Regularity of Horizons and the Area Theorem

Abstract

We prove that the area of sections of future event horizons in spacetimes 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 "Hregular" I + ; 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 * Supported in part by KBN grant # 2 P03B 130 16. E-mail : Chrusciel@Univ-Tours.fr † Current adress: Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm. E-mail : Delay@gargan.math.Univ-Tours.fr ‡ Supported in part by NSF grant # DMS-9803566. E-mail : galloway@math.miami.edu § Supported in part by DoD Grant # N00014-97-1-0806 E-mail : howard@math.sc.edu 1 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.

On fine differentiability properties of horizons and applications to Riemannian geometry

“…Let, then, S be as in Theorem 1; since the result is purely local, without loss of generality we may assume that S is the level set {t = 1} of a time function t, with range R, the level sets of which are Cauchy surfaces. We use the constructions and notations of Chruściel et al [4], with Σ 1 = S and Σ 2 = {t = 2}. Let…”

“…and let A, φ be defined as at the beginning of the proof of Theorem 6.1 in [4]. Hence, A is the subset of S 2 = Σ 2 ∩ H consisting of those points in S 2 that are met by the generators of H that meet S 1 = Σ 1 ∩ H, and φ : A → S 1 is the map that moves the points of A back along these generators to S 1 .…”

“…We use the definitions, constructions and notations of the proof of Chruściel et al [4,Proposition 6.16]. Thus, let U ⊂ Σ 2 be a coordinate neighborhood of the form V ×(a, b) with V ⊂ R n−1 and a, b ∈ R, in which U ∩ N is the graph of a C 1,1 function g : V → R, and in which H ∩ U is the graph of a semi-convex function f : V → R. Here, N = N δ is a locally C 1,1 hypersurface in Σ 2 into which A has been embedded.…”

“…Let pr A denote the projection onto V of A ∩ U, thus A ∩ U is the graph of g over pr A. Now, let x 0 ∈B ∩ prÂ, whereB is the set of full measure in pr A constructed in the proof of Chruściel et al [4,Proposition 6.16], and pr is the projection onto V of ∩ U. Since g is Lipschitz, the graph of g overB ∩ pr has full measure inÂ.…”

“…In the terminology of Chruściel et al [4] "Alexandrov points propagate to the past along generators" of horizons. In the Riemannian setting Alexandrov points of a distance function ρ C propagate away from C along C-minimizing segments.…”