Killing Vector Fields, Maxwell Equations and Lorentzian Spacetimes

In this paper we first analyze the structure of Maxwell equations in a Lorentzian spacetime when the potential is A = eK, i.e., proportional to a 1-form K physically equivalent Killing vector field. We show that A necessarily obeys the Lorenz gauge δA = 0. Moreover we determine the form of the current associated with this potential showing that it is of a superconducting type, i.e., proportional to the potential and given by 2RβA β , where the Rβ are the Ricci 1-form fields. Finally we study the structure of the spacetime generated by the coupled system consisting of a electromagnetic field F = dA (with A = eK), an ideal charged fluid with dynamics described by an action function S and the gravitational field. We show that Einstein equations is then equivalent to Maxwell equations with a current given by fFAF (the product meaning the Clifford product of the corresponding fields), where f is a scalar function which satisfies a well determined algebraic quadratic equation.