The Spacetime Geometry of a Null Electromagnetic Field

TL;DR

This paper characterizes the geometric conditions on spacetime metrics necessary and sufficient for them to represent null electrovacuum solutions to Einstein-Maxwell equations, extending Rainich conditions to null electromagnetic fields.

Contribution

It provides a set of local geometric conditions involving derivatives of the metric that precisely identify null electrovacuum spacetimes, and offers a method to construct the electromagnetic field from the metric.

Findings

Derived necessary and sufficient geometric conditions for null electrovacua.

Provided a procedure to reconstruct electromagnetic fields from the metric.

Illustrated conditions using pure radiation spacetimes.

Abstract

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from the metric. Null electrovacuum geometry is illustrated using some pure radiation spacetimes taken from the literature.