\hat{x} - \hat{x}^0) A_i)}{\partial x^l} + (\Gamma^i_{km} (\hat{x}) A_i) \frac{\partial \Phi (\hat{x} - \hat{x}^0)}{\partial x^l} - \right. \\
&\quad \left. - \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i)}{\partial x^m} - (\Gamma^i_{kl} (\hat{x}) A_i) \frac{\partial \Phi (\hat{x} - \hat{x}^0)}{\partial x^m} \right] \frac{\Delta f^{lm}}{2} = \\
&\quad \left[ \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{km} (\hat{x}) A_i)}{\partial x^l} - \Phi (\hat{x} - \hat{x}^0) \frac{\partial (\Gamma^i_{kl} (\hat{x}) A_i)}{\partial x^m} - \right. \\
&\quad \left. A_i (\hat{x}) \Phi (\hat{x} - \hat{x}^0) \frac{2\delta \Gamma^i_{km} (\hat{x})}{x_l - x_l^0} - A_i (\hat{x}) \Phi (\hat{x} - \hat{x}^0) \frac{2\delta \Gamma^i_{kl} (\hat{x})}{x_m - x_m^0} \right] \frac{\Delta f^{lm}}{2}.
\end{aligned} 
\quad (1.10.5) $$

Substituting the values of the derivatives (1.4) into Eq.(1.5), we get finally:

$$ \widetilde{\Delta A_k} = \tilde{R}_{klm}^i \frac{A_i(\hat{x}) \Phi(\hat{x} - \hat{x}^0) \Delta f^{lm}}{2}, \quad (1.10.6) $$

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

$$ \tilde{R}_{klm}^i = R_{klm}^i + 2\delta \left[ \frac{\Gamma_{km}^i(\hat{x})}{x_l - x_l^0} - \frac{\Gamma_{kl}^i(\hat{x})}{x_m - x_m^0} \right]. \quad (1.10.7) $$

Here $R_{klm}^i$ is the classical Riemann curvature tensor. That $\tilde{R}_{klm}^i$ is a tensor is clear from the fact that in (1.10.6) the left side is a vector—the difference $\widetilde{\Delta A}_k$ between the values of vectors at one and the same point. Note that a similar results was obtained by many authors [5]-[17] by using Colombeau nonlinear generalized functions [1]-[2].

**Definition1.10.1.** The tensor $\tilde{R}_{klm}^i$ is called the generalized curvature tensor or the generalized Riemann tensor.

**Definition1.10.2.** The generalized Ricci curvature tensor $\tilde{R}_{km}$ is defined as

$$ \tilde{R}_{km} = \tilde{R}_{kim}^i. \quad (1.10.8) $$

**Definition1.10.3.** The generalized Ricci scalar $\tilde{R}$ is defined as

$$ \tilde{R} = g^{km} \tilde{R}_{km}. \quad (1.10.9) $$

**Definition1.10.3.** The generalized Einstein tensor $\tilde{G}_{km}$ is defined as

$$ \tilde{G}_{km} = \tilde{R}_{km} - \frac{1}{2} g_{km} \tilde{R}. \quad (1.10.10) $$

**Remark 1.10.3.** (I) Note that the Schwarzschild spacetime is well defined only for $r > 2m$

The boundary of the manifold $\{r > 2m\}$ in $\mathbb{R}^3 \times \mathbb{R}$ is the submanifold $\{r = 2m\}$ of

$\mathbb{R}^3 \times \mathbb{R}$, diffeomorphic to a product $S^2 \times \mathbb{R}$. This submanifold is colled the event horizon, or simply the horizon [33],[34].
Thus obviously by consideration above (see Remark 1.1-Remark 1.10.2) this extension given by Eq.(1.10.15) has no any sense in respect of the canonical Riemannian geometry.

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

**Remark 1.10.4.** Note that in physical literature the spacetime singularity usually is defined as location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the classical scalar invariant curvatures of singular spacetime, which includes a measure of the density of matter.

**Remark 1.10.5.** In general relativity, many investigations have been derived with regard to singular exact vacuum solutions of the Einstein equation and the singularity structure of spacetime. Such solutions have been formally derived under condition $T_{\mu}^{\nu}(x) = 0$, where $T_{\mu}^{\nu}(x)$ represent the energy-momentum densities of the gravity source. This for example is the case for the well-known Schwarzschild solution, which is given by, in the Schwarzschild coordinates $(x^0, r, \theta, \phi)$,

$$ds^2 = -h(r)(dx^0)^2 + h^{-1}(r)(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], h(r) = 1 - \frac{r_s}{r}, \quad (1.10.12)$$

Where, $r_s$ is the Schwarzschild radius $r_s = 2GM/c^2$ with $G, M$ and $c$ being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. The metric (1.3.12) describe the gravitational field produced by a point-like particle located at $r = 0$.

**Remark 1.10.6.** Note that when we say, on the basis of the canonical expression of the curvature square

$$R^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu}(r) = \frac{12r_s^2}{r^6} \qquad (1.10.13)$$

formally obtained from the metric (1.3.12), that $r = 0$ is a singularity of the Schwarzschild spacetime, the source is considered to be point-like and this metric is regarded as *meaningful everywhere* in space-time.

**Remark 1.10.7.** From the metric (1.10.12), the calculation of the canonical Einstein tensor proceeds in a straightforward manner gives for $r \neq 0$

$$G_t^t(r) = G_r^r(r) = -\frac{h'(r)}{\hat{r}} - \frac{1+h(r)}{\hat{r}^2} \equiv 0, G_\theta^\theta(r) = G_\varphi^\varphi(r) = -\frac{h''(r)}{2} - \frac{h(r)}{\hat{r}^2} \equiv 0, \quad (1.10.14)$$

Where $h(r) = -1 + r_s/r$. Using Eq. (1.10.14) one formally obtains a boundary conditions

$$\left\{ \begin{array}{l} G_t^t(0) \triangleq \lim_{r \to 0} G_t^t(r) = 0, G_r^r(0) \triangleq \lim_{r \to 0} G_r^r(r) = 0, \\ G_\theta^\theta(0) \triangleq \lim_{r \to 0} G_\theta^\theta(r) = 0, G_\varphi^\varphi(0) \triangleq \lim_{r \to 0} G_\varphi^\varphi(r) = 0. \end{array} \right. \qquad (1.10.15)$$

However as pointed out above the canonical expression of the Einstein tensor in a sufficiently small neighborhood $\Omega$ of the point $r = 0$ and must be replaced by the generalized Einstein tensor $\tilde{G}_{km}$ (1.10.10). By simple calculation easy to see that
Thus obviously by consideration above (see Remark 1.1-Remark 1.10.2) this extension given by Eq.(1.10.15) has no any sense in respect of the canonical Riemannian geometry.

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

**Remark 1.10.4.** Note that in physical literature the spacetime singularity usually is defined as location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the classical scalar invariant curvatures of singular spacetime, which includes a measure of the density of matter.

**Remark 1.10.5.** In general relativity, many investigations have been derived with regard to singular exact vacuum solutions of the Einstein equation and the singularity structure of spacetime. Such solutions have been formally derived under condition $T_{\mu}^{\nu}(x) = 0$, where $T_{\mu}^{\nu}(x)$ represent the energy-momentum densities of the gravity source. This for example is the case for the well-known Schwarzschild solution, which is given by, in the Schwarzschild coordinates $(x^0, r, \theta, \phi)$,

$$ds^2 = -h(r)(dx^0)^2 + h^{-1}(r)(dr)^2 + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], h(r) = 1 - \frac{r_s}{r}, \quad (1.10.12)$$

Where, $r_s$ is the Schwarzschild radius $r_s = 2GM/c^2$ with $G, M$ and $c$ being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. The metric (1.3.12) describe the gravitational field produced by a point-like particle located at $r = 0$.

**Remark 1.10.6.** Note that when we say, on the basis of the canonical expression of the curvature square

$$R^{\rho\sigma\mu\nu}(r)R_{\rho\sigma\mu\nu}(r) = \frac{12r_s^2}{r^6} \qquad (1.10.13)$$

formally obtained from the metric (1.3.12), that $r = 0$ is a singularity of the Schwarzschild spacetime, the source is considered to be point-like and this metric is regarded as *meaningful everywhere* in space-time.

**Remark 1.10.7.** From the metric (1.10.12), the calculation of the canonical Einstein tensor proceeds in a straightforward manner gives for $r \neq 0$

$$G_t^t(r) = G_r^r(r) = -\frac{h'(r)}{\hat{r}} - \frac{1+h(r)}{\hat{r}^2} \equiv 0, G_\theta^\theta(r) = G_\varphi^\varphi(r) = -\frac{h''(r)}{2} - \frac{h(r)}{\hat{r}^2} \equiv 0, \quad (1.10.14)$$

Where $h(r) = -1 + r_s/r$. Using Eq. (1.10.14) one formally obtains a boundary conditions

$$\left\{ \begin{array}{l} G_t^t(0) \triangleq \lim_{r \to 0} G_t^t(r) = 0, G_r^r(0) \triangleq \lim_{r \to 0} G_r^r(r) = 0, \\ G_\theta^\theta(0) \triangleq \lim_{r \to 0} G_\theta^\theta(r) = 0, G_\varphi^\varphi(0) \triangleq \lim_{r \to 0} G_\varphi^\varphi(r) = 0. \end{array} \right. \qquad (1.10.15)$$

However as pointed out above the canonical expression of the Einstein tensor in a sufficiently small neighborhood $\Omega$ of the point $r = 0$ and must be replaced by the generalized Einstein tensor $\tilde{G}_{km}$ (1.10.10). By simple calculation easy to see that
$$ \left\{ \begin{array}{l} \tilde{G}_t^t (0) \triangleq \lim_{r \to 0} \tilde{G}_t^t (r) = -\infty, \tilde{G}_r^r (0) \triangleq \lim_{r \to 0} \tilde{G}_r^r (r) = -\infty, \\ \tilde{G}_\theta^\theta (0) \triangleq \lim_{r \to 0} \tilde{G}_\theta^\theta (r) = -\infty, \tilde{G}_\varphi^\varphi (0) \triangleq \lim_{r \to 0} \tilde{G}_\varphi^\varphi (r) = -\infty. \end{array} \right. \qquad (1.10.16) $$

and therefore the boundary conditions (1.15) is completely wrong. But other hand as pointed out by many authors [5]-[17] that the canonical representation of the Einstein tensor, valid only in a weak (distributional) sense,i.e. [12]:

$$ G_b^a(\hat{x}) = -8\pi m \delta_0^a \delta_b^0 \delta^3(\hat{x}) \qquad (1.10.17) $$

and therefore again we obtain $G_b^a(0) = -\infty \times (\delta_0^a \delta_b^0)$. Thus canonical definition of the Einstein tensor is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin $r = 0$.

### 1.11.2 The distributional Schwarzschild geometry

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

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

The two singular functions we will work with throughout this paper (namely the singular components of the Schwarzschild metric) are $\frac{1}{r}$ and $\frac{1}{r-r_s}$, $r_s \ge 0$. Since $\frac{1}{r} \in L_{loc}^1(\mathbb{R}^3)$, it obviously gives the regular distribution $\frac{1}{r} \in D'(\mathbb{R}^3)$.

By convolution with a mollifier $3c1(x)$ (adapted to the symmetry of the spacetime, i.e. chosen radially symmetric) we embed it into the Colombeau algebra $G(\mathbb{R}^3)$ [22]:

$$ \frac{1}{r} \to \iota \left(\frac{1}{r}\right) \triangleq \left(\frac{1}{r}\right) * \rho_{\varepsilon} \triangleq \left(\frac{1}{r}\right)_{\varepsilon}, \rho_{\varepsilon} = \frac{1}{\varepsilon^3} 3c1\left(\frac{r}{\varepsilon}\right), \varepsilon \in (0, 1]. \qquad (1.10.18) $$

Inserting (1.10.18) into (1.3.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:

$$ \left\{ \begin{array}{l} (ds_{\varepsilon}^2)_{\varepsilon} = (h_{\varepsilon}(r)(dt)^2)_{\varepsilon} - (h_{\varepsilon}^{-1}(r)(dr)^2)_{\varepsilon} + r^2 [(d\theta)^2 + \sin^2\theta(d\phi)^2], \\ h_{\varepsilon}(r) = -1 + r_s \left(\frac{1}{r}\right)_{\varepsilon}, \varepsilon \in (0, 1]. \end{array} \right. \qquad (1.10.19) $$

**Remark 1.10.8.** Note that under regularization (1.10.18) for any $\varepsilon \in (0, 1]$ the metric

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

obviously is a classical Riemannian object and there no exist an the breakdown of canonical formalism of Riemannian geometry for these metrics, even at origin $r = 0$. It has been suggested by many authors to describe singularity at the origin as an internal point, where the Einstein field equations are satisfied in a distributional sense [5]-[22]. From the Colombeau metric (1.10.19) one obtains in a distributional sense [22]:
$$
\left\{
\begin{array}{l}
\tilde{G}_t^t (0) \triangleq \lim_{r \to 0} \tilde{G}_t^t (r) = -\infty, \tilde{G}_r^r (0) \triangleq \lim_{r \to 0} \tilde{G}_r^r (r) = -\infty, \\
\tilde{G}_\theta^\theta (0) \triangleq \lim_{r \to 0} \tilde{G}_\theta^\theta (r) = -\infty, \tilde{G}_\varphi^\varphi (0) \triangleq \lim_{r \to 0} \tilde{G}_\varphi^\varphi (r) = -\infty.
\end{array}
\right.
\qquad (1.10.16)
$$

and therefore the boundary conditions (1.15) is completely wrong. But other hand as pointed out
by many authors [5]-[17] that the canonical representation of the Einstein tensor, valid only in a
weak (distributional) sense,i.e. [12]:

$$
G_b^a(\hat{x}) = -8\pi m \delta_0^a \delta_b^0 \delta^3(\hat{x}) \qquad (1.10.17)
$$

and therefore again we obtain $G_b^a(0) = -\infty \times (\delta_0^a \delta_b^0)$. Thus canonical definition of the Einstein tensor is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin $r=0$.

1.11.2 The distributional Schwarzschild geometry

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

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

The two singular functions we will work with throughout this paper (namely the singular components of the Schwarzschild metric) are $\frac{1}{r}$ and $\frac{1}{r-r_s}$, $r_s \ge 0$. Since $\frac{1}{r} \in L_{loc}^1(\mathbb{R}^3)$, it obviously gives the regular distribution $\frac{1}{r} \in D'(\mathbb{R}^3)$.

