Regularity of Horizons and The Area Theorem

Piotr T. Chruściel^{∗} Erwann Delay^{†} Département de Mathématiques Faculté des Sciences Parc de Grandmont F37200 Tours, France

Gregory J. Galloway^{‡}

Department of Mathematics and Computer Science

University of Miami Coral Gables FL 33124, USA

Ralph Howard^{§} Department of Mathematics University of South Carolina Columbia S.C. 29208, USA

December 12, 2000

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 “ℋ– regular” 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 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.

∗Supported in part by KBN grant # 2 P03B 130 16. E–mail: Chrusciel@Univ-Tours.fr †Supported in part by the EU TMR project Stochastic Analysis and its Application, ERB-FMRX-CT96-0075. 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

Contents

1 Introduction 2 2 Horizons 7 3 Sections of horizons 13 4 Non-negativity of θ_{Al} 20

4.1 Causally regular conformal completions . . . . . . . . . . . . . . . 20 4.2 Complete generators . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Propagation of Alexandrov points along generators, optical

equation 30 6 Area monotonicity 35 7 Conclusions 51 A The Geometry of C^{2} Null Hypersurfaces 52 B Some comments on the area theorem of Hawking and Ellis 61 C Some comments on the area theorems of Królak 63 D Proof of Theorem 5.6 64 E Proof of Proposition 6.6 65

1 Introduction

The thermodynamics of black holes rests upon Hawking’s area theorem [37] which asserts that in appropriate space–times the area of cross–sections of black hole horizons is non–decreasing towards the future. In the published proofs of this result [37, 64] there is considerable vagueness as to the hypotheses of differentiability of the event horizon (see, however, [35]). Indeed, it is known that black hole horizons can be pretty rough [15], and it is not immediately clear that the area of their cross–sections can even be defined. The reading of the proofs given in [37, 64] suggests that those authors have assumed the horizons under consideration to be piecewise C^{2}. Such a hypothesis is certainly incompatible with the examples constructed in [15] which are nowhere C^{2}. The aim of this paper is to show that the monotonicity theorem holds without any further differentiability hypotheses on the horizon, in appropriate space–times, for a large class of cross–sections of the horizon. More precisely we show the following:

Theorem 1.1 (The area theorem) Let ℋ be a black hole event horizon in a smooth space–time (M, g). Suppose that either

a) (M, g) is globally hyperbolic, and there exists a conformal completion (M̄, ḡ) = (M ∪ ℐ^{+}, Ω^{2}g) of (M, g) with a ℋ–regular^{1} ℐ^{+}. Further the null energy condition holds on the past I^{−}(ℐ^{+}; M̄) ∩ M of ℐ^{+} in M, or

b) the generators of ℋ are future complete and the null energy condition holds on ℋ, or

c) there exists a globally hyperbolic conformal completion (M̄, ḡ) = (M ∪ ℐ^{+}, Ω^{2}g) of (M, g), with the null energy condition holding on I^{−}(ℐ^{+}; M̄)∩ M.

Let Σ_{a}, a = 1, 2 be two achronal spacelike embedded hypersurfaces of C^{2} differentiability class, set S_{a} = Σ_{a} ∩ ℋ. Then:

1. The area Area(S_{a}) of S_{a} is well defined.

2. If

S_{1} ⊂ J^{−}(S_{2}) ,

then the area of S_{2} is larger than or equal to that of S_{1}. (Moreover, this is true even if the area of S_{1} is counted with multiplicity of generators provided that S_{1} ∩ S_{2} = ∅, see Theorem 6.1.)

Point 1. of Theorem 1.1 is Proposition 3.3 below, see also Proposition 3.4. Point 2. above follows immediately from Proposition 4.2, Corollary 4.14 and Proposition 4.17, as a special case of the first part of Theorem 6.1 below.^{2} The question of how to define the area of sections of the horizon is discussed in detail in Section 3. It is suggested there that a notion of area, appropriate for the identification of area with the entropy, should include the multiplicity of generators of the horizon.

1See Section 4.1 for definitions.

2The condition S_{1} ∩ S_{2} = ∅ of Theorem 6.1 is needed for monotonicity of “area with multiplicity”, but is not needed if one only wants to compare standard areas, see Remark 6.2.

We stress that we are not assuming that ℐ^{+} is null — in fact it could be even changing causal type from point to point — in particular Theorem 1.1 also applies when the cosmological constant does not vanish. In point c) global hyperbolicity of (M̄, ḡ) should be understood as that of a manifold with boundary, cf. Section 4.1. Actually in points a) and c) of Theorem 1.1 we have assumed global hyperbolicity of (M, g) or that of (M̄, ḡ) for simplicity only: as far as Lorentzian causality hypotheses are concerned, the assumptions of Proposition 4.1 are sufficient to obtain the conclusions of Theorem 1.1. We show in Section 4.1 that those hypotheses will hold under the conditions of Theorem 1.1. Alternative sets of causality conditions, which do not require global hyperbolicity of (M, g) or of its conformal completion, are given in Propositions 4.8 and 4.10.

It seems useful to compare our results to other related ones existing in the literature [37, 47, 64]. First, the hypotheses of point a) above are fulfilled under the conditions of the area theorem of [64] (after replacing the space–time M considered in [64] by an appropriate subset thereof), and (disregarding questions of differentiability of ℋ) are considerably weaker than the hypotheses of [64]. Consider, next, the original area theorem of [37], which we describe in some detail in Appendix B for the convenience of the reader. We note that we have been unable to obtain a proof of the area theorem without some condition of causal regularity of ℐ^{+} (e.g. the one we propose in point a) of Theorem 1.1), and we do not know^{3} whether or not the area theorem holds under the original conditions of [37] without the modifications indicated above, or in Appendix B; see Appendix B for some comments concerning this point.

Let us make a few comments about the strategy of the proof of Theorem 1.1. It is well known that event horizons are Lipschitz hypersurfaces, and the examples constructed in [8, 15] show that much more cannot be expected. We start by showing that horizons are semi–convex^{4}. This, together with Alexandrov’s theorem concerning the regularity of convex functions shows that they are twice differentiable in an appropriate sense (cf. Proposition 2.1 below) almost everywhere. This allows one to define notions such as the divergence θ_{Al} of the generators of the horizon ℋ, as well as the divergence θ_{Al}^{ℋ∩𝒮} of sections ℋ ∩ 𝒮 of ℋ. The existence of the second order expansions at Alexandrov points leads further to the proof that, under appropriate conditions, θ_{Al} or θ_{Al}^{ℋ∩𝒮} have the right sign. Next, an approximation result of Whitney type, Proposition 6.6, allows one to embed certain subsets of the horizon into C^{1,1} manifolds. The area theorem then follows from the change–of–variables theorem for Lipschitz maps proved in [23]. We note that some further effort is required to convert the information that θ_{Al} has the correct sign into an inequality concerning the Jacobian that appears in the change of variables formula.

3The proof of Proposition 9.2.1 in [37] would imply that the causal regularity condition assumed here holds under the conditions of the area theorem of [37]. However, there are problems with that proof (this has already been noted by Newman [53]).

4Actually this depends upon the time orientation: future horizons are semi–convex, while past horizons are semi–concave.

Various authors have considered the problem of defining black holes in settings more general than standard asymptotically flat spacetimes or spacetimes admitting a conformal infinity; see especially [34, 48, 49] and references cited therein. It is likely that proofs of the area theorem given in more general settings, for horizons assumed to be piecewise C^{2}, which are based on establishing the positivity of the (classically defined) expansion θ of the null generators can be adapted, using the methods of Section 4 (cf. especially Proposition 4.1 and Lemma 4.15), to obtain proofs which do not require the added smoothness. (We show in Appendix C that this is indeed the case for the area theorems of Królak [47–49].) In all situations which lead to the positivity of θ in the weak Alexandrov sense considered here, the area theorem follows from Theorem 6.1 below.

It is of interest to consider the equality case: as discussed in more detail in Section 7, this question is relevant to the classification of stationary black holes, as well as to the understanding of compact Cauchy horizons. Here we prove the following:

Theorem 1.2 Under the hypotheses of Theorem 1.1, suppose that the area of S_{1} equals that of S_{2}. Then

(J^{+}(S_{1}) \ S_{1}) ∩ (J^{−}(S_{2}) \ S_{2})

is a smooth (analytic if the metric is analytic) null hypersurface with vanishing null second fundamental form. Moreover, if γ is a null generator of ℋ with γ(0) ∈ S_{1} and γ(1) ∈ S_{2}, then the curvature tensor of (M, g) satisfies R(X, γ′(t))γ′(t) = 0 for all t ∈ [0, 1] and X ∈ T_{γ(t)}ℋ.

Theorem 1.2 follows immediately from Corollary 4.14 and Proposition 4.17, as a special case of the second part of Theorem 6.1 below. The key step of the proof here is Theorem 6.18, which has some interest in its own. An application of those results to stationary black holes is given in Theorem 7.1, Section 7.

As already pointed out, one of the steps of the proof of Theorem 1.1 is to establish that a notion of divergence θ_{Al} of the generators of the horizon, or of sections of the horizon, can be defined almost everywhere, and that θ_{Al} so defined is positive. We note that θ_{Al} coincides with the usual divergence θ for horizons which are twice differentiable. Let us show, by means of an example, that the positivity of θ might fail to hold in space–times (M, g) which do not satisfy the hypotheses of Theorem 1.1: Let t be a standard time coordinate on the three dimensional Minkowski space–time ℝ^{1,2}, and let K ⊂ {t = 0} be an open conditionally compact set with smooth boundary ∂K. Choose K so that the mean curvature H of ∂K has changing sign. Let M = I^{−}(K), with the metric conformal to the Minkowski metric by a conformal factor which is one in a neighborhood of ∂𝒟^{−}(K; ℝ^{1,2})\K, and which makes ∂J^{−}(K; ℝ^{1,2}) into ℐ^{+} in the completion M̄ ≡ M ∪ ∂J^{−}(K; ℝ^{1,2}). We have M \ J^{−}(ℐ^{+}; M̄) = 𝒟^{−}(K; ℝ^{1,2}) ≠ ∅, thus M contains a black hole region, with the event horizon being the Cauchy horizon ∂𝒟^{−}(K; ℝ^{1,2}) \ K. The generators of the event horizon coincide with the generators of ∂𝒟^{−}(K; ℝ^{1,2}), which are null geodesics normal to ∂K. Further for t negative and close to zero the divergence θ of those generators is well defined in a classical sense (since the horizon is smooth there) and approaches, when t tends to zero along the generators, the mean curvature H of ∂K. Since the conformal factor equals one in a neighborhood of the horizon the null energy condition holds there, and θ is negative near those points of ∂K where H is negative. This implies also the failure of the area theorem for some (local) sections of the horizon. We note that condition b) of Theorem 1.1 is not satisfied because the generators of ℋ are not future geodesically complete. On the other hand condition a) does not hold because g will not satisfy the null energy condition throughout I^{−}(ℐ^{+}; M̄) whatever the choice of the conformal factor^{5}.

This paper is organized as follows: In Section 2 we show that future horizons, as defined there, are always semi–convex (Theorem 2.2). This allows us to define such notions as the Alexandrov divergence θ_{Al} of the generators of the horizon, and their Alexandrov null second fundamental form. In Section 3 we consider sections of horizons and their geometry, in particular we show that sections of horizons have a well defined area. We also discuss the ambiguities which arise when defining the area of those sections when the horizon is not globally smooth. In fact, those ambiguities have nothing to do with “very low” differentiability of horizons and arise already for piecewise smooth horizons. In Section 4 we prove positivity of the Alexandrov divergence of generators of horizons — in Section 4.1 this is done under the hypothesis of existence of a conformal completion satisfying a regularity condition, together with some global causality assumptions on the space–time; in Section 4.2 positivity of θ_{Al} is established under the hypothesis that the generators of the horizon are future complete. In Section 5 we show that Alexandrov points “propagate to the past” along the generators of the horizon. This allows one to show that the optical equation holds on “almost all” generators of the horizon. We also present there a theorem (Theorem 5.6) which shows that “almost all generators are Alexandrov”; while this theorem belongs naturally to Section 5, its proof uses methods which are developed in Section 6 only, therefore it is deferred to Appendix D. In Section 6 we prove our main result – the monotonicity theorem, Theorem 6.1. This is done under the assumption that θ_{Al} is non–negative. One of the key elements of the proof is a new (to us) extension result of Whitney type (Proposition 6.6), the hypotheses of which are rather different from the usual ones; in particular it seems to be much easier to work with in some situations. Section 7 discusses the relevance of the rigidity part of Theorem 6.1 to the theory of black holes and to the differentiability of compact Cauchy horizons. Appendix A reviews the geometry of C^{2} horizons, we also prove there a new result concerning the relationship between the (classical) differentiability of a horizon vs the (classical) differentiability of sections thereof, Proposition A.3. In Appendix B some comments on the area theorem of [37] are made.

5This follows from Theorem 1.1.

2 Horizons

Let (M, g) be a smooth spacetime, that is, a smooth paracompact Hausdorff timeoriented Lorentzian manifold, of dimension n + 1 ≥ 3, with a smooth Lorentzian metric g. Throughout this paper hypersurfaces are assumed to be embedded. A hypersurface ℋ ⊂ M will be said to be future null geodesically ruled if every point p ∈ ℋ belongs to a future inextensible null geodesic Γ ⊂ ℋ; those geodesics will be called the generators of ℋ. We emphasize that the generators are allowed to have past endpoints on ℋ, but no future endpoints. Past null geodesically ruled hypersurfaces are defined by changing the time orientation. We shall say that ℋ is a future (past) horizon if ℋ is an achronal, closed, future (past) null geodesically ruled topological hypersurface. A hypersurface ℋ will be called a horizon if ℋ is a future or a past horizon. Our terminology has been tailored to the black hole setting, so that a future black hole event horizon ∂J^{−}(ℐ^{+}) is a future horizon in the sense just described [37, p. 312]. The terminology is somewhat awkward in a Cauchy horizon setting, in which a past Cauchy horizon 𝒟^{−}(Σ) of an achronal edgeless set Σ is a future horizon in our terminology [56, Theorem 5.12].

Let dimM = n + 1 and suppose that 𝒪 is a domain in ℝ^{n}. Recall that a continuous function f: 𝒪 → ℝ is called semi–convex iff each point p has a convex neighborhood U in 𝒪 so that there exists a C^{2} function ϕ: U → ℝ such that f + ϕ is convex in U. We shall say that the graph of f is a semi–convex hypersurface if f is semi–convex. A hypersurface ℋ in a manifold M will be said to be semi– convex if ℋ can be covered by coordinate patches 𝒰_{α} such that ℋ ∩ 𝒰_{α} is a semi–convex graph for each α. The interest of this notion stems from the specific differentiability properties of such hypersurfaces:

Proposition 2.1 (Alexandrov [26, Appendix E]) Let B be an open subset of ℝ^{p} and let f: B → ℝ be semi–convex. Then there exists a set B_{Al} ⊂ B such that:

1. the p dimensional Lebesgue measure 𝔏^{p}(B \ B_{Al}) of B \ B_{Al} vanishes.

2. f is differentiable at all points x ∈ B_{Al}, i.e.,

∀ x ∈ B_{Al} ∃x^{∗} ∈ (ℝ^{p})^{∗} such that
∀ y ∈ B f(y) − f(x) = x^{∗}(y − x) + r_{1}(x, y) ,  (2.1)

with r_{1}(x, y) = o(|x−y|). The linear map x^{∗} above will be denoted by df(x).

3. f is twice–differentiable at all points x ∈ B_{Al} in the sense that

∀ x ∈ B_{Al} ∃ Q ∈ (ℝ^{p})^{∗} ⊗ (ℝ^{p})^{∗} such that ∀ y ∈ B

f(y) − f(x) − df(x)(y − x) = Q(x − y, x − y) + r_{2}(x, y) , (2.2)

with r_{2}(x, y) = o(|x − y|^{2}). The symmetric quadratic form Q above will be denoted by \frac{1}{2}D^{2}f(x), and will be called the second Alexandrov derivative of f at x.^{6}

The points q at which Equations (2.1)–(2.2) hold will be called Alexandrov points of f, while the points (q, f(q)) will be called Alexandrov points of the graph of f (it will be shown in Proposition 2.5 below that if q is an Alexandrov point of f, then (q, f(q)) will project to an Alexandrov point of any graphing function of the graph of f, so this terminology is meaningful).

We shall say that ℋ is locally achronal if for every point p ∈ ℋ there exists an open neighborhood 𝒪 of p such that ℋ ∩ 𝒪 is achronal in 𝒪. We have the following:

Theorem 2.2 Let ℋ ≠ ∅ be a locally achronal future null geodesically ruled hypersurface. Then ℋ is semi–convex.

Remark 2.3 An alternative proof of Theorem 2.2 can be given using a variational characterization of horizons (compare [4, 33, 57]).

Remark 2.4 Recall for a real valued function f: B → ℝ on a set the epigraph of f is {(x, y) ∈ B × ℝ : y ≥ f(x)}. We note that while the notion of convexity of a function and its epigraph is coordinate dependent, that of semi–convexity is not; indeed, the proof given below is based on the equivalence of semi–convexity and of existence of lower support hypersurfaces with locally uniform one side Hessian bounds. The latter is clearly independent of the coordinate systems used to represent f, or its graph, as long as the relevant orientation is preserved (changing y^{n+1} to −y^{n+1} transforms a lower support hypersurface into an upper one). Further, if the future of ℋ is represented as an epigraph in two different ways,

J^{+}(ℋ) = {x^{n+1} ≥ f(x^{1}, …, x^{n})}
= {y^{n+1} ≥ g(y^{1}, …, y^{n})} ,  (2.3)

then semi–convexity of f is equivalent to that of g. This follows immediately from the considerations below. Thus, the notion of a semi–convex hypersurface is not tied to a particular choice of coordinate systems and of graphing functions used to represent it.

6Caffarelli and Cabré [9] use the term “second punctual differentiability of f at x” for (2.2); Fleming and Soner [26, Definition 5.3, p. 234] use the name “point of twice–differentiability”

for points at which (2.2) holds.

Proof: Let 𝒪 be as in the definition of local achronality; passing to a subset of 𝒪 we can without loss of generality assume that 𝒪 is globally hyperbolic. Replacing the space–time (M, g) by 𝒪 with the induced metric we can without loss of generality assume that (M, g) is globally hyperbolic. Let t be a time function on 𝒪 which induces a diffeomorphism of 𝒪 with ℝ × Σ in the standard way [32, 59], with the level sets Σ_{τ} ≡ {p | t(p) = τ} of t being Cauchy surfaces. As usual we identify Σ_{0} with Σ, and in the identification above the curves ℝ×{q}, q ∈ Σ, are integral curves of ∇t. Define

Σ_{ℋ} = {q ∈ Σ | ℝ × {q} intersects ℋ} .  (2.4)

For q ∈ Σ_{ℋ} the set (ℝ × {q}) ∩ ℋ is a point by achronality of ℋ, we shall denote this point by (f(q), q). Thus an achronal hypersurface ℋ in a globally hyperbolic space–time is a graph over Σ_{ℋ} of a function f. The map which to a point p ∈ ℋ assigns q ∈ Σ such that p = (f(q), q) is injective, so that the hypothesis that ℋ is a topological hypersurface together with the invariance of the domain theorem (cf., e.g., [18, Prop. 7.4, p. 79]) imply that Σ_{ℋ} is open. We wish to use [1, Lemma 3.2]^{7} to obtain semi–convexity of the (local) graphing function f, this requires a construction of lower support hypersurfaces at p. Let Γ be a generator of ℋ passing through p and let p_{+} ∈ Γ ∩ J^{+}(p). By achronality of Γ there are no points on J^{−}(p_{+}) ∩ J^{+}(p) which are conjugate to p_{+}, cf. [37, Prop. 4.5.12, p. 115] or [5, Theorem 10.72, p. 391]. It follows that the intersection of a sufficiently small neighborhood of p with the past light cone ∂J^{−}(p_{+}) of p_{+} is a smooth hypersurface contained in J^{−}(ℋ). This provides the appropriate lower support hypersurface at p. In particular, for suitably chosen points p_{+}, these support hypersurfaces have null second fundamental forms (see Appendix A) which are locally (in the point p) uniformly bounded from below. This in turn implies that the Hessians of the associated graphing functions are locally bounded from below, as is needed to apply Lemma 3.2 in [1]. □

Theorem 2.2 allows us to apply Proposition 2.1 to f to infer twice Alexandrov differentiability of ℋ almost everywhere, in the sense made precise in Proposition 2.1. Let us denote by 𝔏_{h_{0}}^{n} the n dimensional Riemannian measure^{8} on Σ ≡ Σ_{0}, where h_{0} is the metric induced on Σ_{0} by g, then there exists a set Σ_{ℋ_{Al}} ⊂ Σ_{ℋ} on which f is twice Alexandrov differentiable, and such that 𝔏_{h_{0}}^{n}(Σ_{ℋ} \ Σ_{ℋ_{Al}} ) = 0. Set

ℋ_{Al} ≡ graph of f over Σ_{ℋ_{Al}} .  (2.5)

Point 1 of Proposition 2.1 shows that ℋ_{Al} has a tangent space at every point p ∈ ℋ_{Al}. For those points define

k(p) = k_{μ}(p)dx^{μ} = −dt + df(q) ,  p = (f(q), q),  q ∈ Σ_{ℋ_{Al}} .  (2.6)

7The result in that Lemma actually follows from [2, Lemma 2.15] together with [20, Prop. 5.4, p. 24].

8In local coordinates, d𝔏_{h_{0}}^{n} = \sqrt{det h_{0}}d^{n}x.

A theorem of Beem and Królak shows that ℋ is differentiable precisely at those points p which belong to exactly one generator Γ [6, 14], with T_{p}ℋ ⊂ T_{p}M being the null hyper-plane containing Γ̇_{p}. It follows that

K ≡ g^{μν}k_{μ}∂_{ν}  (2.7)

is null, future pointing, and tangent to the generators of ℋ, wherever defined.

We define the generators of ℋ_{Al} as the intersections of the generators of ℋ with ℋ_{Al}. Point 2 of Proposition 2.1 allows us to define the divergence θ_{Al} of those generators, as follows: Let e_{i}, i = 1, …, n be a basis of T_{p}ℋ such that

g(e_{1}, e_{1}) = ⋯ = g(e_{n−1}, e_{n−1}) = 1 ,  g(e_{i}, e_{j}) = 0 , i ≠ j ,  e_{n} = K . (2.8)

We further choose the e_{a}’s, a = 1, …, n − 1 to be orthogonal to ∇t. It follows that the vectors e_{a} = e_{a}^{μ}∂_{μ}’s have no ∂/∂t components in the coordinate system used in the proof of Theorem 2.2, thus e_{a}^{0} = 0. Using this coordinate system for p ∈ ℋ_{Al} we set

i, j = 1, …, n ∇_{i}k_{j} = D_{ij}^{2}f − Γ_{ij}^{μ}k_{μ} ,  (2.9)
θ_{Al} = (e_{1}^{i}e_{1}^{j} + ⋯ + e_{n−1}^{i}e_{n−1}^{j})∇_{i}k_{j} .  (2.10)

It is sometimes convenient to set θ_{Al} = 0 on ℋ \ℋ_{Al}. The function θ_{Al} so defined on ℋ will be called the divergence (towards the future) of both the generators of ℋ_{Al} and of ℋ. It coincides with the usual divergence of the generators of ℋ at every set 𝒰 ⊂ ℋ on which the horizon is of C^{2} differentiability class: Indeed, in a space–time neighborhood of 𝒰 we can locally extend the e_{i}’s to C^{1} vector fields, still denoted e_{i}, satisfying (2.8). It is then easily checked that the set of Alexandrov points of 𝒰 is 𝒰, and that the divergence θ of the generators as defined in [37, 64] or in Appendix A coincides on 𝒰 with θ_{Al} as defined by (2.10).

