## 3 Results
### Equations on the Surface. Coupling Conditions  
In general relativity, there are coupling conditions known as Darmois conditions [41]. The ultimate purpose of these conditions is to avoid the appearance of singular behavior of physical variables on the coupling surface [18; 19]. According to [PERSON] [41], the necessary and sufficient condition for the coupling of two space-time regions on a hypersurface is that both the metrics induced on it (first fundamental form). The inner region, characterized by the line element and the outer region, described by the Reissner-Nordstrom-Vaidya metric [16], are separated by the hypersurface \(r\ =\ \Sigma(\mu)\). Therefore, the coupling of the inner and outer solutions of the field equations must be considered. Furthermore, to know the evolution of the physical variables inside the distribution, it is necessary to construct a system of ordinary differential equations for quantities evaluated on the surface. To obtain these equations, we proceed as follows:  
* The functions \(m\) and \(\beta\) are required to be continuous across the edge of the distribution. * The material velocity at the surface is evaluated. * The first field Equation (20) is evaluated on both sides of the surface. * The conservation equation is calculated \(T^{\mu}_{,\lambda;\mu}\ =\ 0\). The energy conservation equation leads to  
\[\frac{\partial p}{\partial r}\ +\ \frac{\overline{p}\ +\ \overline{p}}{1\ -\ 2\mu/r}\Big{[}4\pi r \overline{p}\ +\ \mu/r^{2}\Big{]}\ -\ e^{-2\beta}(\frac{\overline{p}\ +\ \overline{p}}{1\ -2\mu/r})_{,u}\ =\ \frac{2}{r}(p_{t}\ -\ \overline{p}) \tag{39}\]  
\[-\ \beta_{,u}e^{-2\beta}\ +\ \left(1\ -\ \frac{2m}{r}\right)\beta_{,r}\ -\ \frac{m_{,r}}{2r}\ +\ \frac{C^{2}T}{4r^{3}}\ =\ 0 \tag{40}\]  
evaluated on the \(\Sigma\) surface. Once the above is done, two ordinary differential equations are obtained with two functions of time to be determined, corresponding to  
* The velocity of the edge of the distribution. * The function \(r=\Sigma\). These are the surface radius \(r_{\Sigma}\), the distribution edge velocity, and the function evaluated on \(r=\Sigma\). It is noticeable that the metric functions and, thus, the electric charge and physical variables are obtained from the time dependence of the variables at the surface \(m_{\Sigma}\) y \(r_{\Sigma}\), for which the differential equations on the surface are required, which are obtained from Equations (40) and (41) evaluated at \(\Sigma\)  
\[\dot{m}_{\Sigma}\ =\ -\epsilon_{\Sigma}\bigg{(}1\ -\ \frac{2m_{\Sigma}}{r_{ \Sigma}}\bigg{)}\ +\ \dot{r}_{\Sigma}\frac{q^{2}}{2r_{\Sigma}^{2}} \tag{41}\]  
\[\dot{r}_{\Sigma}\ =\ \bigg{(}1\ -\ \frac{2m}{r_{\Sigma}}\bigg{)}\frac{\omega_{ \Sigma}}{1\ -\ \omega_{\Sigma}} \tag{42}\]  
where the point on the variable indicates time derivative. In addition, the radiation flux at the surface:  
\[\epsilon_{\Sigma}\ =\ \dot{e}(1\ +\ \omega_{\Sigma})/(1\ -\ \omega_{\Sigma}) \tag{43}\]  
The field Equation (22) evaluated on the surface gets to know the total mass of the distribution of matter  
\[m_{\Sigma}\ =\ \frac{1}{\psi}\bigg{(}\xi r_{\Sigma}\ -\ \frac{C^{2}}{r_{ \Sigma}}\bigg{)} \tag{44}\]  
where \(\xi=(l+1)^{2}\) and \(\psi=2\xi+k(4+3l+k)\). This allows us to obtain \(\epsilon_{\Sigma}\) from Equations (22) and (23) and the surface equations. \[\epsilon\ =\ \Bigg{\{}\frac{C^{2}}{r_{\Sigma}^{2}}\bigg{[}\frac{1}{2}\ +\ \frac{1}{\psi}\bigg{]}\ -\ \frac{\xi}{\psi}\bigg{\}}\frac{\omega_{\Sigma}}{1-\omega_{\Sigma}} \tag{45}\]  
Such that one has a single equation on the surface that is numerically integrated with the specification of the total mass \(m_{\Sigma}\) and the radius of the distribution \(r_{\Sigma}=\mathrm{R}\) and the radius of the distribution \(r\Sigma=\mathrm{R}\). Once the dynamics on \(\Sigma\) have been obtained, the following procedure is followed to calculate the values of the physical variables:  
* Specifying the initial electric charge distribution allows for finding the physical variables at any radius (or any layer) of the sphere.