Maxwell Fields in Spacetimes Admitting Non-Null Killing Vectors

TL;DR

This paper shows that in certain spacetimes with symmetries, solving Einstein's equations suffices to determine electromagnetic fields, simplifying the analysis of electrovac solutions with non-null Killing vectors.

Contribution

It demonstrates that for spacetimes with a non-null Killing vector, the Maxwell equations reduce to a single condition, allowing Einstein's equations alone to determine electromagnetic fields under specified conditions.

Findings

Maxwell's equations are equivalent to a divergence-free energy-momentum tensor in these spacetimes.

Solving Einstein's equations suffices for electrovac solutions with invariant electromagnetic fields.

The relation $ ^aT_{ab}=0$ plays a central role in simplifying the equations.

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.