The null second fundamental form B_{Al} of ℋ, or of ℋ_{Al}, is defined as follows: in the basis above, if X = X^{a}e_{a}, Y = Y^{b}e_{b} (the sums are from 1 to n − 1), then at Alexandrov points we set

B_{Al}(X, Y) = X^{a}Y^{b}e_{a}^{i}e_{b}^{j}∇_{i}k_{j}.  (2.11)

with ∇_{i}k_{j} defined by (2.9). This coincides^{9} with the usual definition of B as discussed e.g. in Appendix A on any subset of ℋ which is C^{2}. In this definition of the null second fundamental form B with respect to the null direction K, B measures expansion as positive and contraction as negative.

The definitions of θ_{Al} and B_{Al} given in Equations (2.10)–(2.11) have, so far, been only given for horizons which can be globally covered by an appropriate coordinate system. As a first step towards a globalization of those notions one needs to find out how those objects change when another representation is chosen. We have the following:

9More precisely, when ℋ is C^{2} equation (2.11) defines, in local coordinates, a tensor field which reproduces b defined in Appendix A when passing to the quotient ℋ/K.

Proposition 2.5 1. Let f and g be two locally Lipschitz graphing functions representing ℋ in two coordinate systems {x^{i}}_{i=1,...,n+1} and {y^{i}}_{i=1,...,n+1}, related to each other by a C^{2} diffeomorphism ϕ. If (x_{0}^{1}, … x_{0}^{n}) is an Alexandrov point of f and

(y_{0}^{1}, …, y_{0}^{n}, g(y_{0}^{1}, …, y_{0}^{n})) = ϕ(x_{0}^{1}, …, x_{0}^{n}, f(x_{0}^{1}, …, x_{0}^{n})) ,

then (y_{0}^{1}, …, y_{0}^{n}) is an Alexandrov point of g.

2. The null second fundamental form B_{Al} of Equation (2.11) is invariantly defined modulo a point dependent multiplicative factor. In particular the sign of θ_{Al} defined in Equation (2.10) does not depend upon the choice of the graphing function used in (2.10).

Remark 2.6 Recall that the standard divergence θ of generators is defined up to a multiplicative function (constant on the generators) only, so that (essentially) the only geometric invariant associated to θ is its sign.

Remark 2.7 We emphasize that we do not assume that ∂/∂x^{n+1} and/or ∂/∂y^{n+1} are timelike.

Proof: 1. Let y⃗ = (y^{1}, …, y^{n}), x⃗_{0} = (x_{0}^{1}, …, x_{0}^{n}), etc., set f_{0} = f(x⃗_{0}), g_{0} = g(y⃗_{0}). Without loss of generality we may assume (x⃗_{0}, f(x⃗_{0})) = (y⃗_{0}, f(y⃗_{0})) = 0. To establish (2.2) for g, write ϕ as (ϕ⃗, ϕ^{n+1}), and let

ψ⃗(x⃗) := ϕ⃗(x⃗, f(x⃗)) .

As (id, f) and (id, g), where id is the identity map, are bijections between neighborhoods of zero and open subsets of the graph, invariance of domain shows that ψ⃗ is a homeomorphism from a neighborhood of zero to its image. Further, ψ⃗ admits a second order Alexandrov expansion at the origin:

ψ⃗(x⃗) = L_{1}x⃗ + α_{1}(x⃗, x⃗) + o(|x⃗|^{2}) ,  (2.12)

where

L_{1} = Dx⃗ϕ⃗(0) + D_{x^{n+1}} ϕ⃗(0) Df(0) .

Let us show that L_{1} is invertible: suppose, for contradiction, that this is not the case, let x⃗ ≠ 0 be an element of the kernel ker L_{1}, and for |s| << 1 let y⃗_{s} := ψ⃗(sx⃗) = O(s^{2}). As y⃗_{s} = ϕ⃗(sx⃗, f(sx⃗)) we have

g(y⃗_{s}) = ϕ^{n+1}(sx⃗, f(sx⃗)) = s[Dx⃗ϕ^{n+1}(0) + D_{x^{n+1}}ϕ^{n+1}(0)Df(0)]x⃗ + o(s) .

The coefficient of the term linear in s does not vanish, otherwise (x⃗, Df(0)x⃗) would be a non–zero vector of ker Dϕ(0). We then obtain |g(y⃗_{s})|/|y⃗_{s}| ≥ C/|s| →_{s→0} ∞, which contradicts the hypothesis that g is locally Lipschitz.

To finish the proof, for y⃗ close to the origin let x⃗ := ψ^{−1}(y⃗), thus y⃗ = ψ(x⃗), and from (2.12) we infer

x⃗ = L_{1}^{−1}y⃗ − L_{1}^{−1}α_{1}(L_{1}^{−1}y⃗, L_{1}^{−1}y⃗) + o(|y⃗|^{2}) .  (2.13)

Equation (2.2) and twice differentiability of ϕ show that ϕ^{n+1}(·, f(·)) has a second order Alexandrov expansion at the origin:

ϕ^{n+1}(x⃗, f(x⃗)) = L_{2}x⃗ + α_{2}(x⃗, x⃗) + o(|x⃗|^{2}) .  (2.14)

Finally from g(y⃗) = ϕ^{n+1}(x⃗, f(x⃗)) and Equations (2.13) and (2.14) we get

g(y⃗) = L_{2}L_{1}^{−1}y⃗ + (α_{2} − L_{2}L_{1}^{−1}α_{1})(L_{1}^{−1}y⃗, L_{1}^{−1}y⃗) + o(|y⃗|^{2}) .

2. The proof of point 1. shows that under changes of graphing functions the Alexandrov derivatives D_{ij}f transform into the D_{ij}g’s exactly as they would if f and g were twice differentiable. The proof of point 2. reduces therefore to that of the analogous statement for C^{2} functions, which is well known. □

Proposition 2.5 shows that Equation (2.11) defines an equivalence class of tensors B_{Al} at every Alexandrov point of ℋ, where two tensors are identified when they are proportional to each other with a positive proportionality factor. Whenever ℋ can not be covered by a global coordinate system required in Equation (2.11), one can use Equation (2.11) in a local coordinate patch to define a representative of the appropriate equivalence class, and the classes so obtained will coincide on the overlaps by Proposition 2.5. From this point of view θ_{Al}(p) can be thought as the assignment, to an Alexandrov point of the horizon p, of the number 0, ±1, according to the sign of θ_{Al}(p).

The generators of a horizon ℋ play an important part in our results. The following shows that most points of a horizon are on just one generator.

Proposition 2.8 The set of points of a horizon ℋ that are on more than one generator has vanishing n dimensional Hausdorff measure.

Proof: Let T be the set of points of ℋ that are on two or more generators of ℋ. By a theorem of Beem and Królak [6, 14] a point of ℋ is differentiable if and only if it belongs to exactly one generator of ℋ. Therefore no point of T is differentiable and thus no point of T is an Alexandrov point of ℋ. Whence Theorem 2.2 and Proposition 2.1 imply ℌ^{n}(T ) = 0. □

Remark 2.9 For a future horizon ℋ in a spacetime (M, g) let E be the set of past endpoints of generators of ℋ. Then the set of points T of ℋ that are on two or more generators is a subset of E. From the last Proposition we know that ℌ^{n}(T ) = 0 and it is tempting to conjecture ℌ^{n}(E) = 0. However this seems to be an open question. We note that there are examples [15] of horizons ℋ so

that E is dense in ℋ. Added in proof: This problem has been affirmatively solved in P.T. Chruściel, J. Fu, G. Galloway, R. Howard, “On fine differentiability properties of horizons and applications to Riemannian geometry”, gr-qc/0011067.

3 Sections of horizons

Recall that in the standard approach to the area theorem [37] one considers sections ℋ_{τ} of a black hole event horizon ℋ obtained by intersecting ℋ with the level sets Σ_{τ} of a time function t,

ℋ_{τ} = ℋ ∩ Σ_{τ} .  (3.1)

We note that Theorem 2.2 and Proposition 2.1 do not directly yield any information about the regularity of all the sections ℋ_{τ}, since Proposition 2.1 gives regularity away from a set of zero measure, but those sections are expected to be of zero measure anyway. Now, because the notion of semi–convexity is independent of the choice of the graphing function, and it is preserved by taking restrictions to lower dimensional subsets, we have:

Proposition 3.1 Let Σ be an embedded C^{2} spacelike hypersurface and let ℋ be a horizon in (M, g), set

S = Σ ∩ ℋ .

There exists a semi–convex topological submanifold S_{reg} ⊂ S of co–dimension two in M which contains all points p ∈ S at which ℋ is differentiable.

Remark 3.2 “Most of the time” S_{reg} has full measure in S. There exist, however, sections of horizons for which this is not the case; cf. the discussion at the end of this section, and the example discussed after Equation (3.13) below.

Proof: Let p ∈ S be such that ℋ is differentiable at p, let 𝒪_{p} be a coordinate neighborhood of p such that Σ ∩ 𝒪_{p} is given by the equation t = 0, and such that ℋ ∩ 𝒪_{p} is the graph t = f of a semi–convex function f. If we write p = (f(q), q), then f is differentiable at q. We wish, first, to show that S can be locally represented as a Lipschitz graph, using the Clarke implicit function theorem [17, Chapter 7]. Consider the Clarke differential ∂f of f at q (cf. [17, p. 27]); by [17, Prop. 2.2.7] and by standard properties of convex (hence of semi–convex) functions we have ∂f(q) = {df(q)}. By a rigid rotation of the coordinate axes we may assume that df(q)(∂/∂x^{n}) ≠ 0. If we write q = (0, q^{1}, …, q^{n}), then Clarke’s implicit function theorem [17, Corollary, p. 256] shows that there exists a neighborhood 𝒲_{q} ⊂ {(t, x^{1}, …, x^{n−1}) ∈ ℝ^{n}} of (0, q^{1}, …, q^{n−1}) and a Lipschitz function g such that, replacing 𝒪_{p} by a subset thereof if necessary, ℋ ∩ 𝒪_{p} can be written as a graph of g over 𝒲_{q}. Remark 2.4 establishes semi–convexity of g. Now, semi–convexity is a property which is preserved when restricting a function to a smooth lower dimensional submanifold of its domain of definition, which shows that h ≡ g|_{t=0} is semi–convex. It follows that the graph 𝒱_{q} ⊂ ℋ of h over 𝒲_{q} ∩ {t = 0} is a semi–convex topological submanifold of ℋ. The manifold

S_{reg} ≡ ∪_{q}𝒱_{q} ,

where the q’s run over the set of differentiability points of ℋ in Σ, has all the desired properties. □

Let S, S_{reg}, ℋ and Σ be as in Proposition 3.1. Now S_{reg} and ℋ are both semi–convex, and have therefore their respective Alexandrov points; as S_{reg} ⊂ ℋ, it is natural to inquire about the relationship among those. It should be clear that if p ∈ S is an Alexandrov point of ℋ, then it also is an Alexandrov point of S. The following example shows that the converse is not true: Let ℋ ⊂ ℝ^{1,2} be defined as the set ℋ = {(t, x, y) | t = |x|}, and let Σ_{τ} be the level sets of the Minkowskian time in ℝ^{1,2}, Σ_{τ} = {(t, x, y) | t = τ}. Each section S_{τ} = ℋ ∩ Σ_{τ} is a smooth one dimensional submanifold of ℝ^{1,2} (not connected if τ > 0, empty if τ < 0). In particular all points of S_{0} = {t = x = 0} are Alexandrov points thereof, while none of them is an Alexandrov point of ℋ. In this example the horizon ℋ is not differentiable on any of the points of S_{0}, which is a necessary condition for being an Alexandrov point of ℋ. It would be of some interest to find out whether or not Alexandrov points of sections of ℋ which are also points of differentiability of ℋ are necessarily Alexandrov points of ℋ.

Following [29], we shall say that a differentiable embedded hypersurface 𝒮 meets ℋ properly transversally if for each point p ∈ 𝒮 ∩ ℋ for which T_{p}ℋ exists the tangent hyperplane T_{p}𝒮 is transverse to T_{p}ℋ. If 𝒮 is spacelike and intersects ℋ proper transversality will always hold; on the other hand if 𝒮 is timelike this might, but does not have to be the case. If ℋ and 𝒮 are C^{1} and 𝒮 is either spacelike or timelike intersecting ℋ transversely then ℋ ∩ 𝒮 is a C^{1} spacelike submanifold. Therefore when 𝒮 is timelike there is a spacelike 𝒮_{1} so that ℋ ∩ 𝒮 = ℋ ∩ 𝒮_{1}. It would be interesting to know if this was true (even locally) for the transverse intersection of general horizons ℋ with timelike C^{2} hypersurfaces 𝒮.

Let 𝒮 be any spacelike or timelike C^{2} hypersurface in M meeting ℋ properly transversally.^{10} Suppose, first, that 𝒮 is covered by a single coordinate patch such that 𝒮 ∩ ℋ is a graph x^{n} = g(x^{1}, …, x^{n−1}) of a semi–convex function g. Let n be the field of unit normals to 𝒮; at each point (x^{1}, …, x^{n−1}) at which g is differentiable and for which T_{p}ℋ exists, where

p ≡ (x^{1}, …, x^{n−1}, g(x^{1}, …, x^{n−1})) ,

there is a unique number a(p) ∈ ℝ and a unique future pointing null vector K ∈ T_{p}ℋ such that

⟨K − an, ·⟩ = −dx^{n} + dg .  (3.2)

10Our definitions below makes use of a unit normal to 𝒮, whence the restriction to spacelike or timelike, properly transverse 𝒮’s. Clearly one should be able to give a definition of θ_{Al}^{𝒮∩ℋ} , etc., for any hypersurface 𝒮 intersecting ℋ properly transversally. Now for a smooth ℋ and for smooth properly transverse 𝒮’s the intersection 𝒮 ∩ ℋ will be a smooth spacelike submanifold of M, and it is easy to construct a spacelike 𝒮′ so that 𝒮′ ∩ ℋ = 𝒮 ∩ ℋ. Thus, in the smooth case, no loss of generality is involved by restricting the 𝒮’s to be spacelike. For this reason, and because the current setup is sufficient for our purposes anyway, we do not address the complications which arise when 𝒮 is allowed to be null, or to change type.

Here we have assumed that the coordinate x^{n} is chosen so that J^{−}(ℋ) ∩ 𝒮 lies under the graph of g. We set

k_{μ}dx^{μ} = ⟨K, ·⟩ ∈ T_{p}^{∗}M .

Assume moreover that (x^{1}, …, x^{n−1}) is an Alexandrov point of g, we thus have the Alexandrov second derivatives D^{2}g at our disposal. Consider e_{i} — a basis of T_{p}ℋ as in Equation (2.8), satisfying further e_{i} ∈ T_{p}𝒮 ∩ T_{p}ℋ, i = 1, …, n − 1. In a manner completely analogous to (2.9)–(2.10) we set

i, j = 1, …, n − 1 ∇_{i}k_{j} = D_{ij}^{2}g − Γ_{ij}^{μ}k_{μ} ,  (3.3)
θ_{Al}^{𝒮∩ℋ} = (e_{1}^{i}e_{1}^{j} + ⋯ + e_{n−1}^{i}e_{n−1}^{j})∇_{i}k_{j} .  (3.4)

(Here, as in (2.9), the Γ_{ij}^{μ}’s are the Christoffel symbols of the space–time metric g.) For X, Y ∈ T_{p}𝒮 ∩ T_{p}ℋ we set, analogously to (2.11),

B_{Al}^{𝒮∩ℋ} (X, Y) = X^{a}Y^{b}e_{a}^{i}e_{b}^{j}∇_{i}k_{j}.  (3.5)

Similarly to the definitions of θ_{Al} and B_{Al}, the vector K with respect to which θ_{Al}^{𝒮∩ℋ} and B_{Al}^{𝒮∩ℋ} have been defined has been tied to the particular choice of coordinates used to represent 𝒮 ∩ ℋ as a graph. In order to globalize this definition it might be convenient to regard B_{Al}^{𝒮∩ℋ} (p) as an equivalence class of tensors defined up to a positive multiplicative factor. Then θ_{Al}^{𝒮∩ℋ} (p) can be thought as the assignment to a point p of the number 0, ±1, according to the sign of θ_{Al}^{𝒮∩ℋ} (p). This, together with Proposition 2.5, can then be used to define B_{Al}^{𝒮∩ℋ} and θ_{Al}^{𝒮∩ℋ} for 𝒮 which are not globally covered by a single coordinate patch.

As already pointed out, if p ∈ 𝒮 ∩ ℋ is an Alexandrov point of ℋ then p will also be an Alexandrov point of 𝒮∩ℋ. In such a case the equivalence class of B_{Al}, defined at p by Equation (2.11), will coincide with that defined by (3.5), when (2.11) is restricted to vectors X, Y ∈ T_{p}𝒮 ∩ T_{p}ℋ. Similarly the sign of θ_{Al}(p) will coincide with that of θ_{Al}^{𝒮∩ℋ} (p), and θ_{Al}(p) will vanish if and only if θ_{Al}^{𝒮∩ℋ} (p) does.

Let us turn our attention to the question, how to define the area of sections of horizons. The monotonicity theorem we prove in Section 6 uses the Hausdorff measure, so let us start by pointing out the following:

Proposition 3.3 Let ℋ be a horizon and let 𝒮 be an embedded hypersurface in M. Then 𝒮 ∩ ℋ is a Borel set, in particular it is ν–Hausdorff measurable for any ν ∈ ℝ^{+}.

Proof: Let σ be any complete Riemannian metric on M, we can cover 𝒮 by a countable collection of sets O_{i} ⊂ 𝒮 of the form O_{i} = B_{σ}(p_{i}, r_{i}) ∩ 𝒮, where the B_{σ}(p_{i}, r_{i}) are open balls of σ–radius r_{i} centered at p_{i} with compact closure. We have O_{i} = (\overline{B_{σ}(p_{i}, r_{i}) ∩ 𝒮})\∂B_{σ}(p_{i}, r_{i}), which shows that the O_{i}’s are Borel sets. Since 𝒮 ∩ ℋ = ∪_{i} (O_{i} ∩ ℋ), the Borel character of 𝒮 ∩ ℋ ensues. The Hausdorff measurability of 𝒮 ∩ ℋ follows now from the fact that Borel sets are Hausdorff measurable^{11}. □

Proposition 3.3 is sufficient to guarantee that a notion of area of sections of horizons — namely their (n − 1)–dimensional Hausdorff area — is well defined. Since the Hausdorff area is not something very handy to work with in practice, it is convenient to obtain more information about regularity of those sections. Before we do this let us shortly discuss how area can be defined, depending upon the regularity of the set under consideration. When S is a piecewise C^{1}, (n − 1) dimensional, paracompact, orientable submanifold of M this can be done by first defining

d^{n−1}μ = e_{1} ∧ ⋯ ∧ e_{n−1} ,

where the e_{a}’s form an oriented orthonormal basis of the cotangent space T^{∗}S; obviously d^{n−1}μ does not depend upon the choice of the e_{a}’s. Then one sets

Area(S) = ∫_{S} d^{n−1}μ .  (3.6)

Suppose, next, that S is the image by a Lipschitz map ψ of a C^{1} manifold N. Let h_{S} denote any complete Riemannian metric on N; by [61, Theorem 5.3] for every ϵ > 0 there exists a C^{1} map ψ_{ϵ} from N to M such that

𝔏_{h_{S}}^{n−1}({ψ ≠ ψ_{ϵ}}) ≤ ϵ .  (3.7)

Here and throughout 𝔏_{h_{S}}^{n−1} denotes the (n − 1) dimensional Riemannian measure associated with a metric h_{S}. One then sets S_{ϵ} = ψ_{ϵ}(N) and

Area(S) = lim_{ϵ→0} Area(S_{ϵ}) .  (3.8)

(It is straightforward to check that Area(S) so defined is independent of the choice of the sequence ψ_{ϵ}. In particular if ψ is C^{1} on N one recovers the definition (3.6) using ψ_{ϵ} = ψ for all ϵ.)

It turns out that for general sections of horizons some more work is needed. Throughout this paper σ will be an auxiliary Riemannian metric such that (M, σ) is a complete Riemannian manifold. Let S be a Lipschitz (n − 1) dimensional submanifold of M such that S ⊂ ℋ; recall that, by Rademacher’s theorem, S is differentiable ℌ_{σ}^{n−1} almost everywhere. Let us denote by ℌ_{σ}^{s} the s dimensional Hausdorff measure [24] defined using the distance function of σ. Recall that S is called n rectifiable [24] iff S is the image of a bounded subset of ℝ^{n} under a Lipschitz map. A set is countably n rectifiable iff it is a countable union of n rectifiable sets. (Cf. [24, p. 251].) (Instructive examples of countably rectifiable sets can be found in [52].) We have the following result [42] (The proof of the first part of Proposition 3.4 is given in Remark 6.13 below):

11Let (X, d) be a metric space. Then an outer measure μ defined on the class of all subsets of X is a metric outer measure iff μ(A ∪ B) = μ(A) + μ(B) whenever the distance (i.e.

inf_{a∈A,b∈B} d(a, b)) is positive. For a metric outer measure, μ, the Borel sets are all μ-measurable (cf. [36, p. 48 Prob. 8] or [39, p. 188 Exercise 1.48]). The definition of the Hausdorff outer measures implies they are metric outer measures (cf. [39, p. 188 Exercise 1.49]). See also [61, p. 7] or [19, p. 147].

Proposition 3.4 Let S be as in Proposition 3.1, then S is countably (n−1) rectifiable. If S is compact then it is (n − 1) rectifiable. □

Consider, then, a family of sets 𝒱_{q_{i}} which are Lipschitz images and form a partition of S up to a set of ℌ_{h}^{n−1} Hausdorff measure zero, where h is the metric on Σ induced by g. One sets

Area(S) = ∑_{i} Area(𝒱_{q_{i}}) .  (3.9)

We note that Area(S) so defined again does not depend on the choices made, and reduces to the previous definitions whenever applicable. Further, Area(S) so defined is^{12} precisely the (n − 1) dimensional Hausdorff measure ℌ_{h}^{n−1} of S:

Area(S) = ∫_{S} dℌ_{h}^{n−1}(p) = ℌ_{h}^{n−1}(S) .  (3.10)

As we will see, there is still another quantity which appears naturally in the area theorem, Theorem 6.1 below: the area counting multiplicities. Recall that the multiplicity N(p) of a point p belonging to a horizon ℋ is defined as the number (possibly infinite) of generators passing through p. Similarly, given a subset S of ℋ, for p ∈ ℋ we set

N(p, S) = the number of generators of ℋ passing through

p which meet S when followed to the (causal) future .(3.11)

Note that N(p, S) = N(p) for p ∈ S. Whenever N(p, S_{2}) is ℌ_{h_{1}}^{n−1} measurable on the intersection S_{1} of ℋ with a spacelike hypersurface Σ_{1} we set

Area_{S_{2}}(S_{1}) = ∫_{S_{1}} N(p, S_{2}) dℌ_{h_{1}}^{n−1}(p) ,  (3.12)

where h_{1} is the metric induced by g on Σ_{1}. Note that N(p, S_{2}) = 0 at points of S_{1} which have the property that the generators through them do not meet S_{2}; thus the area Area_{S_{2}}(S_{1}) only takes into account those generators that are seen from S_{2}. If S_{1} ⊂ J^{−}(S_{2}), as will be the case e.g. if S_{2} is obtained by intersecting ℋ with a Cauchy surface Σ_{2} lying to the future of S_{1}, then N(p, S_{2}) ≥ 1 for all p ∈ S_{1} (actually in that case we will have N(p, S_{2}) = N(p)), so that

12The equality (3.10) can be established using the area formula. In the case of C^{1} submanifolds of Euclidean space this is done explicitly in [61, p. 48]. This can be extended to countable n rectifiable sets by use of more general version of the area formula [61, p. 69] (for subsets of Euclidean space) or [23, Theorem 3.1] (for subsets of Riemannian manifolds).

Area_{S_{2}}(S_{1}) ≥ Area(S_{1}) .

