Regularity and temperature of stationary black hole event horizons

TL;DR

This paper proves that stationary black hole event horizons are regular, totally geodesic null hypersurfaces under weak causality and energy conditions, and establishes the constancy of surface gravity and temperature without strong assumptions.

Contribution

It introduces a quotient bundle approach to prove horizon regularity directly and shows that surface gravity is constant under broad conditions, extending the zeroth law of black hole thermodynamics.

Findings

Horizon regularity is established under weak causality conditions.

Surface gravity is shown to be constant without non-degeneracy assumptions.

Black hole temperature is constant, confirming the zeroth law under general conditions.

Abstract

Available proofs of the regularity of stationary black hole event horizons rely on certain assumptions on the existence of sections that imply a $C^1$ differentiability assumption. By using a quotient bundle approach, we remedy this problem by proving directly that, indeed, under the null energy condition event horizons of stationary black holes are totally geodesic null hypersurfaces as regular as the metric. Only later, by using this result, we show that the cross-sections, whose existence was postulated in previous works, indeed exist.These results hold true under weak causality conditions. Subsequently, we prove that under the dominant energy condition stationary black hole event horizons indeed admit constant surface gravity, a result that does not require any non-degeneracy assumption, requirements on existence of cross-sections or a priori smoothness conditions. We are able to make sense of the angular velocity and of the value (not just sign) of surface gravity as quantities related to the horizon, without the need of assuming Einstein's vacuum equations and the Killing extension. Physically, this implies that under very general conditions every stationary black hole has indeed a constant temperature (the zeroth law of black hole thermodynamics).