Is There a General Area Theorem for Black Holes?

TL;DR

This paper examines the conditions under which the black hole area law holds, showing that black hole horizons typically develop cusps, challenging the assumption of smoothness and calling for a measure-theoretic proof of the area theorem.

Contribution

It provides detailed proofs under smoothness assumptions and demonstrates that black hole horizons generally contain cusps, questioning the physical plausibility of smooth horizon models.

Findings

Black hole surfaces often contain cusps before caustics appear.

Piecewise C^2 smoothness assumptions are likely physically unnatural.

A measure-theoretic approach to the area theorem is advocated.

Abstract

The general validity of the area law for black holes is still an open problem. We first show in detail how to complete the usually incompletely stated text-book proofs under the assumption of piecewise $C^2$-smoothness for the surface of the black hole. Then we prove that a black hole surface necessarily contains points where it is not $C^1$ (called ``cusps'') at any time before caustics of the horizon generators show up, like e.g. in merging processes. This implies that caustics never disappear in the past and that black holes without initial cusps will never develop such. Hence black holes which will undergo any non-trivial processes anywhere in the future will always show cusps. Although this does not yet imply a strict incompatibility with piecewise $C^2$ structures, it indicates that the latter are likely to be physically unnatural. We conclude by calling for a purely measure-theoretic proof of the area theorem.