Let us show, by means of an example, that the inequality can be strict in some cases. Consider a black hole in a three dimensional space–time, suppose that its section by a spacelike hypersurface t = 0 looks as shown in Figure 1.

Figure 1: A section of the event horizon in a 2 + 1 dimensional space–time with “two black holes merging”.

As we are in three dimensions area should be replaced by length. (A four dimensional analogue of Figure 1 can be obtained by rotating the curve from Figure 1 around a vertical axis.) When a slicing by spacelike hypersurfaces is appropriately chosen, the behavior depicted can occur when two black holes merge together^{13}. When measuring the length of the curve in Figure 1 one faces various options: a) discard the middle piece altogether, as it has no interior; b) count it once; c) count it twice — once from each side. The purely differential geometric approach to area, as given by Equation (3.6) does not say which choice should be made. The Hausdorff area approach, Equation (3.10), counts the middle piece once. The prescription (3.12) counts it twice. We wish to argue that the most reasonable prescription, from an entropic point of view, is to use the prescription (3.12). In order to do that, let (M, g) be the three dimensional Minkowski space– time and consider a thin long straw R lying on the y axis in the hypersurface t = 0:

R = {t = x = 0, y ∈ [−10, 10]} .  (3.13)

13The four dimensional analogue of Figure 1 obtained by rotating the curve from Figure 1 around a vertical axis can occur in a time slicing of a black–hole space–time in which a “temporarily toroidal black hole” changes topology from toroidal to spherical. Ambiguities related to the definition of area and/or area discontinuities do occur in this example. On the other hand, the four dimensional analogue of Figure 1 obtained by rotating the curve from Figure 1 around a horizontal axis can occur in a time slicing of a black–hole space–time in which two black holes merge together. There are neither obvious ambiguities related to the definition of the area nor area discontinuities in this case.

Set K = J^{+}(R) ∩ {t = 1}, then K is a convex compact set which consists of a strip of width 2 lying parallel to the straw with two half–disks of radius 1 added at the ends. Let M̂ = M \ K, equipped with the Minkowski metric, still denoted by g. Then M̂ has a black hole region ℬ which is the past domain of dependence of K in M (with K removed). The sections ℋ_{τ} of the event horizon ℋ defined as ℋ ∩ {t = τ} are empty for τ < 0 and τ ≥ 1. Next, ℋ_{0} consists precisely of the straw. Finally, for t ∈ (0, 1) ℋ_{t} is the boundary of the union of a strip of width 2t lying parallel to the straw with two half–disks of radius t added at the ends of the strip. Thus

Area(ℋ_{t}) = ∫_{ℋ_{t}} dℌ_{h}^{1} = \{ \begin{matrix}0 , & t < 0 , \\ 10 , & t = 0 , \\ 20 + 2πt , & 0 < t < 1 , \end{matrix}  (3.14)

while

∫_{ℋ_{t}} N(p) dℌ_{h}^{1} = \{ \begin{matrix}0 , & t < 0 , \\ 20 , & t = 0 , \\ 20 + 2πt , & 0 < t < 1 . \end{matrix}  (3.15)

(The end points of the straw, at which the multiplicities are infinite, do not contribute to the above integrals with t = 0, having vanishing measure.) We see that Area(ℋ_{t}) jumps once when reaching zero, and a second time immediately thereafter, while the “area counting multiplicities” (3.15) jumps only once, at the time of formation of the black hole. From an entropy point of view the existence of the first jump can be explained by the formation of a black hole in a very unlikely configuration: as discussed below events like that can happen only for a negligible set of times. However, the second jump of Area(ℋ_{t}) does not make any sense, and we conclude that (3.15) behaves in a more reasonable way. We note that the behavior seen in Figure 1 is obtained by intersecting ℋ by a spacelike surface which coincides with t = 0 in a neighborhood of the center of the straw R and smoothly goes up in time away from this neighborhood.

As already mentioned in footnote 13, this example easily generalizes to 3 + 1 dimensions: To obtain a 3 + 1 dimensional model with a similar discontinuity in the cross sectional area function, for each fixed t, rotate the curve H_{t} about a vertical axis in x–y–z space. This corresponds to looking at the equidistant sets to a disk. The resulting model is a flat space model for the Hughes et al. temporarily toroidal black hole [43], cf. also [30, 44, 60].

We now show that the behavior exhibited in Figure 1, where there is a set of points of positive measure with multiplicity N(p) ≥ 2, can happen for at most a negligible set of times.

Proposition 3.5 Let M be a spacetime with a global C^{1} time function τ: M → ℝ and for t ∈ ℝ let Σ_{t} = {p ∈ M : τ(p) = t} be a level set of τ. Then for any past or future horizon ℋ ⊂ M

ℌ_{h_{t}}^{n−1}({p ∈ Σ_{t} : N(p) ≥ 2}) = 0

for almost all t ∈ ℝ. (Where h_{t} is the induced Riemannian metric on Σ_{t}.) For these t values

∫_{ℋ∩Σ_{t}} N(p) dℌ_{h_{t}}^{n−1}(p) = ℌ_{h_{t}}^{n−1}(ℋ ∩ Σ_{t}) ,

so that for almost all t the area of ℋ ∩ Σ_{t} counted with multiplicity is the same as the n − 1 dimensional Hausdorff measure of ℋ ∩ Σ_{t}.

Proof: Let ℋ_{sing} be the set of points of ℋ that are on more than one generator. By Proposition 2.8 ℌ_{σ}^{n}(ℋ_{sing}) = 0 (where σ is an auxiliary complete Riemannian metric σ on M). The Hausdorff measure version of Fubini’s theorem, known as the “co–area formula” [61, Eq. 10.3, p. 55], gives

∫_{ℝ} ℌ_{σ}^{n−1}(ℋ_{sing} ∩ Σ_{t}) dt = ∫_{ℝ} ℌ_{σ}^{n−1}(τ|_{ℋ}^{−1}[t]) dt = ∫_{ℋsing} J(τ|_{ℋ} ) dℌ_{σ}^{n} = 0 . (3.16)

Here J(τ|_{ℋ} ) is the Jacobian of the function τ restricted to ℋ and the integral over ℋ_{sing} vanishes as ℌ_{σ}^{n}(ℋ_{sing}) = 0.

Equation (3.16) implies that for almost all t ∈ ℝ we have ℌ_{σ}^{n−1}(ℋ_{sing}∩Σ_{t}) = 0. But for any such t we also have ℌ_{h_{t}}^{n−1}(ℋ_{sing} ∩Σ_{t}) = 0. (This can be seen by noting that the identity map between Σ_{t} with the metric of σ restricted to Σ_{t} and (Σ_{t}, h_{t}) and is locally Lipschitz. And locally Lipschitz maps send sets of n−1 dimensional Hausdorff measure zero to sets with n − 1 dimensional Hausdorff measure zero.) This completes the proof. □

We note that the equation ℌ_{σ}^{n−1}(ℋ_{sing} ∩ Σ_{t}) = 0 for almost all t’s shows that the set (S_{t})_{reg} ⊂ S_{t} ≡ Σ_{t} ∩ ℋ given by Proposition 3.1 has full measure in S_{t} for almost all t’s. It is not too difficult to show (using the Besicovitch covering theorem) that (S_{t})_{reg} is countably rectifiable which gives a proof, alternative to Proposition 3.4, of countable rectifiablity (up to a negligible set) of S_{t} for almost all t’s. We emphasize, however, that Proposition 3.4 applies to all sections of ℋ.

4 Non-negativity of θ_{Al}

The proof of the area theorem consists of two rather distinct steps: the first is to show the non-negativity of the divergence of the generators of event horizons under appropriate conditions, the other is to use this result to conclude that the area of sections is nondecreasing towards the future. In this section we shall establish non-negativity of θ_{Al}.

4.1 Causally regular conformal completions

A pair (M̄, ḡ) will be called a conformal completion of (M, g) if M̄ is a manifold with boundary such that M is the interior of M̄. The boundary of M̄ will be called Scri and denoted ℐ. We shall further suppose that there exists a function Ω, positive on M and differentiable on M̄, which vanishes on ℐ, with dΩ nowhere vanishing on ℐ. We emphasize that no assumptions about the causal nature of Scri are made. We shall also require that the metric Ω^{2}g extends by continuity to ℐ, in such a way that the resulting metric ḡ on M̄ is differentiable. We set

ℐ^{+} = {p ∈ ℐ | I^{−}(p; M̄) ∩ M ≠ ∅} .

In the results presented below ℐ^{+} is not required to be connected.

Recall that a space–time is said to satisfy the null energy condition, or the null convergence condition if

Ric(X, X) ≥ 0  (4.1)

for all null vectors X. The following Proposition spells out some conditions which guarantee non-negativity of the Alexandrov divergence of the generators of ℋ; as already pointed out, we do not assume that ℐ^{+} is null:

Proposition 4.1 Let (M, g) be a smooth space–time with a conformal completion (M̄, ḡ) = (M ∪ ℐ^{+}, Ω^{2}g) of (M, g) and suppose that the null energy condition holds on the past I^{−}(ℐ^{+}; M̄) ∩ M of ℐ^{+} in M. Set

ℋ = ∂J^{−}(ℐ^{+}; M̄) ∩ M .

Suppose that there exists a neighborhood O of ℋ with the following property: for every compact set C ⊂ O that meets I^{−}(ℐ^{+}; M̄) there exists a future inextendible (in M) null geodesic η ⊂ ∂J^{+}(C; M) starting on C and having future end point on ℐ^{+}. Then

θ_{Al} ≥ 0 .

Further, if 𝒮 is any C^{2} spacelike or timelike hypersurface which meets ℋ properly transversally, then

θ_{Al}^{𝒮∩ℋ} ≥ 0 on 𝒮 ∩ ℋ .

Proof: Suppose that the result does not hold. Let, first, p_{0} be an Alexandrov point of ℋ with θ_{Al} < 0 at p_{0} and consider a neighborhood N of p_{0} of the form Σ × ℝ, constructed like the set 𝒪 (not to be confused with the set 𝒪 in the statement of the present proposition) in the proof of Theorem 2.2, so that ℋ ∩ N is a graph over Σ of a function f: Σ → ℝ, define x_{0} by p_{0} = (x_{0}, f(x_{0})). By point 3. of Proposition 2.1 in a coordinate neighborhood 𝒰 ⊂ Σ the function f can be written in the form

f(x) = f(x_{0}) + df(x_{0})(x − x_{0}) + \frac{1}{2}D^{2}f(x_{0})(x − x_{0}, x − x_{0}) + o(|x − x_{0}|^{2}) . (4.2)

After a translation, a rotation and a rescaling we will have

x_{0} = 0 , f(x_{0}) = 0 , df(x_{0}) = dx^{n} .  (4.3)

Let B^{n−1}(δ) ⊂ ℝ^{n−1} be the (n − 1) dimensional open ball of radius δ centered at the origin, for q ∈ B^{n−1}(δ), set x = (q, 0) and

f_{ϵ,η}(q) = f(x_{0})+df(x_{0})(x−x_{0})+ \frac{1}{2}D^{2}f(x_{0})(x−x_{0}, x−x_{0})+ϵ|x−x_{0}|^{2} +η . (4.4)

Define

S_{ϵ,η,δ} = { graph over (B^{n−1}(δ) × {0}) ∩ 𝒰 of f_{ϵ,η}} .  (4.5)

If its parameters are chosen small enough, S_{ϵ,η,δ} will be a smooth spacelike submanifold of M of co–dimension two. Let θ_{ϵ,η,δ} be the mean curvature of S_{ϵ,η,δ} with respect to the null vector field K, normal to S_{ϵ,η,δ}, defined as in (3.2) with g there replaced by f_{ϵ,η} (compare (A.1)). From (4.4) when all the parameters are smaller in absolute value than some thresholds one will have

|θ_{ϵ,0,δ}(p_{0}) − θ_{Al}(p_{0})| ≤ Cϵ

for some constant C, so that if ϵ is chosen small enough we will have

θ_{ϵ,η,δ} < 0  (4.6)

at p_{0} and η = 0; as θ_{ϵ,η,δ} is continuous in all its relevant arguments, Equation (4.6) will hold throughout S_{ϵ,η,δ} if δ is chosen small enough, for all sufficiently small ϵ’s and η’s. If δ is small enough and η = 0 we have

\overline{S_{ϵ,η=0,δ}} ⊂ J^{+}(ℋ) , \overline{S_{ϵ,η=0,δ}} ∩ ℋ = \{p_{0}\} .

Here \overline{S_{ϵ,η,δ}} denotes the closure of S_{ϵ,η,δ}. It follows that for all sufficiently small strictly negative η’s we will have

S_{ϵ,η,δ} ∩ J^{−}(ℐ^{+}) ≠ ∅ ,  (4.7)
\overline{S_{ϵ,η,δ}} \ S_{ϵ,η,δ} ⊂ J^{+}(ℋ) ,  (4.8)

Making δ smaller if necessary so that S_{ϵ,η,δ} ⊂ O, our condition on O implies that there exists a null geodesic Γ: [0, 1] → M̄ such that Γ(0) ∈ \overline{S_{ϵ,η,δ}}, Γ|_{[0,1)} ⊂ ∂J^{+}(S_{ϵ,η,δ}; M), and Γ(1) ∈ ℐ^{+}. The behavior of null geodesics under conformal rescalings of the metric (cf., e.g., [37, p. 222]) guarantees that Γ|_{[0,1)} is a complete geodesic in M. Suppose that Γ(0) ∈ \overline{S_{ϵ,η,δ}} \ S_{ϵ,η,δ}, then, by Equation (4.8), Γ would be a causal curve from J^{+}(ℋ) to ℐ^{+}, contradicting the hypothesis that ℋ = ∂J^{−}(ℐ^{+}). It follows that Γ(0) ∈ S_{ϵ,η,δ}. As the null energy condition holds along Γ Equation (4.6) implies that S_{ε,η,δ} will have a focal point, Γ(t_{0}), to the future of Γ(0). But then (cf. [55, Theorem 51, p. 298]) the points of Γ to the future of Γ(t_{0}) are in I^{+}(S_{ε,η,δ}), and thus Γ does not lie in ∂J^{+}(S_{ϵ,η,δ}; M). This is a contradiction and the non-negativity of θ_{Al} follows.

The argument to establish non-negativity of θ_{Al}^{𝒮∩ℋ} is an essentially identical (and somewhat simpler) version of the above. Suppose, thus, that the claim about the sign of θ_{Al}^{𝒮∩ℋ} is wrong, then there exists a point q_{0} ∈ 𝒮 which is an Alexandrov point of 𝒮 ∩ ℋ and at which θ_{Al}^{𝒮∩ℋ} is negative. Consider a coordinate patch around q_{0} = 0 such that 𝒮 ∩ ℋ is a graph x^{n} = g(x^{1}, …, x^{n−1}) of a semi– convex function g. Regardless of the spacelike/timelike character of 𝒮 we can choose the coordinates, consistently with semi–convexity of g, so that J^{−}(ℋ) ∩ 𝒮 lies under the graph of g . The definition (4.4) is replaced by

f_{ϵ,η}(q) = g(q_{0}) + dg(q_{0})(q − q_{0}) + \frac{1}{2}D^{2}g(q_{0})(q − q_{0}, q − q_{0}) + ϵ|q − q_{0}|^{2} + η , (4.9)

while (4.5) is replaced by

S_{ϵ,η,δ} = { graph over B^{n−1}(δ) of f_{ϵ,η}} .  (4.10)

The other arguments used to prove that θ_{Al} ≥ 0 go through without modifications. □

Proposition 4.1 assumes the existence of a neighborhood of the horizon with some precise global properties, and it is natural to look for global conditions which will ensure that such a neighborhood exists. The simplest way to guarantee that is to assume that the conformal completion (M̄, ḡ) is globally hyperbolic, perhaps as a manifold with boundary. To be precise, we shall say that a manifold (M̄, ḡ), with or without boundary, is globally hyperbolic if there exists a smooth time function t on M̄, and if J^{+}(p) ∩ J^{−}(q) is compact for all p, q ∈ M̄ (compare [30]). For example, Minkowski space–time with the standard ℐ^{+} attached is a globally hyperbolic manifold with boundary. (However, if both ℐ^{+} and ℐ^{−} are attached, then it is not. Note that if ℐ^{+} and ℐ^{−} and i^{0} are attached to Minkowski space–time, then it is not a manifold with boundary any more. Likewise, the conformal completions considered in [64] are not manifolds with boundary.) Similarly Schwarzschild space–time with the standard ℐ^{+} attached to it is a globally hyperbolic manifold with boundary. Further, the standard conformal completions of de–Sitter, or anti–de Sitter space–time [34] as well as those of the Kottler space–times [46] (sometimes called Schwarzschild – de Sitter and Schwarzschild – anti–de Sitter space–times) and their generalizations considered in [7] are globally hyperbolic manifolds with boundary.

It is well known that in globally hyperbolic manifolds the causal hypotheses of Proposition 4.1 are satisfied, therefore we have proved:

Proposition 4.2 Under the condition c) of Theorem 1.1, the conclusions of Proposition 4.1 hold. □

The hypothesis of global hyperbolicity of (M̄, ḡ) is esthetically unsatisfactory, as it mixes conditions concerning the physical space–time (M, g) together with conditions concerning an artificial boundary one attaches to it. On the other hand it seems sensible to treat on a different footing the conditions concerning (M, g) and those concerning ℐ^{+}. We wish to indicate here a possible way to do this. In order to proceed further some terminology will be needed:

Definition 4.3 1. A point q in a set A ⊂ B is said to be a past point^{14} of A with respect to B if J^{−}(q; B) ∩ A = {q}.

2. We shall say that ℐ^{+} is ℋ–regular if there exists a neighborhood O of ℋ such that for every compact set C ⊂ O satisfying I^{+}(C; M̄)∩ℐ^{+} ≠ ∅ there exists a past point with respect to ℐ^{+} in ∂I^{+}(C; M̄) ∩ ℐ^{+}.

We note, as is easily verified^{15}, that ∂I^{+}(C; M̄) ∩ ℐ^{+} = ∂(I^{+}(C; M̄) ∩ ℐ^{+}), where the boundary on the right hand side is meant as a subset of ℐ^{+}.

Remark 4.4 In the null and timelike cases the purpose of the condition is to exclude pathological situations in which the closure of the domain of influence of a compact set C in M contains points which are arbitrarily far in the past on ℐ^{+} in an uncontrollable way. A somewhat similar condition has been first introduced in [31] for null ℐ^{+}’s in the context of topological censorship, and has been termed “the i^{0} avoidance condition” there. We shall use the term “ℋ–regular” instead, to avoid the misleading impression that we assume existence of an i^{0}.

Remark 4.5 When ℐ^{+} is null throughout the condition of ℋ–regularity is equivalent to the following requirement: there exists a neighborhood O of ℋ such that for every compact set C ⊂ O satisfying I^{+}(C; M̄) ∩ ℐ^{+} ≠ ∅ there exists at least one generator of ℐ^{+} which intersects \overline{I^{+}(C; M̄)} and leaves it when followed to the past^{16}. This condition is satisfied by the standard conformal completions of Minkowski space–time, or of the Kerr–Newman space–times. In fact, in those examples, when O is suitably chosen, for every compact set C ⊂ O satisfying I^{+}(C; M̄) ∩ ℐ^{+} ≠ ∅ it holds that every generator of ℐ^{+} intersects \overline{I^{+}(C; M̄)} and leaves it when followed to the past.

Remark 4.6 When ℐ^{+} is timelike throughout the condition of ℋ–regularity is equivalent to the following requirement: there exists a neighborhood O of ℋ such that for every compact set C ⊂ O satisfying I^{+}(C; M̄) ∩ ℐ^{+} ≠ ∅, there is a point x in \overline{I^{+}(C; M̄)} ∩ ℐ^{+} such that every past inextendible causal curve in ℐ^{+} from x leaves \overline{I^{+}(C; M̄)}. (The standard conformal completions of anti–de Sitter space–time [34] as well as those of the negative Λ Kottler space–times [46] and their generalizations considered in [7] all satisfy this condition.) Under this condition the existence of a past point in ∂I^{+}(C; M̄) ∩ ℐ^{+} can be established as follows: Let p ∈ ∂I^{+}(C; M̄)∩J^{−}(x; ℐ^{+}), let γ be a causal curve entirely contained in ∂I^{+}(C; M̄) ∩ ℐ^{+} through p; if no such curves exist then p is a past point of ∂I^{+}(C; M̄)∩ℐ^{+} and we are done; if such curves exist, by a standard construction that involves Zorn’s lemma we can without loss of generality assume that γ is past inextendible in ∂I^{+}(C; M̄) ∩ ℐ^{+}. Let Γ be a past inextendible causal curve in ℐ^{+} which contains γ. The current causal regularity condition on ℐ^{+} implies that Γ has an end point q on ∂I^{+}(C; M̄) ∩ ℐ^{+}. If q were not a past point of ∂I^{+}(C; M̄)∩ℐ^{+} one could extend γ as a causal curve in ∂I^{+}(C; M̄)∩ℐ^{+}, which would contradict the maximality of γ. Hence q is a past point of ∂I^{+}(C; M̄)∩ℐ^{+}.

14A very similar notion has already been used in [11, p. 102].

15The inclusion “⊂” makes use of the fact that if q ∈ ℐ^{+}, then for any p ∈ M near q there exists p′ ∈ ℐ^{+}∩I^{+}(p; M̄) near q. If ℐ^{+} is null (and hence possibly type changing) at q one sees this as follows. From the definition of ℐ^{+}, there exists at q a future directed outward pointing timelike vector X_{0}. This can be extended in a neighborhood U of q to a future directed timelike vector field X everywhere transverse to ℐ^{+}. By flowing along the integral curves of X, we see that any point p ∈ M sufficiently close to q is in the timelike past of a point p′ ∈ ℐ^{+} close to q.

16Those points at which the generators of ℐ^{+} exit \overline{I^{+}(C; M̄)} are past points of ∂I^{+}(C; M̄) ∩ ℐ^{+}.

Remark 4.7 When ℐ^{+} is spacelike throughout the condition of ℋ–regularity is equivalent to the requirement of existence of a neighborhood O of ℋ such that for every compact set C ⊂ O satisfying I^{+}(C; M̄) ∩ ℐ^{+} ≠ ∅, the set \overline{I^{+}(C; M̄)} does not contain all of ℐ^{+}. We note that the standard conformal completions of de Sitter space–time, as well as those of the positive-Λ generalized Kottler space–times [34], satisfy our ℋ–regularity condition.

Let us present our first set of conditions which guarantees that the causal hypotheses of Proposition 4.1 hold:

Proposition 4.8 Let (M, g) be a spacetime with a ℋ–regular ℐ^{+}, and suppose that there exists in M a partial Cauchy surface Σ such that

(i) I^{+}(Σ; M) ∩ I^{−}(ℐ^{+}; M̄) ⊂ D^{+}(Σ; M), and

(ii) ℐ^{+} ⊂ I^{+}(Σ; M̄).

Then there is a neighborhood O of ℋ such that if C is a compact subset of J^{+}(Σ; M) ∩ O that meets I^{−}(ℐ^{+}; M̄), there exists a future inextendible (in M) null geodesic η ⊂ ∂I^{+}(C; M) starting at a point on C and having future end point on ℐ^{+}.

Remark 4.9 The conditions (i) and (ii) in Proposition 4.8 form a version of “asymptotic predictability”. They do not imply that global hyperbolicity extends to the horizon or ℐ^{+}. These conditions are satisfied in the set-up of [37].

