Consequences of a Killing symmetry in spacetime's local structure

TL;DR

This paper explores how a Killing symmetry influences the local geometric structure of four-dimensional spacetimes, extending previous vacuum results to include arbitrary energy-momentum contents and electromagnetic fields.

Contribution

It introduces a formalism based on the exterior derivative of the Killing vector, applicable to general matter fields, and clarifies its advantages over other methods.

Findings

Extended the formalism to non-vacuum spacetimes

Applied the framework to electromagnetic field scenarios

Highlighted advantages of the exterior derivative approach

Abstract

In this paper we discuss the consequences of a Killing symmetry on the local geometrical structure of four-dimensional spacetimes. We have adopted the point of view introduced in recent works where the exterior derivative of the Killing plays a fundamental role. Then, we study some issues related with this approach and clarify why in many circumstances its use has advantages with respect to other approaches. We also extend the formalism developed in the case of vacuum spacetimes to the general case of an arbitrary energy-momentum content. Finally, we illustrate our framework with the case of spacetimes with a gravitating electromagnetic field.