Killing Vector Fields, Maxwell Equations and Lorentzian Spacetimes

Abstract

In this paper we first analyze the structure of Maxwell equations in a Lorentzian spacetime where the potential A is proportional to 1-form K physically equivalent to a Killing vector field (supposed to exist). We show that such A obeys the Lorenz gauge and also a wave equation that can be written in terms of the covariant D'Alembertian or the Ricci operator. Moreover, we determine the correct current defined by that potential showing that it is of superconducting type, being two times the product of the components of A by the Ricci 1-form fields. We also study the structure of the spacetime generated by the coupled system consisting of a electromagnetic field F = dA (A, as above), an ideal charged fluid with dynamics described by an action function S and the gravitational field. We show that Einstein equations in this situation is then equivalent to Maxwell equations with a current givn by fFAF (the product meaning the Clifford product of the corresponding form fields), where f is a scalar function which satisfies a well determined algebraic quadratic equation.