On the Papapetrou field in vacuum

TL;DR

This paper investigates the algebraic and differential properties of Papapetrou electromagnetic fields generated by Killing vectors in vacuum spacetimes, providing tools for solving Maxwell's and Einstein's equations and analyzing specific solutions like Kerr and Kasner.

Contribution

It offers an explicit covariant method to find principal null directions of Papapetrou fields and studies their geodesic and optical properties in vacuum solutions.

Findings

Explicit procedure for principal null directions

Conditions for alignment with Weyl tensor

Application to Kerr, Kasner, and pp wave spacetimes

Abstract

In this paper we study the electromagnetic fields generated by a Killing vector field in vacuum space-times (Papapetrou fields). The motivation of this work is to provide new tools for the resolution of Maxwell's equations as well as for the search, characterization, and study of exact solutions of Einstein's equations. The first part of this paper is devoted to an algebraic study in which we give an explicit and covariant procedure to construct the principal null directions of a Papapetrou field. In the second part, we focus on the main differential properties of the principal directions, studying when they are geodesic, and in that case we compute their associated optical scalars. With this information we get the conditions that a principal direction of the Papapetrou field must satisfy in order to be aligned with a multiple principal direction of the Weyl tensor in the case of algebraically special vacuum space-times. Finally, we illustrate this study using the Kerr, Kasner and pp waves space-times.