Are electromagnetic phenomena derivable from extended Einstein equations?

A new term describing interactions between charge and potentials may be added to the right hand side of the Einstein equations. In the proposed term an additional tensor has been introduced containing a charge density, analogous to the energy-momentum tensor containing a mass density. (The contribution of the electromagnetic fields has not been considered in this work.) The metric components in the new equations may contain charge and mass contributions. In the special relativistic case a set of four vacuum Maxwell equations is obtained from the new equations containing a special relativistic charge density. A related result was recently found by Jefimenko from a special relativistic transformation. Moreover, from the postulated equations the usual special relativistic electric energy and Lorentz force can be calculated, when the approximated electromagnetic metric is used. In the non-relativistic case the usual Maxwell equations with rest charge density are obtained. In the presence of a static gravitational field a set of four Maxwell equations is obtained applying in the general relativistic case. From these equations the generally accepted isotropic velocity of light can be deduced. In addition, calculated generalized Maxwell equations show that vacuum permittivity and permeability may differ from unity value in the presence of a big charge Q or a big mass M.