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Regularity of Horizons and The Area Theorem

Piotr T. Chruściel, Erwann Delay, Gregory J. Galloway, Ralph Howard

TL;DR

This work removes differentiability barriers in Hawking's area theorem by proving area monotonicity for cross-sections of future event horizons under the null energy condition, without assuming horizon smoothness. It builds a robust framework based on horizon semi-convexity and Alexandrov differentiability, introducing the Alexandrov divergence $\theta_{\mathcal{A}l}$ and the Alexandrov second fundamental form $B_{\mathcal{A}l}$, and leverages a Whitney-type $C^{1,1}$ extension to compare areas via a change-of-variables argument. The authors prove non-negativity of $\theta_{\mathcal{A}l}$ under several causal/conformal hypotheses and show that Alexandrov points propagate along horizon generators with the optical equation $b'+b^2+R=0$; this yields area monotonicity and a rigidity description in the equality case. The results extend Hawking's area theorem to rough horizons, with implications for stationary black holes, the differentiability of Cauchy horizons, and the interplay between geometry, causality, and conformal completion in general relativity.

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.

Regularity of Horizons and The Area Theorem

TL;DR

This work removes differentiability barriers in Hawking's area theorem by proving area monotonicity for cross-sections of future event horizons under the null energy condition, without assuming horizon smoothness. It builds a robust framework based on horizon semi-convexity and Alexandrov differentiability, introducing the Alexandrov divergence and the Alexandrov second fundamental form , and leverages a Whitney-type extension to compare areas via a change-of-variables argument. The authors prove non-negativity of under several causal/conformal hypotheses and show that Alexandrov points propagate along horizon generators with the optical equation ; this yields area monotonicity and a rigidity description in the equality case. The results extend Hawking's area theorem to rough horizons, with implications for stationary black holes, the differentiability of Cauchy horizons, and the interplay between geometry, causality, and conformal completion in general relativity.

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.

Paper Structure

This paper contains 14 sections, 38 theorems, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Horizons
  3. Sections of horizons
  4. Non-negativity of $\theta_{\mathcal{A}l}$
  5. Causally regular conformal completions
  6. Complete generators
  7. Propagation of Alexandrov points along generators, optical equation
  8. Area monotonicity
  9. Conclusions
  10. The Geometry of $C^2$ Null Hypersurfaces
  11. Some comments on the area theorem of Hawking and Ellis
  12. Some comments on the area theorems of Królak
  13. Proof of Theorem \ref{['TCfr']}
  14. Proof of Proposition \ref{['C11-extend']}

Key Result

Theorem 1.1

Let ${\mathscr H}$ be a black hole event horizon in a smooth space--time $(M,g)$. Suppose that either Let $\Sigma_a$, $a=1,2$ be two achronal spacelike embedded hypersurfaces of $C^2$ differentiability class, set $S_a= \Sigma_a\cap{\mathscr H}$. Then:

Figures (2)

  • Figure 1: A section of the event horizon in a $2+1$ dimensional space--time with "two black holes merging".
  • Figure 2: The set $\Omega\equiv J^+({\mycal S};{\mycal M} ) \cap \partial{\mycal U}$ and its image under $\psi$.

Theorems & Definitions (94)

  • Theorem 1.1: The area theorem
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • proof
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 84 more