Finally, it is easy to obtain similar formulas for the second derivatives of Colombeau generalized tensors for which this is done most easily by considering, for example, Colombeau generalized tensor of the form $(A_{i,\epsilon}B_{k,\epsilon})_{\epsilon}$, and by using formulas from the Eq. (E11) and Eq. (E1); because of the linearity, the formulas thus obtained must be valid for an arbitrary Colombeau generalized tensor $(A_{ik,\epsilon})_{\epsilon}$. Thus we get

$$ (A_{ik;l;m,\epsilon})_{\epsilon} - (A_{ik;m;l,\epsilon})_{\epsilon} = (A_{in,\epsilon}R_{klm,\epsilon})_{\epsilon} + (A_{nk,\epsilon}R_{ilm,\epsilon})_{\epsilon}. \quad (E13) $$

# APPENDIX F. THE MAIN PROPERTIES OF THE DISTRIBUTIONAL CURVATURE TENSOR

From the expression is given by Eq. (E7) it follows directly that for any $x = (x_{\epsilon})_{\epsilon}$ such that $cl[(x_{\epsilon})_{\epsilon}] \in \mathbb{R}^4$ the distributional curvature tensor is antisymmetric in the indices $l$ and $m$: $(R_{klm,\epsilon}^{i}(x_{\epsilon}))_{\epsilon} = -(R_{kml,\epsilon}^{i}(x_{\epsilon}))_{\epsilon}$. (F1)

Obviously, the following identity holds

$$ (R_{kim,\epsilon}^{l}(x_{\epsilon}))_{\epsilon} + (R_{mkl,\epsilon}^{i}(x_{\epsilon}))_{\epsilon} + (R_{lmk,\epsilon}^{i}(x_{\epsilon}))_{\epsilon} = 0_{\mathbb{R}} \quad (F2) $$

In addition to the mixed distributional curvature tensor $(R_{klm,\epsilon}^{i}(x_{\epsilon}))_{\epsilon}$, one also uses the covariant distributional curvature tensor

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} = (g_{in,\epsilon}(x_{\epsilon})R_{klm,\epsilon}^{n}(x_{\epsilon}))_{\epsilon} = $$

$$ ((g_{in,\epsilon}(x_{\epsilon}))_{\epsilon})((R_{klm,\epsilon}^{n}(x_{\epsilon}))_{\epsilon}) \quad (F3) $$

Obviously by means of simple calculation the following expressions for

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} \quad \text{holds} $$

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} = $$

$$ \frac{1}{2}\left(\left(\frac{\partial^2 g_{im,\epsilon}(x_{\epsilon})}{\partial x_{\epsilon}^k \partial x_{\epsilon}^l}\right) + \left(\frac{\partial^2 g_{kl,\epsilon}(x_{\epsilon})}{\partial x_{\epsilon}^l \partial x_{\epsilon}^m}\right) - \left(\frac{\partial^2 g_{il,\epsilon}(x_{\epsilon})}{\partial x_{\epsilon}^k \partial x_{\epsilon}^m}\right)\right. \\ \left. - \left(\frac{\partial^2 g_{km,\epsilon}(x_{\epsilon})}{\partial x_{\epsilon}^l \partial x_{\epsilon}^l}\right)\right) + $$

$$ + ((g_{np,\epsilon}(x_{\epsilon}))_{\epsilon})\left[\left(\Gamma_{kl,\epsilon}^{n}(x_{\epsilon})\Gamma_{im,\epsilon}^{p}(x_{\epsilon})\right)_{\epsilon} - \left(\Gamma_{km,\epsilon}^{n}(x_{\epsilon})\Gamma_{il,\epsilon}^{p}(x_{\epsilon})\right)_{\epsilon}\right] \quad (F4) $$

From this expressions (F4) it follows directly

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} = -(R_{kilm,\epsilon}(x_{\epsilon}))_{\epsilon} = -(R_{ikml,\epsilon}(x_{\epsilon}))_{\epsilon}, $$

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} = (R_{lmik,\epsilon}(x_{\epsilon}))_{\epsilon}. \quad (F5) $$

For $(R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon}$ the following identities holds

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} = (R_{lmik,\epsilon}(x_{\epsilon}))_{\epsilon} \quad (F6) $$

and

$$ (R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon} + (R_{imkl,\epsilon}(x_{\epsilon}))_{\epsilon} + (R_{ilmk,\epsilon}(x_{\epsilon}))_{\epsilon} = 0_{\mathbb{R}} \quad (F7) $$

