## 3 Solving [PERSON]'s Equations  
We can now express the tensor \(T_{\alpha\beta}\) in (9) in a generic metric \(g_{\alpha\beta}\) and plug it on the right-hand side of [PERSON]'s equation:  
\[G_{\alpha\beta}=R_{\alpha\beta}-\tfrac{1}{2}g_{\alpha\beta}R=\chi T_{\alpha\beta} \tag{18}\]  
where, as usual, \(R_{\alpha\beta}\) is the [PERSON]'s tensor and \(R=g^{\alpha\beta}R_{\alpha\beta}\) denotes the scalar curvature. The magnitude of the dimensional constant \(\chi\) (meters/joules) is suggested by the type of application. Note, however, that \(\chi\) is not in relation with the gravitational constant \(G\), since, as already remarked in the previous section, there are no physical masses involved in our study. The sign of \(\chi\) will be discussed later on. It is known that the trace of \(G_{\alpha\beta}\) is equal to \(-R\). This suggest interesting relations between the scalar curvature and the trace of \(T_{\alpha\beta}\) (see (17) for the case of the metric (4)). We look for a metric of the form:  
\[(ds)^{2}=c^{2}\ \tau^{2}(t,r)(dt)^{2}-\sigma^{2}(t,r)(dr)^{2}-r^{2}(d\theta)^{ 2}-r^{2}\sin^{2}\theta\ (d\phi)^{2} \tag{19}\]where \(\tau\) and \(\sigma\) should be determined as a function of the forcing term \(E\). In other words, the time-varying electric field emanated by the ball produces alterations of the space-time geometry that are linked to the solution of (18). We may call them gravitational waves, though there is no \"gravity\" in our exercise. Similar settings were considered in [21; 22; 23], for non-static spheres of charged dust. Other related papers are for instance [24; 25]. We continue our analysis by observing that \(\sqrt{-g}=r^{2}\sin\theta\ \tau(t,r)\sigma(t,r)\), and by computing the (non-vanishing) [PERSON]'s symbols:  
\[\Gamma^{0}_{00}=-\frac{1}{\sigma}\frac{\partial\sigma}{\partial t} \Gamma^{0}_{11}=\sigma^{3}\ \frac{\partial\sigma}{\partial t} \Gamma^{0}_{01}=\Gamma^{0}_{10}=-\frac{1}{\sigma}\frac{\partial \sigma}{\partial r}\] \[\Gamma^{1}_{00}=-\frac{1}{\sigma^{5}}\frac{\partial\sigma}{ \partial r} \Gamma^{1}_{11}=\frac{1}{\sigma}\frac{\partial\sigma}{\partial r} \Gamma^{1}_{01}=\Gamma^{1}_{10}=\frac{1}{\sigma}\frac{\partial \sigma}{\partial t}\] \[\Gamma^{1}_{22}=-\frac{r}{\sigma^{2}} \Gamma^{1}_{33}=-\frac{r\sin^{2}\theta}{\sigma^{2}} \Gamma^{3}_{31}=\Gamma^{3}_{13}=\Gamma^{2}_{12}=\Gamma^{2}_{21}= \frac{1}{r}\] \[\Gamma^{2}_{33}=-\sin\theta\cos\theta \Gamma^{3}_{23}=\Gamma^{3}_{32}=\frac{\cos\theta}{\sin\theta} \tag{20}\]  
The nonzero entries of [PERSON]'s tensor \(G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R\) take the form:  
\[G_{00}=-\frac{\tau^{2}}{\sigma^{2}}\bigg{(}\frac{1}{r^{2}}-\frac{\sigma^{2}}{ r^{2}}-\frac{2}{r\sigma}\frac{\partial\sigma}{\partial r}\bigg{)} G_{11}=\frac{1}{r^{2}}-\frac{\sigma^{2}}{r^{2}}+\frac{2}{r\tau}\frac{ \partial\tau}{\partial r}\]  
\[G_{22}=-\frac{r}{\sigma^{3}\tau^{3}}\bigg{(}\tau^{3}\ \frac{\partial\sigma}{ \partial r}-\sigma\tau^{2}\ \frac{\partial\tau}{\partial r}-r\tau^{2}\ \frac{\partial^{2}\tau}{\partial r^{2}}+r\sigma^{2}\tau\ \frac{\partial^{2}\sigma}{\partial t^{2}}+r\tau^{2}\ \frac{\partial \tau}{\partial r}\frac{\partial\sigma}{\partial r}-r\sigma^{2}\ \frac{\partial \tau}{\partial t}\frac{\partial\sigma}{\partial t}\bigg{)}\]  
\[G_{01}=G_{10}=\frac{2}{r\sigma}\frac{\partial\sigma}{\partial t} G_{33}=G_{22}\sin^{2}\theta \tag{21}\]  
For a given \(\vec{E}=(E,0,0)\), we build up the electromagnetic stress-energy tensor \(U_{\alpha\beta}\):  
\[U_{00}=\frac{E^{2}}{2\sigma^{2}} U_{11}=-\frac{E^{2}}{2\tau^{2}}, U_{22}=\frac{r^{2}E^{2}}{2\sigma^{2}\tau^{2}} U_{33}=U_{22}\ \sin^{2}\theta \tag{22}\]  
Moreover, we need the velocity 4-vectors: \(V^{\alpha}=(c,v,0,0)\) and \(V_{\alpha}=(c\tau^{2},-v\sigma^{2},0,0)\). Thus, we are ready to set up the mass tensor:  
\[M_{\alpha\beta}=\frac{\varrho}{\mu}\left( \begin{array}{cccc}c^{2}\tau^{4}& -cv\tau^{2}\sigma^{2}&0&0\\ -cv\tau^{2}\sigma^{2}&v^{2}\sigma^{4}&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right)+\frac{1}{\epsilon_{0}}\left(