Proof: Choose O as in the definition of ℋ–regularity; then for compact C ⊂ O satisfying I^{+}(C; M̄) ∩ ℐ^{+} ≠ ∅ there is a past point q on ∂I^{+}(C; M̄) ∩ ℐ^{+}. By [64, Theorem 8.1.6, p. 194] (valid in the present context) there exists a causal curve η ⊂ ∂I^{+}(C; M̄) with future end point q which either is past inextendible in M̄ or has a past end point on C. As q is a past point with respect to ℐ^{+}, η must meet M. Note that, due to the potentially unusual shape of ℐ^{+}, in principle η may meet ℐ^{+} after q when followed backwards in time even infinitely often in a finite interval. The following argument avoids this difficulty altogether: we consider any connected component η_{0} of η ∩ M; η_{0} is a null geodesic with future end point on ℐ^{+}. Now, conditions (i) and (ii) imply that η_{0} enters the interior of D^{+}(Σ; M). If η_{0} is past inextendible in M, then, following η_{0} into the past, η_{0} must meet and enter the timelike past of Σ. This leads to an achronality violation of Σ: Choose p ∈ η_{0} ∩I^{−}(Σ; M). Then, since I^{+}(∂I^{+}(C; M); M) ⊂ I^{+}(C; M), by moving slightly to the future of p, we can find a point p′ ∈ I^{+}(C; M)∩I^{−}(Σ; M) ⊂ I^{+}(Σ; M) ∩ I^{−}(Σ; M). We conclude that η_{0} has a past end point on C. □

The conditions of the proposition that follows present an alternative to those of Proposition 4.8; they form a version of “strong asymptotic predictability”:

Proposition 4.10 Let (M, g) be a spacetime with a ℋ–regular ℐ^{+}, and suppose that M contains a causally simple domain V such that

\overline{I^{−}(ℐ^{+}; M̄)} ∩ M ⊂ V .

Then there is a neighborhood O of ℋ such that if C is a compact subset of V ∩ O that meets I^{−}(ℐ^{+}; M̄), there exists a future inextendible (in M) null geodesic η ⊂ ∂I^{+}(C; V) starting at a point on C and having future end point on ℐ^{+}.

Remark 4.11 Recall, an open set V in M is causally simple provided for all compact subsets K ⊂ V, the sets J^{±}(K; V) are closed in V. Causal simplicity is implied by global hyperbolicity; note, however, that the latter is not a sensible assumption in the timelike ℐ^{+} case: For example, anti–de Sitter space is not globally hyperbolic; nevertheless it is causally simple.

Remark 4.12 Proposition 4.10 assumes that causal simplicity extends to the horizon, but not necessarily to ℐ^{+} — it is not assumed that the closure \overline{V} of V in M̄ is causally simple. Its hypotheses are satisfied in the set-up of [64] and in the (full) set-up of [37]. (In both those references global hyperbolicity of V is assumed.) We note that replacing M by V we might as well assume that M is causally simple.

Remark 4.13 If C is a smooth spacelike hypersurface-with-boundary, with ∂C = S, then (i) η meets C at a point on S, (ii) η meets S orthogonally, and (iii) η is outward pointing relative to C.

Proof: Choose O ⊂ V as in the definition of ℋ–regularity, and let q be a past point of ∂I^{+}(C; M̄) ∩ ℐ^{+} with respect to ℐ^{+}. Then q ∈ ∂I^{+}(C; V̄), where V̄ = V ∪ ℐ^{+}. Arguing as in Proposition 4.8 we obtain a causal curve η ⊂ ∂I^{+}(C; V̄) with future end point q ∈ ℐ^{+}, which is past inextendible in ∂I^{+}(C; V̄). As before, η meets M, and we let η_{0} be a component of η ∩ M. Then η_{0} is a null geodesic in V with future end point on ℐ^{+} which is past inextendible in ∂I^{+}(C; V). Since V is causally simple, ∂I^{+}(C; V) = J^{+}(C; V) \ I^{+}(C; V), which implies that η_{0} has past end point on C. □

If M itself is globally hyperbolic, taking V = M in Proposition 4.10 one obtains:

Corollary 4.14 Under the hypotheses of point a) of Theorem 1.1, the conclusions of Proposition 4.1 hold.

4.2 Complete generators

Let us turn our attention to horizons ℋ the generators of which are future complete. We emphasize that ℋ is not necessarily an event horizon, and the space– time does not have to satisfy any causality conditions. We start with some terminology: consider a C^{2} spacelike manifold S ⊂ M of co–dimension two, let p_{0} ∈ S. We can choose coordinates in a globally hyperbolic neighborhood 𝒪 of p_{0} such that the paths s → (x^{1}, …, x^{n}, x^{n+1} = s) are timelike and such that

S ∩ 𝒪 = {x^{n} = x^{n+1} = 0} .  (4.11)

Suppose further that

p_{0} ∈ S ∩ ℋ ,  T_{p_{0}}S ⊂ T_{p_{0}}ℋ ,  (4.12)

and that p_{0} is an Alexandrov point of ℋ. Passing to a subset of 𝒪 if necessary we may assume that 𝒪 ∩ ℋ is a graph of a function f, with p_{0} = (x_{0}, f(x_{0})). We shall say that S is second order tangent to ℋ if the above conditions hold and if in the coordinate system of Equation (4.11) we have

∀ X, Y ∈ T_{p_{0}}S  D^{2}f(X, Y) = 0 .  (4.13)

Here D^{2}f is the Alexandrov second derivative of f at x_{0}, as in Equation (2.2). The notion of S ⊂ 𝒮 being second order tangent to a section 𝒮 ∩ ℋ at an Alexandrov point of this section is defined in an analogous way, with coordinates adapted so that, locally, 𝒮 = {x^{n} = 0} or 𝒮 = {x^{n+1} = 0}.

We note the following result:

Lemma 4.15 Let S be a C^{2} spacelike manifold of co–dimension two which is tangent to a horizon ℋ at p_{0} ∈ ℋ ∩ S. Suppose that

1. p_{0} is an Alexandrov point of ℋ, and S is second order tangent to ℋ there, or

2. p_{0} is an Alexandrov point of 𝒮 ∩ ℋ, with 𝒮 = {x^{n} = 0} in the coordinate system of (4.11), and S is second order tangent to 𝒮 ∩ ℋ there, or 3. p_{0} is an Alexandrov point of 𝒮 ∩ ℋ, with 𝒮 = {x^{n+1} = 0} in the coordinate

system of (4.11), and S is second order tangent to 𝒮 ∩ ℋ there.

Let Γ be a null geodesic containing a generator γ of ℋ through p_{0}, with Γ(0) = p_{0}. If Γ(1) is a focal point of S, then there exists a ∈ [0, 1] such that Γ(a) is an endpoint of γ on ℋ.

Remark 4.16 We stress that we haven’t assumed anything about the time orientation of Γ.

Proof: If Γ(0) is an end point of ℋ there is nothing to prove, so we can suppose that p_{0} is an interior point of the generator γ. Suppose that Γ(1) is a focal point of S, it is well known^{17} that for any b > 1 there exists a one parameter family of timelike paths Γ_{v}: [0, b] → M, |v| ≤ v_{0}, such that

Γ_{v}(0) ∈ S , Γ_{v}(b) = Γ(b) ,
∀u ∈ [0, b] d_{σ}(Γ_{v}(u), Γ(u)) ≤ C|v| ,  (4.14)
g(\frac{∂Γ_{v}}{∂u} , \frac{∂Γ_{v}}{∂u} ) ≤ −Cv^{2} ,  (4.15)

with some constant C. Here d_{σ} denotes the distance on M measured with respect to some auxiliary complete Riemannian metric σ.

Let X be any smooth vector field which vanishes at Γ(b) and which equals ∂/∂x^{n+1} in a neighborhood of p_{0}, where x^{n+1} is the (n + 1)st coordinate in the coordinate system of Equation (4.11). Let ϕ_{s} denote the (perhaps locally defined) flow of X, for |s| ≤ 1 we have the straightforward estimate

|g(\frac{∂(ϕ_{s}(Γ_{v}(u)))}{∂u}, \frac{∂(ϕ_{s}(Γ_{v}(u)))}{∂u}) − g(\frac{∂Γ_{v}(u)}{∂u}, \frac{∂Γ_{v}(u)}{∂u})| ≤ C_{1}|s| ,  (4.16)

for some constant C_{1}, whenever ϕ_{s}(Γ_{v})(u) is defined. Set

η(δ) = sup_{q∈B^{n−1}(0,δ)} \frac{|f(q)|}{(d_{σ}(q, 0))^{2}} ,

where f is again the graphing function of ℋ ∩ 𝒪, and B^{n−1}(0, δ) is as in the proof of Proposition 4.1; d_{σ}(q, 0) should be understood in the obvious way. Under the hypotheses 1. and 2., by Equations (2.2) and (4.13) we have

η(δ) →_{δ→0} 0 .  (4.17)

17The only complete proof of this fact known to us is to be found in [28, Section 2]. For the case at hand one should use the variation defined there with c_{ℓ} = 0. We note the following misprints in [28, Section 2]: In Equation (2.7) η′_{0} should be replaced by N_{0} and η′_{ℓ} should be replaced by N_{ℓ}; in Equation (2.10) N_{0} should be replaced by η′_{0} and N_{ℓ} should be replaced by η′_{ℓ}.

On the other hand, suppose that 3. holds, let g be the graphing function of 𝒮 ∩ ℋ in the coordinate system of (4.11), the hypothesis that S is second order tangent to 𝒮 ∩ ℋ gives

g(x) = o(|x|^{2}) ,

where x is a shorthand for (x^{1}, …, x^{n−1}). Now (x, g(x)) ∈ ℋ so that f(x, g(x)) = 0, hence

|f(x, 0)| = |f(x, 0) − f(x, g(x))| ≤ L|g(x)| = o(|x|^{2}) ,

where L is the Lipschitz continuity constant of f on B^{n}(0, δ). Thus Equation (4.17) holds in all cases.

Equation (4.14) shows that for |v| ≤ δ/C_{2}, for some constant C_{2}, we will have Γ_{v}(0) ∈ B^{n−1}(0, δ) × {0} × {0} ⊂ S ∩ 𝒪, so that Γ_{v}(0) can be written as

Γ_{v}(0) = (x(v), x^{n+1} = 0) ,  x(v) = (q(v), x^{n} = 0) ,  q(v) ∈ B^{n−1}(0, δ) .

Consider the point p(v) ≡ (x(v), f(x(v))); by definition of X we have

p(v) = ϕ_{f(x(v))}(Γ_{v}(0)) ∈ ℋ .

From Equations (4.14), (4.15) and (4.16), together with the definition of η we obtain

∀u ∈ [0, b] g(\frac{∂(ϕ_{f(x(v))}(Γ_{v}(u)))}{∂u}, \frac{∂(ϕ_{f(x(v))}(Γ_{v}(u)))}{∂u}) ≤ −Cv^{2} + C_{1}C^{2}η(δ)v^{2} .

This and Equation (4.17) show that for δ small enough ϕ_{s}(Γ_{v}) will be a timelike path from p(v) ∈ ℋ to Γ_{v}(b). Achronality of ℋ implies that Γ_{v}(b) ∉ ℋ, thus Γ leaves ℋ somewhere on [0, b). As b is arbitrarily close to 1, the result follows. □

As a Corollary of Lemma 4.15 one immediately obtains:

Proposition 4.17 Under the hypothesis b) of Theorem 1.1, we have

θ_{Al} ≥ 0 .

Further, if 𝒮 is any spacelike or timelike C^{2} hypersurface that meets ℋ properly transversally, then

θ_{Al}^{𝒮∩ℋ} ≥ 0 .

Proof: Suppose that θ_{Al}(p_{0}) < 0, for δ small enough let S = S(δ) be the manifold S_{0,0,δ} defined in Equation (4.5). Consider the future directed maximally extended null geodesic Γ normal to S which coincides, for some values of its parameters, with a generator γ of ℋ through p_{0}. Now, γ is complete to the future by hypothesis, thus so must be the case with Γ. By well known results^{17} [55, Theorem 43, p. 292], S has a focal point along Γ at finite affine distance, say 1. By Lemma 4.15 the generator γ has a future end point, which contradicts the definition of a future horizon. Finally, if 𝒮 is a spacelike hypersurface, or a timelike hypersurface properly transverse to ℋ, then the same argument with S_{0,0,δ} given by Equation (4.10) establishes θ_{Al}^{𝒮∩ℋ} ≥ 0. □

5 Propagation of Alexandrov points along generators, optical equation

The aim of this section is to show that, roughly speaking, Alexandrov points “propagate to the past” along the generators. Recall, now, that the Weingarten map b of a smooth null hypersurface S satisfies a Ricatti equation

b′ + b^{2} + R = 0.  (5.1)

Here ′ and R are defined in Appendix A, see Equation (A.3) and the paragraph that follows there. Equation (5.1) leads to the well known Raychaudhuri equation in general relativity: by taking the trace of (5.1) we obtain the following formula for the derivative of the null mean curvature θ = θ(s) along η,

θ′ = −Ric(η′, η′) − σ^{2} − \frac{1}{n − 2}θ^{2},  (5.2)

where σ, the shear scalar^{18}, is the trace of the square of the trace free part of b. Equation (5.2) is the well-known Raychaudhuri equation (for an irrotational null geodesic congruence) of relativity. This equation shows how the Ricci curvature of spacetime influences the null mean curvature of a null hypersurface. We will refer to Equation (5.1) as the optical equation.

Let σ be any auxiliary complete smooth Riemannian metric on M and define UM ⊂ T M to be the bundle of σ–unit vectors tangent to M. Following [15] for p ∈ ℋ we define 𝒩_{p} ⊂ U_{p}M as the collection of σ–unit, future directed vectors tangent to generators of ℋ through p. Those vectors are necessarily lightlike and will be called semi–tangents to ℋ. We set

𝒩 = ∪_{p∈ℋ} 𝒩_{p} .

The main result of this section is the following:

Theorem 5.1 Let Γ_{int} denote the set of interior points of a generator Γ of ℋ ( i.e., Γ without its end–point, if any). If p_{0} ∈ Γ_{int} is an Alexandrov point of a section 𝒮 ∩ ℋ, where 𝒮 is an embedded spacelike or timelike C^{2} hypersurface that intersects ℋ properly transversally, then

Γ_{int} ∩ J^{−}(p_{0}) \ {p_{0}} ⊂ ℋ_{Al} .  (5.3)

Further the Alexandrov derivative D^{2}f of any graphing function of ℋ varies smoothly over Γ_{int} ∩ J^{−}(p_{0}), and the null Weingarten map b_{Al} constructed out of D^{2}f in a way analogous to that presented in Appendix A satisfies the optical equation (5.1) and the Raychaudhuri equation (5.2) there.

18This is one of the very few occurrences of the shear scalar (traditionally denoted by σ in the physics literature) in our paper, we hope that this conflict of notation with the auxiliary Riemannian metric also denoted by σ will not confuse the reader.

Remark 5.2 It would be of interest to find out whether or not the inclusion (5.3) can be strengthened to Γ_{int} ⊂ ℋ_{Al}. It follows from Theorem 5.6 below that this last inclusion will hold for ℌ_{σ}^{n−1} almost all generators passing through any given section 𝒮 ∩ ℋ of ℋ, when the generators are counted by counting the points at which they meet 𝒮 ∩ ℋ.

Remark 5.3 There exist horizons containing generators on which no points are Alexandrov. As an example, let ℋ be the boundary of the future of a square lying in the t = 0 hypersurface of three dimensional Minkowski space–time. Then ℋ is a union of portions of null planes orthogonal to the straight segments lying on the boundary of the square together with portions of light cones emanating from the corners of the square. Consider those generators of ℋ at which the null planes meet the light cones: it is easily seen that no point on those generators is an Alexandrov point of ℋ.

As constructing support functions (or support hypersurfaces) to the horizon ℋ is generally easier than doing analysis directly on ℋ we start by giving a criteria in terms of upper and lower support functions for a function to have a second order expansion (2.2).

Lemma 5.4 Let U ⊂ ℝ^{n} be an open neighborhood of x_{0} and f: U → ℝ a function. Assume for ℓ = 1, 2, … that there are open neighborhoods U_{ℓ}^{+} and U_{ℓ}^{−} of x_{0} and C^{2} functions f_{ℓ}^{±}: U_{ℓ}^{±} → ℝ so that f_{ℓ}^{−} ≤ f on U_{ℓ}^{−} and f ≤ f_{ℓ}^{+} on U_{ℓ}^{+}, with f(x_{0}) = f_{ℓ}^{±}(x_{0}). Also assume that there is a symmetric bilinear form Q so that

lim_{ℓ→∞} D^{2}f_{ℓ}^{+}(x_{0}) = lim_{ℓ→∞} D^{2}f_{ℓ}^{−}(x_{0}) = Q .

Then f has a second order expansion at x_{0}:

f(x) = f(x_{0}) + df(x_{0})(x − x_{0}) + \frac{1}{2}Q(x − x_{0}, x − x_{0}) + o(|x − x_{0}|^{2}) ,

where df(x_{0}) = df_{ℓ}^{±}(x_{0}) (this is independent of ℓ and the choice of + or −). Thus the Alexandrov second derivative of f exists at x = x_{0} and is given by D^{2}f(x_{0}) = Q.

Remark 5.5 The extra generality will not be needed here, but we remark that the proof shows that the hypothesis the functions f_{ℓ}^{+} are C^{2} can be weakened to only requiring that they all have a second order expansion at x_{0} with no regularity being needed away from x = x_{0}.

Proof: Without loss of generality we may assume that x_{0} = 0. By replacing U_{ℓ}^{+} and U_{ℓ}^{−} by U_{ℓ} := U_{ℓ}^{+} ∩ U_{ℓ}^{−} we assume for each ℓ that f_{ℓ}^{+} and f_{ℓ}^{−} have the same domain. For any k, ℓ we have f_{ℓ}^{+} − f_{k}^{−} ≥ 0 has a local minimum at x_{0} = 0 and thus a critical point there. Whence df_{ℓ}^{+}(0) = df_{k}^{−}(0) for all k and l. This shows the linear functional df(0) = df_{ℓ}^{+}(0) is well defined. We now replace f by x ↦ f(x) − df(0)x − \frac{1}{2}Q(x, x) and f_{ℓ}^{±} by x ↦ f_{ℓ}^{±}(x) − df(0)x − \frac{1}{2}Q(x, x). Then df(0) = df_{ℓ}^{±}(0) = 0, lim_{ℓ→0} D^{2}f_{ℓ}^{±}(0) = 0 and to finish the proof it is enough to show that f(x) = o(|x|^{2}). That is, we need to find for every ε > 0 a δ_{ε} > 0 so that |x| < δ_{ε} implies |f(x)| ≤ ε|x|^{2}. To do this choose any r_{0} > 0 so that B(0, r_{0}) ⊂ U. Then choose an ℓ large enough that

−ε|x|^{2} ≤ D^{2}f_{ℓ}^{−}(0)(x, x) , D^{2}f_{ℓ}^{+}(0)(x, x) ≤ ε|x|^{2} ,

for all x ∈ B(0, r_{0}). Now choose r_{1} ≤ r_{0} so that B(0, r_{1}) ⊂ U_{ℓ}. For this ℓ we can use the Taylor expansions of f_{ℓ}^{+} and f_{ℓ}^{−} at 0 to find a δ_{ε} > 0 with 0 < δ_{ε} ≤ r_{1} so that

|f_{ℓ}^{±}(x) − \frac{1}{2}Df_{ℓ}^{±}(0)(x, x)| ≤ \frac{1}{2}ε|x|^{2}

for all x with |x| < δ_{1}. Then f ≤ f_{ℓ}^{+} implies that if |x| < δ_{ε} then

f(x) ≤ \frac{1}{2}D^{2}f_{ℓ}^{+}(0)(x, x) + (f_{ℓ}^{+}(x) − \frac{1}{2}D^{2}f_{ℓ}^{+}(0)(x, x)) ≤ \frac{ε}{2}|x|^{2} + \frac{ε}{2}|x|^{2} = ε|x|^{2} ,

with a similar calculation, using f_{ℓ}^{−} ≤ f, yielding −ε|x|^{2} ≤ f(x). This completes the proof. □

Proof of Theorem 5.1: This proof uses, essentially, the same geometric facts about horizons that are used in the proof of Proposition A.1. Let p_{0} ∈ Γ_{int} be an Alexandrov point of a section 𝒮 ∩ ℋ, where 𝒮 is a spacelike or timelike C^{2} hypersurface that intersects ℋ properly transversally, and with p_{0} ∈ Γ_{int}. By restricting to a suitable neighborhood of p_{0} we can assume without loss of generality that M is globally hyperbolic, that Γ maximizes the distance between any two of its points, and that ℋ is the boundary of J^{+}(ℋ) and J^{−}(ℋ) (that is ℋ divides M into two open sets, its future I^{+}(ℋ) and its past I^{−}(ℋ)).

We will simplify notation a bit and assume that 𝒮 is spacelike, only trivial changes are required in the case 𝒮 is timelike. We can find C^{2} coordinates x^{1}, …, x^{n} on 𝒮 centered at p_{0} so that in these coordinates 𝒮 ∩ ℋ is given by a graph x^{n} = h(x^{1}, …, x^{n−1}). By restricting the size of 𝒮 we can assume that these coordinates are defined on all of 𝒮 and that their image is B^{n−1}(r) × (−δ, δ) for some r, δ > 0 and that h: B^{n−1}(r) → (−δ, δ). As p_{0} is an Alexandrov point of 𝒮 ∩ ℋ and h(0) = 0 the function h has a second order expansion

h(x⃗) = dh(0)x⃗ + \frac{1}{2}D^{2}h(0)(x⃗, x⃗) + o(|x⃗|^{2}) ,

where x⃗ = (x^{1}, …, x^{n−1}). By possibly changing the sign of x^{n} we may assume that

{(x⃗, y) ∈ 𝒮 | y ≥ h(x⃗)} ⊂ J^{+}(ℋ) , {(x⃗, y) ∈ 𝒮 | y ≤ h(x⃗)} ⊂ J^{−}(ℋ) .

For ε ≥ 0 define

h_{ε}^{±}(x⃗) = dh(0)x⃗ + \frac{1}{2}D^{2}h(0)(x⃗, x⃗) ± \frac{ε}{2}|x⃗|^{2} .

Then h_{ε}^{±} is a C^{2} function on B^{n−1}(r), and for ε = 0 the function h_{0} = h_{0}^{+} = h_{0}^{−} is just the second order expansion of h at x⃗ = 0. For each ε > 0 the second order Taylor expansion for h at 0 implies that there is an open neighborhood V_{ε} of 0 in B^{n−1}(r) so that

h_{ε}^{−} ≤ h ≤ h_{ε}^{+} on V_{ε} .

Set

N := {(x⃗, h_{0}(x⃗)) | x⃗ ∈ B^{n−1}(r)} , N_{ε}^{±} := {(x⃗, h_{ε}^{±}(x⃗)) | x⃗ ∈ V_{ε}} .

Let n be the future pointing timelike unit normal to 𝒮 and let η: (a, b) → Γ_{int} be the affine parameterization of Γ_{int} with η(0) = p_{0} and ⟨η′(0), n(p_{0})⟩ = −1 (which implies that η is future directed). Let k be the unique C^{1} future directed null vector field along N so that k(p_{0}) = η′(0) and ⟨k, n⟩ = −1. Likewise let k_{ε}^{±} be the C^{1} future directed null vector field along N_{ε}^{±} with k(p_{0}) = η′(0) and ⟨k_{ε}^{±}, n⟩ = −1.

Let p be any point of Γ_{int} ∩ J^{−}(p_{0}) \ {p_{0}}. Then p = η(t_{0}) for some t_{0} ∈ (a, b). To simplify notation assume that t_{0} ≤ 0. By Lemma 4.15 N has no focal points in Γ_{int} and in particular no focal points on η|_{(a,0]}. Therefore if we fix a t_{1} with t_{1} < t_{0} < b then there will be an open neighborhood W of 0 in B^{n−1}(r) so that

ℋ̃ := {exp(tk(x⃗, h_{0}(x⃗))) | x⃗ ∈ W, t ∈ (t_{1}, 0)}

is an embedded null hypersurface of M. By Proposition A.3 the hypersurface is of smoothness class C^{2}. The focal points depend continuously on the second fundamental form so there is an ε_{0} > 0 so that if ε < ε_{0} then none of the submanifolds N_{ε}^{±} have focal points along η|_{[t_{1},0]}. Therefore if 0 < ε < ε_{0} there is an open neighborhood W_{ε} and such that

ℋ̃_{ε}^{±} := {exp(tk(x⃗, h(x⃗))) | x⃗ ∈ W_{ε}, t ∈ (t_{1}, 0)}

is a C^{2} embedded hypersurface of M.