The Bianchi identity holds
$$ (R_{ikl;m,\epsilon}^{n}(x_{\epsilon}))_{\epsilon} + (R_{ilm;k,\epsilon}^{n}(x_{\epsilon}))_{\epsilon} + (R_{ilm;k,\epsilon}^{n}(x_{\epsilon}))_{\epsilon} = 0_{\mathbb{R}}. \quad (F8) $$

From the Colombeau generalized curvature tensor we can, by contraction, construct Colombeau generalized tensor of the second rank. This contraction can be carried out in only one way: contraction of the Colombeau generalized tensor $(R_{iklm,\epsilon}(x_{\epsilon}))_{\epsilon}$ on the indices $i$ and $k$ or $l$ and $m$ gives zero because of the antisymmetry in these indices, while contraction on any other pair always gives the same result, except for sign. We define the Colombeau generalized tensor $(R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon}$ (the Colombeau generalized Ricci tensor) as

$$ (R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon} = (g_{\epsilon}^{lm}(x_{\epsilon})R_{ilm,k}(x_{\epsilon}))_{\epsilon} = (R_{ilk,\epsilon}^{l}(x_{\epsilon}))_{\epsilon}. \quad (F9) $$

From Eq.(E7) we get

$$ \begin{aligned} (R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon} &= \left(\frac{\partial(\Gamma_{ik,\epsilon}^{l}(x_{\epsilon}))}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} - \left(\frac{\partial(\Gamma_{il,\epsilon}^{l}(x_{\epsilon}))}{\partial x_{\epsilon}^{m}}\right)_{\epsilon} + \\ & \qquad (\Gamma_{ik,\epsilon}^{l}(x_{\epsilon})\Gamma_{lm,\epsilon}^{m}(x_{\epsilon}))_{\epsilon} - (\Gamma_{il,\epsilon}^{m}(x_{\epsilon})\Gamma_{km,\epsilon}^{l}(x_{\epsilon}))_{\epsilon}. \end{aligned} \quad (F12) $$

This Colombeau generalized tensor $(R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon}$ obviously is symmetric:

$$ (R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon} = (R_{ki,\epsilon}(x_{\epsilon}))_{\epsilon}. \quad (F12) $$

From Eq.(E10)-Eq.(E11) we get

$$ \begin{aligned} (R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon} &= \left(\frac{\partial(\Gamma_{ik,\epsilon}^{l}(x_{\epsilon}))}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} - \left(\frac{\partial(\Gamma_{il,\epsilon}^{l}(x_{\epsilon}))}{\partial x_{\epsilon}^{m}}\right)_{\epsilon} + \\ & \qquad (\Gamma_{ik,\epsilon}^{l}(x_{\epsilon})\Gamma_{lm,\epsilon}^{m}(x_{\epsilon}))_{\epsilon} - (\Gamma_{il,\epsilon}^{m}(x_{\epsilon})\Gamma_{km,\epsilon}^{l}(x_{\epsilon}))_{\epsilon} \end{aligned} \quad (F13) $$

and

$$ (R_{ik,\epsilon}(x_{\epsilon}))_{\epsilon} = (R_{ki,\epsilon}(x_{\epsilon}))_{\epsilon}. \quad (F14) $$

# APPENDIX G. THE ACTION FUNCTIONAL FOR THE SINGULAR GRAVITATIONAL FIELD

The action $S[(g_{\epsilon})_{\epsilon}]$ obviously must be expressed in terms of an scalar Colombeau integral taken over all space and over the time coordinate $(x_{\epsilon}^{0})_{\epsilon}$ between two given values,i.e.

$$ S[(g_{\epsilon})_{\epsilon}] = \left( \int G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon} \right)_{\epsilon} = \int (G_{\epsilon})_{\epsilon} \sqrt{-(g_{\epsilon})_{\epsilon}} (d\Omega_{\epsilon})_{\epsilon}. \quad (G1) $$