By convolution with a mollifier 3c1 (x) (adapted to the symmetry of the spacetime, i.e. chosen
radially symmetric) we embed it into the Colombeau algebra G (R³) [22]:

$$
\frac{1}{r} \to \iota \left(\frac{1}{r}\right) \triangleq \left(\frac{1}{r}\right) * \rho_{\varepsilon} \triangleq \left(\frac{1}{r}\right)_{\varepsilon}, \rho_{\varepsilon} = \frac{1}{\varepsilon^3} 3c1\left(\frac{r}{\varepsilon}\right), \varepsilon \in (0,1]. \quad (1.10.18)
$$

Inserting (1.10.18) into (1.3.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:

$$
\left\{
\begin{array}{l}
 ds_{\epsilon}^2 = (h_{\epsilon}(r)(dt)^2)_{\epsilon} - (h_{\epsilon}^{-1}(r)(dr)^2)_{\epsilon} + r^2 [(d\theta)^2 + \sin^2\theta(d\phi)^2], \\
 h_{\epsilon}(r) = -1 + r_s \left(\frac{1}{r}\right)_{\epsilon}, \epsilon \in (0,1].
\end{array}
\right.
\qquad (1.10.19)
$$

Remark 1.10.8. Note that under regularization (1.10.18) for any $\epsilon \in (0, 1]$ the metric

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

obviously is a classical Riemannian object and there no exist an the breakdown of canonical
formalism of Riemannian geometry for these metrics, even at origin $r = 0$. It has been suggested
by many authors to describe singularity at the origin as an *internal point*, where the Einstein field
equations are satisfied in a distributional sense [5]-[22]. From the Colombeau metric (1.10.19) one
obtains in a distributional sense [22]:
$$ (R_2^2(r, \varepsilon))_{\varepsilon} = (R_3^3)_{\varepsilon} = \left(\frac{h_{\varepsilon}^\prime(r)}{r} + \frac{1+h_{\varepsilon}(r)}{r^2}\right)_{\varepsilon} = 8\pi m \frac{\delta(r)}{r^2}, \\ (R_0^0(r, \varepsilon))_{\varepsilon} = (R_1^1)_{\varepsilon} = \frac{1}{2}\left(\frac{h_{\varepsilon}^{\prime\prime}(r)}{2} + \frac{h_{\varepsilon}^\prime(r)}{r}\right)_{\varepsilon} = -4\pi m \delta \frac{\delta(r)}{r^2}. \qquad (1.10.20) $$

Hence, the distributional Ricci tensor and the distributional curvature scalar $(R_\varepsilon(r))_\varepsilon$ are of 3b4-type, i.e. $(R_\varepsilon(r))_\varepsilon = \pi m \frac{\delta(r)}{r^2}$.

**Remark 1.10.9.** Note that the formulae (1.10.20) should be contrasted with what is the expected result $G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x)$ given by Eq.(1.10.17). However the equations (1.10..20) are obviously given in spherical coordinates and therefore strictly speaking this is not correct, because the basis fields $\left\{\frac{\partial}{\partial r}, \frac{\partial}{\partial \varphi}, \frac{\partial}{\partial \theta}\right\}$ are not globally defined. Representing distributions concentrated at the origin requires a basis regular at the origin. Transforming the formulae for $(R_{ij}(\varepsilon))_{3b5}$ into Cartesian coordinates associated with the spherical ones, i.e., $\{r, 3b8, 3c6\} \leftrightarrow \{x^i\}$, we obtain, e.g., for the Einstein tensor the expected result $G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x)$ given by Eq.(1.17), see [22].

### 1.11.4 The nonsmooth regularization of the singularity at the origin

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

$$ (h_{\varepsilon}(r))_{\varepsilon} = -1 + \left(\frac{r_s}{r}\Theta(r-\varepsilon)\right)_{\varepsilon}, \quad \varepsilon \in (0, 1], r < r_s. \qquad (1.10.21) $$

Here $\Theta(u)$ is the Heaviside function and the limit $\varepsilon \to 0$ is understood in a distributional sense. Equation (1.10.19) with $h_\varepsilon$ as given in (1.10.21) can be considered as an regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.10.21), the calculation of the distributional Einstein tensor proceeds in a straightforward manner. By simple calculation it gives [12]:

$$ \left\{ \begin{aligned} (G_t^t(r, \varepsilon))_{\varepsilon} &= (G_r^r(r, \varepsilon))_{\varepsilon} = -\left(\frac{h_{\varepsilon}^\prime(r)}{r}\right)_{\varepsilon} - \left(\frac{1+h_{\varepsilon}(r)}{r^2}\right)_{\varepsilon} = \\ &= -r_s \left(\frac{\delta(r-\varepsilon)}{r^2}\right)_{\varepsilon} = -r_s \frac{\delta(r)}{r^2} \end{aligned} \right. \qquad (1.10.22) $$

and

$$ \left\{ \begin{aligned} (G_{\theta}^{\theta}(r, \varepsilon))_{\varepsilon} &= (G_{\varphi}^{\varphi}(r, \varepsilon))_{\varepsilon} = -\left(\frac{h_{\varepsilon}^{\prime\prime}(r)}{2}\right)_{\varepsilon} - \left(\frac{h_{\varepsilon}(r)}{r^2}\right)_{\varepsilon} = \\ &r_s \left(\frac{\delta(r-\varepsilon)}{r^2}\right)_{\varepsilon} - r_s \left(\frac{\varepsilon}{r^2} \frac{d}{dr} \delta(r-\varepsilon)\right)_{\varepsilon} = -r_s \frac{\delta(r)}{r^2}. \end{aligned} \right. \qquad (1.10.23) $$

Which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function $398(r-\varepsilon)$. Transforming now the formulae for $(G_b^a(r, \varepsilon))_{3b5}$ into Cartesian coordinates associated with the spherical ones, i.e., $\{r, 3b8, 3c6\} \leftrightarrow \{x^i\}$, we obtain for the generalized Einstein tensor the expected result given by Eq.(1.10.17)

$$ G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x), \qquad (1.10.24) $$

see Remark 1.10.9.
$$ (R_2^2(r, \epsilon))_\epsilon = (R_3^3)_\epsilon = \left(\frac{h'_\epsilon(r)}{r} + \frac{1+h_\epsilon(r)}{r^2}\right)_\epsilon = 8\pi m \frac{\delta(r)}{r^2}, \\ (R_0^0(r, \epsilon))_\epsilon = (R_1^1)_\epsilon = \frac{1}{2}\left(\frac{h''_\epsilon(r)}{2} + \frac{h'_\epsilon(r)}{r}\right)_\epsilon = -4\pi m \delta \frac{\delta(r)}{r^2}. \qquad (1.10.20) $$

Hence, the distributional Ricci tensor and the distributional curvature scalar $(R_\epsilon(r))_\epsilon$ are of 3b4-type, i.e. $(R_\epsilon(r))_\epsilon = \pi m \frac{\delta(r)}{r^2}$.

**Remark 1.10.9.** Note that the formulae (1.10.20) should be contrasted with what is the expected result $G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x)$ given by Eq.(1.10.17). However the equations (1.10..20) are obviously given in spherical coordinates and therefore strictly speaking this is not correct, because the basis fields $\{\frac{\partial}{\partial r}, \frac{\partial}{\partial \varphi}, \frac{\partial}{\partial \theta}\}$ are not globally defined. Representing distributions concentrated at the origin requires a basis regular at the origin. Transforming the formulae for $(R_{ij}(\epsilon))_{3b5}$ into Cartesian coordinates associated with the spherical ones, i.e., $\{r, 3b8, 3c6\} \leftrightarrow \{x^i\}$, we obtain, e.g., for the Einstein tensor the expected result $G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x)$ given by Eq.(1.17), see [22].

### 1.11.4 The nonsmooth regularization of the singularity at the origin

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

$$ (h_{\epsilon}(r))_{\epsilon} = -1 + \left(\frac{r_s}{r}\Theta(r-\epsilon)\right)_{\epsilon}, \quad \epsilon \in (0, 1], r < r_s. \qquad (1.10.21) $$

Here $\Theta(u)$ is the Heaviside function and the limit $\epsilon \to 0$ is understood in a distributional sense. Equation (1.10.19) with $h_\epsilon$ as given in (1.10.21) can be considered as an regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.10.21), the calculation of the distributional Einstein tensor proceeds in a straightforward manner. By simple calculation it gives [12]:

$$ \left\{ \begin{aligned} (G_t^t(r, \epsilon))_\epsilon &= (G_r^r(r, \epsilon))_\epsilon = -\left(\frac{h'_\epsilon(r)}{r}\right)_\epsilon - \left(\frac{1+h_\epsilon(r)}{r^2}\right)_\epsilon = \\ &= -r_s \left(\frac{\delta(r-\epsilon)}{r^2}\right)_\epsilon = -r_s \frac{\delta(r)}{r^2} \end{aligned} \right. \qquad (1.10.22) $$

and

$$ \left\{ \begin{aligned} (G_{\theta}^{\theta}(r, \epsilon))_{\epsilon} &= (G_{\varphi}^{\varphi}(r, \epsilon))_{\epsilon} = -\left(\frac{h''_{\epsilon}(r)}{2}\right)_{\epsilon} - \left(\frac{h_{\epsilon}(r)}{r^2}\right)_{\epsilon} = \\ &r_s \left(\frac{\delta(r-\epsilon)}{r^2}\right)_{\epsilon} - r_s \left(\frac{\epsilon}{r^2} \frac{d}{dr} \delta(r-\epsilon)\right)_{\epsilon} = -r_s \frac{\delta(r)}{r^2}. \end{aligned} \right. \qquad (1.10.23) $$

Which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function $398(r - \epsilon)$. Transforming now the formulae for $(G_b^a(r, \epsilon))_{3b5}$ into Cartesian coordinates associated with the spherical ones, i.e., $\{r, 3b8, 3c6\} \leftrightarrow \{x^i\}$, we obtain for the generalized Einstein tensor the expected result given by Eq.(1.10.17)

$$ G_b^a(x) = -8\pi m \delta_0^a \delta_b^0 \delta^3(x), \qquad (1.10.24) $$

see Remark 1.10.9.
### 1.11.5 The smooth regularization via Horizon

The smooth regularization via Horizon is considered by J.M.Heinzle and R.Steinbauer in paper [22].

Note that $\frac{1}{r-r_s} \notin L_{loc}^1(\mathbb{R}^3)$. An canonical regularization is the principal value $\mathbf{vp}(\frac{1}{r-r_s}) \in D'(\mathbb{R}^3)$ which can be embedded into $\mathcal{G}(\mathbb{R}^3)$ [22]:

$$ \frac{1}{r-r_s} \xrightarrow{\mathbf{vp}} \mathbf{vp}\left(\frac{1}{r-r_s}\right) \xrightarrow{\iota} \left[\rho_{\varepsilon} * \mathbf{vp}\left(\frac{1}{r-r_s}\right)\right] \triangleq \left(\frac{1}{r-r_s}\right)_{\varepsilon} \in \mathcal{G}(\mathbb{R}^3). \quad (1.10.25) $$

Inserting now (1.10.25) into (1.10.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:

$$ \left\{ \begin{array}{l} (ds_{\varepsilon})_{\varepsilon} = (h(r)(dt)^2)_{\varepsilon} - (h_{\varepsilon}^{-1}(r)(dr)^2)_{\varepsilon} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], \\ h(r) = -1 + \frac{r_s}{r}, h_{\varepsilon}^{-1}(r) = -1 - r_s \left(\frac{1}{r-r_s}\right)_{\varepsilon}, \varepsilon \in (0,1]. \end{array} \right. \quad (1.10.26) $$

**Remark 1.10.10.** Note that obviously Colombeau object, (1.26) is degenerate at $r = r_s$, because $h(r)$ is zero at the horizon. However, this does not come as a surprise. Both $h(r)$ and $h^{-1}(r)$ are positive outside of the black hole and negative in the interior. As a consequence any smooth regularization of $h(r)$ (or $h^{-1}(r)$) must pass through zero somewhere and, additionally, this zero must converge to $r = r_s$ as the regularization parameter goes to zero.

**Remark 1.10.11.** Note that due to the degeneracy of Colombeau object (1.26), even the distributional Levi-Civita connection obviously is not available by using the smooth regularization via horizon [22].

### 1.11.6 The nonsmooth regularization via Gorizon

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

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

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

The emphasis of the present work lies on mathematical rigor. We derive the “physically expected” result for the distributional energy momentum tensor of the Schwarzschild geometry, i.e., $T_0^0 = 8\pi m\delta^{(3)}(\vec{x})$, in a conceptually satisfactory way. Additionally, we set up a unified language to comment on the respective merits of some of the approaches taken so far. In particular, we discuss questions of differentiable structure as well as smoothness and degeneracy problems of the
### 1.11.5 The smooth regularization via Horizon

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

$$ \frac{1}{r-r_s} \xrightarrow{\mathbf{vp}} \mathbf{vp}\left(\frac{1}{r-r_s}\right) \to \iota\left[\rho_{\varepsilon} * \mathbf{vp}\left(\frac{1}{r-r_s}\right)\right] \triangleq \left(\frac{1}{r-r_s}\right)_{\varepsilon} \in \mathcal{G}(\mathbb{R}^3). \quad (1.10.25) $$

Inserting now (1.10.25) into (1.10.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:

$$ \left\{ \begin{array}{l} (ds_{\varepsilon})_{\varepsilon} = (h(r)(dt)^2)_{\varepsilon} - (h_{\varepsilon}^{-1}(r)(dr)^2)_{\varepsilon} + r^2[(d\theta)^2 + \sin^2\theta(d\phi)^2], \\ h(r) = -1 + \frac{r_s}{r}, h_{\varepsilon}^{-1}(r) = -1 - r_s \left(\frac{1}{r-r_s}\right)_{\varepsilon}, \varepsilon \in (0,1]. \end{array} \right. \quad (1.10.26) $$

**Remark 1.10.10.** Note that obviously Colombeau object, (1.26) is degenerate at $r = r_s$, because $h(r)$ is zero at the horizon. However, this does not come as a surprise. Both $h(r)$ and $h^{-1}(r)$ are positive outside of the black hole and negative in the interior. As a consequence any smooth regularization of $h(r)$ (or $h^{-1}$) must pass through zero somewhere and, additionally, this zero must converge to $r = r_s$ as the regularization parameter goes to zero.

**Remark 1.10.11.** Note that due to the degeneracy of Colombeau object (1.26), even the distributional Levi-Civita connection obviously is not available by using the smooth regularization via horizon [22].