We now choose smooth local coordinates y^{1}, …, y^{n+1} for M centered at p = η(t_{0}) so that ∂/∂y^{n+1} is a future pointing timelike vector field and the level sets y^{n+1} = Const. are spacelike. Then there will be an open neighborhood U of 0 in ℝ^{n} so that near p the horizon ℋ is given by a graph y^{n+1} = f(y^{1}, …, y^{n}). Near p the future and past of ℋ are given by J^{+}(ℋ) = {y^{n+1} ≥ f(y^{1}, …, y^{n})} and J^{−}(ℋ) = {y^{n+1} ≤ f(y^{1}, …, y^{n})}. There will also be open neighborhoods U_{0} and U_{ε}^{±} of 0 in ℝ^{n} and functions f_{0} and f_{ε}^{±} defined on these sets so that near p

ℋ̃ = {(y^{1}, …, y^{n}, f_{0}(y^{1}, …, y^{n})) | (y^{1}, …, y^{n}) ∈ U_{0}} ,
ℋ̃_{ε}^{±} = {(y^{1}, …, y^{n}, f_{ε}^{±}(y^{1}, …, y^{n})) | (y^{1}, …, y^{n}) ∈ U_{ε}^{±}} .

The hypersurfaces ℋ̃ and ℋ̃_{ε}^{±} are C^{2} which implies the functions f_{0} and f_{ε}^{±} are all C^{2}.

Since N_{ε}^{−} ⊂ J^{−}(ℋ), a simple achronality argument shows that ℋ̃_{ε}^{−} ⊂ J^{−}(ℋ). (This uses the properties of ℋ described in the first paragraph of the proof.) By choosing N_{ε}^{+} small enough, we can assume that ℋ̃_{ε}^{+} is achronal. We now show that, relative to some neighborhood of p, ℋ ⊂ J^{−}(ℋ̃_{ε}^{+}). Let O_{ε} be a neighborhood of p, disjoint from 𝒮, such that ℋ̃_{ε}^{+} ∩ O_{ε} separates O_{ε} into the disjoint open sets I^{+}(ℋ̃_{ε}^{+} ∩ O_{ε}; O_{ε}) and I^{−}(ℋ̃_{ε}^{+} ∩ O_{ε}; O_{ε}). Now, by taking O_{ε} small enough, we claim that ℋ does not meet I^{+}(ℋ̃_{ε}^{+} ∩ O_{ε}; O_{ε}). If that is not the case, there is a sequence {p_{ℓ}} such that p_{ℓ} ∈ ℋ ∩ I^{+}(ℋ̃_{ε}^{+} ∩ O_{ε}; O_{ε}) and p_{ℓ} → p. For each ℓ, let Γ_{ℓ} be a future inextendible null generator of ℋ starting at p_{ℓ}. Since p is an interior point of Γ, the portions of the Γ_{ℓ}’s to the future of p_{ℓ} must approach the portion of Γ to the future of p. Hence for ℓ sufficiently large, Γ_{ℓ} will meet 𝒮 at a point q_{ℓ} ∈ 𝒮 ∩ ℋ such that q_{ℓ} → p_{0}. For such ℓ, the segment of Γ_{ℓ} from p_{ℓ} to q_{ℓ} approaches the segment of Γ from p to p_{0} uniformly as ℓ → ∞. Since ℋ̃_{ε}^{+} separates a small neighborhood of the segment of Γ from p to p_{0}, it follows that for ℓ large enough, the segment of Γ_{ℓ} from p_{ℓ} to q_{ℓ} will meet ℋ̃_{ε}^{+}. But this implies for such ℓ that p_{ℓ} ∈ I^{−}(N_{ε}^{+}), which contradicts the achronality of ℋ̃_{ε}^{+}. We conclude, by choosing O_{ε} small enough, that ℋ ∩ I^{+}(ℋ̃_{ε}^{+} ∩ O_{ε}; O_{ε}) = ∅, and hence that ℋ ∩ O_{ε} ⊂ J^{−}(ℋ̃_{ε}^{+} ∩ O_{ε}; O_{ε}). Shrinking U_{ε}^{+} if necessary, this inclusion, and the inclusion ℋ̃_{ε}^{−} ⊂ J^{−}(ℋ) imply that on U_{ε}^{−} we have f_{ε}^{−} ≤ f and that on U_{ε}^{+} we have f ≤ f_{ε}^{+}. Therefore if we can show that lim_{ε↘0} D^{2}f_{ε}^{±}(0) = D^{2}f_{0}(0) then Lemma 5.4 implies that 0 is an Alexandrov point of f with Alexandrov second derivative given by D^{2}f(0) = D^{2}f_{0}(0).

To see that lim_{ε↘0} D^{2}f_{ε}^{±}(0) = D^{2}f_{0}(0) let b_{0} and b_{ε}^{±} be the Weingarten maps of ℋ̃ and ℋ̃_{ε}^{±} along η. By Proposition A.1 they all satisfy the same Ricatti equation (A.2). The initial condition for b_{0} is calculated algebraically from the second fundamental form of N at p_{0} (cf. Section 3) and likewise the initial condition for b_{ε}^{±} is calculated algebraically from the second fundamental form of N_{ε}^{±} at p_{0}. ¿From the definitions we clearly have

Second Fundamental Form of N_{ε}^{±} at p_{0} ⟶ Second Fundamental Form of N at p_{0}

as ε ↘ 0. Therefore continuous dependence of solutions to ODE’s on initial conditions implies that lim_{ε↘0} b_{ε}^{±} = b_{0} at all points of η|_{[0,t_{0}]}. As D^{2}f_{ε}^{±}(0) and D^{2}f_{0}(0) are algebraic functions of b_{ε}^{±} and b_{0} at the point p = η(t_{0}) this implies that lim_{ε↘0} D^{2}f_{ε}^{±}(0) = D^{2}f_{0}(0), and completes the proof that p is an Alexandrov point of ℋ.

Finally, it follows from the argument that at all points of Γ_{int} the null Weingarten map b for ℋ is the same as the Weingarten for the C^{2} null hypersurface ℋ̃. So b will satisfy (A.2) by Proposition A.1. □

We end this section with one more regularity result, Theorem 5.6 below. Its proof requires some techniques which are introduced in Section 6 only — more precisely, an appropriate generalization of Lemma 6.11 is needed; this, in turn, relies on an appropriate generalization of Lemma 6.9. For this reason we defer the proof of Theorem 5.6 to an appendix, Appendix D.

Theorem 5.6 Let 𝒮 be any C^{2} hypersurface intersecting ℋ properly transversally, and define

S_{0} = {q ∈ 𝒮 ∩ ℋ : q is an interior point of a generator of ℋ} , (5.4)
S_{1} = {q ∈ S_{0} : all interior points of the generator
through q are Alexandrov points of ℋ} ,  (5.5)
S_{2} = {q ∈ S_{0} : q is an Alexandrov point of ℋ} ,  (5.6)

Then S_{1} and S_{2} have full n − 1 dimensional Hausdorff measure in S_{0}.

Remark 5.7 S_{0} does not have to have full measure in 𝒮 ∩ ℋ, it can even be empty. This last case occurs indeed when 𝒮 = {t = 0} in the example described around Equation (3.13) in Section 3. Note, however, that if 𝒮 is a level set of a properly transverse foliation 𝒮(t), then (as already mentioned) for almost all t’s the sets S(t)_{0} (as defined in (5.4) with 𝒮 replaced by 𝒮(t)) will have full measure in 𝒮(t). We shall call a generator an Alexandrov generator if all its interior points are Alexandrov points. It follows that for generic sections (in the measure sense above) for almost all points through those sections the corresponding generators will be Alexandrov. The discussion here thus gives a precise meaning to the statement that almost all generators of a horizon are Alexandrov generators.

6 Area monotonicity

In this section we shall show the monotonicity of the area, assuming that the Alexandrov divergence θ_{Al} of the generators of ℋ, or that of a section of ℋ, is non–negative. The result is local in the sense that it only depends on the part of the event horizon ℋ that is between the two sections S_{1} and S_{2} whose area we are trying to compare. We consider a spacetime (M, g) of dimension n + 1. We have the following:

Theorem 6.1 Suppose that (M, g) is an (n + 1) dimensional spacetime with a future horizon ℋ. Let Σ_{a}, a = 1, 2 be two embedded achronal spacelike hypersurfaces of C^{2} differentiability class, set S_{a} = Σ_{a} ∩ ℋ. Assume that

S_{1} ⊂ J^{−}(S_{2}) ,  S_{1} ∩ S_{2} = ∅ ,  (6.1)

and that either 1. the divergence θ_{Al} of ℋ defined (ℌ_{σ}^{n}–almost everywhere) by Equation (2.10)

is non–negative on J^{+}(S_{1}) ∩ J^{−}(S_{2}), or

2. the divergence θ_{Al}^{S_{2}} of S_{2} defined (ℌ_{σ}^{n−1}–almost everywhere) by Equation (3.4) is non–negative, with the null energy condition (4.1) holding on J^{+}(S_{1}) ∩ J^{−}(S_{2}).

Then

1.

Area_{S_{2}}(S_{1}) ≤ Area(S_{2}) ,  (6.2)

with Area_{S_{2}}(S_{1}) defined in Equations (3.11)–(3.12) and Area(S_{2}) defined in Equation (3.10). This implies the inequality

Area(S_{1}) ≤ Area(S_{2}) .

2. If equality holds in (6.2), then (J^{+}(S_{1}) \ S_{1}) ∩ (J^{−}(S_{2}) \ S_{2}) (which is the part of ℋ between S_{1} and S_{2}) is a smooth totally geodesic (analytic if the metric is analytic) null hypersurface in M. Further, if γ is a null generator of ℋ with γ(0) ∈ S_{1} and γ(1) ∈ S_{2}, then the curvature tensor of (M, g) satisfies R(X, γ′(t))γ′(t) = 0 for all t ∈ [0, 1] and X ∈ T_{γ(t)}ℋ.

Remark 6.2 Note that if S_{1}∩S_{2} ≠ ∅ then the inequality Area_{S_{2}}(S_{1}) ≤ Area(S_{2}) need not hold. (The inequality Area(S_{1}) ≤ Area(S_{2}) will still hold.) For example, if Area_{S_{1}}(S_{1}) > Area(S_{1}), then letting S_{2} = S_{1} gives a counterexample. If S_{1} ∩ S_{2} ≠ ∅ then the correct inequality is Area_{S_{2}}(S_{1} \ S_{2}) ≤ Area(S_{2} \ S_{1}).

Proof: Let us start with an outline of the proof, without the technical details — these will be supplied later in the form of a series of lemmas. Let N_{ℋ} (S_{1}) be the collection of generators of ℋ that meet S_{1}. Let A ⊆ S_{2} be the set of points that are of the form S_{2} ∩ γ with γ ∈ N_{ℋ} (S_{1}); replacing Σ_{2} by an appropriate submanifold thereof if necessary, A will be a closed subset of S_{2} (Proposition 6.3). The condition (6.1) together with achronality of ℋ imply that every q ∈ A is on exactly one of the generators γ ∈ N_{ℋ} (S_{1}), we thus have a well defined function ϕ: A → S_{1} given by

ϕ(q) = S_{1} ∩ γ where γ ∈ N_{ℋ} (S_{1}) and q = S_{2} ∩ γ.  (6.3)

Let us for simplicity assume that the affine distance from S_{1} to S_{2} on the generators passing through S_{1} is bounded from below, and that the affine existence time of those generators to the future of S_{2} is also bounded from below (in what follows we will see how to reduce the general case to this one). In this case A can be embedded in a C^{1,1} hypersurface N in Σ_{2} (Lemma 6.9) and ϕ can be extended to a locally Lipschitz function ϕ̂, from N to Σ_{1} (Lemma 6.11). A is ℌ_{h_{2}}^{n−1} measurable (closed subsets of manifolds are Hausdorff measurable^{11}), so we can apply the generalization of the change–of–variables theorem known as the area formula [23, Theorem 3.1] (with m and k there equal to n − 1) to the extension ϕ̂ of ϕ to get

∫_{S_{1}} #ϕ^{−1}[p] dℌ_{h_{1}}^{n−1}(p) = ∫_{A} J(ϕ)(q) dℌ_{h_{2}}^{n−1}(q) ,  (6.4)

where J(ϕ) is the restriction of the Jacobian^{19} of ϕ̂ to A, and h_{a}, a = 1, 2, denotes the metric induced on Σ_{a} by g. We observe that

#ϕ^{−1}[p] = N(p, S_{2})

for all q ∈ S_{1}. Indeed, if γ is a null generator of ℋ and p ∈ γ ∩ S_{1} then the point q ∈ S_{2} with q ∈ γ satisfies ϕ(q) = p. Thus there is a bijective correspondence between ϕ^{−1}[p] and the null generators of ℋ passing through p. (This does use that S_{1} ∩ S_{2} = ∅.) This implies that

∫_{S_{1}} #ϕ^{−1}[p] dℌ_{h_{1}}^{n−1}(p) = ∫_{S_{1}} N(p, S_{2}) dℌ_{h_{1}}^{n−1}(p) ,

which can be combined with the area formula (6.4) to give that

Area_{S_{2}}(S_{1}) = ∫_{A} J(ϕ)(q) dℌ_{h_{2}}^{n−1}(q).  (6.5)

We note that [24, Theorem 3.2.3, p. 243] also guarantees^{20} that A ∋ p → N(p, S_{2}) is measurable, so that Area_{S_{2}}(S_{1}) is well defined. Now a calculation, that is straightforward when ℋ is smooth (Proposition A.5, Appendix A), shows that having the null mean curvatures θ_{Al} – or θ_{Al}^{S_{2}} nonnegative together with the null energy condition – implies that J(ϕ) ≤ 1 almost everywhere with respect to ℌ_{h_{2}}^{n−1} on A — this is established in Proposition 6.16. Using this in (6.5) completes the proof.

Our first technical step is the following:

Proposition 6.3 Let the setting be as in Theorem 6.1. Then there exists an open submanifold Σ′_{2} of Σ_{2} such that A is a closed subset of S′_{2} = Σ′_{2} ∩ ℋ. Replacing Σ_{2} with Σ′_{2} we can thus assume that A is closed in S_{2}.

19It should be pointed out that the Jacobian J(ϕ̂) is not the usual Jacobian which occurs in the change–of–variables theorem for Lebesgue measure on ℝ^{n}, but contains the appropriate \sqrt{det g_{ij}} factors occurring in the definition of the measure associated with a Riemannian metric, see [23, Def. 2.10, p. 423]. A clear exposition of the Jacobians that occur in the area and co–area formulas can be found in [22, Section 3.2] in the flat ℝ^{n} case. See also [41, Appendix p. 66] for the smooth Riemannian case.

20The result in [24, Theorem 3.2.3, p. 243] is formulated for subsets of ℝ^{q}, but the result generalizes immediately to manifolds by considering local charts, together with a partition of unity argument.

Proof: Let W_{2} be the subset of S_{2} consisting of all points p ∈ S_{2} for which there exists a semi–tangent X at p of ℋ such that the null geodesic η starting at p in the direction X does not meet Σ_{1} when extended to the past (whether or not it remains in ℋ). Let {p_{k}} be a sequence of points in W_{2} such that p_{k} → p ∈ S_{2}. We show that p ∈ W_{2}, and hence that W_{2} is a closed subset of S_{2}. For each k, let X_{k} be a semi–tangent at p_{k} such that the null geodesic η_{k} starting at p_{k} in the direction X_{k} does not meet Σ_{1} when extended to the past. Since the collection of σ-normalized null vectors is locally compact, by passing to a subsequence if necessary, we may assume that X_{k} → X, where X is a future pointing null vector at p. Let η be the null geodesic starting at p in the direction X. Since (p_{k}, X_{k}) → (p, X), η_{k} → η in the strong sense of geodesics. Since, further, the η_{k}’s remain in ℋ to the future and ℋ is closed, it follows that η is a null generator of ℋ and X is a semi-tangent of ℋ at p. Since η_{k} → η, if η meets Σ_{1} when extended to the past then so will η_{k} for k large enough. Hence, η does not meet Σ_{1}, so p ∈ W_{2}, and W_{2} is a closed subset of S_{2}. Then S′_{2} := S_{2} \ W_{2} is an open subset of S_{2}. Thus, there exists an open set Σ′_{2} in Σ_{2} such that S′_{2} = S_{2}∩Σ′_{2} = Σ′_{2}∩ℋ. Note that p ∈ S′_{2} iff for each semi-tangent X of ℋ at p, the null geodesic starting at p in the direction X meets Σ_{1} when extended to the past. In particular, A ⊂ S′_{2}. To finish, we show that A is a closed subset of S′_{2}. Let {p_{k}} be a sequence in A such that p_{k} → p ∈ S′_{2}. Let η_{k} be the unique null generator of ℋ through p_{k}. Let X_{k} be the σ-normalized tangent to η_{k} at p_{k}. Let q_{k} be the point where η_{k} meets S_{1}. Again, by passing to a subsequence if necessary, we may assume X_{k} → X, where X is a semi–tangent of ℋ at p. Let η be the null geodesic at p in the direction X. We know that when extended to the past, η meets Σ_{1} at a point q, say. Since η_{k} → η we must in fact have q_{k} → q and hence q ∈ S_{1}. It follows that η starting from q is a null generator of ℋ. Hence p ∈ A, and A is closed in S′_{2}. □

We note that in the proof above we have also shown the following:

Lemma 6.4 The collection 𝒩 = ∪_{p∈ℋ} 𝒩_{p} of semi–tangents is a closed subset of T M. □

Recall that we have fixed a complete Riemannian metric σ on M. For each δ > 0 let

A_{δ} : = {p ∈ A | σ-dist(p, S_{1}) ≥ δ , and the generator γ through
p can be extended at least a σ–distance δ to the future} . (6.6)

We note that if the σ distance from S_{1} to S_{2} on the generators passing through S_{1} is bounded from below, and if the length of the portions of those generators which lie to the future of S_{2} is also bounded from below (which is trivially fulfilled when the generators are assumed to be future complete), then A_{δ} will coincide with A for δ small enough.

Lemma 6.5 Without loss of generality we may assume

\overline{S}_{1} ∩ S_{2} = ∅  (6.7)

and

A = ∪_{δ>0}A_{δ} .  (6.8)

Proof: We shall show how to reduce the general situation in which S_{1} ∩S_{2} = ∅ to one in which Equation (6.7) holds. Assume, first, that Σ_{1} is connected, and introduce a complete Riemannian metric on Σ_{1}. With respect to this metric Σ_{1} is a complete metric space such that the closed distance balls \overline{B}(p, r) (closure in Σ_{1}) are compact in Σ_{1}, and hence also in M. Choose a p in Σ_{1} and let

Σ_{1,i} = B(p, i)

(B(p, i) – open balls). Then Σ_{1} = ∪_{i}Σ_{1,i} = ∪_{i}\overline{Σ}_{1,i}) (closure either in Σ_{1} or in M) is an increasing union of compact sets. The Σ_{1,i}’s are spacelike achronal hypersurfaces which have the desired property

\overline{S}_{1,i} ∩ S_{2} ⊂ Σ_{1} ∩ Σ_{2} = ∅

(closure in M), S_{1,i} ≡ Σ_{1,i} ∩ ℋ. Suppose that we have shown that point 1 of Theorem 6.1 is true for S_{1,i}, thus

Area_{S_{2}}(S_{1,i}) ≤ Area(S_{2}) .

As S_{1,i} ⊂ S_{1,i+1}, S_{1} = ∪S_{1,i}, the monotone convergence theorem gives

lim_{i→∞} Area_{S_{2}}(S_{1,i}) = Area_{S_{2}}(S_{1}) ,

whence the result.

If Σ_{1} is not connected, one can carry the above procedure out on each component (at most countably many), obtaining a sequence of sets Σ_{1,i} for each component of Σ_{1}. The resulting collection of sets is countable, and an obvious modification of the above argument establishes that (6.7) holds.

But if Equation (6.7) holds, that is if \overline{S}_{1} ∩ S_{2} = ∅, we have σ-dist(p, S_{1}) = σ-dist(p, \overline{S}_{1}) > 0 for all p ∈ S_{2}. Therefore (6.8) holds. This completes the proof. □

In order to continue we need the following variation of the Whitney extension theorem. As the proof involves very different ideas than the rest of this section we postpone the proof to Appendix E.

Proposition 6.6 Let A ⊂ ℝ^{n} and f: A → ℝ. Assume there is a constant C > 0 and for each p ∈ A there is a vector a_{p} ∈ ℝ^{n} so that the inequalities

f(p) + ⟨x − p, a_{p}⟩ − \frac{C}{2} ∥x − p∥^{2} ≤ f(x) ≤ f(p) + ⟨x − p, a_{p}⟩ + \frac{C}{2} ∥x − p∥^{2} (6.9)

hold for all x ∈ A. Also assume that for all p, q ∈ A and all x ∈ ℝ^{n} the inequality

f(p) + ⟨x − p, a_{p}⟩ − \frac{C}{2} ∥x − p∥^{2} ≤ f(q) + ⟨x − q, a_{q}⟩ + \frac{C}{2} ∥x − q∥^{2}  (6.10)

holds. Then there is a function F: ℝ^{n} → ℝ of class C^{1,1} so that f is the restriction of F to A.

Remark 6.7 Following [9, Prop. 1.1 p. 7] the hypothesis (6.9) of Proposition 6.6 will be stated by saying f has global upper and lower support paraboloids of opening C in K. The condition (6.10) can be expressed by saying the upper and lower support paraboloids of f are disjoint.

Remark 6.8 Unlike most extension theorems of Whitney type this result does not require that the set being extend from be closed (here A need not even be measurable) and there is no continuity assumption on f or on the map that sends p to the vector a_{p} (cf. [62, Chap. VI Sec. 2], [40, Vol. 1, Thm 2.3.6, p. 48] where to get a C^{1,1} extension the mapping p ↦ a_{p} is required to be Lipschitz.) The usual continuity assumptions are replaced by the “disjointness” condition (6.10) which is more natural in the geometric problems considered here.

Proposition 6.6 is a key element for the proof of the result that follows [42]:

Lemma 6.9 For any δ > 0 there is a C_{loc}^{1,1} hypersurface N_{δ} of Σ_{2} (thus N_{δ} has co–dimension two in M) with

A_{δ} ⊆ N_{δ} .

Remark 6.10 We note that N_{δ} does not have to be connected.

Proof: The strategy of the proof is as follows: We start by showing that all points q in A_{δ} possess space–time neighborhoods in which all the generators of ℋ passing through A_{δ} are contained in a C^{1,1} hypersurface in M. This hypersurface is not necessarily null, but is transverse to Σ_{2} on A_{δ}. N_{δ} will be obtained by a globalization argument that uses the intersections of those locally defined hypersurfaces with Σ_{2}.

By the local character of the arguments that follow, one can assume without loss of generality that M is globally hyperbolic; otherwise, where relevant, one could restrict to a globally hyperbolic neighborhood and define the pasts, futures, etc., with respect to this neighborhood.

For q ∈ A_{δ} define q^{±} ∈ J^{±}(q) ∩ ℋ as those points on the generator γ_{q} of ℋ through q which lie a σ–distance δ away from q. Achronality of ℋ implies that γ_{q} has no conjugate points, hence the hypersurfaces ∂J^{∓}(q^{±}) are smooth in a neighborhood of q, tangent to ℋ there.

Let q_{0} in A_{δ} and choose a basis B = {E_{1}, …, E_{n}, E_{n+1}} of T_{q_{0}^{+}}M such that g(E_{i}, E_{j}) = δ_{ij} and g(E_{i}, E_{n+1}) = 0 for i, j = 1, …, n, while g(E_{n+1}, E_{n+1}) = −1.

