On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions

TL;DR

This paper classifies six-dimensional vacuum spacetimes with specific algebraic and optical properties, showing they are all Kerr--Schild type D solutions related to Kerr-NUT-(A)dS metrics, thus establishing their uniqueness.

Contribution

It provides the most general form of such six-dimensional spacetimes under certain conditions, demonstrating they are all locally isometric to known Kerr-NUT-(A)dS solutions.

Findings

All solutions are Kerr--Schild type D spacetimes.

Solutions are locally isometric to Kerr-NUT-(A)dS metrics.

Recovered subclasses include Kerr-(A)dS and its extensions.

Abstract

We study the class of six-dimensional $\Lambda$-vacuum spacetimes which admit a non-degenerate multiple Weyl aligned null direction l (thus being of Weyl type~II or more special) with a ``generic'' optical matrix. Subject to an additional assumption on the asymptotic fall-off of the Weyl tensor, we obtain the most general metric of this class, which is specified by one discrete (normalized) and three continuous parameters. All solutions turn out to be Kerr--Schild spacetimes of type~D and, in passing, we comment on their Kerr--Schild double copy. We further show that the obtained family is locally isometric to the general doubly-spinning Kerr-NUT-(A)dS metric with the NUTs parameters switched off. In particular, the Kerr-(A)dS subclass and its extensions (i.e., analytic continuation and ``infinite-rotation'' limit) are recovered when certain polynomial metric functions are assumed to be fully factorized. As a side result, a unified metric form which encompasses all three branches of the extended Kerr-(A)dS family in all even dimensions is presented in an appendix.