On the equations of electrodynamics in a flat or a curved spacetime and a possible interaction energy

Abstract

We investigate which are the independent equations of continuum electrodynamics and what is their number, beginning with the standard equations used in special and in general relativity. We check by using differential identities that there are as much independent equations as there are unknowns, for the case with given sources as well as for the general case where the motion of the charged medium producing the field is unknown. Then we study that problem in an alternative theory of gravity with a preferred reference frame, in order to constrain an additional, "interaction" energy tensor that has to be postulated in this theory, and that would be present also outside usual matter. In order that the interaction tensor be Lorentz-invariant in special relativity, it has to depend only on a scalar field $p$. Since the system of electrodynamics of the theory is closed in the absence of the interaction tensor, just one scalar equation more is needed to close it again in the presence of $p$. We add the equation for charge conservation. We derive equations that will allow one to determine the field $p$ in a given weak gravitational field and in a given electromagnetic field. Keywords: Maxwell equations, special relativity, general relativity, alternative theory of gravitation, preferred reference frame