We shall denote by (y_{1}, …, y_{n+1}) the normal coordinates associated with this basis.

We note that A_{δ′} ⊂ A_{δ} for δ < δ′ so that it is sufficient to establish our claims for δ small. Choosing δ small enough the cone

\{y_{n+1} = −\sqrt{y_{1}^{2} + ⋯ + y_{n}^{2}} | y_{i} ∈ ℝ, i = 1, …, n\}

coincides with ∂J^{−}(q_{0}^{+}) in a neighborhood of q_{0}. Reducing δ again if necessary we can suppose that q_{0}^{−} belongs to a normal coordinate neighborhood of q_{0}^{+}. Let y_{n+1}^{0} be the (n + 1)^{st} coordinate of q_{0} and let y_{n+1}^{0−} be the (n + 1)^{st} coordinate of q_{0}^{−}. Denote by C^{−} the smooth hypersurface with compact closure obtained by intersecting the previous cone by the space–time slab lying between the hypersurfaces y_{n+1} = \frac{1}{2}y_{n+1}^{0} and y_{n+1} = \frac{1}{2}(y_{n+1}^{0} + y_{n+1}^{0−}). Let us use the symbol ω to denote any smooth local parameterization ω ⟶ n(ω) ∈ T_{q_{0}^{+}}M of this hypersurface. We then obtain a smooth parameterization by ω of ∂J^{−}(q_{0}^{+}) in a neighborhood of q_{0} by using the map ω ⟶ exp_{q_{0}^{+}}(n(ω)). Reducing δ again if necessary, without loss of generality we may assume that there are no conjugate points on exp_{q_{0}^{+}}(n(ω)) ⊂ C^{−}.

Using parallel transport of B along radial geodesics from q_{0}^{+}, we can then use ω to obtain a smooth parameterization of ∂J^{−}(m) in a neighborhood of q_{0}, for all m in a neighborhood of q_{0}^{+}. For all m in this neighborhood let us denote by n_{m}(ω) the corresponding vector in T_{m}M. The map (m, ω) ⟶ exp_{m}(n_{m}(ω)) is smooth for m close to q_{0}^{+} and for ω ∈ C^{−}. This implies the continuity of the map which to a couple (m, ω) assigns the second fundamental form II(m, ω), defined with respect to an auxiliary Riemannian metric σ, of ∂J^{−}(m) at exp_{m}(n_{m}(ω)). If we choose the m’s in a sufficiently small compact coordinate neighborhood of q_{0}^{+}, compactness of C^{−} implies that the σ-norm of II can be bounded in this neighborhood by a constant, independently of (m, ω).

Let, now (x_{1}, …, x_{n+1}) be a coordinate system covering a globally hyperbolic neighborhood of q_{0}, centered at q_{0}, and of the form 𝒪 = B^{n}(2r) × (−a, a). We can further require that g(∂_{n+1}, ∂_{n+1}) < C < 0 over 𝒪. Transversality shows that, reducing r and a if necessary, the hypersurfaces exp_{m}(n_{m}(ω)) are smooth graphs above B^{n}(2r), for m close to q_{0}^{+}; we shall denote by f_{m}^{−} the corresponding graphing functions. Now, the first derivatives of the f_{m}^{−}’s can be bounded by an m–independent constant: the vectors V_{i} = ∂_{i} + D_{i}f_{m}^{−}∂_{n+1} are tangent to the graph, hence non–timelike, and the result follows immediately from the equation g(V_{i}, V_{i}) ≥ 0. Next, the explicit formula for the second fundamental form of a graph (cf., e.g., [2, Eqns. (3.3) and (3.4), p. 604]) gives a uniform bound for the second derivatives of the f_{m}^{−}’s over B^{n}(2r). We make a similar argument, using future light cones issued from points m near to q_{0}^{−}, and their graphing functions over B^{n}(2r) denoted by f_{m}^{+}. Let f be the graphing function of ℋ over B^{n}(2r); we can choose a constant C such that, for all p in

G_{δ} := {x ∈ Γ | Γ is a generator of ℋ passing through A_{δ}} ∩ 𝒪 ,  (6.11)

the graph of the function

f_{p}(x) = f(x_{p}) + Df(x_{p})(x − x_{p}) − C||x − x_{p}||^{2}

lies under the graph of f_{p^{+}}^{−}, hence in the past of p^{+}. Here we write x_{p} for the space coordinate of p, thus p = (x_{p}, f(x_{p})), etc. Similarly, for all q in G_{δ} the graph of

f_{q}(x) = f(x_{q}) + Df(x_{q})(x − x_{q}) + C||x − x_{q}||^{2}

lies above the graph of f_{q^{−}}^{+}, thus is to the future of q^{−}. Achronality of the horizon implies then that the inequality (6.10) holds for x ∈ B^{n}(2r). Increasing C if necessary, Equation (6.10) will hold for all x_{p}, x_{q} ∈ B^{n}(r) and x ∈ ℝ^{n}: Indeed, let us make C large enough so that

(C − 1)r^{2} ≥ Lip_{B^{n}(r)}^{2}(f) − inf_{B^{n}(r)}(f) + sup_{B^{n}(r)}(f) ,

where Lip_{B^{n}(r)}(f) is the Lipschitz continuity constant of f on B^{n}(r). For x ∉ B^{n}(2r) we then have

f(x_{q}) − f(x_{p}) + \frac{C}{2} (||x − x_{q}||^{2} + ||x − x_{p}||^{2})

+⟨x − x_{q}, a_{q}⟩ − ⟨x − x_{p}, a_{p}⟩ ≥

f(x_{q}) − f(x_{p}) + \frac{C}{2} (||x − x_{q}||^{2} + ||x − x_{p}||^{2})

−\frac{1}{2}(||x − x_{q}||^{2} + ||x − x_{p}||^{2} + ||a_{q}||^{2} + ||a_{p}||^{2}) ≥
inf_{B^{n}(r)}(f) − sup_{B^{n}(r)}(f) + (C − 1)r^{2} − Lip_{B^{n}(r)}^{2}(f) ≥ 0 .

Here a_{p} = df(x_{p}), a_{q} = df(x_{q}), and we have used Lip_{B^{n}(r)}(f) to control ||a_{p}|| and ||a_{q}||.

Let the set A of Proposition 6.6 be the projection on B^{n}(r) of G_{δ}; by that proposition there exists a C^{1,1} function from B^{n}(r) to (−a, a), the graph of which contains G_{δ}. (It may be necessary to reduce r and 𝒪 to obtain this). Note that this graph contains A_{δ} ∩ 𝒪 and is transverse to Σ_{2} there. From the fact that a transverse intersection of a C^{2} hypersurface with a C^{1,1} hypersurface is a C^{1,1} manifold we obtain that A_{δ} ∩ 𝒪 is included in a C^{1,1} submanifold of Σ_{2}, which has space–time co–dimension two.

Now A_{δ} is a closed subset of the manifold Σ′_{2} defined in Proposition 6.3, and can thus be covered by a countable union of compact sets. Further, by definition of A_{δ} any point thereof is an interior point of a generator. Those facts and the arguments of the proof of Proposition 3.1 show that A_{δ} can be covered by a countable locally finite collection of relatively compact coordinate neighborhoods 𝒰_{i} of the form 𝒰_{i} = B^{n−1}(ϵ_{i}) × (−η_{i}, η_{i}) such that 𝒰_{i} ∩ ℋ is a graph of a semi– convex function g_{i}:

𝒰_{i} ∩ ℋ = {x^{n} = g_{i}(x^{A}) , (x^{A}) ≡ (x^{1}, …, x^{n−1}) ∈ B^{n−1}(ϵ_{i})} .  (6.12)

The ϵ_{i}’s will further be restricted by the requirement that there exists a C^{1,1} function

h_{i}: B^{n−1}(ϵ_{i}) → ℝ  (6.13)

such that the graph of h_{i} contains 𝒰_{i} ∩ 𝒜_{δ}. The h_{i}’s are the graphing functions of the C^{1,1} manifolds just constructed in a neighborhood of each point of A_{δ}.

At those points at which S_{2} is differentiable let m denote a h_{2}–unit vector field normal to S_{2} pointing towards J^{−}(ℋ) ∩ Σ_{2}. Choosing the orientation of x^{n} appropriately we can assume that at those points at which m is defined we have

m(x^{n} − g_{i}) > 0 .  (6.14)

In order to globalize this construction we use an idea of [42]. Let ϕ_{i} be a partition of unity subordinate to the cover {𝒰_{i}}_{i∈ℕ}, define

χ_{i}(q) = \{ \begin{matrix}(x^{n} − h_{i}(x^{A}))ϕ_{i}(q) , & q = (x^{A}, x^{n}) ∈ 𝒰_{i} , \\ 0 , & otherwise, \end{matrix}  (6.15)
χ_{δ} = ∑_{i} χ_{i} .  (6.16)

Define a hypersurface N_{δ} ⊂ ∪_{i}𝒰_{i} via the equations

N_{δ} ∩ 𝒰_{i} = {χ_{δ} = 0 , dχ_{δ} ≠ 0} .  (6.17)

(We note that N_{δ} does not have to be connected, but this is irrelevant for our purposes.) If q ∈ A_{δ} we have χ_{i}(q) = 0 for all i’s hence A_{δ} ⊂ {χ_{δ} = 0}. Further, for q ∈ A_{δ} ∩ 𝒰_{i} we have m(χ_{i}) = ϕ_{i}m(x^{n} − g_{i}) ≥ 0 from Equation (6.14), and since for each q ∈ A_{δ} there exists an i at which this is strictly positive we obtain

m(χ_{δ}) = ∑_{i} m(χ_{i}) > 0 .

It follows that A_{δ} ⊂ N_{δ}, and the result is established. □

Lemma 6.11 Under the condition (6.8), the map ϕ: A → S_{1} is locally Lipschitz.

Proof: Let q_{0} ∈ A, we need to find a neighborhood 𝒰_{A} ≡ 𝒰 ∩ A in A so that ϕ is Lipschitz in this neighborhood. By the condition (6.8) there exists δ > 0 such that q_{0} ∈ A_{δ}. By lower semi–continuity of the existence time of geodesics we can choose a neighborhood 𝒰 of q_{0} in Σ_{2} small enough so that

𝒰 ∩ A_{δ} = 𝒰 ∩ A .  (6.18)

Denote by N the C_{loc}^{1,1} hypersurface N_{δ} (corresponding to the chosen small value of δ) given by Lemma 6.9, so that

A ∩ 𝒰 ⊆ N ∩ 𝒰 .

Let n be the future pointing unit normal to Σ_{2}. (So n is a unit timelike vector.) Let m be the unit normal to N in Σ_{2} that points to J^{−}(ℋ) ∩ Σ_{2}. Let k := −(n + m). Then k is a past pointing null vector field along N and if q ∈ A_{δ} then k(q) is tangent to the unique generator of ℋ through q. As Σ_{2} is C^{2} then the vector field n is C^{1}, while the C_{loc}^{1,1} character of N implies that m is locally Lipschitz. It follows that k is a locally Lipschitz vector field along N. Now for each q ∈ A ∩ 𝒰 there is a unique positive real number r(q) so that ϕ(q) = exp(r(q)k(q)) ∈ S_{1}. Lower semi–continuity of existence time of geodesics implies that, passing to a subset of 𝒰 if necessary, for each q ∈ N ∩ 𝒰 there is a unique positive real number r̂(q) so that ϕ̂(q) = exp(r̂(q)k(q)) ∈ Σ_{1}. As Σ_{1} is a Lipschitz hypersurface, Clarke’s implicit function theorem [17, Corollary, p. 256] implies that q ↦ r̂(q) is a locally Lipschitz function near q_{0}. Thus ϕ̂ is Lipschitz near q_{0} and the restriction of ϕ̂ to A_{δ} is ϕ. □

Corollary 6.12 The section S_{1} is countably (n − 1) rectifiable.

Remark 6.13 By starting with an arbitrary section S = S_{1} of the form S = ℋ ∩ Σ, where Σ is a C^{2} spacelike hypersurface or timelike hypersurface that meets ℋ properly transversely, and then constructing a section S_{2} = ℋ ∩ Σ_{2} for a C^{2} spacelike hypersurface Σ_{2} so that the hypotheses of Theorem 6.1 hold, one obtains that every such section S is countably (n − 1) rectifiable. In the case that Σ_{1} ≡ Σ is spacelike a more precise version of this is given in [42] (where it is shown that S is countably (n − 1) rectifiable of class C^{2}).

Proof: From Lemma 6.9 it is clear for each δ > 0 that A_{δ} is countably (n − 1) rectifiable. By Lemma 6.11 the map ϕ|_{A_{δ}}: A_{δ} → S_{1} is locally Lipschitz and therefore ϕ[A_{δ}] is locally countable (n − 1) rectifiable. But S_{1} = ⋃_{k=1}^{∞} ϕ[A_{1/k}] so S_{1} is a countable union of countably (n − 1) rectifiable sets and therefore is itself countably (n − 1) rectifiable. □

It follows from the outline given at the beginning of the proof of Theorem 6.1 that we have obtained:

Proposition 6.14 The formula (6.5) holds.

Corollary 6.15 For any section S = ℋ∩Σ where Σ is a C^{2} spacelike or timelike hypersurface that intersects ℋ properly transversely the set of points of S that are on infinitely many generators has vanishing (n − 1) dimensional Hausdorff measure.

Proof: For any point p of S choose a globally hyperbolic neighborhood 𝒪 of the point and then choose a neighborhood S_{1} of p in S small enough that the closure \overline{S}_{1} is compact, satisfies \overline{S}_{1} ⊂ S∩𝒪, and so that there is a C^{∞} Cauchy hypersurface Σ_{2} of 𝒪 such that \overline{S}_{1} ⊂ I^{−}(Σ_{2}; 𝒪). Let S_{2} = ℋ∩Σ_{2} and let A be the set of points of S_{2} that are of the form S_{2} ∩γ where γ is a generator of ℋ that meets \overline{S}_{1}. Compactness of \overline{S}_{1} together with the argument of the proof of Proposition 6.3 show that the set of generators of ℋ that meet \overline{S}_{1} is a compact subset of the bundle of null geodesic rays of M, and that A is a compact subset of S_{2}. Then Lemma 6.9 implies that A is a compact set in a C^{1,1} hypersurface of Σ_{2} and so ℌ^{n−1}(A) < ∞. Lemma 6.11 and the compactness of A yields that ϕ: A → \overline{S}_{1} given by (6.3) (with \overline{S}_{1} replacing S_{1}) is Lipschitz. Therefore the Jacobian J(ϕ) is bounded on

A. By Proposition 6.14 ∫_{\overline{S}_{1}} N(p, S_{2}) dℌ^{n−1}(p) = ∫_{A} J(ϕ)(q) dℌ_{h_{2}}^{n−1}(q) < ∞ and so

N(p, S_{2}) < ∞ except on a set of ℌ^{n−1} measure zero. But N(p, S_{2}) is the number of generators of ℋ through p so this implies the set of points of S_{1} that are on infinitely many generators has vanishing (n − 1) dimensional Hausdorff measure. Now S can be covered by a countable collection of such neighborhoods S_{1} and so the set of points on S that are on infinitely many generators has vanishing (n − 1) dimensional Hausdorff measure. This completes the proof. □

To establish (6.2) it remains to show that J(ϕ) ≤ 1 ℌ_{h_{2}}^{n−1}-almost everywhere. To do this one would like to use the classical formula that relates the Jacobian of ϕ to the divergence θ of the horizon, cf. Proposition A.5, Appendix A below. However, for horizons which are not C^{2} we only have the Alexandrov divergence θ_{Al} at our disposal, and it is not clear whether or not this formula holds with θ replaced by θ_{Al} for general horizons. The proof below consists in showing that this formula remains true after such a replacement for generators passing through almost all points of A.

Proposition 6.16 J(ϕ) ≤ 1 ℌ_{h_{2}}^{n−1}–almost everywhere on A.

Proof: The argument of the paragraph preceding Equation (6.18) shows that it is sufficient to show J(ϕ) ≤ 1 ℌ_{h_{2}}^{n−1}–almost everywhere on A_{δ} for each δ > 0. Let N_{δ} be the C^{1,1} manifold constructed in Lemma 6.9, and 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_{δ} 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. By [24, Theorem 3.1.5, p. 227] for every ϵ > 0 there exists a twice differentiable function g_{1/ϵ}: 𝒱 → ℝ such that

𝔏_{h_{𝒱}}^{n−1}({g ≠ g_{1/ϵ}}) < ϵ .  (6.19)

Here 𝔏_{h_{𝒱}}^{n−1} denotes the (n − 1) dimensional Riemannian measure on 𝒱 associated with the pull–back h_{𝒱} of the space–time metric g to 𝒱. Let prA_{δ} denote the projection on 𝒱 of A_{δ} ∩ 𝒰, thus A_{δ} ∩ 𝒰 is the graph of g over prA_{δ}. For q ∈ 𝒱 let θ_{∗}^{n−1}(𝔏_{h_{𝒱}}^{n−1}, prA_{δ}, q) be the density function of prA_{δ} in 𝒱 with respect to the measure 𝔏_{h_{𝒱}}^{n−1}, defined as in [61, page 10] using geodesic coordinates centered at q with respect to the metric h_{𝒱}. Define

B = {q ∈ prA_{δ} | θ_{∗}^{n−1}(𝔏_{h_{𝒱}}^{n−1}, prA_{δ}, q) = 1} ⊂ 𝒱 .  (6.20)

By [52, Corollary 2.9] or [24, 2.9.12] the function θ_{∗}^{n−1}(𝔏_{h_{𝒱}}^{n−1}, prA_{δ}, ·) differs from the characteristic function χ_{prA_{δ}} of prA_{δ} by a function supported on a set of vanishing measure, which implies that B has full measure in prA_{δ}. Let

B_{1/ϵ} = B ∩ {g = g_{1/ϵ}} ;

Equation (6.19) shows that

𝔏_{h_{𝒱}}^{n−1}(B \ B_{1/ϵ}) < ϵ .  (6.21)

We define B̃_{1/ϵ} ⊂ B_{1/ϵ} ⊂ prA_{δ} ⊂ 𝒱 as follows:

B̃_{1/ϵ} = {q ∈ B_{1/ϵ} | θ_{∗}^{n−1}(𝔏_{h_{𝒱}}^{n−1}, B_{1/ϵ}, q) = 1} .  (6.22)

Similarly the set B̃_{1/ϵ} has full measure in B_{1/ϵ}, hence

𝔏_{h_{𝒱}}^{n−1}(B \ B̃_{1/ϵ}) < ϵ .  (6.23)

Let (prA_{δ})_{Al} denote the projection on 𝒱 of the set of those Alexandrov points of ℋ ∩Σ_{2} which are in A_{δ} ∩𝒰; (prA_{δ})_{Al} has full measure in prA_{δ}. Let, further, 𝒱_{Rad} be the set of points at which g is twice differentiable; by Rademacher’s theorem (cf., e.g., [22, p. 81]) 𝒱_{Rad} has full measure in 𝒱. We set

B̂_{1/ϵ} = B̃_{1/ϵ} ∩ (prA_{δ})_{Al} ∩ 𝒱_{Rad} ,  (6.24)
B̂ = ⋃_{i∈ℕ} B̂_{i}  (6.25)

It follows from (6.23) that B̂ has full measure in prA_{δ}. Since g is Lipschitz we obtain that the graph of g over B̂ has full ℌ_{h_{2}}^{n−1}–measure in A_{δ} ∩ 𝒰.

Consider any x_{0} ∈ B̂, then x_{0} is an Alexandrov point of f so that we have the expansion

f(x) = f(x_{0}) + df(x_{0})(x − x_{0}) + \frac{1}{2}D^{2}f(x_{0})(x − x_{0}, x − x_{0}) + o(|x − x_{0}|^{2}) .(6.26)

Next, g is twice differentiable at x_{0} so that we also have

g(x) = g(x_{0}) + dg(x_{0})(x − x_{0}) + \frac{1}{2}D^{2}g(x_{0})(x − x_{0}, x − x_{0}) + o(|x − x_{0}|^{2}) , (6.27)
dg(x) − dg(x_{0}) = D^{2}g(x_{0})(x − x_{0}, ·) + o(|x − x_{0}|) .  (6.28)

Further there exists j ∈ ℕ such that x_{0} ∈ B̂_{j} ⊂ B_{j}; by definition of B_{j} we then have

g_{j}(x_{0}) = g(x_{0}) = f(x_{0}) .  (6.29)

We claim that the set of directions n⃗ ∈ B^{n−1}(1) ⊂ T_{x_{0}}𝒱 for which there exists a sequence of points q_{i} = x_{0} + r_{i}n⃗ with r_{i} → 0 and with the property that f(q_{i}) = g_{j}(q_{i}) is dense in B^{n−1}(1). Indeed, suppose that this is not the case, then there exists ϵ > 0 and an open set Ω ⊂ B^{n−1}(1) ⊂ T_{x_{0}}𝒱 of directions n⃗ ∈ B^{n−1}(1) such that the solid cone K_{ϵ} = {x_{0}+rn⃗ | n⃗ ∈ Ω , r ∈ [0, ϵ]} contains no points from B_{j}. It follows that the density function θ_{∗}^{n−1}(𝔏_{h_{𝒱}}^{n−1}, B_{j}, x_{0}) is strictly smaller than one, which contradicts the fact that x_{0} is a density point of prA_{δ} (x_{0} ∈ B̂_{j} ⊂ B, cf. Equation (6.20)), and that x_{0} is a density point of B_{j} (x_{0} ∈ B̂_{j} ⊂ B̃_{j}, cf. Equation (6.22)).

