General approach to the study of vacuum space-times with an isometry

TL;DR

This paper introduces a new formalism based on the algebraic structure of the Papapetrou field to study vacuum space-times with an isometry, enabling classification and complete determination of the Weyl curvature.

Contribution

It develops a novel algebraic formalism linking isometries and Petrov types, providing a new classification and explicit curvature expressions for vacuum space-times.

Findings

New classification of vacuum space-times with isometry

Complete determination of Weyl curvature from integrability conditions

Formalism applicable to various problems involving isometries

Abstract

In vacuum space-times the exterior derivative of a Killing vector field is a 2-form (named here as the Papapetrou field) that satisfies Maxwell's equations without electromagnetic sources. In this paper, using the algebraic structure of the Papapetrou field, we will set up a new formalism for the study of vacuum space-times with an isometry, which is suitable to investigate the connections between the isometry and the Petrov type of the space-time. This approach has some advantages, among them, it leads to a new classification of these space-times and the integrability conditions provide expressions that determine completely the Weyl curvature. These facts make the formalism useful for application to any problem or situation with an isometry and requiring the knowledge of the curvature.