On higher dimensional black holes with abelian isometry group

TL;DR

This paper investigates the geometric and causal properties of higher-dimensional black holes with abelian symmetries, establishing positivity of orbit areas, timelike orbits, and conditions for static horizons, aiding classification efforts.

Contribution

It proves positivity of orbit areas, timelike nature of isometry group orbits, and non-existence of static Killing vector zeros on horizons in higher dimensions with abelian symmetries.

Findings

Orbit area is positive on the domain of outer communications.

Orbits of the connected isometry group are timelike throughout the domain.

Zeros of static Killing vectors do not occur on degenerate horizons.

Abstract

We consider (n+1)--dimensional, stationary, asymptotically flat, or Kaluza-Klein asymptotically flat black holes, with an abelian $s$--dimensional subgroup of the isometry group satisfying an orthogonal integrability condition. Under suitable regularity conditions we prove that the area of the group orbits is positive on the domain of outer communications, vanishing only on its boundary and on the "symmetry axis". We further show that the orbits of the connected component of the isometry group are timelike throughout the domain of outer communications. Those results provide a starting point for the classification of such black holes. Finally, we show non-existence of zeros of static Killing vectors on degenerate Killing horizons, as needed for the generalisation of the static no-hair theorem to higher dimensions.