Equation (6.29) leads to the following Taylor expansions at x_{0}:

g_{j}(x) = f(x_{0}) + dg_{j}(x_{0})(x − x_{0}) + \frac{1}{2}D^{2}g_{j}(x_{0})(x − x_{0}, x − x_{0}) + o(|x − x_{0}|^{2}) , (6.30)

dg_{j}(x) − dg_{j}(x_{0}) = D^{2}g_{j}(x_{0})(x − x_{0}, ·) + o(|x − x_{0}|) .  (6.31)

Subtracting Equation (6.26) from Equation (6.30) at points q_{i} = x_{0}+r_{i}n⃗ at which f(q_{i}) = g_{j}(q_{i}) we obtain

dg_{j}(x_{0})(n⃗) − df(x_{0})(n⃗) = O(r_{i}) .  (6.32)

Density of the set of n⃗’s for which (6.32) holds implies that

dg_{j}(x_{0}) = df(x_{0}) .  (6.33)

Again comparing Equation (6.26) with Equation (6.30) at points at which f(q_{i}) = g_{j}(q_{i}) it now follows that

D^{2}g_{j}(x_{0})(n⃗, n⃗) − D^{2}f(x_{0})(n⃗, n⃗) = 2\frac{g_{j}(x_{0} + r_{i}n⃗) − f(x_{0} + r_{i}n⃗)}{r_{i}^{2}} + o(1) = o(1) ,

(6.34)

and density of the set of n⃗’s together with the polarization identity gives

D^{2}g_{j}(x_{0}) = D^{2}f(x_{0}) .  (6.35)

Similarly one obtains

dg(x_{0}) = df(x_{0}) ,  (6.36)
D^{2}g(x_{0}) = D^{2}f(x_{0}) .  (6.37)

Define S_{j} to be the graph (over 𝒱) of g_{j}; Equations (6.33) and (6.36) show that both S_{j} and N_{δ} are tangent to ℋ ∩ Σ_{2} at p_{0} ≡ (x_{0}, f(x_{0})):

T_{p_{0}}(ℋ ∩ Σ_{2}) = T_{p_{0}}N_{δ} = T_{p_{0}}S_{j} .  (6.38)

Let n_{j} denote the (C^{1}) field of σ–unit future directed null normals to S_{j} such that

n_{j}(p_{0}) = n(p_{0}) ,

where n(p_{0}) is the semi–tangent to ℋ at p_{0}. Let ϕ_{j}: S_{j} → Σ_{1} be the map obtained by intersecting the null geodesics passing through points q ∈ S_{j} with tangent parallel to n_{j}(q) there. Equation (6.38) shows that

ϕ_{j}(p_{0}) = ϕ(p_{0}) = ϕ̂(p_{0}) .  (6.39)

The lower semi–continuity of the existence time of geodesics shows that, passing to a subset of 𝒰 if necessary, ϕ_{j} is well defined on S_{j}. By an argument similar to the one leading to (6.33), it follows from Equations (6.28), (6.31), (6.33) and (6.35)–(6.37) that the derivatives of ϕ_{j} and of ϕ̂ coincide at p_{0}, in particular

J(ϕ)(p_{0}) ≡ J(ϕ̂)(p_{0}) = J(ϕ_{j})(p_{0}) .  (6.40)

Equation (6.35) further shows that S_{j} is second order tangent to ℋ at p_{0} in the sense defined before Lemma 4.15, we thus infer from that lemma that there are no focal points of S_{j} along the segment of the generator Γ of ℋ passing through p_{0} which lies to the future of Σ_{1}. Consider the set ℋ_{j} obtained as the union of null geodesics passing through S_{j} and tangent to n_{j} there; standard considerations show that there exists a neighborhood of Γ∩I^{+}(Σ_{1})∩I^{−}(Σ_{2}) in which ℋ_{j} is a C^{1} hypersurface. It is shown in Appendix A that 1) ℋ_{j} is actually a C^{2} hypersurface, cf. Proposition A.3, and 2) the null Weingarten map b = b_{ℋ_{j}} of ℋ_{j} satisfies the Ricatti equation

b′ + b^{2} + R = 0 .  (6.41)

Here a prime denotes a derivative with respect to an affine parameterization s ↦ η(s) of Γ that makes Γ future directed. Theorem 5.1 implies that ℋ has a null Weingarten map b_{Al} defined in terms of the Alexandrov second derivatives of ℋ on all of the segment Γ ∩ I^{+}(Σ_{1}) ∩ I^{−}(Σ_{2}) and that this Weingarten map also satisfies the Ricatti equation (6.41). As the null Weingarten map can be expressed in terms of the first and second derivatives of the graphing function of a section Equations (6.33) and (6.35) imply that b_{ℋ_{j}} (p_{0}) = b_{Al}(p_{0}). Therefore uniqueness of solutions to initial value problems implies that b_{ℋ_{j}} = b_{Al} on all of Γ ∩ I^{+}(Σ_{1}) ∩ I^{−}(Σ_{2}). But the divergence (or null mean curvature) of a null hypersurface is the trace of its null Weingarten map and thus on the segment Γ ∩ I^{+}(Σ_{1}) ∩ I^{−}(Σ_{2}) we have θ_{ℋ_{j}} = trace b_{ℋ_{j}} = trace b_{Al} = θ_{Al}.

We now finish the proof under the first of hypothesis of Theorem 6.1, that is that the divergence the θ_{Al} ≥ 0 on J^{+}(S_{1}) ∩ J^{−}(S_{2}). Then θ_{ℋ_{j}} = θ_{Al} ≥ 0 and Proposition A.5 implies that J(ϕ)(p_{0}) = J(ϕ_{j})(p_{0}) ≤ 1 as required.

The other hypothesis of Theorem 6.1 is that θ_{Al}^{S_{2}} ≥ 0 and the null energy condition holding on J^{+}(S_{1}) ∩ J^{−}(S_{2}). Recall that p_{0} = Γ ∩ S_{2} is an Alexandrov point of ℋ; thus Theorem 5.1 applies and shows that θ = θ_{Al} exists along the segment Γ ∩ I^{+}(Σ_{1}) ∩ I^{−}(Σ_{2}), and satisfies the Raychaudhuri equation

θ′ = −Ric(η′, η′) − σ^{2} − \frac{1}{n − 2}θ^{2} .

Here σ^{2} is the norm squared of the shear (and should not be confused with the auxiliary Riemannian metric which we have also denoted by σ). But the null energy condition implies Ric(η′, η′) ≥ 0 so this equation and θ_{Al}^{S_{2}} ≥ 0 implies θ = θ_{Al} ≥ 0 on Γ ∩ I^{+}(Σ_{1}) ∩ I^{−}(Σ_{2}). Then the equality θ_{ℋ_{j}} = θ_{Al} yields θ_{ℋ_{j}} ≥ 0 and again we can use Proposition A.5 to conclude J(ϕ)(p_{0}) = J(ϕ_{j})(p_{0}) ≤ 1. This completes the proof. □

To finish the proof of Theorem 6.1, we need to analyze what happens when

Area_{S_{2}}(S_{1}) = Area(S_{2}) .  (6.42)

In this case Equation (6.4) together with Proposition 6.16 show that A has full measure in S_{2}, that N(p, S_{2}) = 1 ℌ_{h_{1}}^{n−1}–almost everywhere on S_{1}, and that J(ϕ) = 1 ℌ_{h_{2}}^{n−1}–almost everywhere on S_{2}. Next, the arguments of the proof of that Proposition show that

θ_{Al} = 0  (6.43)

ℌ_{σ}^{n}–almost everywhere on J^{−}(A) ∩ J^{+}(S_{1}) = J^{−}(S_{2}) ∩ J^{+}(S_{1}). The proof of Proposition 6.16 further shows that Ric(η′, η′) = 0 ℌ_{σ}^{n}–almost everywhere on ℋ ∩ J^{+}(S_{1}) ∩ J^{−}(S_{2}) (cf. Equation (6.41)), hence everywhere there as the metric is assumed to be smooth and the distribution of semi–tangents is a closed set (Lemma 6.4). Here η′ is any semi–tangent to ℋ. We first note the following observation, the proof of which borrows arguments from [6, Section IV]:

Lemma 6.17 Under the hypotheses of Theorem 6.1, suppose further that the equality (6.42) holds. Then there are no end points of generators of ℋ on

Ω ≡ (J^{+}(S_{1}) \ S_{1}) ∩ (J^{−}(S_{2}) \ S_{2}) .  (6.44)

Proof: Suppose that there exists q ∈ Ω which is an end point of a generator Γ of ℋ, set {p} = Γ ∩ S_{2}, extending Γ beyond its end point and parameterizing it appropriately we will have

Γ(0) = q , Γ(1) = p , Γ(a) ∈ I^{−}(ℋ) ,

for any a < 0 for which Γ(a) is defined. Now p is an interior point of a generator, and semi–tangents at points in a sufficiently small neighborhood of p are arbitrarily close to the semi–tangent X_{p} at p. Since I^{−}(ℋ) is open it follows from continuous dependence of solutions of ODE’s upon initial values that there exists a neighborhood 𝒱 ⊂ S_{2} of p such that every generator of ℋ passing through 𝒱 leaves ℋ before intersecting Σ_{1} when followed backwards in time from S_{2}, hence A ∩ 𝒱 = ∅, and A does not have full measure in S_{2}. □

To finish the proof we shall need the following result, which seems to be of independent interest:

Theorem 6.18 Let Ω be an open subset of a horizon ℋ which contains no end points of generators of ℋ, and suppose that the divergence θ_{Al} of ℋ defined (ℌ_{σ}^{n}– almost everywhere) by Equation (2.10) vanishes ℌ_{σ}^{n}–almost everywhere. Then Ω is a smooth submanifold of M (analytic if the metric is analytic).

Remark 6.19 The condition on Ω is equivalent to Ω being a C^{1} hypersurface, cf. [6].

Proof: Let p_{0} ∈ Ω and choose a smooth local foliation {Σ_{λ} | −ε < λ < ε} of an open neighborhood U of p_{0} in M by spacelike hypersurfaces so that \overline{U ∩ Ω} ⊂ Ω and so that p_{0} ∈ Σ_{0}. Letting σ be a the auxiliary Riemannian metric, by possibly making U smaller we can assume that the σ-distance of \overline{U ∩ Ω} to \overline{Ω} \ Ω is < δ for some δ > 0. Let Ω_{Al} be the set of Alexandrov points of Ω and let B = Ω \ Ω_{Al} be the set of points of Ω where the Alexandrov second derivatives do not exist. We view λ as a function λ: U → ℝ in the natural way. Now λ is smooth on U and by Remark 6.19 Ω is a C^{1} manifold so the restriction λ|_{Ω} is a C^{1} function. Letting h_{σ} be the pull back of our auxiliary Riemannian metric σ to Ω we apply the co-area formula to λ|_{Ω} and use that by Alexandrov’s theorem ℌ_{h_{σ}}^{n}(B) = 0 to get

∫_{−ε}^{ε} ℌ_{h_{σ}}^{n−1}(B ∩ Σ_{λ}) dλ = ∫_{B} J(λ|_{Ω}) dℌ_{h_{σ}}^{n} = 0 .

This implies that for almost all λ ∈ (−ε, ε) that ℌ_{h_{σ}}^{n−1}(B ∩ Σ_{λ}) = 0. Therefore we can choose a λ just a little bigger than 0 with ℌ_{h_{σ}}^{n−1}(B ∩ Σ_{λ}) = 0 and so that p_{0} ∈ J^{−}(Ω ∩ Σ_{λ}). To simplify notation we denote Σ_{λ} by Σ. Then from the choice of Σ we have that ℌ_{h_{σ}}^{n−1} almost every point of Σ is an Alexandrov point of Ω.

By transversality and that Ω is C^{1} the set Σ ∩ Ω is a C^{1} submanifold of Σ. Recalling that δ is less than the σ-distance of \overline{U ∩ Ω} to \overline{Ω} \ Ω we see that for any p ∈ Σ the unique (because Ω is C^{1}) generator Γ of Ω through p extends in Ω a σ-distance of at least δ both to the future and to the past of Σ. Letting A = Σ ∩ Ω and using the notation of Equation (6.6), this implies that A = A_{δ}. As A = Σ∩Ω is already a C^{1} submanifold of Σ Lemma 6.9 implies that Σ∩Ω is a C^{1,1} hypersurface in Σ. Let g by any Lipschitz local graphing function of A in Σ. From Rademacher’s theorem (cf., e.g., [22, p. 81]) it follows that C^{1,1} = W_{loc}^{2,∞}, further the Alexandrov second derivatives of g coincide with the classical ones almost everywhere. By [22, p. 235] the second distributional derivatives of g equal the second classical derivatives of g almost everywhere. It follows that the equation

θ_{Al} = 0  (6.45)

can be rewritten, by freezing the coefficients of the second derivatives at the solution g, as a linear elliptic weak (distributional) equation with Lipschitz continuous coefficients for the graphing function g ∈ W_{loc}^{2,∞}. Elliptic regularity shows that g is, locally, of C^{2,α} differentiability class for any α ∈ (0, 1). Further, Equation (6.45) is a quasi–linear elliptic equation for g (cf., e.g., [29]), a standard bootstrap argument shows that g is smooth (analytic if the metric is analytic) and it easily follows that Ω in a neighborhood of Σ ∩ Ω containing p_{0} is smooth (or analytic). As p_{0} was an arbitrary point of Ω this completes the proof. □

Returning to the proof of Theorem 6.1, we note that Lemma 6.17 shows that all points of Σ ∩ Ω, where Ω is given by Equation (6.44), are interior points of generators of ℋ. Simple arguments together with the invariance of the domain theorem (cf., e.g., [18, Prop. 7.4, p. 79]) show that Ω is an open submanifold of ℋ, and Equation (6.43) shows that we can use Theorem 6.18 to conclude. □

7 Conclusions

Let us present here some applications of Theorems 1.2 and 6.18, proved above. The first one is to the theory of stationary black holes (cf., e.g., [12, 38] and references therein): in that theory the question of differentiability of event horizons arises at several key places. Recall that smoothness of event horizons has been established in 1) static [10, 63] and 2) [10] stationary–axisymmetric space–times. However, staticity or stationarity–axisymmetry are often not known a priori — that is indeed the case in Hawking’s rigidity theorem [37]^{21}. Now, the rigidity theorem asserts that a certain class of stationary black holes have axi–symmetric domains of outer communication; its hypotheses include that of analyticity of the metric and of the event horizon. The examples of black holes (in analytic vacuum space–times) the horizons of which are nowhere C^{2} constructed in [15] show that the hypothesis of analyticity of the event horizon and that of analyticity of the metric are logically independent. It is thus of interest to note the following result, which is a straightforward corollary of Theorem 1.2 and of the fact that isometries preserve area:

Theorem 7.1 Let ϕ be an isometry of a black–hole space–time (M, g) satisfying the hypotheses of Theorem 1.2. If ϕ maps ℋ into ℋ, then for every spacelike hypersurface Σ such that

ϕ(Σ ∩ ℋ) ⊂ J^{+}(Σ ∩ ℋ)  (7.1)

the set

(J^{−}(ϕ(Σ ∩ ℋ)) \ ϕ(Σ ∩ ℋ)) ∩ (J^{+}(Σ ∩ ℋ) \ (Σ ∩ ℋ)) ⊂ ℋ  (7.2)

is a smooth (analytic if the metric is analytic) null submanifold of M with vanishing null second fundamental form.

21This theorem is actually wrong as stated in [37]; a corrected version, together with a proof, can be found in [13].

As already pointed out, the application we have in mind is that to stationary black holes, where ϕ actually arises from a one parameter group of isometries ϕ_{t}. We note that in such a setting the fact that isometries preserve the event horizon, as well as the existence of hypersurfaces Σ for which (7.1) holds with ϕ replaced by ϕ_{t} (for some, or for all t’s), can be established under various standard conditions on the geometry of stationary black holes, which are of no concern to us here.

Recall, next, that the question of differentiability of Cauchy horizons often arises in considerations concerning cosmic censorship issues (cf., e.g., [3, 58]). An interesting result in this context, indicating non–genericity of occurrence of compact Cauchy horizons, is the Isenberg–Moncrief theorem, which asserts that analytic compact Cauchy horizons with periodic generators in analytic, electro– vacuum space-times are Killing horizons, for a Killing vector field defined on a neighborhood of the Cauchy horizon [45]. We note that if all the generators of the horizon are periodic, then the horizon has no end–points, and analyticity follows^{22} from Theorem 6.18. Hence the hypothesis of analyticity of the event horizon is not needed in [45]. We also note that there exists a (partial) version of the Isenberg–Moncrief theorem, due to Friedrich, Rácz and Wald [27], in which the hypotheses of analyticity of [45] are replaced by those of smoothness both of the metric and of the Cauchy horizon. Theorem 6.18 again shows that the hypothesis of smoothness of the Cauchy horizon is not necessary in [27].

To close this section let us note an interesting theorem of Beem and Królak [6, Section IV], which asserts that if a compact Cauchy horizon in a space–time satisfying the null energy condition contains, roughly speaking, an open dense subset 𝒪 which is a C^{2} manifold, then there are no end points of the generators of the event horizon, and the divergence of the event horizon vanishes. Theorem 6.18 again applies to show that the horizon must be as smooth as the metric allows. Our methods here could perhaps provide a proof of a version of the Beem–Królak theorem in which the hypothesis of existence of the set 𝒪 will not be needed; this remains to be seen.

A The Geometry of C^{2} Null Hypersurfaces

In this appendix we prove a result concerning the regularity of null hypersurfaces normal to a C^{k} submanifold in space–time. We also review some aspects of the geometry of null hypersurfaces, with the presentation adapted to our needs. We follow the exposition of [29].

Let (M, g) be a spacetime, i.e., a smooth, paracompact time-oriented Lorentzian manifold, of dimension n+1 ≥ 3. We denote the Lorentzian metric on M by g or ⟨ , ⟩. A (C^{2}) null hypersurface in M is a C^{2} co-dimension one embedded submanifold ℋ of M such that the pullback of the metric g to ℋ is degenerate. Each such hypersurface ℋ admits a C^{1} non-vanishing future directed null vector field K ∈ ΓT ℋ such that the normal space of K at a point p ∈ ℋ coincides with the tangent space of ℋ at p, i.e., K_{p}^{⊥} = T_{p}ℋ for all p ∈ ℋ. (If ℋ is C^{2} the best regularity we can require for K is C^{1}.) In particular, tangent vectors to ℋ not parallel to K are spacelike. It is well-known that the integral curves of K, when suitably parameterized, are null geodesics. These integral curves are called the null geodesic generators of ℋ. We note that the vector field K is unique up to a positive scale factor.

22The proof proceeds as follows: Theorem 5.1 shows that the optical equations hold on almost all generators of the Cauchy horizon; periodicity of the generators together with the Raychaudhuri equation shows then that θ_{Al} = 0 almost everywhere, hence Theorem 6.18 applies.

Since K is orthogonal to ℋ we can introduce the null Weingarten map and null second fundamental form of ℋ with respect K in a manner roughly analogous to what is done for spacelike hypersurfaces or hypersurfaces in a Riemannian manifold, as follows: We start by introducing an equivalence relation on tangent vectors: for X, X′ ∈ T_{p}ℋ, X′ = X mod K if and only if X′ − X = λK for some λ ∈ ℝ. Let \overline{X} denote the equivalence class of X. Simple computations show that if X′ = X mod K and Y′ = Y mod K then ⟨X′, Y′⟩ = ⟨X, Y ⟩ and ⟨∇_{X′}K, Y′⟩ = ⟨∇_{X}K, Y ⟩, where ∇ is the Levi-Civita connection of M. Hence, for various quantities of interest, components along K are not of interest. For this reason one works with the tangent space of ℋ modded out by K, i.e., T_{p}ℋ/K = \{\overline{X} | X ∈ T_{p}ℋ\} and T ℋ/K = ∪_{p∈ℋ} T_{p}ℋ/K. T ℋ/K is a rank n − 1 vector bundle over ℋ. This vector bundle does not depend on the particular choice of null vector field K. There is a natural positive definite metric h in T ℋ/K induced from ⟨, ⟩: For each p ∈ ℋ, define h: T_{p}ℋ/K × T_{p}ℋ/K → ℝ by h(\overline{X}, \overline{Y}) = ⟨X, Y ⟩. From remarks above, h is well-defined.

The null Weingarten map b = b_{K} of ℋ with respect to K is, for each point p ∈ ℋ, a linear map b: T_{p}ℋ/K → T_{p}ℋ/K defined by b(\overline{X}) = \overline{∇_{X}K}. It is easily verified that b is well-defined and, as it involves taking a derivative of K, which is C^{1} the tensor b will be C^{0} but no more regularity can be expected. Note if K̃ = fK, f ∈ C^{1}(ℋ), is any other future directed null vector field tangent to ℋ, then ∇_{X} K̃ = f∇_{X}K mod K. Thus b_{fK} = fb_{K}. It follows that the Weingarten map b of ℋ is unique up to positive scale factor and that b at a given point p ∈ ℋ depends only on the value of K at p when we keep ℋ fixed but allow K to vary while remaining tangent to the generators of ℋ.

A standard computation shows, h(b(\overline{X}), \overline{Y}) = ⟨∇_{X}K, Y ⟩ = ⟨X, ∇_{Y} K⟩ = h(\overline{X}, b(\overline{Y})). Hence b is self-adjoint with respect to h. The null second fundamental form B = B_{K} of ℋ with respect to K is the bilinear form associated to b via h: For each p ∈ ℋ, B: T_{p}ℋ/K × T_{p}ℋ/K → ℝ is defined by B(\overline{X}, \overline{Y}) = h(b(\overline{X}), \overline{Y}) = ⟨∇_{X}K, Y ⟩. Since b is self-adjoint, B is symmetric. In a manner analogous to the second fundamental form for spacelike hypersurfaces, a null hypersurface is totally geodesic if and only if B vanishes identically [51, Theorem 30].

The null mean curvature of ℋ with respect to K is the continuous scalar field θ ∈ C^{0}(ℋ) defined by θ = tr b; in the general relativity literature θ is often referred to as the convergence or divergence of the horizon. Let e_{1}, e_{2}, …, e_{n−1} be n − 1 orthonormal spacelike vectors (with respect to ⟨, ⟩) tangent to ℋ at p. Then \{\overline{e}_{1}, \overline{e}_{2}, …, \overline{e}_{n−1}\} is an orthonormal basis (with respect to h) of T_{p}ℋ/K. Hence at p,

θ = tr b = ∑_{i=1}^{n−1} h(b(\overline{e}_{i}), \overline{e}_{i}) = ∑_{i=1}^{n−1}⟨∇_{e_{i}}K, e_{i}⟩.  (A.1)

Let Σ be the intersection, transverse to K, of a hypersurface in M with ℋ. Then Σ is a C^{2} (n − 1) dimensional spacelike submanifold of M contained in ℋ which meets K orthogonally. From Equation (A.1), θ|_{Σ} = div_{Σ}K, and hence the null mean curvature gives a measure of the divergence of the null generators of ℋ. Note that if K̃ = fK then θ̃ = fθ. Thus the null mean curvature inequalities θ ≥ 0, θ ≤ 0, are invariant under positive scaling of K. In Minkowski space, a future null cone ℋ = ∂I^{+}(p) \ {p} (respectively, past null cone ℋ = ∂I^{−}(p) \ {p}) has positive null mean curvature, θ > 0 (respectively, negative null mean curvature, θ < 0).

The null second fundamental form of a null hypersurface obeys a well-defined comparison theory roughly similar to the comparison theory satisfied by the second fundamental forms of a family of parallel spacelike hypersurfaces (cf. Eschenburg [21], which we follow in spirit).

Let η: (a, b) → M, s → η(s), be a future directed affinely parameterized null geodesic generator of ℋ. For each s ∈ (a, b), let

b(s) = b_{η′(s)} : T_{η(s)}ℋ/η′(s) → T_{η(s)}ℋ/η′(s)

be the Weingarten map based at η(s) with respect to the null vector K = η′(s). Recall that the null Weingarten map b of a smooth null hypersurface ℋ satisfies a Ricatti equation (cf. [5, p. 431]; for completeness we indicate the proof below).

b′ + b^{2} + R = 0.  (A.2)

Here ′ denotes covariant differentiation in the direction η′(s), with η – an affinely parameterized null geodesic generator of ℋ; more precisely, if X = X(s) is a vector field along η tangent to ℋ, then^{23}

b′(\overline{X}) = b(\overline{X})′ − b(\overline{X′}).  (A.3)

23Here b(\overline{X}) is an equivalence class of vectors, so it might be useful to give a practical prescription how its derivative b(\overline{X})′ can be calculated. Let s → c(s) be a null generator of ℋ.

Let s → V (s) be a T ℋ/K–vector field along c, i.e., for each s, V (s) is an element of T_{c(s)}ℋ/K.

Say s → V (s) is smooth if (at least locally) there is a smooth — in the usual sense — vector field s → Y (s) along c such that V (s) = Ȳ (s) for each s. Then define the covariant derivative of s → V (s) along c by: V′(s) = \overline{Y′(s)}, where Y′ is the usual covariant derivative. It is easily shown, using the fact that ∇_{K}K is proportional to K, that V′ so defined is independent of the choice of Y. This definition applies in particular to b(X̄).

Finally R: T_{η(s)}ℋ/η′(s) → T_{η(s)}ℋ/η′(s) is the curvature endomorphism defined by R(\overline{X}) = \overline{R(X, η′(s))η′(s)}, where (X, Y, Z) → R(X, Y)Z is the Riemann curvature tensor of M (in our conventions, R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z).

We indicate the proof of Equation (A.2). Fix a point p = η(s_{0}), s_{0} ∈ (a, b), on η. On a neighborhood U of p in ℋ we can scale the null vector field K so that K is a geodesic vector field, ∇_{K}K = 0, and so that K, restricted to η, is the velocity vector field to η, i.e., for each s near s_{0}, K_{η(s)} = η′(s). Let X ∈ T_{p}M. Shrinking U if necessary, we can extend X to a smooth vector field on U so that [X, K] = ∇_{X}K − ∇_{K}X = 0. Then, R(X, K)K = ∇_{X}∇_{K}K − ∇_{K}∇_{X}K − ∇_{[X,K]}K = −∇_{K}∇_{K}X. Hence along η we have, X′′ = −R(X, η′)η′ (which implies that X, restricted to η, is a Jacobi field along η). Thus, from Equation (A.3), at the point p we have,

b′(\overline{X}) = \overline{∇_{X}K}′ − b(\overline{∇_{K}X}) = \overline{∇_{K}X}′ − b(\overline{∇_{X}K})
= \overline{X′′} − b(b(\overline{X})) = −\overline{R(X, η′)η′} − b^{2}(\overline{X})
= −R(\overline{X}) − b^{2}(\overline{X}),  (A.4)

which establishes Equation (A.2).

