## 1 Preliminary Considerations  
We discuss an elementary electrodynamics problem: the surface of a given ball is subject to variation of charge and we would like to study how the signal develops at its exterior. We will work in simplified conditions of perfect radial symmetry. For this reason, the approach may sound mainly academic, though there are examples of applications in several areas. From the theoretical viewpoint, the analysis of black holes is a natural referring topic. From the practical viewpoint, we just mention the case of our Sun. This has an effective small charge (less than 100 Coulombs [1]) and presents periodic movements on its surface (see, e.g., [2]). Real-life problems are indeed very challenging; however, we believe that it is important to understand the difficulties hidden in the study of the most basic formulation. As a matter of fact, finding the modeling equations ruling the behavior of the fields outside the ball is not a trivial question, as one would expect, especially if the surrounding space is void (i.e. : the ball is immersed in a vacuum). It is implicitly assumed that we look for solutions that develop at a finite speed (commonly that of light). Expedients suggested by [PERSON]'s law are not taken into account, since they are based on an infinite velocity of propagation. We briefly formalize here the problem as follows. We work in the spherical coordinate system \((r,\phi,\phi)\), and we assume that the fields do not depend on the variables \(\theta\) and \(\phi\). This will ensure perfect central symmetry. As usual, we denote by \(c\) the speed of light. Time-dependent boundary conditions are imposed on a ball of fixed radius. These are homogeneously distributed on the surface in such a way that the electric field \(E=(E,0,0)\) is of radial type. As we specified, outside the ball, the function \(E\) will continue to depend only on \(t\) and \(r\). Clearly, no magnetic field can be generated in this circumstance. This means that, if the ball is surrounded by a pure vacuum, we cannot invoke the help of [PERSON]'s equations. Indeed, the Ampere law: \(\partial\vec{E}/\partial t=c^{2}\mathrm{curl}\vec{B}=0\) implies that only a stationary electric field can be taken into account. In this trivial circumstance, we get the expression \(Q/r^{2}\), where \(Q\) is the fixed charge. A solution of the type \(Q(t)/r^{2}\) is not acceptable because the whole three-dimensional space would be simultaneously affected by what happens at the surface of the ball. We claimed instead that the information has to develop with finite velocity. Thus, it is necessary to build up the proper modeling equations in order to study the phenomenon. We start by introducing the divergence \(\rho\) in spherical coordinates:  
\[\rho=\frac{\partial E}{\partial r}+\frac{2E}{r}=\frac{1}{r^{2}}\frac{\partial(Er^ {2})}{\partial r}=\text{div}\vec{E} \tag{1}\]  
We then assume that the field \(\vec{V}=(v,0,0)\) represents the radial velocity of propagation of the signal. The vectors \(\vec{E}\) and \(\vec{V}\) are parallel, therefore the information develops longitudinally. This sounds unconventional, but represents the reason why the investigation is not so easy. On the other hand, there are no other ways to proceed. For simplicity, we take \(v\) constant with \(v\leq c\). If \(\rho\) is allowed to be different from zero, the Ampere law (with \(\vec{B}=0\)) now becomes:  
\[\frac{\partial\vec{E}}{\partial t}=-\rho\vec{V}\qquad\Rightarrow\qquad\frac{ \partial E}{\partial t}=-\rho v \tag{2}\]  
This admits solutions of the type \(r^{-2}g(vt-r)\), where \(g\) is totally arbitrary. When \(g=Q\) is a constant we return to the classical stationary case. These positions are not extraordinary if the transmission of the signal happens inside a certain medium allowing for \(\rho\) to be different from zero. They may be considered unorthodox in empty space. We show in Section 2 that (2) descends naturally from the construction of a suitable stress-energy tensor, and needs to be coupled with another equation (see (15)).