Force-free electrodynamics and foliations in an arbitrary spacetime

Abstract

In this paper, we formulate the relationship between force-free electrodynamics and foliations. The background metric is considered predetermined and electrically neutral, but otherwise arbitrary. As it turns out, solutions to force-free electrodynamics are intimately connected to the existence of foliations of spacetime with prescribed properties. We prove a local existence and uniqueness theorem concerning foliations when the field is non-null. In the null case, we show that there will always be a class of solutions with exactly two degrees of freedom for every admissible foliation. We are also able to also prove a singularity theorem for when non-null solutions approach the null limit. All new results are clarified with examples. For a variety in the discussion, we conclude with a solution to Maxwell’s equations in FRW cosmology.