Spacetime Ehlers group: Transformation law for the Weyl tensor

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.