## 2 Development  
### The Metric, Energy-Momentum Tensor, and Field Equations  
The metric for defining the shape of the distribution on which the profiles of geometrical and physical variables in general relativity will be evaluated; the choice of a suitable coordinate system greatly simplifies the analysis of the system under study. In particular, in the study of radioactive systems, the description provided by the so-called radiation coordinates or Bondi metrics [25] is adequate. Therefore, for the description of the geometry of the interior of the distribution, the Bondi metrics [25] are used. Considering a spherically symmetric matter distribution, which is given by a charged fluid, with values of pressure, energy, electric energy density, and radiation energy flux given by \(p,\rho,\varrho,\) and \(\epsilon\), respectively, and measured from the perspective of an observer in a Minkowski representation moving at par (comoving) with the fluid (with velocity \(-\omega\)), the metric is expressed in a radiation coordinate system [26; 27; 28] as Equation (3):  
\[ds^{2}\ =\ e^{2\beta}\left(\frac{V}{r}du^{2}\ +\ 2 du\ dr\right)\ -\ r^{2}\left(d\theta^{2}\ +\ sin^{2}\theta d\phi^{2}\right) \tag{3}\]  
of which \(\beta\) is a function \(u\), and \(V\) is a function of \(r\), the Bondi time being \(u=x^{0}\)[25] at null infinity, where \(u\) is constant. Where \(r=x^{1}\) is the radial coordinate tangent to the null cones. Moreover, both \(\theta=x^{2}\) as well as \(\varphi=x^{3}\) correspond to angular coordinates. Geometric units will be used throughout this work. \(G=c=1\). From the metric function \(V\), the function \(m\) (\(u,\)\(r\)) is defined, which represents the mass term in the interior of the distribution. This quantity was introduced by [PERSON] [29] as a generalization of the mass in the static case to the radiation case and is related to \(\beta\) and for the case of charged distributions by the following expression:  
\[V\ =\ e^{2\beta}\bigg{(}r\ -\ 2m\bigg{(}1\ -\ \frac{2C}{r}\bigg{)}\bigg{)} \tag{4}\]  
where \(C\) represents the charge inside the distribution. Knowing that, in spherical symmetry, the only components with nonzero values of the electromagnetic field tensor are \(F^{\mu\
u}\ =\ -F^{\mu}\) (Equation (4)). Then, a function \(C(u,r)\) can be defined by the correspondence observed in [30; 31; 32].  
\[F\ =\ \frac{C}{r^{2}}e^{-2\beta} \tag{5}\]  
The nonhomogeneous Maxwell equations are Equations (6) and (7)  
\[C_{\gamma}\ =\ 4\pi r^{2}J^{\mu}e^{2\beta} \tag{6}\]  
\[\text{and}\ C_{,\mu}\ =\ -4\pi r^{2}J^{\
u}e^{2\beta} \tag{7}\]  
The partial derivative as a function of the indicated coordinate is denoted by the comma symbol (\(\,\)); then, the time component is represented by \(J^{\mu}\) and the radial component by \(J^{\
u}\), both belonging to the electric current density quadrivector; then, the idea is that \(C\ (u,r)\) shows a function of charge at radius \(r\) at a given time \(u\). In this way, the conservation of charge in the inhomogeneous electromagnetic equations is expressed as well as observed in Equation (8)  
\[U^{\mu}C_{\mu}\ =\ 0 \tag{8}\]  
where the quadravelocity of the fluid is given by (9).  
\[U^{\mu}\ =\ e^{-\beta}\bigg{[}\sqrt{\frac{r}{V}}\bigg{(}\frac{1\ -\ \omega}{1\ +\ \omega}\bigg{)}\delta^{\mu}_{u}\ +\ \sqrt{\frac{V}{r}}\frac{\ \ \omega}{\ (1\ -\ \omega^{2})^{1/2}}\delta^{\mu}_{r}\bigg{]} \tag{9}\]