### 1.11.6 The nonsmooth regularization via Gorizon

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

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

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

The emphasis of the present work lies on mathematical rigor. We derive the “physically expected” result for the distributional energy momentum tensor of the Schwarzschild geometry, i.e., $T_0^0 = 8\pi m\delta^{(3)}(\vec{x})$, in a conceptually satisfactory way. Additionally, we set up a unified language to comment on the respective merits of some of the approaches taken so far. In particular, we discuss questions of differentiable structure as well as smoothness and degeneracy problems of the
regularized metrics, and present possible refinements and workarounds. These aims are accomplished using the framework of nonlinear supergeneralized functions (supergeneralized Colombeau algebras $\tilde{G}(\mathbb{R}^3, \Sigma)$). Examining the Schwarzschild metric (1.12) in a neighborhood of the horizon, we see that, whereas $h(r)$ is smooth, $h^{-1}(r)$ is not even $L_{\text{loc}}^1$ (note that the origin is now always excluded from our considerations; the space we are working on is $\mathbb{R}^3\setminus\{0\}$). Thus, regularizing the Schwarzschild metric amounts to embedding $h^{-1}$ into $\tilde{G}(\mathbb{R}^3, \Sigma)$ (as done in (3.2)). Obviously, (3.1) is degenerate at $r = 2m$, because $h(r)$ is zero at the horizon. However, this does not come as a surprise. Both $h(r)$ and $h^{-1}(r)$ are positive outside of the black hole and negative in the interior. As a consequence any (smooth) regularization $h_{\epsilon}^{+}(r)$ ($h_{\epsilon}^{-}(r)$) [above (below) horizon] of $h(r)$ must pass through small enough vicinity $O_{\epsilon}^{+}(2m) = \{\vec{x} \in \mathbb{R}^3 \mid \|\vec{x}\| > 2m, \|\vec{x} - 2m\| \le \epsilon\}$ ($O_{\epsilon}^{-}(2m) = \{\vec{x} \in \mathbb{R}^3 \mid \|\vec{x}\| < 2m, \|\vec{x} - 2m\| \le \epsilon\}$) of zeros set $O_0(2m) = \{\vec{y} \in \mathbb{R}^3 \mid \|\vec{y}\| = 2m\}$ somewhere and, additionally, this vicinity $O_{\epsilon}^{+}(2m)$ ($O_{\epsilon}^{-}(2m)$) must converge to $O_0(2m)$ as the regularization parameter $\epsilon$ goes to zero. Due to the degeneracy of (1.10.12), the Levi-Civita connection is not available. By appropriate nonsmooth regularization (see section 3) we obtain a Colombeau generalized object modeling the singular Schwarzschild metric above and below horizon, i.e.,

$$ egin{cases} (ds_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (ds_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \end{cases} \quad (1.10.28) \\ \epsilon \in (0, 1]. $$

Consider corresponding distributional connections

$$ \begin{aligned} (\Gamma_{kj}^{+l}(\epsilon))_{\epsilon} &= (\Gamma_{kj}^{-l}[h_{\epsilon}^{+}])_{\epsilon} \in \tilde{G}(\mathbb{R}^3, \Sigma) \text{ and } (\Gamma_{kj}^{-l}(\epsilon))_{\epsilon} = (\Gamma_{kj}^{+l}[h_{\epsilon}^{+}])_{\epsilon} \in \tilde{G}(\mathbb{R}^3, \Sigma): \\ (\Gamma_{kj}^{+l}(\epsilon))_{\epsilon} &= \frac{1}{2}((g_{\epsilon}^{+lm})[(g_{\epsilon}^{+})_{mk,j} + (g_{\epsilon}^{+})_{mj,k} - (g_{\epsilon}^{+})_{kj,m}])_{\epsilon}, \\ (\Gamma_{kj}^{-l}(\epsilon))_{\epsilon} &= \frac{1}{2}((g_{\epsilon}^{-lm})[(g_{\epsilon}^{-})_{mk,j} + (g_{\epsilon}^{-})_{mj,k} - (g_{\epsilon}^{-})_{kj,m}])_{\epsilon}. \end{aligned} \quad (1.10.29) $$

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

$$ \left\{ \begin{aligned} (\mathbf{[R}_{\epsilon}^{+}]_{1})_{\epsilon} &= (\mathbf{[R}_{\epsilon}^{+}]_{0})^{0} \approx -m\tilde{\Phi}(2m), \\ (\mathbf{[R}_{\epsilon}^{-}]_{1})_{\epsilon} &= (\mathbf{[R}_{\epsilon}^{-}]_{0})^{0} \approx m\tilde{\Phi}(2m). \end{aligned} \right. \quad (1.10.30) $$

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

$$ \tilde{\Phi}(r) = \pi_0 \sin \theta d\theta_0^{2\pi} d\phi \Phi(x) $$

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

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

$$ \left\{ \begin{aligned} w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} &= w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} = m \langle \tilde{\delta} | \Phi \rangle = -m \tilde{\Phi}(2m), \\ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} &= w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} = m \langle \tilde{\delta} | \Phi \rangle = m \tilde{\Phi}(2m), \end{aligned} \right. \quad (1.10.31) $$
regularized metrics, and present possible refinements and workarounds. These aims are accomplished using the framework of nonlinear supergeneralized functions (supergeneralized Colombeau algebras $\tilde{G}(\mathbb{R}^3, \Sigma)$). Examining the Schwarzschild metric (1.12) in a neighborhood of the horizon, we see that, whereas $h(r)$ is smooth, $h^{-1}(r)$ is not even $L_{\text{loc}}^1$ (note that the origin is now always excluded from our considerations; the space we are working on is $\mathbb{R}^3\setminus\{0\}$). Thus, regularizing the Schwarzschild metric amounts to embedding $h^{-1}$ into $\tilde{G}(\mathbb{R}^3, \Sigma)$ (as done in (3.2)). Obviously, (3.1) is degenerate at $r = 2m$, because $h(r)$ is zero at the horizon. However, this does not come as a surprise. Both $h(r)$ and $h^{-1}(r)$ are positive outside of the black hole and negative in the interior. As a consequence any (smooth) regularization $h_{\epsilon}^{+}(r)$ ($h_{\epsilon}^{-}(r)$) [above (below) horizon] of $h(r)$ must pass through small enough vicinity $O_{\epsilon}^{+}(2m) = \{\vec{x} \in \mathbb{R}^3 \mid \|\vec{x}\| > 2m, \|\vec{x} - 2m\| \le \epsilon\}$ ($O_{\epsilon}^{-}(2m) = \{\vec{x} \in \mathbb{R}^3 \mid \|\vec{x}\| < 2m, \|\vec{x} - 2m\| \le \epsilon\}$) of zeros set $O_0(2m) = \{\vec{y} \in \mathbb{R}^3 \mid \|\vec{y}\| = 2m\}$ somewhere and, additionally, this vicinity $O_{\epsilon}^{+}(2m)$ ($O_{\epsilon}^{-}(2m)$) must converge to $O_0(2m)$ as the regularization parameter $\epsilon$ goes to zero. Due to the degeneracy of (1.10.12), the Levi-Cività connection is not available. By appropriate nonsmooth regularization (see section 3) we obtain an Colombeau generalized object modeling the singular Schwarzschild metric above and below horizon, i.e.,

$$ egin{cases} (ds_{\epsilon}^{+2})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (ds_{\epsilon}^{-2})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \end{cases} \quad (1.10.28) \\ \epsilon \in (0, 1]. $$

Consider corresponding distributional connections

$$ \begin{aligned} (\Gamma_{kj}^{+l}(\epsilon))_{\epsilon} &= (\Gamma_{kj}^{-l}[h_{\epsilon}^{+}])_{\epsilon} \in \tilde{G}(\mathbb{R}^3, \Sigma) \text{ and } (\Gamma_{kj}^{-l}(\epsilon))_{\epsilon} = (\Gamma_{kj}^{+l}[h_{\epsilon}^{+}])_{\epsilon} \in \tilde{G}(\mathbb{R}^3, \Sigma): \\ (\Gamma_{kj}^{+l}(\epsilon))_{\epsilon} &= \frac{1}{2}((g_{\epsilon}^{+lm})[(g_{\epsilon}^{+})_{mk,j} + (g_{\epsilon}^{+})_{mj,k} - (g_{\epsilon}^{+})_{kj,m}])_{\epsilon}, \\ (\Gamma_{kj}^{-l}(\epsilon))_{\epsilon} &= \frac{1}{2}((g_{\epsilon}^{-lm})[(g_{\epsilon}^{-})_{mk,j} + (g_{\epsilon}^{-})_{mj,k} - (g_{\epsilon}^{-})_{kj,m}])_{\epsilon}. \end{aligned} \quad (1.10.29) $$

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

$$ \left\{ \begin{aligned} ( [\mathbf{R}_{\epsilon}^{+}]_{1} )_{\epsilon} &= ( [\mathbf{R}_{\epsilon}^{+}]_{0} )^{0} \approx -m \tilde{\Phi}(2m), \\ ( [\mathbf{R}_{\epsilon}^{-}]_{1} )_{\epsilon} &= ( [\mathbf{R}_{\epsilon}^{-}]_{0} )^{0} \approx m \tilde{\Phi}(2m). \end{aligned} \right. \quad (1.10.30) $$

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

$$ \tilde{\Phi}(r) = \pi_0 \sin \theta d\theta_0^{2\pi} d\phi \Phi(x) $$

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

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

$$ \left\{ \begin{aligned} w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{1}^{1} &= w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{+}]_{0}^{0} = m \langle \tilde{\delta} | \Phi \rangle = -m \tilde{\Phi}(2m), \\ w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{1}^{1} &= w - \lim_{\epsilon \to 0} [\mathbf{R}_{\epsilon}^{-}]_{0}^{0} = m \langle \tilde{\delta} | \Phi \rangle = m \tilde{\Phi}(2m), \end{aligned} \right. \quad (1.10.31) $$
i.e., the Schwarzschild spacetime is weakly Ricci-nonflat (the origin was excluded from our considerations). Furthermore, the Tolman formula [3],[4] for the total energy of a static and asymptotically flat spacetime with g the determinant of the four dimensional metric and $d^3x$ the coordinate volume

element, gives

$$E_T = (\mathbf{T}_r^r + \mathbf{T}_\theta^\theta + \mathbf{T}_\phi^\phi + \mathbf{T}_t^t) \sqrt{-g} d^3x = m, \qquad (1.10.32)$$

as it should be.

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

### 1.11.7 Distributional Eddington-Finkelstein spacetime

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

$$\nu^+ = t + r + 2m \ln \left( \frac{r}{2m} - 1 \right). \qquad (1.10.33)$$

From (1.10.31) it follows for $r > 2m$

$$dt = - \frac{dr}{1 - \frac{2m}{r}} + dv^+. \qquad (1.10.34)$$

The Schwarzschild metric (1.10.12) above horizon $ds^{+2}$ (see section 3) in this coordinate obviously takes the form

$$ds^{+2} = - \left(1 - \frac{2m}{r}\right) dv^{+2} + 2dr dv^{-} + r^2 \left[(d\theta)^2 + \sin^2\theta (d\phi)^2\right]. \qquad (1.10.35)$$

When $r < 2m$ we replace (1.10.33) below horizon by

$$\nu^{-} = t + r + 2m \ln \left(1 - \frac{r}{2m}\right) \qquad (1.10.36)$$

From (1.36) follows for $r < 2m$

$$dt = \frac{dr}{\frac{2m}{r} - 1} + dv^{-}. \qquad (1.10.37)$$

The Schwarzschild metric (1.10.12) below horizon $ds^{-}$ (see section 3) in this coordinate obviously
i.e., the Schwarzschild spacetime is weakly Ricci-nonflat (the origin was excluded from our considerations). Furthermore, the Tolman formula [3],[4] for the total energy of a static and asymptotically flat spacetime with g the determinant of the four dimensional metric and $d^3x$ the coordinate volume

element, gives

$$E_T = (\mathbf{T}_r^r + \mathbf{T}_\theta^\theta + \mathbf{T}_\phi^\phi + \mathbf{T}_t^t) \sqrt{-g} d^3x = m, \qquad (1.10.32)$$

as it should be.

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

### 1.11.7 Distributional Eddington-Finkelstein spacetime

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

$$\nu^+ = t + r + 2m \ln \left( \frac{r}{2m} - 1 \right). \qquad (1.10.33)$$

From (1.10.31) it follows for $r > 2m$

$$dt = - \frac{dr}{1 - \frac{2m}{r}} + dv^+. \qquad (1.10.34)$$

The Schwarzschild metric (1.10.12) above horizon $ds^{+2}$ (see section 3) in this coordinate obviously takes the form

$$ds^{+2} = - \left(1 - \frac{2m}{r}\right) dv^{+2} + 2dr dv^{-} + r^2 \left[(d\theta)^2 + \sin^2\theta (d\phi)^2\right]. \qquad (1.10.35)$$

When $r < 2m$ we replace (1.10.33) below horizon by

$$\nu^{-} = t + r + 2m \ln \left(1 - \frac{r}{2m}\right) \qquad (1.10.36)$$

From (1.36) follows for $r < 2m$

$$dt = \frac{dr}{\frac{2m}{r} - 1} + dv^{-}. \qquad (1.10.37)$$

The Schwarzschild metric (1.10.12) below horizon $ds^{-}$ (see section 3) in this coordinate obviously
takes the form

$$ds^{-2} = \left(\frac{2m}{r} - 1\right) dv^{-2} - 2drdv^{-} + r^2 [(d\theta)^2 + \sin^2\theta(d\phi)^2]. \qquad (1.10.38)$$

**Remark 1.10.12. (i)** Note that the metric (1.10.33) is defined on the manifold $S^2 \times (r > 0) \times \mathbb{R}$ and obviously it is regular Lorentzian metric: its coefficients are smooth.

