Regularity of Horizons and The Area Theorem

TL;DR

This paper proves that the area of sections of future event horizons in certain space-times is non-decreasing towards the future, extending Hawking's theorem without assuming smoothness, and explores conditions for smoothness or analyticity when equality holds.

Contribution

It extends Hawking's area theorem to less restrictive conditions, proving non-decreasing horizon area without assuming smoothness and establishing new differentiability results.

Findings

Non-decreasing horizon area under various conditions.

Smoothness or analyticity when area equality is attained.

New differentiability properties of horizons.

Abstract

We prove that the area of sections of future event horizons in space-times satisfying the null energy condition is non-decreasing towards the future under any one of the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic space-time and there exists a conformal completion with a ``H-regular'' Scri plus; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. No assumptions about the cosmological constant or its sign are made. We prove smoothness or analyticity of the relevant part of the event horizon when equality in the area inequality is attained - this has applications to the theory of stationary black holes, as well as to the structure of compact Cauchy horizons. In the course of the proof we establish several new results concerning the differentiability properties of horizons.