A spacetime characterization of the Kerr-NUT-(A)de Sitter and related metrics

TL;DR

This paper provides a geometric characterization of the Kerr-NUT-(A)de Sitter spacetime among four-dimensional vacuum solutions with a cosmological constant, extending previous results and introducing new geometric tools.

Contribution

It introduces a novel characterization based on the proportionality of the self-dual Weyl tensor and a self-dual double two-form, and constructs a second Killing vector from geometric data.

Findings

Characterization of Kerr-NUT-(A)de Sitter metrics using Weyl tensor properties

Extension of previous Kerr metric characterizations

Identification of a geometric structure involving Riemannian submersion

Abstract

A characterization of the Kerr-NUT-(A)de Sitter metric among four dimensional \Lambda-vacuum spacetimes admitting a Killing vector is obtained in terms of the proportionality of the self-dual Weyl tensor and a natural self-dual double two-form constructed from the Killing vector. This result recovers and extends a previous characterization of the Kerr and Kerr-NUT metrics. The method of proof is based on (i) the presence of a second Killing vector field which is built in terms of geometric information arising from the previous Killing vector exclusively, and (ii) the existence of an interesting underlying geometric structure involving a Riemannian submersion of a conformally related metric, both of which may be of independent interest.