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

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.

25 pages. v2: added explicit form of the KS vector in various coordinates, improved comments in footnote 15 and in the last paragraph of app. C.1.2, typos fixed

References in corpus (30)

• Black holes and the double copy
• General Kerr-NUT-AdS Metrics in All Dimensions
• The Kerr-Schild double copy in curved spacetime
• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Robinson-Trautman spacetimes in higher dimensions
• Algebraic classification of higher dimensional spacetimes based on null alignment
• Generalization of the Geroch-Held-Penrose formalism to higher dimensions
• Type D Einstein spacetimes in higher dimensions
• Electromagnetic Properties of Kerr-Anti-de Sitter Black Holes
• Kerr-NUT-de Sitter Curvature in All Dimensions
• Higher dimensional Kerr-Schild spacetimes
• Kerr-Schild spacetimes with (A)dS background
• Robinson-Trautman spacetimes with an electromagnetic field in higher dimensions
• Gyromagnetic Ratio of Charged Kerr-Anti-de Sitter Black Holes
• Kerr-Schild Structure and Harmonic 2-forms on (A)dS-Kerr-NUT Metrics
• On a five-dimensional version of the Goldberg-Sachs theorem
• Electromagnetic Field in Higher-Dimensional Black-Hole Spacetimes
• Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
• Bel-Debever criteria for the classification of the Weyl tensors in higher dimensions
• Aligned Fields Double Copy to Kerr-NUT-(A)dS
• On higher dimensional Einstein spacetimes with a warped extra dimension
• Charging Kerr-Schild spacetimes in higher dimensions
• Weakly charged generalized Kerr-NUT-(A)dS spacetimes
• On the Goldberg-Sachs theorem in higher dimensions in the non-twisting case
• Classification of Kerr-de Sitter-like spacetimes with conformally flat $\mathcal{I}$ in all dimensions
• Twisting non-shearing congruences of null geodesics, almost CR structures, and Einstein metrics in even dimensions
• Type II Einstein spacetimes in higher dimensions
• The limit of Kerr-de Sitter spacetime with infinite angular-momentum parameter $a$
• Covariant classification of conformal Killing vectors of locally conformally flat $n$-manifolds with an application to Kerr-de Sitter
• Kerr-AdS type higher dimensional black holes with non-spherical cross-sections of horizons

Cited by in corpus (1)

• On type II(D) Einstein spacetimes in six dimensions