Maxwell Fields in Spacetimes Admitting Non-Null Killing Vectors

Abstract

We consider source-free electromagnetic fields in spacetimes possessing a non-null Killing vector field, $\xi^a$. We assume further that the electromagnetic field tensor, $F_{ab}$, is invariant under the action of the isometry group induced by $\xi^a$. It is proved that whenever the two potentials associated with the electromagnetic field are functionally independent the entire content of Maxwell's equations is equivalent to the relation $\n^aT_{ab}=0$. Since this relation is implied by Einstein's equation we argue that it is enough to solve merely Einstein's equation for these electrovac spacetimes because the relevant equations of motion will be satisfied automatically. It is also shown that for the exceptional case of functionally related potentials $\n^aT_{ab}=0$ implies along with one of the relevant equations of motion that the complementary equation concerning the electromagnetic field is satisfied.