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

(iii) Due to the nondegeneracy of the metric (1.10.32) the Levi-Civita connection

$$\{ \Gamma_{kj}^{+l} (\{\}) = \frac{1}{2} [g^{lm} (\{\})] [(g_{mk,j} (\{\}) + g_{mj,k} (\{\}) - g_{kj,m} (\{\})] \qquad (1.10.39)$$

obviously now available and therefore nonsingular on horizon in contrast with Schwarzschild metric (1.10.12) one obtains [3]:

$$\begin{align}
\Gamma_{\nu\nu}^{\nu} &= \frac{r_s}{2r^2}, \quad \Gamma_{\nu\nu}^{r} = \frac{r_s(r-r_s)}{2r^3}, \quad \Gamma_{\nu r}^{r} = -\frac{r_s}{2r^2}, \quad \Gamma_{r\theta}^{\theta} = \frac{1}{r}, \nonumber \\
\Gamma_{r\varphi}^{\varphi} &= \frac{1}{r}, \quad \Gamma_{\theta\theta}^{\nu} = -r, \quad \Gamma_{\theta\theta}^{r} = -r(r-r_s), \quad \Gamma_{\theta\varphi}^{\varphi} = \cot\theta, \tag{1.10.40} \\
\Gamma_{\varphi\varphi}^{\nu} &= -r \sin^2\theta, \quad \Gamma_{\varphi\varphi}^{r} = -r(r-r_s)\sin^2\theta, \quad \Gamma_{\varphi\varphi}^{\theta} = -\sin\theta\cos\theta. \nonumber
\end{align}$$

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

**Remark 1.10.13.** Let us consider now the coordinates: (i) $v^+, r' = r, \theta' = \theta, \varphi' = \varphi$ and (ii) $v^-, r' = r, \theta' = \theta, \varphi' = \varphi$. Obviously both transformations given by Eq.(1.10.33) and Eq.(1.10.36) are singular because the both Jacobian of these transformations is singular at $r = 2m$:

$$\left(\begin{array}{cc} \frac{\partial v^{+}}{\partial t} & \frac{\partial v^{+}}{\partial r} \\ \frac{\partial r^{+}}{\partial t} & \frac{\partial r^{+}}{\partial r} \end{array}\right) = \left(\begin{array}{cc} 1 & \frac{r}{r-2m} \\ 0 & 1 \end{array}\right) \qquad (1.10.41) \text{ and}$$

$$\left(\begin{array}{cc} \frac{\partial v^{-}}{\partial t} & \frac{\partial v^{-}}{\partial r} \\ \frac{\partial r^{-}}{\partial t} & \frac{\partial r^{-}}{\partial r} \end{array}\right) = \left(\begin{array}{cc} 1 & -\frac{r}{2m-r} \\ 0 & 1 \end{array}\right). \qquad (1.10.42)$$

**Remark 1.10.14.** Note first (i) such singular transformations not allowed in conventional Lorentzian geometry and second (ii) both Eddington-Finkelstein metrics given by Eq.(1.10.35) and by Eq.(1.10.38) well defined in rigorous mathematical sense at $r = 2m$.

**Remark 1.10.15. (I)** From consideration above follows that Schwarzschild spacetime $(S^2 \times \{r > 2m\}) \times \mathbb{R}$ is not extendible, in the sense that it can be immersed in a larger spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with $r > 2m$. Thus Eddington-Finkelstein spacetime cannot be considered as an extension of the Schwarzschild spacetime in natural way in respect with conventional Lorentzian geometry. Such "extension" are the extension by abnormal definition and nothing more. (II) However distributional Eddington-Finkelstein spacetime (1.10.53) is equivalent of the distributional Schwarzschild spacetime in natural way.

**Remark 1.10.16.** From consideration above follows that it is necessary an regularization of the Eq.(1.10.34) and Eq.(1.10.37) on horizon. However obviously only nonsmooth regularization via horizon $r = 2m$ possible. Under nonsmooth regularization (see section 3) Eq.(1.10.34) and
takes the form

$$ds^{-2} = \left(\frac{2m}{r} - 1\right) dv^{-2} - 2drdv^{-} + r^2 [(d\theta)^2 + \sin^2\theta(d\phi)^2]. \qquad (1.10.38)$$

**Remark 1.10.12. (i)** Note that the metric (1.10.33) is defined on the manifold $S^2 \times (r > 0) \times \mathbb{R}$ and obviously it is regular Lorentzian metric: its coefficients are smooth.

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

(iii) Due to the nondegeneracy of the metric (1.10.32) the Levi-Civita connection

$$\{ \Gamma_{kj}^{+l} (\{\}) = \frac{1}{2} [g^{lm} (\{\})] [(g_{mk,j} (\{\}) + g_{mj,k} (\{\}) - g_{kj,m} (\{\})] \qquad (1.10.39)$$

obviously now available and therefore nonsingular on horizon in contrast with Schwarzschild metric (1.10.12) one obtains [3]:

$$\begin{align}
\Gamma_{\nu\nu}^{\nu} &= \frac{r_s}{2r^2}, \quad \Gamma_{\nu\nu}^{r} = \frac{r_s(r-r_s)}{2r^3}, \quad \Gamma_{\nu r}^{r} = -\frac{r_s}{2r^2}, \quad \Gamma_{r\theta}^{\theta} = \frac{1}{r}, \nonumber \\
\Gamma_{r\varphi}^{\varphi} &= \frac{1}{r}, \quad \Gamma_{\theta\theta}^{\nu} = -r, \quad \Gamma_{\theta\theta}^{r} = -r(r-r_s), \quad \Gamma_{\theta\varphi}^{\varphi} = \cot\theta, \tag{1.10.40} \\
\Gamma_{\varphi\varphi}^{\nu} &= -r \sin^2\theta, \quad \Gamma_{\varphi\varphi}^{r} = -r(r-r_s)\sin^2\theta, \quad \Gamma_{\varphi\varphi}^{\theta} = -\sin\theta\cos\theta. \nonumber
\end{align}$$

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

**Remark 1.10.13.** Let us consider now the coordinates: (i) $v^+, r' = r, \theta' = \theta, \varphi' = \varphi$ and (ii) $v^-, r' = r, \theta' = \theta, \varphi' = \varphi$. Obviously both transformations given by Eq.(1.10.33) and Eq.(1.10.36) are singular because the both Jacobian of these transformations is singular at $r = 2m$:

$$\left(\frac{\partial v^{+}}{\partial t}, \frac{\partial v^{+}}{\partial r}, \frac{\partial v^{+}}{\partial r'}\right) = \begin{pmatrix} 1 & \frac{r}{r-2m} \\ 0 & 1 \end{pmatrix} \qquad (1.10.41) \text{ and}$$

$$\left(\frac{\partial v^{-}}{\partial t}, \frac{\partial v^{-}}{\partial r}, \frac{\partial v^{-}}{\partial r'}\right) = \begin{pmatrix} 1 & -\frac{r}{2m-r} \\ 0 & 1 \end{pmatrix}. \qquad (1.10.42)$$

**Remark 1.10.14.** Note first (i) such singular transformations not allowed in conventional Lorentzian geometry and second (ii) both Eddington-Finkelstein metrics given by Eq.(1.10.35) and by Eq.(1.10.38) well defined in rigorous mathematical sense at $r = 2m$.

**Remark 1.10.15. (I)** From consideration above follows that Schwarzschild spacetime $(S^2 \times \{r > 2m\}) \times \mathbb{R}$ is not extendible, in the sense that it can be immersed in a larger spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with $r > 2m$. Thus Eddington-Finkelstein spacetime cannot be considered as an extension of the Schwarzschild spacetime in natural way in respect with conventional Lorentzian geometry. Such "extension" are the extension by abnormal definition and nothing more. (II) However distributional Eddington-Finkelstein spacetime (1.10.53) is equivalent of the distributional Schwarzschild spacetime in natural way.

**Remark 1.10.16.** From consideration above follows that it is necessary an regularization of the Eq.(1.10.34) and Eq.(1.10.37) on horizon. However obviously only nonsmooth regularization via horizon $r = 2m$ possible. Under nonsmooth regularization (see section 3) Eq.(1.10.34) and
Eq.(1.10.37) takes the form

$$ dt = - \frac{dr}{\frac{1}{r}\sqrt{(r-2m)^2 + \epsilon^2}} + dv_{\epsilon}^{+}, \qquad (1.10.43) $$
$\epsilon \in (0, 1]$

and

$$ dt = \frac{dr}{\frac{1}{r}\sqrt{(2m-r)^2 + \epsilon^2}} + dv_{\epsilon}^{-}, \qquad (1.10.44) $$
$\epsilon \in (0, 1]$

Correspondingly. Therefore Eq.(1.10.41)-Eq.(1.10.42) takes the form

$$ \left( \begin{array}{cc} \frac{\partial v_{\epsilon}^{+}}{\partial t'} & \frac{\partial v_{\epsilon}^{+}}{\partial r'} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right) \qquad (1.10.45) $$

and

$$ \left( \begin{array}{cc} \frac{\partial v_{\epsilon}^{-}}{\partial t'} & \frac{\partial v_{\epsilon}^{-}}{\partial r'} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & -\frac{r}{\sqrt{(2m-r)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right). \qquad (1.10.46) $$

From Eq.(1.10.43)-Eq.(1.10.44) one obtain generalized Eddington-Finkelstein transformation such that

$$ dt = - \frac{rdr}{\left(\sqrt{(r-2m)^2 + \epsilon^2}\right)_{\epsilon}} + (dv_{\epsilon}^{+})_{\epsilon}, \qquad (1.10.47) $$
$\epsilon \in (0, 1]$

and

$$ dt = \frac{rdr}{\left(\sqrt{(2m-r)^2 + \epsilon^2}\right)_{\epsilon}} + dv_{\epsilon}^{-}, \qquad (1.10.48) $$
$\epsilon \in (0, 1]$

Therefore Eq.(1.10.45)-Eq.(1.10.46) takes the form

$$ \left( \begin{array}{cc} \left( \frac{\partial v_{\epsilon}^{+}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{+}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right) \qquad (1.10.49) $$

and

$$ \left( \begin{array}{cc} \left( \frac{\partial v_{\epsilon}^{-}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{-}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & -\frac{r}{\sqrt{(2m-r)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right). \qquad (1.10.50) $$

At point $r = 2m$ one obtain
Eq.(1.10.37) takes the form

$$dt = - \frac{dr}{\frac{1}{r} \sqrt{(r-2m)^2 + \epsilon^2}} + dv_{\epsilon}^{+}, \qquad (1.10.43)$$
$$\epsilon \in (0, 1]$$

and

$$dt = \frac{dr}{\frac{1}{r} \sqrt{(2m-r)^2 + \epsilon^2}} + dv_{\epsilon}^{-}, \qquad (1.10.44)$$
$$\epsilon \in (0, 1]$$

Correspondingly. Therefore Eq.(1.10.41)-Eq.(1.10.42) takes the form

$$\left( \begin{array}{cc} \frac{\partial v_{\epsilon}^{+}}{\partial t'} & \frac{\partial v_{\epsilon}^{+}}{\partial r'} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right) \qquad (1.10.45)$$

and

$$\left( \begin{array}{cc} \frac{\partial v_{\epsilon}^{-}}{\partial t'} & \frac{\partial v_{\epsilon}^{-}}{\partial r'} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & -\frac{r}{\sqrt{(2m-r)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right). \qquad (1.10.46)$$

From Eq.(1.10.43)-Eq.(1.10.44) one obtain generalized Eddington-Finkelstein transformation such that

$$dt = - \frac{rdr}{\left( \sqrt{(r-2m)^2 + \epsilon^2} \right)_{\epsilon}} + (dv_{\epsilon}^{+})_{\epsilon}, \qquad (1.10.47)$$
$$\epsilon \in (0, 1]$$

and

$$dt = \frac{rdr}{\left( \sqrt{(2m-r)^2 + \epsilon^2} \right)_{\epsilon}} + dv_{\epsilon}^{-}, \qquad (1.10.48)$$
$$\epsilon \in (0, 1].$$

Therefore Eq.(1.10.45)-Eq.(1.10.46) takes the form

$$\left( \begin{array}{cc} \left( \frac{\partial v_{\epsilon}^{+}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{+}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right) \qquad (1.10.49)$$

and

$$\left( \begin{array}{cc} \left( \frac{\partial v_{\epsilon}^{-}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{-}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{array} \right) = \left( \begin{array}{cc} 1 & -\frac{r}{\sqrt{(2m-r)^2 + \epsilon^2}} \\ 0 & 1 \end{array} \right). \qquad (1.10.50)$$

At point $r = 2m$ one obtain
$$ \left. \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{+}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{+}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} \right|_{r=2m} = \begin{pmatrix} 1 & r (\epsilon^{-1})_{\epsilon} \\ 0 & 1 \end{pmatrix} \qquad (1.10.51) $$

and

$$ \left. \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{-}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{-}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} \right|_{r=2m} = \begin{pmatrix} 1 & -r (\epsilon^{-1})_{\epsilon} \\ 0 & 1 \end{pmatrix}, \qquad (1.10.52) $$

Where $(\epsilon^{-1})_{\epsilon} \in \tilde{\mathbb{R}}$.

Thus generalized Eddington-Finkelstein transformations (1.10.47)-(1.10.48) well defined in sense of Colombeau generalized functions. Therefore Colombeau generalized object modeling the classical Eddington-Finkelstein metric (1.10.35)-(1.10.36) above and below horizon takes the form

$$ \begin{aligned} (ds_{\epsilon}^{+2})_{\epsilon} &= \frac{1}{r} \sqrt{(r-2m)^2 + \epsilon^2 dv_{\epsilon}^{+2} + 2dr dv_{\epsilon}^{+} + r^2 [(d\theta)^2 + \sin^2 \theta(d\phi)^2]} . \\ (ds_{\epsilon}^{-2})_{\epsilon} &= \frac{1}{r} \sqrt{(2m-r)^2 + \epsilon^2 dv_{\epsilon}^{-2} + 2dr dv_{\epsilon}^{-} + r^2 [(d\theta)^2 + \sin^2 \theta(d\phi)^2]} . \end{aligned} \qquad (1.10.53) $$

It easily to verify by using formula A.2 (see appendix) that the distributional curvature scalar $(\mathbf{R}(\epsilon))_{\epsilon}$ again singular at $r = 2m$ as in the case of the distributional Schwarzschild spacetime given by Eq.(1.10.28). However this is not surprising because the classical Eddington- Finkelstein spacetime and generalized Eddington-Finkelstein spacetime given by Eq.(1.10.53) that is essentially different geometrical objects.

# 2 Generalized Colombeau Calculus

## 2.1 Notation and Basic Notions from Standard Colombeau Theory

We use [1],[2],[7] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper $\Omega$ will denote an open subset of $\mathbb{R}^n$.

Stanford Colombeau generalized functions on $\Omega$ are defined as equivalence classes $u = [(u_{\epsilon})_{\epsilon}]$ of nets of smooth functions $u_{\epsilon} \in C^{\infty}(\Omega)$ (regularizations) subjected to asymptotic norm conditions with respect to $\epsilon \in (0, 1]$ for their derivatives on compact sets.