In order to determine this Colombeau scalar we assume that the equations of the gravitational field must contain derivatives of the "potentials" no higher than the second order. Since the field equations are obtained by varying the action, it is necessary that the integrand $(G_{\epsilon})_{\epsilon}$ contain derivatives of $(g_{ik,\epsilon})_{\epsilon}$ no higher than first order; thus $(G_{\epsilon})_{\epsilon}$ must contain only the tensor $(g_{ik,\epsilon})_{\epsilon}$ and the quantities $(\Gamma_{kl,\epsilon}^{l})_{\epsilon}$. However, it is impossible to construct an invariant from the quantities $(g_{ik,\epsilon})_{\epsilon}$ and $(\Gamma_{kl,\epsilon}^{l})_{\epsilon}$ alone. This is clear from the fact that by a suitable choice of generalized coordinate system we can always make all the quantities $(\Gamma_{kl,\epsilon}^{l})_{\epsilon}$ are zero at a given point. There is, however, the Colombeau scalar $(R_{\epsilon})_{\epsilon}$ (the generalized curvature of the four-space), which though it contains in addition to the $(g_{ik,\epsilon})_{\epsilon}$ and its first derivatives also the second derivatives of $(g_{ik,\epsilon})_{\epsilon}$, is linear in the second order Colombeau derivatives. The action functional for the gravitational field reads $(\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon})_{\epsilon}$. (G2)
The invariant Colombeau integral (G2) can be transformed by means of generalized Gauss' theorem, [see Remark B.7] to the integral of an expression not containing the second derivatives. Thus Colombeau integral (G2) can be presented in the following

$$ \text{form } \left( \int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon} \right)_{\epsilon} = \left( \int G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon} \right)_{\epsilon} + $$

$$ \left( \int \frac{\partial(\sqrt{-g_{\epsilon}} w_{\epsilon}^{i})}{\partial x^{i}} d\Omega_{\epsilon} \right)_{\epsilon}, \qquad (G3) $$

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 Colombeau generalized quantity $(w_{\epsilon}^{i})_{\epsilon}$. According to generalized 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

$$ \delta\left(\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon}\right)_{\epsilon} = \left(\delta\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon}\right)_{\epsilon} = \\ \left(\delta\int G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon}\right)_{\epsilon}. \quad (G4) $$

The left side is Colombeau scalar and 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 $\delta S[(g_{\epsilon})_{\epsilon}] =$

$$ -\frac{c^3}{16\pi\kappa} \left(\delta\int G_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon}\right)_{\epsilon} = -\frac{c^3}{16\pi\kappa} \left(\delta\int R_{\epsilon} \sqrt{-g_{\epsilon}} d\Omega_{\epsilon}\right)_{\epsilon}. \quad (G5) $$

The constant $\kappa$ is called the gravitational constant. The dimensions of $\kappa$ follow from Eq. (G5). Its numerical value is $\kappa = 6.67 \times 10^{-8} \text{ sm } 3 \times \text{gr} -1 \times \text{sec}^{-2}$.

Let us calculate now the quantity $(G_{\epsilon})_{\epsilon}$ of the Eq. (G5). From the Eq. (F12) for

$(R_{ik,\epsilon})_{\epsilon}$, we get

$$ (\sqrt{-g_{\epsilon}} R_{\epsilon})_{\epsilon} = (\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik} R_{ik,\epsilon})_{\epsilon} = \{(\sqrt{-g_{\epsilon}})_{\epsilon}\} \times $$

$$ \left\{ \left( g_{\epsilon}^{ik} \frac{\partial \Gamma_{ik,\epsilon}^{l}}{\partial x_{\epsilon}^{l}} \right)_{\epsilon} - \left( g_{\epsilon}^{ik} \frac{\partial \Gamma_{il,\epsilon}^{l}}{\partial x_{\epsilon}^{k}} \right)_{\epsilon} + \left( g_{\epsilon}^{ik} \Gamma_{ik,\epsilon}^{l} \Gamma_{lm,\epsilon}^{m} \right)_{\epsilon} + \left( g_{\epsilon}^{ik} \Gamma_{il,\epsilon}^{m} \Gamma_{km,\epsilon}^{l} \right)_{\epsilon} \right\}. \quad (G6) $$

