On the `Stationary Implies Axisymmetric' Theorem for Extremal Black Holes in Higher Dimensions

TL;DR

This paper extends the 'Stationary Implies Axisymmetric' theorem to extremal black holes in higher dimensions, showing that under a diophantine condition, these black holes also possess additional symmetries and constant surface gravity.

Contribution

It generalizes the stationary axisymmetry theorem to extremal black holes in higher dimensions under a diophantine condition, expanding previous results for non-extremal cases.

Findings

The theorem holds for extremal black holes if the angular velocities satisfy a diophantine condition.

The result applies to higher-dimensional black holes, not just four-dimensional cases.

The diophantine condition is satisfied except on a measure-zero set.

Abstract

All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gravity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [gr-qc/0605106] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einstein's equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain "diophantine condition," which holds except for a set of measure zero.