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

**Definition 2.1.** The classical Colombeau's algebra of generalized functions on $M$ is defined as the quotient:

$$ \mathcal{G}(M) \triangleq \mathcal{E}_M(M)/\mathcal{N}(M) \qquad (2.1) $$

of the space $\mathcal{E}_M(M)$ of sequences of moderate growth modulo the space $\mathcal{N}(M)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where $\mathcal{X}(M)$ denoting the space of smooth vector fields on $M$):
$$ \left. \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{+}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{+}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} \right|_{r=2m} = \begin{pmatrix} 1 & r (\epsilon^{-1})_{\epsilon} \\ 0 & 1 \end{pmatrix} \qquad (1.10.51) $$

and

$$ \left. \begin{pmatrix} \left( \frac{\partial v_{\epsilon}^{-}}{\partial t} \right)_{\epsilon} & \left( \frac{\partial v_{\epsilon}^{-}}{\partial r} \right)_{\epsilon} \\ \frac{\partial r'}{\partial t} & \frac{\partial r'}{\partial r} \end{pmatrix} \right|_{r=2m} = \begin{pmatrix} 1 & -r (\epsilon^{-1})_{\epsilon} \\ 0 & 1 \end{pmatrix}, \qquad (1.10.52) $$

Where $(\epsilon^{-1})_{\epsilon} \in \tilde{\mathbb{R}}$.

Thus generalized Eddington-Finkelstein transformations (1.10.47)–(1.10.48) well defined in sense of Colombeau generalized functions. Therefore Colombeau generalized object modeling the classical Eddington-Finkelstein metric (1.10.35)-(1.10.36) above and below horizon takes the form

$$ \begin{aligned} (ds_{\epsilon}^{+2})_{\epsilon} &= \frac{1}{r} \sqrt{(r-2m)^2 + \epsilon^2 dv_{\epsilon}^{+2} + 2dr dv_{\epsilon}^{+} + r^2 [(d\theta)^2 + \sin^2 \theta(d\phi)^2]} . \\ (ds_{\epsilon}^{-2})_{\epsilon} &= \frac{1}{r} \sqrt{(2m-r)^2 + \epsilon^2 dv_{\epsilon}^{-2} + 2dr dv_{\epsilon}^{-} + r^2 [(d\theta)^2 + \sin^2 \theta(d\phi)^2]} . \end{aligned} \qquad (1.10.53) $$

It easily to verify by using formula A.2 (see appendix) that the distributional curvature scalar $(\mathbf{R}(\epsilon))_{\epsilon}$ again singular at $r = 2m$ as in the case of the distributional Schwarzschild spacetime given by Eq.(1.10.28). However this is not surprising because the classical Eddington- Finkelstein spacetime and generalized Eddington-Finkelstein spacetime given by Eq.(1.10.53) that is essentially different geometrical objects.

# 2 Generalized Colombeau Calculus

## 2.1 Notation and Basic Notions from Standard Colombeau Theory

We use [1],[2],[7] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper $\Omega$ will denote an open subset of $\mathbb{R}^n$.

Stanford Colombeau generalized functions on $\Omega$ are defined as equivalence classes $u = [(u_{\epsilon})_{\epsilon}]$ of nets of smooth functions $u_{\epsilon} \in C^{\infty}(\Omega)$ (regularizations) subjected to asymptotic norm conditions with respect to $\epsilon \in (0, 1]$ for their derivatives on compact sets.

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

**Definition 2.1.** The classical Colombeau’s algebra of generalized functions on $M$ is defined as the quotient:

$$ \mathcal{G}(M) \triangleq \mathcal{E}_M(M)/\mathcal{N}(M) \qquad (2.1) $$

of the space $\mathcal{E}_M(M)$ of sequences of moderate growth modulo the space $\mathcal{N}(M)$ of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where $\mathcal{X}(M)$ denoting the space of smooth vector fields on $M$):
$$ \left\{ \begin{array}{l} \mathcal{E}_M(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M) \forall k (k \in \mathbb{N}) \exists N (N \in \mathbb{N}) \\ \quad \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}, \end{array} \right. \quad (2.2) $$

$$ \left\{ \begin{array}{l} \mathcal{N}(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M), \forall k (k \in \mathbb{N}_0) \forall q (q \in N) \\ \quad \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \}. \end{array} \right. \quad (2.3) $$

**Remark 2.1.** In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0 (\epsilon_0 \in (0, 1]) \forall \epsilon (\epsilon < \epsilon_0) [a_\epsilon \le C\psi(\epsilon)]$.

**Definition 2.3.** Elements of $\mathrm{calG}(M)$ are denoted by:

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}(M). \quad (2.4) $$

**Remark 2.2.** With componentwise operations $(\cdot, \pm)$ $\mathcal{G}(M)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \mathrm{cl}[(L_\xi u_\epsilon)_\epsilon]$.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \mathcal{G}(M)$ iff $u \circ \psi_\alpha \in \mathcal{G}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \mathcal{G}(M)$ iff $u \circ \psi_\alpha \in \mathcal{G}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 2.4.** Smooth functions $f \in \mathcal{C}^\infty(M)$ are embedded into $\mathcal{G}(M)$ simply by the “constant” embedding $\sigma$, i.e., $\sigma(f) = \mathrm{cl}[(f)_\epsilon]$, hence $\mathcal{C}^\infty(M)$ is a faithful subalgebra of $\mathcal{G}(M)$.

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

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

Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".

**Definition 2.5.** Inserting $p \in M$ into $u \in \mathcal{G}(M)$ yields a well defined element of the ring of constants (also called generalized numbers) $\mathcal{K}$ (corresponding to $\mathbf{K} = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers $((r_\epsilon)_\epsilon \in \mathbf{K}^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $\mathcal{G}(M)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ \mathcal{K} = \mathcal{E}_c(M) / \mathcal{N}_c(M) \quad (2.5) $$

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

$$ \left\{ \begin{array}{l} \mathcal{E}_c(M) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \exists n (n \in \mathbb{N}) \quad [|r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0] \} \\ \mathcal{N}_c(M) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \forall m (m \in \mathbb{N}) \quad [|r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0] \} \\ I = (0, 1]. \end{array} \right. \quad (2.6) $$
$$ \left\{ \begin{array}{l} \mathcal{E}_M(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M) \forall k (k \in \mathbb{N}) \exists N (N \in \mathbb{N}) \\ \quad \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}) \text{ as } \epsilon \to 0 \right] \}, \end{array} \right. \quad (2.2) $$

$$ \left\{ \begin{array}{l} \mathcal{N}(M) \triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M), \forall k (k \in \mathbb{N}_0) \forall q (q \in N) \\ \quad \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q) \text{ as } \epsilon \to 0 \right] \}. \end{array} \right. \quad (2.3) $$

**Remark 2.1.** In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C(C > 0) \exists \epsilon_0 (\epsilon_0 \in (0, 1]) \forall \epsilon (\epsilon < \epsilon_0) [a_\epsilon \le C\psi(\epsilon)]$.

**Definition 2.3.** Elements of $\mathrm{calG}(M)$ are denoted by:

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}(M). \quad (2.4) $$

**Remark 2.2.** With componentwise operations $(\cdot, \pm)$ $\mathcal{G}(M)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \mathrm{cl}[(L_\xi u_\epsilon)_\epsilon]$.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \mathcal{G}(M)$ iff $u \circ \psi_\alpha \in \mathcal{G}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \mathcal{G}(M)$ iff $u \circ \psi_\alpha \in \mathcal{G}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 2.4.** Smooth functions $f \in \mathcal{C}^\infty(M)$ are embedded into $\mathcal{G}(M)$ simply by the “constant” embedding $\sigma$, i.e., $\sigma(f) = \mathrm{cl}[(f)_\epsilon]$, hence $\mathcal{C}^\infty(M)$ is a faithful subalgebra of $\mathcal{G}(M)$.

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

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

Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".

**Definition 2.5.** Inserting $p \in M$ into $u \in \mathcal{G}(M)$ yields a well defined element of the ring of constants (also called generalized numbers) $\mathcal{K}$ (corresponding to $\mathbf{K} = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers $((r_\epsilon)_\epsilon \in \mathbf{K}^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $\mathcal{G}(M)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ \mathcal{K} = \mathcal{E}_c(M) / \mathcal{N}_c(M) \quad (2.5) $$

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

$$ \left\{ \begin{array}{l} \mathcal{E}_c(M) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \exists n (n \in \mathbb{N}) \quad [|r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0] \} \\ \mathcal{N}_c(M) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \forall m (m \in \mathbb{N}) \quad [|r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0] \} \\ I = (0, 1]. \end{array} \right. \quad (2.6) $$
Generalized functions on M are characterized by their generalized point values, i.e., by their values on points in $\tilde{M}_c$, the space of equivalence classes of compactly supported nets $(p_\epsilon)_\epsilon \in M^{(0,1]}$ with respect to the relation $p_\epsilon \sim p'_\epsilon :\Leftrightarrow d_h(p_\epsilon, p'_\epsilon) = O(\epsilon^m)$ for all $m$, where $d_h$ denotes the distance on M induced by any Riemannian metric.

**Definition 2.6.** For $u \in \mathcal{G}(M)$ and $x_0 \in M$, the point value of $u$ at the point $x_0$, $u(x_0)$, is defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $\mathcal{K}$.

**Definition 2.7.** We say that an element $r \in \mathcal{K}$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

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

**Definition 2.8.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. On a set $\hat{\Omega}$:

$$ \hat{\Omega} = \{(x_\epsilon)_\epsilon \in \Omega^I | \exists p(p>0) \, [|x_\epsilon| = O(\epsilon^p)]\} = \{(x_\epsilon)_\epsilon \in \Omega^I | \exists p(p>0) \, \exists \epsilon_0 (\epsilon_0>0) \, [|x_\epsilon| \le \epsilon^p, \text{ for } 0 < \epsilon < \epsilon_0]\} \quad (2.7) $$

We introduce an equivalence relation:

$$ (x_\epsilon)_\epsilon \sim (y_\epsilon)_\epsilon \Leftrightarrow \forall q(q>0) \forall \epsilon(\epsilon>0) \, [|x_\epsilon - y_\epsilon| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0] \quad (2.8) $$

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

$$ \tilde{\Omega}_c = \{ \tilde{x} = \text{cl}[(x_\epsilon)_\epsilon] \in \tilde{\Omega} | \exists K (K \subset \Omega) \exists \epsilon_0 (\epsilon_0 > 0) [x_\epsilon \in K \text{ for } 0 < \epsilon < \epsilon_0] \} \quad (2.9) $$

**Definition 2.9.** A generalized function $u \in \mathcal{G}(M)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in L.Schwartz sense if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly,i.e.

$$ w - \lim_{\epsilon \to 0} u_\epsilon = 0 \quad (2.10) $$

We shall often write:

$$ u \stackrel{\text{Sch}}{\approx} 0. \quad (2.11) $$

The $\mathcal{G}(M)$-module of generalized sections in vector bundles-especially the space of generalized tensor fields $\mathcal{T}_s^r(M)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

$$ \mathcal{G}_s^r(M) = \mathcal{G}(M) \otimes \mathcal{T}_s^r(M), \quad (2.12) $$

where $\mathcal{T}_s^r(M)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $\mathcal{C}^\infty(M)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [1]-[2], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $\mathcal{G}(M)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $\mathcal{G}(M)$-modules we have
Generalized functions on $M$ are characterized by their generalized point values, i.e., by their values on points in $\tilde{M}_c$, the space of equivalence classes of compactly supported nets $(p_\epsilon)_\epsilon \in M^{(0,1]}$ with respect to the relation $p_\epsilon \sim p'_\epsilon :\Leftrightarrow d_h(p_\epsilon, p'_\epsilon) = O(\epsilon^m)$ for all $m$, where $d_h$ denotes the distance on $M$ induced by any Riemannian metric.

**Definition 2.6.** For $u \in \mathcal{G}(M)$ and $x_0 \in M$, the point value of $u$ at the point $x_0$, $u(x_0)$, is defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $\mathcal{K}$.

**Definition 2.7.** We say that an element $r \in \mathcal{K}$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

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

**Definition 2.8.** Let $\Omega$ be an open subset of $\mathbb{R}^n$. On a set $\hat{\Omega}$:

$$ \hat{\Omega} = \{(x_\epsilon)_\epsilon \in \Omega^I | \exists p(p>0) \, [ |x_\epsilon| = O(\epsilon^p)] \} = \{(x_\epsilon)_\epsilon \in \Omega^I | \exists p(p>0) \, \exists \epsilon_0 (\epsilon_0>0) \, [ |x_\epsilon| \le \epsilon^p, \text{ for } 0 < \epsilon < \epsilon_0 ] \} \quad (2.7) $$

We introduce an equivalence relation:

$$ (x_\epsilon)_\epsilon \sim (y_\epsilon)_\epsilon \Leftrightarrow \forall q(q>0) \forall \epsilon(\epsilon>0) \, [ |x_\epsilon - y_\epsilon| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0 ] \quad (2.8) $$

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

$$ \tilde{\Omega}_c = \{ \tilde{x} = \text{cl}[(x_\epsilon)_\epsilon] \in \tilde{\Omega} | \exists K (K \subset \Omega) \exists \epsilon_0 (\epsilon_0 > 0) [x_\epsilon \in K \text{ for } 0 < \epsilon < \epsilon_0] \} \quad (2.9) $$

**Definition 2.9.** A generalized function $u \in \mathcal{G}(M)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in L.Schwartz sense if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly, i.e.

$$ w - \lim_{\epsilon \to 0} u_\epsilon = 0 \quad (2.10) $$

We shall often write:

$$ u \underset{\text{Sch}}{\approx} 0. \quad (2.11) $$

The $\mathcal{G}(M)$-module of generalized sections in vector bundles especially the space of generalized tensor fields $\mathcal{T}_s^r(M)$ is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

$$ \mathcal{G}_s^r(M) = \mathcal{G}(M) \otimes \mathcal{T}_s^r(M), \quad (2.12) $$

where $\mathcal{T}_s^r(M)$ denotes the space of smooth tensor fields and the tensor product is taken over the module $\mathcal{C}^\infty(M)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [1]-[2], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $\mathcal{G}(M)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $\mathcal{G}(M)$-modules we have
$$ \mathcal{G}_s^r(M) \cong L_{(M)}(\mathcal{G}_1^0(M)^r, \mathcal{G}_0^1(M)^s; \mathcal{G}(M)). \qquad (2.13) $$

In particular a generalized metric is defined to be a symmetric, generalized (0,2)-tensor field $g_{ab} = [((g_\epsilon)_{ab})_\epsilon]$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in $\mathcal{G}(M)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have $\forall K \subset M \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^m]$ for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric iff (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $\mathcal{G}(M)$-linear isomorphism from $\mathcal{G}_0^1(M)$ to $\mathcal{G}_1^0(M)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $\mathcal{G}_0^2(M)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also the generalized Levi-Civita connection as well as the generalized Riemann-, Ricci- and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives.

