# Regularity of Horizons and the Area Theorem

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

Research output: Contribution to journalArticle

44 Citations (Scopus)

### 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 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 "${\cal H}$-regular" ${\cal J}$+; 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. 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.

Original language English (US) 109-178 70 Annales Henri Poincare 2 1 https://doi.org/10.1007/PL00001029 Published - Feb 1 2001

### Fingerprint

event horizon
regularity
horizon
Horizon
theorems
Regularity
Theorem
hyperbolic coordinates
Black Holes
Space-time
Completion
Smoothness
Analyticity
Differentiability
Cauchy
Null
Equality
energy

### Keywords

• Black Hole
• Energy Condition
• Event Horizon
• Future Event
• Relevant Part

### ASJC Scopus subject areas

• Statistical and Nonlinear Physics
• Nuclear and High Energy Physics
• Mathematical Physics

### Cite this

Regularity of Horizons and the Area Theorem. / Chruściel, Piotr T.; Delay, Erwann; Galloway, Gregory J.; Howard, Ralph.

In: Annales Henri Poincare, Vol. 2, No. 1, 01.02.2001, p. 109-178.

Research output: Contribution to journalArticle

Chruściel, PT, Delay, E, Galloway, GJ & Howard, R 2001, 'Regularity of Horizons and the Area Theorem', Annales Henri Poincare, vol. 2, no. 1, pp. 109-178. https://doi.org/10.1007/PL00001029
Chruściel, Piotr T. ; Delay, Erwann ; Galloway, Gregory J. ; Howard, Ralph. / Regularity of Horizons and the Area Theorem. In: Annales Henri Poincare. 2001 ; Vol. 2, No. 1. pp. 109-178.
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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 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 {"}${\cal H}$-regular{"} ${\cal J}$+; 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. 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.",
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N2 - 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 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 "${\cal H}$-regular" ${\cal J}$+; 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. 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.

AB - 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 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 "${\cal H}$-regular" ${\cal J}$+; 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. 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.

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