On type II(D) Einstein spacetimes in six dimensions

TL;DR

This work extends the algebraic classification of Einstein spacetimes to six dimensions under two structural assumptions: a non-degenerate optical matrix and rapid Weyl fall-off. The main result is a complete local description of the general six-dimensional, Weyl type II or more special, Einstein spacetime as a Kerr-Schild, type D metric with one discrete and three continuous parameters, encompassing Kerr-NUT-(A)dS as a special case. A notable specialisation occurs when the governing polynomial ${ m extcal P}(s)$ factorises, yielding the doubly-spinning Kerr-(A)dS family and its generalizations within a unified framework. The analysis extends higher-dimensional Goldberg–Sachs-type insights and clarifies how the six-dimensional case connects to known four- and five-dimensional black-hole families, highlighting the role of twist and the structure of WANDs in even dimensions.

Abstract

After a concise overview of Einstein spacetimes of type II (or more special) in four and five dimensions, we summarize recent results in the six-dimensional case. We assume the optical matrix to be non-degenerate and ``generic'', and the Weyl tensor to fall off sufficiently rapidly at infinity. As it turns out, the most general metric is characterized by one discrete (normalized) and three continuous parameters, is of type D and belongs to the Kerr-Schild class. Its relation to the previously known Kerr-(A)dS and Kerr-NUT-(A)dS metrics is clarified.