Spacetime Ehlers group: Transformation law for the Weyl tensor

TL;DR

This paper explores the spacetime Ehlers group as a symmetry of Einstein vacuum equations, providing a transformation law for the Weyl tensor and applying it to characterize the Kerr-NUT metric.

Contribution

It defines the Ehlers group in a purely spacetime setting and derives the explicit transformation law for the Weyl tensor, extending understanding of spacetime symmetries.

Findings

Ehlers group is well-defined regardless of the Killing vector's causal character.

Explicit transformation law for the Weyl tensor under Ehlers transformations.

Application to local characterization of the Kerr-NUT metric.

Abstract

The spacetime Ehlers group, which is a symmetry of the Einstein vacuum field equations for strictly stationary spacetimes, is defined and analyzed in a purely spacetime context (without invoking the projection formalism). In this setting, the Ehlers group finds its natural description within an infinite dimensional group of transformations that maps Lorentz metrics into Lorentz metrics and which may be of independent interest. The Ehlers group is shown to be well defined independently of the causal character of the Killing vector (which may become null on arbitrary regions). We analyze which global conditions are required on the spacetime for the existence of the Ehlers group. The transformation law for the Weyl tensor under Ehlers transformations is explicitly obtained. This allows us to study where, and under which circumstances, curvature singularities in the transformed spacetime will arise. The results of the paper are applied to obtain a local characterization of the Kerr-NUT metric.