Birkhoff theorem and conformal Killing-Yano tensors

TL;DR

This paper explores the geometric conditions underlying the Jebsen-Birkhoff theorem, demonstrating that a conformal Killing-Yano tensor suffices for the existence of an additional Killing vector, broadening the theorem's applicability.

Contribution

It shows that the presence of a conformal Killing-Yano tensor, rather than a three-dimensional isometry group, guarantees an extra Killing vector in certain spacetimes.

Findings

Existence of a conformal Killing-Yano tensor implies an additional Killing vector.

The traditional requirement of a three-dimensional isometry group is relaxed.

The approach clarifies the geometric origin of symmetries in ${ m D}$-metrics.

Abstract

We analyze the main geometric conditions imposed by the hypothesis of the Jebsen-Birkhoff theorem. We show that the result (existence of an additional Killing vector) does not necessarily require a three-dimensional isometry group on two-dimensional orbits but only the existence of a conformal Killing-Yano tensor. In this approach the (additional) isometry appears as the known invariant Killing vector that the ${\cal D}$-metrics admit.