## 2.3 Generalized Colombeau Calculus

We briefly recall the basic generalized Colombeau construction. Colombeau supergeneralized functions on $\Omega \subseteq \mathbb{R}^n$, where $\dim(\Omega) = n$ are defined as equivalence classes $u = [(u_\epsilon)_\epsilon]$ of nets of smooth functions $u_\epsilon \in C^\infty(\Omega \setminus \Sigma)$, where $\dim(\Sigma) < n$ (regularizations) subjected to asymptotic norm conditions with respect to $\epsilon \in (0, 1]$ for their derivatives on compact sets.

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

**Definition 2.10.** The supergeneralized Colombeau's algebra $\tilde{\mathcal{G}} = \tilde{\mathcal{G}}(M, \Sigma)$ of supergeneralized functions on $M$, where $\Sigma \subset M$, $\dim(M) = n$, $\dim(\Sigma) < n$, is defined as the quotient:

$$ \tilde{\mathcal{G}}(M, \Sigma) \triangleq \mathcal{E}_M(M, \Sigma)/\mathcal{N}(M, \Sigma) \qquad (2.14) $$

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

$$ \begin{align} \mathcal{E}_M(M, \Sigma) &\triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M \setminus \Sigma) \ \forall k (k \in \mathbb{N}) \ \exists N (N \in \mathbb{N}) \notag \\ & \phantom{\triangleq \{ } \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M \setminus \Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}), \epsilon \to 0 \right] \ \& \notag \\ & \phantom{\triangleq \{ } \forall K (K \subsetneq M) \ \forall k (k \in \mathbb{N}) \ \exists N (N \in \mathbb{N}) \ \forall (f \in C^\infty(M)) \ \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \notag \\ & \phantom{\triangleq \{ } \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon\| = \left( \sup_{f \in C^\infty(M)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^{-N}), \epsilon \to 0 \right] \notag \end{align} \qquad (2.15) $$
$$ \mathcal{G}_s^r(M) \cong L_{(M)}(\mathcal{G}_1^0(M)^r, \mathcal{G}_0^1(M)^s; \mathcal{G}(M)). \qquad (2.13) $$

In particular a generalized metric is defined to be a symmetric, generalized (0,2)-tensor field $g_{ab} = [((g_\epsilon)_{ab})_\epsilon]$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in $\mathcal{G}(M)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have $\forall K \subset M \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^m]$ for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric iff (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $\mathcal{G}(M)$-linear isomorphism from $\mathcal{G}_0^1(M)$ to $\mathcal{G}_1^0(M)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $\mathcal{G}_0^2(M)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also the generalized Levi-Civita connection as well as the generalized Riemann-, Ricci- and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives.

## 2.3 Generalized Colombeau Calculus

We briefly recall the basic generalized Colombeau construction. Colombeau supergeneralized functions on $\Omega \subseteq \mathbb{R}^n$, where $\dim(\Omega) = n$ are defined as equivalence classes $u = [(u_\epsilon)_\epsilon]$ of nets of smooth functions $u_\epsilon \in C^\infty(\Omega \setminus \Sigma)$, where $\dim(\Sigma) < n$ (regularizations) subjected to asymptotic norm conditions with respect to $\epsilon \in (0, 1]$ for their derivatives on compact sets.

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

**Definition 2.10.** The supergeneralized Colombeau's algebra $\tilde{\mathcal{G}} = \tilde{\mathcal{G}}(M, \Sigma)$ of supergeneralized functions on $M$, where $\Sigma \subset M$, $\dim(M) = n$, $\dim(\Sigma) < n$, is defined as the quotient:

$$ \tilde{\mathcal{G}}(M, \Sigma) \triangleq \mathcal{E}_M(M, \Sigma)/\mathcal{N}(M, \Sigma) \qquad (2.14) $$

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

$$ \begin{align} \mathcal{E}_M(M, \Sigma) &\triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M \setminus \Sigma) \ \forall k (k \in \mathbb{N}) \ \exists N (N \in \mathbb{N}) \notag \\ &\phantom{\triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M \setminus \Sigma) \ \forall k (k \in \mathbb{N}) \ \exists N (N \in \mathbb{N})} \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M \setminus \Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^{-N}), \epsilon \to 0 \right] \ \& \notag \\ &\phantom{\triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M \setminus \Sigma) \ \forall k (k \in \mathbb{N}) \ \exists N (N \in \mathbb{N}) \ \forall (f \in C^\infty(M)) \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M))} \forall K (K \subsetneq M) \ \forall k (k \in \mathbb{N}) \ \exists N (N \in \mathbb{N}) \ \forall (f \in C^\infty(M)) \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon\| = \left( \sup_{f \in C^\infty(M)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^{-N}), \epsilon \to 0 \right] \right\}, \tag{2.15} \end{align} $$
$$
\begin{equation}
\begin{aligned}
\mathcal{N}(M, \Sigma) &\triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M \setminus \Sigma), \forall k (k \in \mathbb{N}_0) \forall q (q \in N) \\
&\phantom{\triangleq \{} \quad \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M \setminus \Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q), \epsilon \to 0 \right] \} \& \\
&\phantom{\triangleq \{} \quad \forall K (K \subsetneq M) \forall k (k \in \mathbb{N}) \exists N (N \in \mathbb{N}) \forall (f \in C^\infty(M)) \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \\
&\phantom{\triangleq \{} \quad \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon\| = \left( \sup_{f \in C^\infty(M)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^q), \epsilon \to 0 \right] \},
\end{aligned}
\tag{2.16}
\end{equation}
$$

where $L_{\xi_k}^w$ denoting the weak Lie derivative in L.Schwartz sense.In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C (C > 0) \exists \epsilon_0 (\epsilon_0 \in (0, 1]) \forall \epsilon (\epsilon < \epsilon_0) [a_\epsilon \le C\psi(\epsilon)]$.

**Definition 2.11.** Elements of $\tilde{\mathcal{G}}(M, \Sigma)$ are denoted by:

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}(M, \Sigma). \tag{2.17} $$

**Remark 2.5.** With componentwise operations $(\cdot, \pm) \tilde{\mathcal{G}}(M, \Sigma)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \mathrm{cl}[(L_\xi u_\epsilon)_\epsilon]$.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \tilde{\mathcal{G}}(M, \Sigma)$ iff $u \circ \psi_\alpha \in \tilde{\mathcal{G}}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \tilde{\mathcal{G}}(M, \Sigma)$ iff $u \circ \psi_\alpha \in \tilde{\mathcal{G}}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 2.6.** Smooth functions $f \in C^\infty(M \setminus \Sigma)$ are embedded into $\tilde{\mathcal{G}}(M, \Sigma)$ simply by the “constant” embedding $\sigma$, i.e., $\sigma(f) = \mathrm{cl}[(f)_\epsilon]$, hence $C^\infty(M \setminus \Sigma)$ is a faithful subalgebra of $\tilde{\mathcal{G}}(M, \Sigma)$.

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

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

Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".

**Definition 2.12.** Inserting $p \in M$ into $u \in \tilde{\mathcal{G}}(M, \Sigma)$ yields a well defined element of the ring of constants (also called generalized numbers) $\tilde{\mathcal{K}}$ (corresponding to $K = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers $((r_\epsilon)_\epsilon \in K^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $\tilde{\mathcal{G}}(M, \Sigma)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ \tilde{\mathcal{K}}_{\Sigma} = \mathcal{E}_c(M, \Sigma) / \mathcal{N}_c(M, \Sigma) \tag{2.18} $$

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

Where
$$
\begin{equation} \tag{2.16}
\begin{split}
\mathcal{N}(M, \Sigma) &\triangleq \{(u_\epsilon)_\epsilon | \forall K (K \subsetneq M \setminus \Sigma), \forall k (k \in \mathbb{N}_0) \forall q (q \in N) \\
&\phantom{\triangleq \{} \quad \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M \setminus \Sigma)) \left[ \sup_{p \in K} |L_{\xi_1} \dots L_{\xi_k} u_\epsilon(p)| = O(\epsilon^q), \epsilon \to 0 \right] \} \& \\
&\phantom{\triangleq \{} \quad \forall K (K \subsetneq M) \forall k (k \in \mathbb{N}) \exists N (N \in \mathbb{N}) \forall (f \in C^\infty(M)) \forall \xi_1, \dots, \xi_k (\xi_1, \dots, \xi_k \in \mathfrak{X}(M)) \\
&\phantom{\triangleq \{} \quad \left[ \|L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon\| = \left( \sup_{f \in C^\infty(M)} |L_{\xi_1}^w \dots L_{\xi_k}^w u_\epsilon(f)| \right) = O(\epsilon^q), \epsilon \to 0 \right] \},
\end{split}
\end{equation}
$$

where $L_{\xi_k}^w$ denoting the weak Lie derivative in L.Schwartz sense.In the definition the Landau symbol $a_\epsilon = O(\psi(\epsilon))$ appears, having the following meaning: $\exists C (C > 0) \exists \epsilon_0 (\epsilon_0 \in (0, 1]) \forall \epsilon (\epsilon < \epsilon_0) [a_\epsilon \le C\psi(\epsilon)]$.

**Definition 2.11.** Elements of $\tilde{\mathcal{G}}(M, \Sigma)$ are denoted by:

$$ u = \mathrm{cl}[(u_\epsilon)_\epsilon] \triangleq (u_\epsilon)_\epsilon + \mathcal{N}(M, \Sigma). \quad (2.17) $$

**Remark 2.5.** With componentwise operations $(\cdot, \pm) \tilde{\mathcal{G}}(M, \Sigma)$ is a fine sheaf of differential algebras with respect to the Lie derivative defined by $L_\xi u \triangleq \mathrm{cl}[(L_\xi u_\epsilon)_\epsilon]$.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \tilde{\mathcal{G}}(M, \Sigma)$ iff $u \circ \psi_\alpha \in \tilde{\mathcal{G}}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., $u \in \tilde{\mathcal{G}}(M, \Sigma)$ iff $u \circ \psi_\alpha \in \tilde{\mathcal{G}}(\psi_\alpha(V_\alpha))$ for all charts $(V_\alpha, \psi_\alpha)$, where on the open set $\psi_\alpha(V_\alpha) \subset \mathbb{R}^n$ in the respective estimates Lie derivatives are replaced by partial derivatives.

**Remark 2.6.** Smooth functions $f \in C^\infty(M \setminus \Sigma)$ are embedded into $\tilde{\mathcal{G}}(M, \Sigma)$ simply by the “constant” embedding $\sigma$, i.e., $\sigma(f) = \mathrm{cl}[(f)_\epsilon]$, hence $C^\infty(M \setminus \Sigma)$ is a faithful subalgebra of $\tilde{\mathcal{G}}(M, \Sigma)$.

## 2.4 Point Values of a Supergeneralized Functions on M. Supergeneralized Numbers

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

Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these "numbers".

**Definition 2.12.** Inserting $p \in M$ into $u \in \tilde{\mathcal{G}}(M, \Sigma)$ yields a well defined element of the ring of constants (also called generalized numbers) $\tilde{\mathcal{K}}$ (corresponding to $\mathbf{K} = \mathbb{R}$ resp. $\mathbb{C}$), defined as the set of moderate nets of numbers $((r_\epsilon)_\epsilon \in \mathbf{K}^{(0,1]}$ with $|r_\epsilon| = O(\epsilon^{-N})$ for some $N$) modulo negligible nets ($|r_\epsilon| = O(\epsilon^m)$ for each $m$); componentwise insertion of points of $M$ into elements of $\tilde{\mathcal{G}}(M, \Sigma)$ yields well-defined generalized numbers, i.e., elements of the ring of constants:

$$ \tilde{\mathcal{K}}_{\Sigma} = \mathcal{E}_c(M, \Sigma) / \mathcal{N}_c(M, \Sigma) \quad (2.18) $$

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

Where
$$
\left\{
\begin{array}{l}
\mathcal{E}_c(M, \Sigma) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \exists n (n \in \mathbb{N}) \ [|r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0]\}, \\
\mathcal{N}_c(M, \Sigma) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \forall m (m \in \mathbb{N}) \ [|r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0]\} \\
\qquad I = (0, 1].
\end{array}
\right.
\qquad (2.19)
$$

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

**Definition 2.13.** For $u \in \tilde{\mathcal{G}}(M, \Sigma)$ and $x_0 \in M$, the point value of $u$ at the point $x_0$, $u(x_0)$, is defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $\tilde{\mathcal{K}}$.

