## 4 Comments  
We can provide further observation concerning the stationary solution presented at the end of the previous section. In this case, [PERSON]'s equation reduces to:  
\[G_{\alpha\beta}=\chi\epsilon_{0}U_{\alpha\beta} \tag{29}\]  
in which only the electromagnetic stress tensor \(U_{\alpha\beta}\) appears on the right-hand side. As we noted, a general solution has the form: \(\tau(r)=\sqrt{1-K/r+\chi\epsilon_{0}Q^{2}/2r^{2}},\ \sigma(r)=1/\tau(r)\). This effectively corresponds to the Reissner-Nordstrom metric (\(K=2M\)) when \(\chi>0\) (note that the signature of our space is \((+,-,-,-)\)). The existence of \(\tau\) and \(\sigma\) is guaranteed if \(r\) is large enough and \(M^{2}-\chi\epsilon_{0}Q^{2}\geq 0\). This is an effective restriction when \(\chi\) is positive, but can be totally removed by taking \(\chi\) as negative. As a matter of fact, there is no real reason to choose a specific sign accompanying the electromagnetic stress-energy tensor, since unlike astronomical or cosmological applications, there is no sign actually involved in electromagnetic phenomena. This issue was already discussed in ([4], p. 128), and can be further reinforced after examining the Kerr-Newman metric [27], simulating a rotating massive black hole. The metric generalizes the above one and uses a particular coordinates framework, namely the Boyer-Lindquist coordinates. The explicit expression is:  
\[c^{2}(ds)^{2}=-\bigg{(}\frac{(dr)^{2}}{\Delta}+(d\theta)^{2}\bigg{)}x^{2}+ \bigg{(}c\,dt-a\sin^{2}\theta\,d\phi\bigg{)}^{2}\frac{\Delta}{x^{2}}-\bigg{(}( r^{2}+a^{2})d\phi-ac\,dt\bigg{)}^{2}\,\frac{\sin^{2}\theta}{x^{2}} \tag{30}\]  
where \(x^{2}=r^{2}+a^{2}\cos^{2}\theta\), \(a\) is a parameter and \(\Delta=r^{2}-2 Mr+a^{2}+Q^{2}\). For brevity, we omitted various dimensional constants. It turns out that, if \(\chi\) is positive in (29), we get restrictions on \(r\), bringing to the definition of an \"event horizon\". It is also required that \(M^{2}\geq Q^{2}+a^{2}\) (see [19], p. 879), which may enforce the mass \(M\) to be rather large. On the other hand, if we now define \(\Delta=r^{2}-2 Mr+a^{2}-Q^{2}\), we still get solutions to [PERSON]'s equation. They correspond to negative values of \(\chi\), and the restriction becomes milder, i.e.: \(M^{2}+Q^{2}\geq a^{2}\). Our suggestion is to adopt the negative sign concerning all tensors that involve the electromagnetic portion of the phenomenon under study.  
As a final remark, we propose the full set of modeling equations, including the presence of the magnetic field. They are obtained as in (11) by imposing \(\
abla_{\beta}T^{\alpha\beta}=0\), with \(T^{\alpha\beta}\) built on the most general electromagnetic tensor \(F_{\alpha\beta}\). We have:  
\[\frac{\partial\vec{E}}{\partial t}=c^{2}\text{curl}\vec{B}-\rho\vec{V}\qquad \qquad\qquad\frac{\partial\vec{B}}{\partial t}=-\text{curl}\vec{E} \tag{31}\]  
with \(\rho=\text{div}\vec{E}\) and \(\text{div}\vec{B}=0\). In addition, we get:  
\[\rho\bigg{(}\mu^{-1}\frac{D\vec{V}}{Dt}+\vec{E}+\vec{V}\times\vec{B}\bigg{)}=- \epsilon_{0}^{-1}\vec{\
abla}p \tag{32}\]  
Note that the term \(\vec{E}+\vec{V}\times\vec{B}\) recalls [PERSON]'s force. Thus, we are coupling [PERSON]'s equations with those ruling fluid dynamics. This is a very classical approach, especially in the framework of plasma physics (see, e.g., [28]). The crucial difference is that the equations above do not necessarily require the presence of massive particles, and they can survive in a perfect vacuum. One of the advantages is to be able to solve the problem of the radially pulsating charged ball, which cannot otherwise be dealt with with more standard tools.