In the first two terms on the right hands sight of (E6), we get

$$ \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik} \frac{\partial \Gamma_{ik,\epsilon}^{l}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} = \frac{\partial}{\partial x^{l}} \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik} \Gamma_{ik,\epsilon}^{l}\right)_{\epsilon} - \left\{\left(\Gamma_{ik,\epsilon}^{l}\right)_{\epsilon}\right\} \frac{\partial}{\partial x_{\epsilon}^{l}} \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik}\right)_{\epsilon} $$

$$ \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik} \frac{\partial \Gamma_{il,\epsilon}^{l}}{\partial x_{\epsilon}^{k}}\right)_{\epsilon} = \frac{\partial}{\partial x^{k}} \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik} \Gamma_{il,\epsilon}^{l}\right)_{\epsilon} - \left\{\left(\Gamma_{il,\epsilon}^{l}\right)_{\epsilon}\right\} \frac{\partial}{\partial x_{\epsilon}^{k}} \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik}\right)_{\epsilon}. \quad (G7) $$

Dropping the total derivatives in the Eq. (G7), we find

$$ \begin{aligned} \left(\sqrt{-g_{\epsilon}} G_{\epsilon}\right)_{\epsilon} = & \left\{\left(\Gamma_{lm,\epsilon}^{m}\right)_{\epsilon}\right\} \frac{\partial}{\partial x_{\epsilon}^{k}} \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik}\right)_{\epsilon} - \\ & \left\{\left(\Gamma_{ik,\epsilon}^{l}\right)_{\epsilon}\right\} \frac{\partial}{\partial x_{\epsilon}^{l}} \left(\sqrt{-g_{\epsilon}} g_{\epsilon}^{ik}\right)_{\epsilon}. \end{aligned} \quad (G8) $$
Note that the first two terms on the right are equal to $(\sqrt{-g_{\epsilon}})_{\epsilon}$ multiplied by

$$2(\Gamma^l_{ik,\epsilon}\Gamma^i_{lm,\epsilon}g^{mk}_{\epsilon}) - (\Gamma^m_{im,\epsilon}\Gamma^i_{kl,\epsilon}g^{kl}_{\epsilon}) - (\Gamma^l_{ik,\epsilon}\Gamma^m_{lm,\epsilon}g^{ik}_{\epsilon}) =$$

$$2\{(g^{ik}_{\epsilon})\}_{\epsilon}[(\Gamma^m_{il,\epsilon}\Gamma^l_{km,\epsilon} - \Gamma^l_{ik,\epsilon}\Gamma^m_{lm,\epsilon})]_{\epsilon}. \quad (G9)$$

Finally, we get

$$(G_{\epsilon})_{\epsilon} = \{(g^{ik}_{\epsilon})\}_{\epsilon}[(\Gamma^m_{im,\epsilon}\Gamma^l_{km,\epsilon})_{\epsilon} - (\Gamma^l_{ik,\epsilon}\Gamma^m_{lm,\epsilon})_{\epsilon}]. \quad (G10)$$

# APPENDIX H.THE DISTRIBUTIONAL ENERGY-MOMENTUM TENSOR

In this subsection we consider the general rule for calculating the energy-momentum tensor of any physical system whose action is given in the form of the Colombeau integral over four dimensional distributional spacetime. This integral read

$$(S_{m,\epsilon})_{\epsilon} = \frac{1}{c} \left( \int \Lambda_{\epsilon}(x_{\epsilon}) \sqrt{-g_{\epsilon}(x)} d\Omega_{\epsilon} \right)_{\epsilon}. \quad (H1)$$

Using generalized Gauss' theorem, and setting $(\delta g^{ik}_{\epsilon}) = 0_{\tilde{R}}$ at the integration limits, one finds $\delta(S_{m,\epsilon})$ in the form, (see Remark H.2):

$$\delta(S_{m,\epsilon})_{\epsilon} = \frac{1}{c} \int \left\{ \left( \frac{\partial \sqrt{-g_{\epsilon} \Lambda_{\epsilon}}}{\partial g^{ik}_{\epsilon}} \delta g^{ik}_{\epsilon} \right) + \left( \frac{\partial}{\partial x^l} \frac{\partial \sqrt{-g_{\epsilon} \Lambda_{\epsilon}}}{\partial \frac{\partial g^{ik}_{\epsilon}}{\partial x^l}} \delta \frac{\partial g^{ik}_{\epsilon}}{\partial x^l} \right) \right\} (d\Omega_{\epsilon})_{\epsilon} =$$

$$\frac{1}{c} \int \left\{ \left( \frac{\partial \sqrt{-g_{\epsilon} \Lambda_{\epsilon}}}{\partial g^{ik}_{\epsilon}} \right)_{\epsilon} - \left( \frac{\partial}{\partial x^l} \frac{\partial \sqrt{-g_{\epsilon} \Lambda_{\epsilon}}}{\partial \frac{\partial g^{ik}_{\epsilon}}{\partial x^l}} \delta \frac{\partial g^{ik}_{\epsilon}}{\partial x^l} \right)_{\epsilon} \right\} \{(\delta g^{ik}_{\epsilon})_{\epsilon}\} (d\Omega_{\epsilon})_{\epsilon} \quad (H2)$$