**Definition 2.14.** We say that an element $r \in \tilde{\mathcal{K}}$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

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

**Definition 2.15.** Let $\Omega$ be an open subset of $\mathbb{R}^n \setminus \Sigma$. On a set $\hat{\Omega}_\Sigma$:

$$
\left\{
\begin{array}{l}
\hat{\Omega}_{\Sigma} = \{(x_{\epsilon})_{\epsilon} \in (\Omega \setminus \Sigma)^{I} | \exists p (p > 0) \ [|x_{\epsilon}| = O(\epsilon^{p})]\} = \\
\quad \{(x_{\epsilon})_{\epsilon} \in (\Omega \setminus \Sigma)^{I} | \exists p (p > 0) \ \exists \epsilon_{0} (\epsilon_{0} > 0) \ [|x_{\epsilon}| \le \epsilon^{p}, \text{ for } 0 < \epsilon < \epsilon_{0}]\}
\end{array}
\right.
\qquad (2.20)
$$

we introduce an equivalence relation:

$$
(x_\epsilon)_\epsilon \sim (y_\epsilon)_\epsilon \iff \forall q (q > 0) \forall \epsilon (\epsilon > 0) [|x_\epsilon - y_\epsilon| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0] \quad (2.21)
$$

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

$$
\tilde{\Omega}_{\Sigma, c} = \{ \tilde{x} = \text{cl}[(x_{\epsilon})_{\epsilon}] \in \tilde{\Omega}_{\Sigma} | \exists K (K \subset \Omega \setminus \Sigma) \exists \epsilon_{0} (\epsilon_{0} > 0) [x_{\epsilon} \in K \text{ for } 0 < \epsilon < \epsilon_{0}] \} \quad (2.22)
$$

**Definition 2.16.** A supergeneralized function $u \in \tilde{\mathcal{G}}(M, \Sigma)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in L. *Schwartz's sense* if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly,i.e.

$$
w - \lim_{\epsilon \to 0} u_{\epsilon} = 0 \qquad (2.23)
$$

We shall often write:

$$
u \underset{\text{Sch}}{\approx} 0. \qquad (2.24)
$$

**Definition 2.17.** The $\tilde{\mathcal{G}}(M, \Sigma)$-module of supergeneralized sections in vector bundles- especially the space of generalized tensor fields $\mathcal{T}_s^r(M\setminus\Sigma)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

$$
\tilde{\mathcal{G}}_s^r(M, \Sigma) = \tilde{\mathcal{G}}(M, \Sigma) \otimes \mathcal{T}_s^r(M \setminus \Sigma), \qquad (2.25)
$$

where $\mathcal{T}_s^r(M\setminus\Sigma)$ denotes the space of smooth tensor fields and the tensor product is taken over the
$$
\left\{
\begin{array}{l}
\mathcal{E}_c (M, \Sigma) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \exists n (n \in \mathbb{N}) \ [|r_\epsilon| = O(\epsilon^{-n}) \text{ as } \epsilon \to 0]\}, \\
\mathcal{N}_c (M, \Sigma) = \{(r_\epsilon)_\epsilon \in \mathbf{K}^I | \forall m (m \in \mathbb{N}) \ [|r_\epsilon| = O(\epsilon^m) \text{ as } \epsilon \to 0]\} \\
I = (0, 1].
\end{array}
\right.
\qquad (2.19)
$$

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

**Definition 2.13.** For $u \in \tilde{\mathcal{G}}(M, \Sigma)$ and $x_0 \in M$, the point value of $u$ at the point $x_0$, $u(x_0)$, is defined as the class of $(u_\epsilon(x_0))_\epsilon$ in $\tilde{\mathcal{K}}$.

**Definition 2.14.** We say that an element $r \in \tilde{\mathcal{K}}$ is strictly nonzero if there exists a representative $(r_\epsilon)_\epsilon$ and a $q \in \mathbb{N}$ such that $|r_\epsilon| \ge \epsilon^q$ for $\epsilon$ sufficiently small. If $r$ is strictly nonzero, then it is also invertible with the inverse $[(1/r_\epsilon)_\epsilon]$. The converse is true as well.

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

**Definition 2.15.** Let $\Omega$ be an open subset of $\mathbb{R}^n \setminus \Sigma$. On a set $\hat{\Omega}_\Sigma$:

$$
\left\{
\begin{array}{l}
\hat{\Omega}_{\Sigma} = \{(x_{\epsilon})_{\epsilon} \in (\Omega \setminus \Sigma)^{I} | \exists p (p > 0) \ [|x_{\epsilon}| = O(\epsilon^{p})]\} = \\
\quad \{(x_{\epsilon})_{\epsilon} \in (\Omega \setminus \Sigma)^{I} | \exists p (p > 0) \ \exists \epsilon_{0} (\epsilon_{0} > 0) \ [|x_{\epsilon}| \le \epsilon^{p}, \text{ for } 0 < \epsilon < \epsilon_{0}]\}
\end{array}
\right.
\qquad (2.20)
$$

we introduce an equivalence relation:

$$
(x_\epsilon)_\epsilon \sim (y_\epsilon)_\epsilon \iff \forall q (q > 0) \forall \epsilon (\epsilon > 0) [|x_\epsilon - y_\epsilon| \le \epsilon^q, \text{ for } 0 < \epsilon < \epsilon_0] \quad (2.21)
$$

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

$$
\tilde{\Omega}_{\Sigma,c} = \{ \tilde{x} = \text{cl}[(x_{\epsilon})_{\epsilon}] \in \tilde{\Omega}_{\Sigma} | \exists K (K \subset \Omega \setminus \Sigma) \exists \epsilon_{0} (\epsilon_{0} > 0) [x_{\epsilon} \in K \text{ for } 0 < \epsilon < \epsilon_{0}] \} \quad (2.22)
$$

**Definition 2.16.** A supergeneralized function $u \in \tilde{\mathcal{G}}(M, \Sigma)$ is called associated to zero, $u \approx 0$ on $\Omega \subseteq M$ in L. *Schwartz's sense* if one (hence any) representative $(u_\epsilon)_\epsilon$ converges to zero weakly,i.e.

$$
w - \lim_{\epsilon \to 0} u_{\epsilon} = 0 \qquad (2.23)
$$

We shall often write:

$$
u \underset{\text{Sch}}{\approx} 0. \qquad (2.24)
$$

**Definition 2.17.** The $\tilde{\mathcal{G}}(M, \Sigma)$-module of supergeneralized sections in vector bundles- especially the space of generalized tensor fields $\mathcal{T}_s^r(M\setminus\Sigma)$-is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields

$$
\tilde{\mathcal{G}}_s^r(M, \Sigma) = \tilde{\mathcal{G}}(M, \Sigma) \otimes \mathcal{T}_s^r(M \setminus \Sigma), \qquad (2.25)
$$

where $\mathcal{T}_s^r(M\setminus\Sigma)$ denotes the space of smooth tensor fields and the tensor product is taken over the
module $C^\infty(M\setminus\Sigma)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting $[]$, in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $\tilde{G}(M, \Sigma)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $\tilde{G}(M, \Sigma)$-modules we have

$$ \tilde{G}_s^r(M, \Sigma) \cong L_{(M)}(\tilde{G}_1^0(M, \Sigma))^r, \tilde{G}_0^1(M, \Sigma)^s; \tilde{G}(M, \Sigma)). \quad (2.26) $$

In particular a supergeneralized metric is defined to be a symmetric, supergeneralized (0, 2)-tensor field $g_{ab} = [(g_\epsilon)_{ab}]_\epsilon$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in $\tilde{G}(M, \Sigma)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have $\forall K \subset M\setminus\Sigma \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^q]$ for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric iff (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $\tilde{G}(M, \Sigma)$-linear isomorphism from $\tilde{G}_0^1(M, \Sigma)$ to $\tilde{G}_1^0(M, \Sigma)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $\tilde{G}_0^2(M, \Sigma)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also the supergeneralized Levi-Civita connection as well as the supergeneralized Riemann, Ricci and Einstein tensor of a supergeneralized metric are defined simply by the usual coordinate formulae on the level of representatives.

## 2.5 Distributional General Relativity

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

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

of space-time $M$, the metric can be diagonalized as $ds_p^2 = \eta_{\mu\nu}(dX^\mu)_p(dX^\nu)_p$ with $\eta_{\mu\nu} \triangleq (-1, 1, 1, 1)$, by choosing the coordinate system $\{X^\mu; \mu = 0, 1, 2, 3\}$ appropriately.

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

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

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

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

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

The classical smooth curvature tensor is given by
module $C^\infty(M\setminus\Sigma)$. Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting $[]$, in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as $\tilde{G}(M, \Sigma)$-multilinear maps taking generalized vector and covector fields to generalized functions, i.e., as $\tilde{G}(M, \Sigma)$-modules we have

$$ \tilde{G}_s^r(M, \Sigma) \cong L_{(M)}(\tilde{G}_1^0(M, \Sigma))^r, \tilde{G}_0^1(M, \Sigma)^s; \tilde{G}(M, \Sigma)). \quad (2.26) $$

In particular a supergeneralized metric is defined to be a symmetric, supergeneralized (0, 2)-tensor field $g_{ab} = [(g_\epsilon)_{ab}]_\epsilon$ (with its index independent of $\epsilon$ and) whose determinant $\det(g_{ab})$ is invertible in $\tilde{G}(M, \Sigma)$. The latter condition is equivalent to the following notion called strictly nonzero on compact sets: for any representative $\det((g_\epsilon)_{ab})_\epsilon$ of $\det(g_{ab})$ we have $\forall K \subset M\setminus\Sigma \exists m \in \mathbb{N}[\inf_{p \in K} |\det(g_{ab}(\epsilon))| \ge \epsilon^q]$ for all $\epsilon$ small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: $g_{ab}$ is a generalized metric iff (on every relatively compact open subset $V$ of $M$) there exists a representative $((g_\epsilon)_{ab})_\epsilon$ of $g_{ab}$ such that for fixed $\epsilon$ (small enough) $(g_\epsilon)_{ab} = g_{ab}(\epsilon)$ (resp. $(g_\epsilon)_{ab}|_V$) is a classical pseudo-Riemannian metric and $\det(g_{ab})$ is invertible in the algebra of generalized functions. A generalized metric induces a $\tilde{G}(M, \Sigma)$-linear isomorphism from $\tilde{G}_0^1(M, \Sigma)$ to $\tilde{G}_1^0(M, \Sigma)$ and the inverse metric $g^{ab} \triangleq [(g_{ab}^{-1}(\epsilon))_\epsilon]$ is a well defined element of $\tilde{G}_0^2(M, \Sigma)$ (i.e., independent of the representative $((g_\epsilon)_{ab})_\epsilon$). Also the supergeneralized Levi-Civita connection as well as the supergeneralized Riemann, Ricci and Einstein tensor of a supergeneralized metric are defined simply by the usual coordinate formulae on the level of representatives.

## 2.5 Distributional General Relativity

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

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

of space-time $M$, the metric can be diagonalized as $ds_p^2 = \eta_{\mu\nu}(dX^\mu)_p(dX^\nu)_p$ with $\eta_{\mu\nu} \triangleq (-1, 1, 1, 1)$, by choosing the coordinate system $\{X^\mu; \mu = 0, 1, 2, 3\}$ appropriately.

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

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

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

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

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

The classical smooth curvature tensor is given by
$$R^{\rho}_{\sigma\mu\nu} \triangleq \partial_{\mu}\{\frac{\rho}{\sigma\nu}\} - \partial_{\nu}\{\frac{\rho}{\sigma\mu}\} + \{\frac{\rho}{\lambda\mu}\}\{\frac{\lambda}{\sigma\nu}\} - \{\frac{\rho}{\lambda\nu}\}\{\frac{\lambda}{\sigma\mu}\} \quad (2.29)$$

with $\{\frac{\rho}{\sigma\nu}\}$ being the smooth Christoffel symbol. The supergeneralized nonsmooth curvature tensor is given by

$$\left\{ (R^{\rho}_{\sigma\mu\nu}(\epsilon))_{\epsilon} \triangleq \partial_{\mu}\{\{\frac{\rho}{\sigma\nu}\}_{\epsilon}\}_{\epsilon} - \partial_{\nu}\{\{\frac{\rho}{\lambda\mu}\}_{\epsilon}\}_{\epsilon} + \{\{\frac{\rho}{\lambda\mu}\}_{\epsilon}\}_{\epsilon}\{\{\frac{\lambda}{\sigma\nu}\}_{\epsilon}\}_{\epsilon} - \{\{\frac{\rho}{\lambda\nu}\}_{\epsilon}\}_{\epsilon}\{\{\frac{\lambda}{\sigma\mu}\}_{\epsilon}\}_{\epsilon} \right\} \quad (2.30)$$

with $\{\frac{\rho}{\sigma\nu}\}_{\epsilon}$ being the supergeneralized Christoffel symbol. The fundamental classical action integral $\mathcal{I}$ is

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

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

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

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

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

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

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

with

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

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

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

where $(\mathcal{L}_{M}(\epsilon))_{\epsilon}$ is the supergeneralized Lagrangian density of a gravitational source and $(\bar{\mathcal{L}}_{G}(\epsilon))_{\epsilon}$ is the supergeneralized gravitational Lagrangian density given by

$$(\bar{\mathcal{L}}_{G}(\epsilon))_{\epsilon} = \frac{1}{2\kappa} (\mathcal{G}_{\epsilon})_{\epsilon}. \quad (2.37)$$

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

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

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

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

with

$$(\mathcal{D}_{\epsilon}^{\mu})_{\epsilon} = -\sqrt{-(g_{\epsilon})_{\epsilon}} \left( (g_{\epsilon}^{\mu\nu})_{\epsilon} \left\{ \frac{\lambda}{\nu\lambda} \right\}_{\epsilon} - (g_{\epsilon}^{\nu\lambda})_{\epsilon} \left\{ \frac{\mu}{\nu\lambda} \right\}_{\epsilon} \right). \quad (2.40)$$
$$R^{\rho}_{\sigma\mu\nu} \triangleq \partial_{\mu}\{\frac{\rho}{\sigma\nu}\} - \partial_{\nu}\{\frac{\rho}{\sigma\mu}\} + \{\frac{\rho}{\lambda\mu}\}\{\frac{\lambda}{\sigma\nu}\} - \{\frac{\rho}{\lambda\nu}\}\{\frac{\lambda}{\sigma\mu}\} \quad (2.29)$$

with $\{\frac{\rho}{\sigma\nu}\}$ being the smooth Christoffel symbol. The supergeneralized nonsmooth curvature tensor is given by

$$\left\{ (R^{\rho}_{\sigma\mu\nu}(\epsilon))_{\epsilon} \triangleq \partial_{\mu}\{\{\frac{\rho}{\sigma\nu}\}_{\epsilon}\}_{\epsilon} - \partial_{\nu}\{\{\frac{\rho}{\lambda\mu}\}_{\epsilon}\}_{\epsilon} + \{\{\frac{\rho}{\lambda\mu}\}_{\epsilon}\}_{\epsilon}\{\{\frac{\lambda}{\sigma\nu}\}_{\epsilon}\}_{\epsilon} - \{\{\frac{\rho}{\lambda\nu}\}_{\epsilon}\}_{\epsilon}\{\{\frac{\lambda}{\sigma\mu}\}_{\epsilon}\}_{\epsilon} \right. \quad (2.30)$$

with $\{\frac{\rho}{\sigma\nu}\}_{\epsilon}$ being the supergeneralized Christoffel symbol. The fundamental classical action integral $\mathcal{I}$ is

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

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

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

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

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

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

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

with

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

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

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

where $(\mathcal{L}_{M}(\epsilon))_{\epsilon}$ is the supergeneralized Lagrangian density of a gravitational source and $(\bar{L}_{G}(\epsilon))_{\epsilon}$ is the supergeneralized gravitational Lagrangian density given by

