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 is intimately connected to the existence of foliations of a spacetime with prescribed properties. We also prove a local existence and uniqueness theorem and provide a recipe for constructing the unique solution/class of solutions when certain conditions are met. We clarify the theorem with examples. We are also able to also prove a singularity theorem for when non-null solutions approach the null limit. Here too, we construct an explicit example to illustrate the singularity theorem.