We introduce now the notation

$$\frac{1}{2}(\sqrt{-g_{\epsilon}} T_{ik,\epsilon})_{\epsilon} = \left(\frac{\partial \sqrt{-g_{\epsilon}} \Lambda_{\epsilon}}{\partial g^{ik}_{\epsilon}}\right)_{\epsilon} - \left(\frac{\partial}{\partial x^l_{\epsilon}} \frac{\partial \sqrt{-g_{\epsilon}} \Lambda_{\epsilon}}{\partial \frac{\partial g^{ik}_{\epsilon}}{\partial x^l_{\epsilon}}}\right)_{\epsilon}, \quad (H3)$$

then $\delta(S_{m,\epsilon})_{\epsilon}$ reads

$$\delta(S_{m,\epsilon})_{\epsilon} = \frac{1}{2c} \left( \int T_{ik,\epsilon} \sqrt{-g_{\epsilon}} \delta g^{ik}_{\epsilon} d\Omega_{\epsilon} \right)_{\epsilon} = -\frac{1}{2c} \left( \int T^{ik}_{\epsilon} \sqrt{-g_{\epsilon}} \delta g_{ik,\epsilon} d\Omega_{\epsilon} \right)_{\epsilon}, \quad (H4)$$

note that $(T_{ik,\epsilon}\delta g^{ik}_{\epsilon}) = -(T^{ik}_{\epsilon}\delta g_{ik,\epsilon})_{\epsilon}$ since $(g^{ik}_{\epsilon}\delta g_{ik,\epsilon})_{\epsilon} = -(g_{ik,\epsilon}\delta g^{ik}_{\epsilon})_{\epsilon}$.

Remark H.1.In the Eq.(H1) we carry out now a transformation from the coordinates $(x^i_{\epsilon})$ to the coordinates $(x^i_{\epsilon}) = (x^i_{\epsilon}) + (\xi^i_{\epsilon})$, where the $(\xi^i_{\epsilon}) \in \tilde{R}$ are small quantities. Under this transformation the $(g^{ik}_{\epsilon})$ are transformed according to the formulas

$$(g^{ik}_{\epsilon}(x^{i}_{\epsilon}))_{\epsilon} = (g^{im}_{\epsilon}(x^{i}_{\epsilon}) \frac{\partial x^{i}_{\epsilon}}{\partial x^{l}_{\epsilon}} \frac{\partial x^{k}_{\epsilon}}{\partial x^{m}_{\epsilon}})_{\epsilon} =$$

$$\{(g^{im}_{\epsilon})\}_{\epsilon} \left( \delta^{i}_{l} + \left( \frac{\partial \xi^{i}_{\epsilon}}{\partial x^{l}_{\epsilon}} \right)_{\epsilon} \right) \left( \delta^{k}_{m} + \left( \frac{\partial \xi^{k}_{\epsilon}}{\partial x^{m}_{\epsilon}} \right)_{\epsilon} \right) \approx$$

$$(g^{ik}_{\epsilon}(x^{l}_{\epsilon}))_{\epsilon} + \{(g^{im}_{\epsilon})\}_{\epsilon} \left(\frac{\partial \xi^{k}_{\epsilon}}{\partial x^{m}_{\epsilon}}\right)_{\epsilon} + \{(g^{kl}_{\epsilon})\}_{\epsilon} \left(\frac{\partial \xi^{i}_{\epsilon}}{\partial x^{l}_{\epsilon}}\right)_{\epsilon}, \quad (H5)$$

were the tensor $(g^{ik}_{\epsilon}(x^{i}_{\epsilon}))_{\epsilon}$ is a function of the $(x^{i}_{\epsilon})$, while the tensor $(g^{ik}_{\epsilon})_{\epsilon}$ is a
function of the original coordinates $x^l$. In order to represent all terms as functions of one and the same variables, we expand $(g_{\epsilon}^{ik}(x_{\epsilon}^l + \xi_{\epsilon}^l))_{\epsilon}$ in powers of $(\xi_{\epsilon}^l)_{\epsilon}$.

