Field Line Solutions of the Einstein-Maxwell Equations

TL;DR

This paper reviews gravitating electromagnetic fields within the Einstein-Maxwell framework, focusing on local field line solutions that extend classical flat space solutions to curved space-times, highlighting existence conditions and specific examples.

Contribution

It introduces recent results on local field line solutions in curved space-time, generalizing known flat space solutions and analyzing obstructions to global solutions.

Findings

Existence of local field line solutions in curved space-time

Generalization of Ra ilde{a}da solutions to Einstein-Maxwell equations

Discussion of specific solutions like the Kopi ilde{n}ski-Nat ilde{a}rio field

Abstract

In this paper, we are going to review the gravitating electromagnetic field in the 1+3 formalism on a general hyperbolic space-time manifold. We also discuss the recent results on the existence of the local field line solutions of the Einstein-Maxwell equations that generalize the Ra\~{n}ada solutions from the flat space-time. The global field line solutions do not always exist since the space-time manifold could impose obstructions to the global extension of various geometric objects necessary to build the fields. One example of a gravitating field line solution is the Kopi\'{n}ski-Nat\'{a}rio field which is discussed in some detail.