Domain Structure of Black Hole Space-Times with a Cosmological Constant

TL;DR

This paper extends the concept of domain structures to classify and analyze stationary black hole space-times with a cosmological constant, providing new topological invariants and insights into horizon topologies in various dimensions.

Contribution

It generalizes the domain structure framework to include asymptotically de Sitter and Anti-de Sitter space-times, offering a unified approach for higher-dimensional black hole classification.

Findings

Domain structures for asymptotically Anti-de Sitter space-times resemble flat cases.

Domain structures for asymptotically de Sitter space-times are compact.

Horizon topologies are restricted by the domain structure analysis in certain dimensions.

Abstract

We generalize the domain structure for stationary black hole space-times to include asymptotically de Sitter and Anti-de Sitter space-times. Given a set of commuting Killing vector fields of a space-time the domain structure lives on the submanifold of the orbit space on which at least one of the Killing vector fields has zero norm. In general the domain structure provides topological and geometrical invariants of black hole space-times that in specific cases have proven to be a crucial part of a full characterization leading to uniqueness theorems. In four and five dimensions the domain structure generalizes the rod structure. We examine in detail the domain structure for four, five, six and seven-dimensional black hole space-times including a very general class of spherically symmetric and static black hole space-times as well as the exact solutions for Kerr-(Anti)-de Sitter black holes. While for asymptotically Anti-de Sitter space-times the domain structures resemble that of asymptotically flat space-times, the domain structures of asymptotically de Sitter space-times are shown to be compact. Finally, we find under certain assumptions that the horizon topologies for four- and five-dimensional black holes are restricted by our domain structure analysis.