Furthermore, if we neglect terms of higher order in $(\xi_{\epsilon}^l)_{\epsilon}$, we can in all terms containing $(\xi_{\epsilon}^l)_{\epsilon}$, replace $(g_{\epsilon}^{ik})_{\epsilon}$ by $(g_{\epsilon}^{ik})_{\epsilon}$. Therefore we obtain

$$ (g_{\epsilon}^{ik}(x_{\epsilon}^{l}))_{\epsilon} = (g_{\epsilon}^{ik}(x_{\epsilon}^{l}))_{\epsilon} - \{(\xi_{\epsilon}^{l})_{\epsilon}\} \left(\frac{\partial g_{\epsilon}^{ik}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} + \{(g_{\epsilon}^{il})_{\epsilon}\} \left(\frac{\partial \xi_{\epsilon}^{k}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon} + \{(g_{\epsilon}^{kl})_{\epsilon}\} \left(\frac{\partial \xi_{\epsilon}^{i}}{\partial x_{\epsilon}^{l}}\right)_{\epsilon}. \quad (H6) $$

It is easy to verify by direct trial that the last three terms on the right can be written as a sum $(\xi_{\epsilon}^{i;k})_{\epsilon} + (\xi_{\epsilon}^{k;i})_{\epsilon}$ of contravariant derivatives of the $(\xi_{\epsilon}^i)_{\epsilon}$. Thus we finally

obtain the transformation of the $(g_{\epsilon}^{ik})_{\epsilon}$ in the form

$$ (g_{\epsilon}^{ik})_{\epsilon} = (g_{\epsilon}^{ik})_{\epsilon} + (\delta g_{\epsilon}^{ik})_{\epsilon}, \quad (\delta g_{\epsilon}^{ik})_{\epsilon} = (\xi_{\epsilon}^{i;k})_{\epsilon} + (\xi_{\epsilon}^{k;i})_{\epsilon}. \quad (H7) $$

For the covariant components, we obtain

$$ (g_{ik,\epsilon})_{\epsilon} = (g_{ik,\epsilon})_{\epsilon} + (\delta g_{ik,\epsilon})_{\epsilon}, \quad (\delta g_{ik,\epsilon})_{\epsilon} = -(\xi_{i;k,\epsilon})_{\epsilon} - (\xi_{k;i,\epsilon})_{\epsilon}, \quad (H8) $$

so that, to terms of first order we satisfy the condition $(g_{il,\epsilon}g_{\epsilon}^{il})_{\epsilon} = \delta_{i}^{k}$.

Remark H.2. Since the action $(S_{m,\epsilon})_{\epsilon}$ is a Colombeau scalar, it does not change under a transformation of coordinates. On the other hand, the change $\delta(S_{m,\epsilon})_{\epsilon}$ in the action under a transformation of coordinates can be written in the following form: let $(q_{\epsilon})_{\epsilon}$ denote the quantities defining the physical system to which the action $(S_{m,\epsilon})_{\epsilon}$ applies. Under coordinate transformation the quantities $(q_{\epsilon})_{\epsilon}$ change by $(\delta q_{\epsilon})_{\epsilon}$. In calculating $\delta(S_{m,\epsilon})_{\epsilon} = (\delta S_{m,\epsilon})_{\epsilon}$ we need not write terms containing the changes in $(q_{\epsilon})_{\epsilon}$. All such terms must cancel each other by virtue of the "equations of motion" of the physical system, since these equations are obtained by equating to zero the variation of $(S_{m,\epsilon})_{\epsilon}$ with respect to the quantities $(q_{\epsilon})_{\epsilon}$. Therefore it is enough to write the terms associated with changes in the $(g_{ik,\epsilon})_{\epsilon}$. Using generalized Gauss' theorem, and setting $(\delta g_{\epsilon}^{ik})_{\epsilon} = 0_{\bar{R}}$ at the integration limits, one finds $\delta(S_{m,\epsilon})_{\epsilon}$ in the form of the Eq. (H2).

Remark H.3. We note that the equations

$$ (\xi_{\epsilon}^{i;k})_{\epsilon} + (\xi_{\epsilon}^{k;i})_{\epsilon} = 0 \quad (H9) $$

determine the infinitesimal generalized coordinate transformations that do not change the Colombeau generalized metric. These are called the distributional Killing equations.
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