Domain Structure of Black Hole Space-Times

TL;DR

This paper introduces the concept of domain structure for stationary black hole spacetimes, providing a new invariant framework that generalizes previous rod structures and applies across various dimensions.

Contribution

It defines the domain structure for any dimension with multiple Killing vectors, extending the rod structure concept and analyzing specific black hole solutions.

Findings

Domain structure is a useful invariant for classifying black hole spacetimes.

Canonical form of metrics for asymptotically flat spacetimes is established.

Analysis of domain structures for Minkowski, Schwarzschild-Tangherlini, and Myers-Perry black holes.

Abstract

We introduce the domain structure for stationary black hole space-times. Given a set of commuting Killing vector fields of the space-time the domain structure lives on the submanifold where at least one of the Killing vector fields have zero norm. Depending on which Killing vector field has zero norm the submanifold is naturally divided into domains. A domain corresponds either to a set of fixed points of a spatial symmetry or to a Killing horizon, depending on whether the characterizing Killing vector field is space-like or time-like near the domain. The domain structure provides invariants of the space-time, both topological and geometrical. It is defined for any space-time dimension and any number of commuting Killing vector fields. We examine the domain structure for asymptotically flat space-times and find a canonical form for the metric of such space-times. The domain structure generalizes the rod structure introduced for space-times with D-2 commuting Killing vector fields. We analyze in detail the domain structure for Minkowski space, the Schwarzschild-Tangherlini black hole and the Myers-Perry black hole in six and seven dimensions. Finally we consider the possible domain structures for asymptotically flat black holes in six and seven dimensions.