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

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

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

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

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

with

$$(\mathcal{D}_{\epsilon}^{\mu})_{\epsilon} = -\sqrt{-(g_{\epsilon})_{\epsilon}} \left( (g_{\epsilon}^{\mu\nu})_{\epsilon} \left\{ \frac{\lambda}{\nu\lambda} \right\}_{\epsilon} - (g_{\epsilon}^{\nu\lambda})_{\epsilon} \left\{ \frac{\mu}{\nu\lambda} \right\}_{\epsilon} \right). \quad (2.40)$$
Also, we have defined the classical scalar curvature by

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

With the smooth Ricci tensor

$$ R_{\mu\nu} = R^\lambda_{\mu\lambda\nu}. \qquad (2.42) $$

From the action $\mathcal{I}$, the classical Einstein equation

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

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

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

With

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

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

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

With the supergeneralized Ricci tensor

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

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

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

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

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

With

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

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

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

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

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

With the smooth Ricci tensor

$$ R_{\mu\nu} = R^\lambda_{\mu\lambda\nu}. \qquad (2.42) $$

From the action $\mathcal{I}$, the classical Einstein equation

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

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

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

With

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

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

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

With the supergeneralized Ricci tensor

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

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

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

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

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

With

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

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

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

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

Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational singularities, Distributional SAD BHs and AdS BHs in the AdS-BH Reggeized Vacuum Dominance

## 3.1 Calculation of the Stress-tensor by using Nonsmooth

$$ (t_{\mu}^{\nu}(\epsilon) \text{Regulation via Horizon} (\partial_{\nu}L_{\mu\nu}(\epsilon))_{\epsilon} - (\partial_{\sigma\tau}q_{\sigma\tau}(\epsilon) / dx^{\sigma})_{\epsilon}) \quad (2.52) $$

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

$$ \left\{ \begin{array}{ll} \text{above horizon } r > 2m: & ds^{+2} = h^{+}(r)dt^2 - [h^{+}(r)]^{-1}dr^2 + r^2d\Omega^2, \\ h^{+}(r) = -1 + \frac{2m}{r} = -\frac{r-2m}{r} & \\ \text{below horizon } r < 2m: & ds^{-2} = h^{-}(r)dt^2 - h^{-}(r)^{-1}dr^2 + r^2d\Omega^2, \\ h^{-}(r) = -1 + \frac{2m}{r} = \frac{2m-r}{r} & \end{array} \right. \qquad (3.1) $$

**Remark 3.1.** Following the above discussion we consider the metric coefficients $h^{+}(r)$, $[h^{+}(r)]^{-1}h^{-}(r)$, and $[h^{-}(r)]^{-1}$ as an element of $\mathcal{D}'(\mathbb{R}^3)$ and embed it into $(\mathcal{G}(\mathbb{R}^3))$ by replacements above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly

$$ 3mm] \quad r \ge 2m : r - 2m \mapsto \sqrt{(r - 2m)^2 + \epsilon^2}, \\ r \le 2m : 2m - r \mapsto \sqrt{(2m - r)^2 + \epsilon^2}. $$

**Remark 3.2.** Note that, accordingly, we have fixed the differentiable structure of the manifold: the usual Schwarzschild coordinates and the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1) are extended on $r = 2m$ through the horizon. Therefore we have above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly

$$ h(r) = \left\{ \begin{array}{ll} -\frac{r-2m}{r} & \text{if } r \ge 2m \\ 0 & \text{if } r \le 2m \end{array} \right\} \mapsto (h_{\epsilon}^{+}(r))_{\epsilon} = \left( -\frac{\sqrt{(r-2m)^2 + \epsilon^2}}{r} \right)_{\epsilon}, \\ \text{where } (h_{\epsilon}^{+}(r))_{\epsilon} \in \tilde{\mathcal{G}}(\mathbb{R}^3, B^{+}(2m, R)), B^{+}(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}. $$

$$ h^{-1}(r) = \left\{ \begin{array}{ll} -\frac{r}{r-2m}, & r > 2m \\ \infty, & r = 2m \end{array} \right\} \mapsto (h_{\epsilon}^{+})^{-1}(r) = \\ h^{-}(r) = \left\{ \begin{array}{ll} -\frac{r-2m}{r} & \text{if } r \le 2m \\ 0 & \text{if } r \ge 2m \end{array} \right\} \mapsto h_{\epsilon}^{-}(r) = \\ = \left( \frac{\sqrt{(2m-r)^2 + \epsilon^2}}{r} \right) \in \tilde{\mathcal{G}}(\mathbb{R}^3, B^{-}(0, 2m)), \\ \text{where } B^{-}(0, 2m) = \left\{ x \in \mathbb{R}^3 \middle| 0 < \|x\| \le 2m \right\} \\ \left\{ \begin{array}{ll} -\frac{r}{r-2m}, & r < 2m \\ \infty, & r = 2m \end{array} \right\} \mapsto (h_{\epsilon}^{-})^{-1}(r) = \\ = \left( \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \right) \in \tilde{\mathcal{G}}(\mathbb{R}^3, B^{-}(0, 2m)) $$
3 Distributional Schwarzschild Geometry from

Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational singularities, Nonsmooth Regularization via Horizon

3.1 Calculation of the Stress-tensor by using Nonsmooth

$$
(t_{\mu}^{\nu}(\epsilon, \epsilon) \text{Regulation via Horizon}(\partial_{\sigma\tau}(\epsilon), \epsilon))_{\epsilon} \quad (2.52)
$$

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

$$
\left\{
\begin{array}{ll}
\text{above horizon } r > 2m: & \\
\displaystyle ds^2 = h^+(r)dt^2 - [h^+(r)]^{-1}dr^2 + r^2d\Omega^2, & \\
\displaystyle h^+(r) = -1 + \frac{2m}{r} = -\frac{r-2m}{r} & \quad (3.1) \\
\text{below horizon } r < 2m: & \\
\displaystyle ds^2 = h^-(r)dt^2 - h^-(r)^{-1}dr^2 + r^2d\Omega^2, & \\
\displaystyle h^-(r) = -1 + \frac{2m}{r} = \frac{2m-r}{r} &
\end{array}
\right.
$$

**Remark 3.1.** Following the above discussion we consider the metric coefficients $h^+(r)$, $[h^+(r)]^{-1}$ $h^-(r)$, and $[h^-(r)]^{-1}$ as an element of $\mathcal{D}'(\mathbb{R}^3)$ and embed it into $(\mathcal{G}(\mathbb{R}^3))$ by replacements above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly

$$
3mm] \quad r \ge 2m : r - 2m \mapsto \sqrt{(r - 2m)^2 + \epsilon^2}, \\
r \le 2m : 2m - r \mapsto \sqrt{(2m - r)^2 + \epsilon^2}.
$$

**Remark 3.2.** Note that, accordingly, we have fixed the differentiable structure of the manifold: the usual Schwarzschild coordinates and the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1) are extended on $r = 2m$ through the horizon. Therefore we have above horizon $r \ge 2m$ and below horizon $r \le 2m$ correspondingly

$$
h(r) = \left\{ \begin{array}{ll} -\frac{r-2m}{r} & \text{if } r \ge 2m \\ 0 & \text{if } r \le 2m \end{array} \right\} \mapsto (h_{\epsilon}^{+}(r))_{\epsilon} = \left( -\frac{\sqrt{(r-2m)^2 + \epsilon^2}}{r} \right)_{\epsilon}, \\
\text{where } (h_{\epsilon}^{+}(r))_{\epsilon} \in \tilde{\mathcal{G}}(\mathbb{R}^3, B^{+}(2m, R)), B^{+}(2m, R) = \{x \in \mathbb{R}^3 | 2m \le \|x\| \le R\}.
$$

$$
\begin{align}
h^{-1}(r) &= \left\{ \begin{array}{ll} -\frac{r}{r-2m}, & r > 2m \\ \infty, & r = 2m \end{array} \right\} \mapsto (h_{\epsilon}^{+})^{-1}(r) = \nonumber \\
h^{-}(r) &= \left\{ \begin{array}{ll} -\frac{r-2m}{r} & \text{if } r \le 2m \\ 0 & \text{if } r \ge 2m \end{array} \right\} \mapsto h_{\epsilon}^{-}(r) = \nonumber \\
&= \left( \frac{\sqrt{(2m-r)^2 + \epsilon^2}}{r} \right) \in \tilde{\mathcal{G}}(\mathbb{R}^3, B^{-}(0, 2m)), \nonumber \\
\text{where } B^{-}(0, 2m) &= \left\{ x \in \mathbb{R}^3 \middle| 0 < \|x\| \le 2m \right\} \nonumber \\
&\quad \left\{ \begin{array}{ll} -\frac{r}{r-2m}, & r < 2m \\ \infty, & r = 2m \end{array} \right\} \mapsto (h_{\epsilon}^{-})^{-1}(r) = \nonumber \\
&= \left( \frac{r}{\sqrt{(r-2m)^2 + \epsilon^2}} \right) \in \tilde{\mathcal{G}}(\mathbb{R}^3, B^{-}(0, 2m)) \tag{3.2}
\end{align}
$$
Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) horizon, i.e.,

$$ \begin{cases} (ds_{\epsilon}^{+})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (ds_{\epsilon}^{-})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2} \end{cases} \qquad (3.3) $$

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

$$ \left\{ \begin{aligned} \left( [\mathbf{R}_{\epsilon}^{+}]_{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}]_{1} \right)_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{+''})_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{+'})_{\epsilon} \right) \\ \left( [\mathbf{R}_{\epsilon}^{+}]_{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}]_{3} \right)_{\epsilon} = \frac{(h_{\epsilon}^{+'})_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{+})_{\epsilon}}{r^2}. \end{aligned} \right. \qquad (3.4) $$

From (3.2) we obtain

$$ \begin{aligned} h_{\epsilon}^{+'}(r) &= -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2}, \\ r \left\{ -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2} \right\} + 1 - \frac{\sqrt{[(r-2m)^2 + \epsilon^2]}}{r} &= \\ -\frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r} + 1 - \frac{\sqrt{[(r-2m)^2 + \epsilon^2]}}{r} &= \\ -\frac{r-2m}{[(r-2m)^2 + \epsilon^2]^{1/2}} + 1. \end{aligned} \\ h_{\epsilon}''(r) = -\left(\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}}\right)' + \left(\frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2}\right)' = \\ = -\frac{1}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \\ + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r^3}. \end{aligned} \qquad (3.5) \\ r^2 \left\{ -\frac{1}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \\ + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r^3} \right\} + \\ + 2r \left\{ -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2} \right\} = \\ -\frac{r}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \\ + \frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r} + \\ - \frac{2(r-2m)}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r} = \\ - \frac{r}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{3/2}}. \end{aligned} $$

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

$$ \tilde{\Phi}(r) = \pi_0 \sin \theta d\theta_0^{2\pi} d\phi \Phi(x) $$
Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) horizon, i.e.,

$$ \begin{cases} (ds_{\epsilon}^{+})_{\epsilon} = (h_{\epsilon}^{+}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{+}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2}, \\ (ds_{\epsilon}^{-})_{\epsilon} = (h_{\epsilon}^{-}(r)dt^{2})_{\epsilon} - ([h_{\epsilon}^{-}(r)]^{-1}dr^{2})_{\epsilon} + r^{2}d\Omega^{2} \end{cases} \quad (3.3) $$

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

$$ \left\{ \begin{aligned} \left( [\mathbf{R}_{\epsilon}^{+}]_{0} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}]_{1} \right)_{\epsilon} = \frac{1}{2} \left( (h_{\epsilon}^{+''})_{\epsilon} + \frac{2}{r} (h_{\epsilon}^{+'})_{\epsilon} \right) \\ \left( [\mathbf{R}_{\epsilon}^{+}]_{2} \right)_{\epsilon} &= \left( [\mathbf{R}_{\epsilon}^{+}]_{3} \right)_{\epsilon} = \frac{(h_{\epsilon}^{+'})_{\epsilon}}{r} + \frac{1 + (h_{\epsilon}^{+})_{\epsilon}}{r^2}. \end{aligned} \right. \quad (3.4) $$

From (3.2) we obtain

$$ \begin{aligned} h_{\epsilon}^{+'}(r) &= -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2}, \\ r \left\{ -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2} \right\} + 1 - \frac{\sqrt{[(r-2m)^2 + \epsilon^2]}}{r} &= \\ -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r} + 1 - \frac{\sqrt{[(r-2m)^2 + \epsilon^2]}}{r} &= \\ -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + 1. \end{aligned} \\ h_{\epsilon}''(r) = -\left(\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}}\right)' + \left(\frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2}\right)' = \\ = -\frac{1}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \\ + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r^3}. \end{aligned} \quad (3.5) \\ r^2 \left\{ -\frac{1}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \\ + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r^3} \right\} + \\ + 2r \left\{ -\frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{[(r-2m)^2 + \epsilon^2]^{1/2}}{r^2} \right\} = \\ -\frac{r}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \\ + \frac{r-2m}{r[(r-2m)^2 + \epsilon^2]^{1/2}} - \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r} + \\ - \frac{2(r-2m)}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{2[(r-2m)^2 + \epsilon^2]^{1/2}}{r} = \\ - \frac{r}{r[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r(r-2m)^2}{r[(r-2m)^2 + \epsilon^2]^{3/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{1/2}} + \frac{r-2m}{r^2[(r-2m)^2 + \epsilon^2]^{3/2}}. \end{aligned} $$

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

$$ \tilde{\Phi}(r) = \pi_0 \sin \theta d\theta_0^{2\pi} d\phi \Phi(x) $$
and let $\Phi(x)$ be the function $\Phi(x) \in S_{2m}^{+}(\mathbb{R}^3)$, where by $S_{2m}^{+}(\mathbb{R}^3)$ we denote the class of all functions $\Phi(x)$ with compact support such that:

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

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

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

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

$$ \left\{ \begin{aligned} & \lim_{\epsilon \to 0} \mathbf{I}_3^+(\epsilon) = \lim_{\epsilon \to 0} \mathbf{I}_2^+(\epsilon) = \\ & \lim_{\epsilon \to 0} \left\{ -\epsilon \tilde{\Phi}(2m) \left[ \sqrt{\left(\frac{R-2m}{\epsilon}\right)^2 + 1} + 1 - \frac{R-2m}{\epsilon} \right] \right\} + \\ & + \lim_{\epsilon \to 0} \left\{ -\frac{\epsilon^2}{1 0} \frac{R-2m}{\epsilon} \left[ \frac{\eta}{(\eta^2+1)^{1/2}} - 1 \right] \tilde{\Phi}^{(1)}(\xi) \eta d\eta \right\} = 0 \end{aligned} \right. \quad (3.11) $$