Equation (A.2) leads to the well known Raychaudhuri equation for an irrotational null geodesic congruence in general relativity: by taking the trace of (A.2) we obtain the following formula for the derivative of the null mean curvature θ = θ(s) along η,

θ′ = −Ric(η′, η′) − σ^{2} − \frac{1}{n − 2}θ^{2},  (A.5)

where σ, the shear scalar, is the trace of the square of the trace free part of b. This equation shows how the Ricci curvature of spacetime influences the null mean curvature of a null hypersurface. We note the following:

Proposition A.1 Let ℋ be a C^{2} null hypersurface in the (n + 1) dimensional spacetime (M, g) and let b be the one parameter family of Weingarten maps along an affine parameterized null generator η. Then the covariant derivative b′ defined by Equation (A.3) exists and satisfies Equation (A.2).

Remark A.2 When ℋ is smooth this is a standard result, proved by the calculation (A.4). However when ℋ is only C^{2} all we know is that b is a C^{0} tensor field so that there is no reason a priori that the derivative b′ should exist. A main point of the proposition is that it does exist and satisfies the expected differential equation. As the function s ↦ R_{η(s)} is C^{∞} then the Riccati equation implies that actually the dependence of b_{η(s)} on s is C^{∞}. This will be clear from the proof below for other reasons.

Proof: Let η: (a, b) → ℋ be an affinely parameterized null generator of ℋ. To simplify notation we assume that 0 ∈ (a, b) and choose a C^{∞} spacelike hypersurface Σ of M that passes through p = η(0) and let N = ℋ ∩ Σ. Then N is a C^{2} hypersurface in Σ. Now let Ñ be a C^{∞} hypersurface in Σ so that Ñ has second order contact with N at p. Let K̃ be a smooth null normal vector field along Ñ such that at p, K̃ = η′(0). Consider the hypersurface ℋ̃ obtained by exponentiating normally along Ñ in the direction K̃; by Lemma 4.15 there are no focal points along η as long as η stays on ℋ. Passing to a subset of Ñ if necessary to avoid cut points, ℋ̃ will then be a C^{∞} null hypersurface in a neighborhood of η. Let B(s) and B̃(s) be the null second fundamental forms of ℋ and ℋ̃, respectively, at η(s) in the direction η′(s). We claim that B̃(s) = B(s) for all s ∈ (a, b). Since the null Weingarten maps b̃ = b̃(s) associated to B̃ = B̃(s) satisfy Equation (A.2), this is sufficient to establish the lemma.

We first show that B̃(s) = B(s) for all s ∈ [0, c] for some c ∈ (0, b). By restricting to a suitable neighborhood of p we can assume without loss of generality that M is globally hyperbolic. Let X ∈ T_{p}Σ be the projection of η′(0) ∈ T_{p}M onto T_{p}Σ. By an arbitrarily small second order deformation of Ñ ⊂ Σ (depending on a parameter ϵ in a fashion similar to Equation (4.4)) we obtain a C^{∞} hypersurface Ñ_{ϵ}^{+} in Σ which meets N only in the point p and lies to the side of N into which X points. Similarly, we obtain a C^{∞} hypersurface Ñ_{ϵ}^{−} in Σ which meets N only in the point p and lies to the side of N into which −X points. Let K̃_{ϵ}^{±} be a smooth null normal vector field along Ñ_{ϵ}^{±} which agrees with η′(0) at p. By exponentiating normally along Ñ_{ϵ}^{±} in the direction K̃_{ϵ}^{±} we obtain, as before, in a neighborhood of η|_{[0,c]} a C^{∞} null hypersurface ℋ̃_{ϵ}^{±}, for some c ∈ (0, b). Let B̃_{ϵ}^{±}(s) be the null second fundamental form of ℋ̃_{ϵ}^{±} at η(s) in the direction η′(s). By restricting the size of Σ if necessary we find open sets W, W_{ϵ}^{±} in Σ, with W_{ϵ}^{−} ⊂ W ⊂ W_{ϵ}^{+}, such that N ⊂ ∂_{Σ}W and Ñ_{ϵ}^{±} ⊂ ∂_{Σ}W_{ϵ}^{±}. Restricting to a sufficiently small neighborhood of η|_{[0,c]}, we have ℋ ∩ J^{+}(Σ) ⊂ ∂J^{+}(W) and ℋ̃_{ϵ}^{±} ∩ J^{+}(Σ) ⊂ ∂J^{+}(W_{ϵ}^{±}). Since J^{+}(\overline{W_{ϵ}^{−}}) ⊂ J^{+}(\overline{W}) ⊂ J^{+}(\overline{W_{ϵ}^{+}}), it follows that ℋ̃_{ϵ}^{−} is to the future of ℋ near η(s) and ℋ is to the future of ℋ̃_{ϵ}^{+} near η(s), s ∈ [0, c]. Now if two null hypersurfaces ℋ_{1} and ℋ_{2} are tangent at a point p, and ℋ_{2} is to the future of ℋ_{1}, then the difference of the null second fundamental forms B_{2} − B_{1} is positive semidefinite at p. We thus obtain B̃_{ϵ}^{−}(s) ≥ B(s) ≥ B̃_{ϵ}^{+}(s). Letting ϵ → 0, (i.e., letting the deformations go to zero), we obtain B̃(s) = B(s) for all s ∈ [0, c]. A straightforward continuation argument implies, in fact, that B̃(s) = B(s) for all s ∈ [0, b). A similar argument establishes equality for s ∈ (a, 0]. □

In the last result above the hypersurface ℋ had to be of at least C^{2} differentiability class. Now, in our applications we have to consider hypersurfaces ℋ obtained as a collection of null geodesics normal to a C^{2} surface. A naive inspection of the problem at hand shows that such ℋ’s could in principle be of C^{1} differentiability only. Let us show that one does indeed have C^{2} differentiability of the resulting hypersurface:

Proposition A.3 Consider a C^{k+1} spacelike submanifold N ⊂ M of co– dimension two in an (n + 1) dimensional spacetime (M, g), with k ≥ 1. Let k be a non-vanishing C^{k} null vector field along N, and let U ⊆ ℝ × N → M be the set of points where the function

f(t, p) := exp_{p}(tk(p))

is defined. If f_{(t_{0},p_{0})∗} is injective then there is an open neighborhood O of (t_{0}, p_{0}) so that the image f[O] is a C^{k+1} embedded hypersurface in M.

Remark A.4 In our application we only need the case k = 1. This result is somewhat surprising as the function p ↦ k(p) used in the definition of f is only C^{k}. We emphasize that we are not assuming that f is injective. We note that f will not be of C^{k+1} differentiability class in general, which can be seen as follows: Let t → r(t) be a C^{k+1} curve in the x-y plane of Minkowski 3-space which is not of C^{k+2} differentiability class. Let t → n(t) be the spacelike unit normal field along the curve in the x-y plane, then t → n(t) is C^{k} and is not C^{k+1}. Let T = (0, 0, 1) be the unit normal to the x-y plane. Then K(t) = n(t) + T is a C^{k} normal null field along t → r(t). The normal exponential map f: R^{2} → ℝ^{3} in the direction K is given by f(s, t) = r(t) + s[n(t) + T], and hence df/dt = r′(t) + sn′(t), showing explicitly that the regularity of f can be no greater than the regularity of n(t), and hence no greater than the regularity of r′(t).

Proof: This result is local in N about p_{0} so there is no loss of generality, by possibly replacing N by a neighborhood of p_{0} in N, in assuming that N is a embedded submanifold of M. The map f is of class C^{k} and the derivative f_{(t_{0},p_{0})∗} is injective so the implicit function theorem implies f[U] is a C^{k} hypersurface near f(t_{0}, p_{0}). Let η be any nonzero timelike C^{∞} vector field on M defined near p_{0} (some restrictions to be put on η shortly) and let Φ_{s} be the flow of η. Then for sufficiently small ε the map f̃: (−ε, ε) × N → M given by

f̃(s, p) := Φ_{s}(p)

is injective and of class C^{k+1}. Extend k to any C^{k} vector field k̃ along f̃. (It is not assumed that the extension k̃ is null.) That is k̃: (−ε, ε) × N → T M is a C^{k} map and k̃(s, p) ∈ T_{f̃(s,p)}M. Note that we can choose k̃(s, p) so that the covariant derivative \frac{∇k̃}{∂s}(0, p_{0}) has any value we wish at the one point (0, p_{0}). Define a map F: (t_{0} − ε, t_{0} + ε) × (−ε, ε) × N → M by

F (t, s, p) = exp(tk̃(s, p)).

We now show that F can be chosen to be a local diffeomorphism near (t_{0}, 0, p_{0}). Note that F (t, 0, p) = f(t, p) and by assumption f_{∗(t_{0},p_{0})} is injective. Therefore the restriction of F_{∗(t_{0},0,p_{0})} to T_{(t_{0},p_{0})}(ℝ × N) ⊂ T_{(t_{0},0,p_{0})}(ℝ × ℝ × N) is injective.

Thus by the inverse function theorem it is enough to show that F_{∗(t_{0},0,p_{0})}(∂/∂s) is linearly independent of the subspace F_{∗(t_{0},0,p_{0})}[T_{(t_{0},p_{0})}(ℝ × N)]. Let

V (t) = \frac{∂F}{∂s} (t, s, p_{0})|_{s=0} .

Then V (t_{0}) = F_{∗(t_{0},0,p_{0})}(∂/∂s) and our claim that F is a local diffeomorphism follows if V (t_{0}) ∉ F_{∗(t_{0},0,p_{0})}[T_{(t_{0},p_{0})}(ℝ × N)]. For each s, p the map t ↦ F (s, t, p) is a geodesic and therefore V is a Jacobi field along t ↦ F (0, t, p_{0}). (Those geodesics might change type as s is varied at fixed p_{0}, but this is irrelevant for our purposes.) The initial conditions of this geodesic are

V (0) = \frac{∂}{∂s}F (0, s, p_{0})|_{s=0} = \frac{∂}{∂s}Φ_{s}(p_{0})|_{s=0} = η(p_{0})

and

\frac{∇V}{∂t} (0) = \frac{∇}{∂t} \frac{∇}{∂s}F (t, s, p_{0})|_{s=0,t=0} = \frac{∇}{∂s} \frac{∇}{∂t}F (t, s, p_{0})|_{s=0,t=0} = \frac{∇k̃}{∂s}(0, p_{0}) .

¿From our set up we can choose η(p_{0}) to be any timelike vector and \frac{∇k̃}{∂s}(0, p_{0}) to be any vector. As the linear map from T_{p_{0}}M × T_{p_{0}}M → T_{f(t_{0},p_{0})}M which maps the initial conditions V (0), \frac{∇V}{∂t} (0) of a Jacobi field V to its value V (t_{0}) is surjective^{24} it is an open map. Therefore we can choose η(p_{0}) and \frac{∇k̃}{∂s}(0, p_{0}) so that V (t_{0}) is not in the nowhere dense set F_{∗(t_{0},0,p_{0})}[T_{(t_{0},p_{0})}(ℝ × N)]. Thus we can assume F is a local C^{k} diffeomorphism on some small neighborhood A of (t_{0}, 0, p_{0}) onto a small neighborhood B := F [A] of F (t_{0}, 0, p_{0}) as claimed.

Consider the vector field F_{∗}(∂/∂t) = ∂F/∂t along F. Then the integral curves of this vector field are the geodesics t ↦ F (t, s, p) = exp(tk̃(s, p)). (This is true even when F is not injective on its entire domain.) These geodesics and their velocity vectors depend smoothly on the initial data. In the case at hand the initial data is C^{k} so ∂F/∂t is a C^{k} vector field along F. Therefore the one form α defined by α(X) := ⟨X, ∂F/∂t⟩ on the neighborhood B of q_{0} is C^{k}. The definition of F implies that f(t, p) = F (t, 0, p) and therefore the vector field ∂F/∂t is tangent to f[O] and the null geodesics t ↦ f(t, p) = F (t, 0, p) rule f[O] so that f[O] is a null hypersurface. Therefore for any vector X tangent to f[O] we have α(X) = ⟨X, ∂F/∂t⟩ = 0. Thus f[O] is an integral submanifold for the distribution {X | α(X) = 0} defined by α. But, as is easily seen by writing out the definitions in local coordinates, an integral submanifold of a C^{k} distribution is a C^{k+1} submanifold. (Note that in general there is no reason to believe that the distribution defined by α is integrable. However, we have shown directly that f[O] is an integral submanifold of that distribution.) □

24If v ∈ T_{f(t_{0},p_{0})}N there is a Jacobi field with V (t_{0}) = v and \frac{∇V}{∂t} (t_{0}) = 0, which implies subjectivity.

We shall close this appendix with a calculation, needed in the main body of the paper, concerning Jacobians. Let us start by recalling the definition of the Jacobian needed in our context. Let ϕ: M → N be a C^{1} map between Riemannian manifolds, with dim M ≤ dim N. Let n = dim M and let e_{1}, …, e_{n} be an orthonormal basic of T_{p}M then the Jacobian of ϕ at p is J(ϕ)(p) = ∥ϕ_{∗p}e_{1} ∧ ϕ_{∗p}e_{2} ∧ ⋯ ∧ ϕ_{∗p}e_{n}∥. When dim M = dim N and both M and N are oriented with ω_{M} being the volume form on M, and ω_{N} being the volume form on N, then J(ϕ) can also be described as the positive scalar satisfying: ϕ^{∗}(ω_{N}) = ±J(ϕ) ω_{M}.

Let S be a C^{2} co-dimension two acausal spacelike submanifold of a smooth spacetime M, and let K be a past directed C^{1} null vector field along S. Consider the normal exponential map in the direction K, Φ: ℝ × S → M, defined by Φ(s, x) = exp_{x} sK. (Φ need not be defined on all of ℝ × S.) Suppose the null geodesic η: s → Φ(s, p) meets a given acausal spacelike hypersurface Σ at η(1). Then there is a neighborhood W of p in S such that each geodesic s → Φ(s, x), x ∈ W meets Σ, and so determines a C^{1} map ϕ: W → Σ, which is the projection into Σ along these geodesics. Let J(ϕ) denote the Jacobian determinant of ϕ at p. J(ϕ) may be computed as follows. Let {X_{1}, X_{2}, …, X_{k}} be an orthonormal basis for the tangent space T_{p}S. Then,

J(ϕ) = ∥ϕ_{∗p}X_{1} ∧ ϕ_{∗p}X_{2} ∧ ⋯ ∧ ϕ_{∗p}X_{k}∥.

Suppose there are no focal points to S along η|_{[0,1]}. Then by shrinking W and rescaling K if necessary, Φ: [0, 1] × W → M is a C^{1} embedded null hypersurface N such that Φ({1} × W) ⊂ Σ. Extend K to be the C^{1} past directed null vector field, K = Φ_{∗}(\frac{∂}{∂s}) on N. Let θ = θ(s) be the null mean curvature of N with respect to −K along η. For completeness let us give a proof of the following, well known result:

Proposition A.5 With θ = θ(s) as described above,

1. If there are no focal points to S along η|_{[0,1]}, then

J(ϕ) = exp ( − ∫_{0}^{1} θ(s)ds) .  (A.6)

2. If η(1) is the first focal point to S along η|_{[0,1]}, then

J(ϕ) = 0 .

Remark A.6 In particular, if N has nonnegative null mean curvature with respect to the future pointing null normal, i.e., if θ ≥ 0, we obtain that J(ϕ) ≤ 1.

Remark A.7 Recall that θ was only defined when a normalization of K has been chosen. We stress that in (A.6) that normalization is so that K is tangent to an affinely parameterized geodesic, with s being an affine distance along η, and with p corresponding to s = 0 and ϕ(p) corresponding to s = 1.

Proof: 1. To relate J(ϕ) to the null mean curvature of N, extend the orthonormal basis {X_{1}, X_{2}, …, X_{k}} to Lie parallel vector fields s → X_{i}(s), i = 1, …, k, along η, L_{K}X_{i} = 0 along η. Then by a standard computation,

J(ϕ) = ∥ϕ_{∗p}X_{1} ∧ ϕ_{∗p}X_{2} ∧ ⋯ ∧ ϕ_{∗p}X_{k}∥
= ∥X_{1}(1) ∧ X_{2}(1) ∧ ⋯ ∧ X_{k}(1)∥
= \sqrt{g} |_{s=1} ,

where g = det[g_{ij}], and g_{ij} = g_{ij}(s) = ⟨X_{i}(s), X_{j}(s)⟩. We claim that along η,

θ = −\frac{1}{\sqrt{g}} \frac{d}{ds}\sqrt{g} .

The computation is standard. Set b_{ij} = B(\overline{X}_{i}, \overline{X}_{j}), where B is the null second fundamental form of N with respect to −K, h_{ij} = h(\overline{X}_{i}, \overline{X}_{j}) = g_{ij}, and let g^{ij} be the i, j’th entry of the inverse matrix [g_{ij}]^{−1}. Then θ = g^{ij}b_{ij}. Differentiating g_{ij} along η we obtain,

\frac{d}{ds}g_{ij} = K⟨X_{i}, X_{j}⟩ = ⟨∇_{K}X_{i}, X_{j}⟩ + ⟨X_{i}, ∇_{K}X_{j}⟩
= ⟨∇_{X_{i}}K, X_{j}⟩ + ⟨X_{i}, ∇_{X_{j}}K⟩
= −(b_{ij} + b_{ji}) = −2b_{ij} .

Thus,

θ = g^{ij}b_{ij} = −\frac{1}{2}g^{ij} \frac{d}{ds}g_{ij} = −\frac{1}{2} \frac{1}{g} \frac{dg}{ds} = −\frac{1}{\sqrt{g}} \frac{d}{ds}\sqrt{g} ,

as claimed. Integrating along η from s = 0 to s = 1 we obtain,

J(ϕ) = \sqrt{g} |_{s=1} = \sqrt{g} |_{s=0} · exp ( − ∫_{0}^{1} θ ds) = exp ( − ∫_{0}^{1} θ ds) .

2. Suppose now that η(1) is a focal point to S along η, but that there are no focal points to S along η prior to that. Then we can still construct the C^{1} map Φ: [0, 1] × W → M, with Φ({1} × W) ⊂ Σ, such that Φ is an embedding when restricted to a sufficiently small open set in [0, 1]×W containing [0, 1)×{p}. The vector fields s → X_{i}(s), s ∈ [0, 1), i = 1, .., k, may be constructed as above, and are Jacobi fields along η|_{[0,1)}, which extend smoothly to η(1). Since η(1) is a focal point, the vectors ϕ_{∗}X_{1} = X_{1}(1), …, ϕ_{∗}X_{k} = X_{k}(1) must be linearly dependent, which implies that J(ϕ) = 0. □

B Some comments on the area theorem of Hawking and Ellis

In this appendix we wish to discuss the status of our ℋ–regularity condition with respect to the conformal completions considered by Hawking and Ellis [37] in their treatment of the area theorem. For the convenience of the reader let us recall here the setting of [37]. One of the conditions of the Hawking–Ellis area theorem [37, Proposition 9.2.7, p. 318] is that spacetime (ℳ , g) is weakly asymptotically simple and empty (“WASE”, [37, p. 225]). This means that there exists an open set 𝒰 ⊂ ℳ which is isometric to 𝒰′ ∩ℳ′, where 𝒰′ is a neighborhood of null infinity in an asymptotically simple and empty (ASE) spacetime (ℳ′, g′) [37, p. 222]. It is further assumed that ℳ admits a partial Cauchy surface 𝒮 with respect to which ℳ is future asymptotically predictable ([37], p. 310). This is defined by the requirement that ℐ^{+} is contained in closure of the future domain of dependence D^{+}(𝒮; ℳ) of 𝒮, where the closure is taken in the conformally completed manifold ℳ = ℳ ∪ℐ^{+}∪ℐ^{−}, with both ℐ^{+} and ℐ^{−} being null hypersurfaces. Next, one says that (ℳ , g) is strongly future asymptotically predictable ([37], p. 313) if it is future asymptotically predictable and if J^{+}(𝒮) ∩ J̄^{−}(ℐ^{+}; ℳ ) is contained in D^{+}(𝒮; ℳ). Finally ([37], p. 318), (ℳ , g) is said to be a regular predictable space if (ℳ , g) is strongly future asymptotically predictable and if the following three conditions hold:

(α) 𝒮 ∩ J̄^{−}(ℐ^{+}; ℳ ) is homeomorphic to ℝ^{3}\(an open set with compact closure).

(β) 𝒮 is simply connected.

(γ) the family of hypersurfaces 𝒮 (τ) constructed in [37, Proposition 9.2.3, p. 313] has the property that for sufficiently large τ the sets 𝒮 (τ) ∩ J̄^{−}(ℐ^{+}; ℳ ) are contained in J̄^{+}(ℐ^{−}; ℳ ).

It is then asserted in [37, Proposition 9.2.7, p. 318] that the area theorem holds for regular predictable spaces satisfying the null energy condition.

Now in the proof of [37, Proposition 9.2.1, p. 311] (which is one of the results used in the proof of [37, Proposition 9.2.7, p. 318]) Hawking and Ellis write: “This shows that if 𝒲 is any compact set of 𝒮, every generator of ℐ^{+} leaves J^{+}(𝒲 ; ℳ ).” The justification of this given in [37] is wrong^{25}. If one is willing to

25In the proof of [37, Proposition 9.2.1, p. 311] it is claimed that “... Then 𝒮′ \ 𝒰′ is compact...”. This statement is incorrect in general, as shown by the example (ℳ , g) = (ℳ′, g′) = (ℝ^{4}, diag(−1, +1, +1, +1)), 𝒮 = 𝒮′ = {t = 0}, 𝒰 = 𝒰′ = {t ≠ 0}. This example does not show that the claim is wrong, but that the proof is; we do not know whether the claim in Proposition 9.2.1 is correct as stated under the hypothesis of future asymptotic predictability of (ℳ , g) made there. Let us note that the conditions (α)–(γ) do not seem to be used anywhere in the proof of Proposition 9.2.7 as presented in [37], and it is conceivable that the

K 𝒮  C  C′ ψ ℐ_{ℳ}^{+}  ℐ_{ℳ′}^{+}  ℐ_{ℳ′}^{+} ℐ_{ℳ}^{+} 𝒰  𝒰′ 𝒰  ψ(Ω) Ω  𝒰′

Figure 2: The set Ω ≡ J^{+}(𝒮; ℳ) ∩ ∂𝒰 and its image under ψ.

impose this as a supplementary hypothesis, then this condition can be thought of as the Hawking–Ellis equivalent of our condition of ℋ–regularity of ℐ^{+}. When such a condition is imposed in addition to the hypothesis of strong asymptotic predictability and weak asymptotic simplicity (“WASE”) of (ℳ , g), then the hypotheses of Proposition 4.1 hold, and the conclusions of our version of the area theorem, Theorem 1.1, apply.

An alternative way to guarantee that the hypotheses of Proposition 4.1 will hold in the “future asymptotically predictable WASE” set–up of [37] (for those sets C which lie to the future of 𝒮) is to impose some mild additional conditions on 𝒰 and 𝒮. There are quite a few possibilities, one such set of conditions is as follows: Let ψ : 𝒰 → 𝒰′∩ℳ′ denote the isometry arising in the definition of the WASE spacetime ℳ. First, we require that ψ can be extended by continuity to a continuous map, still denoted by ψ, defined on \overline{𝒰} . Next, suppose there exists a compact set K ⊂ ℳ′ such that,

ψ(J^{+}(𝒮; ℳ) ∩ ∂𝒰 ) ⊂ J^{+}(K; ℳ′) ,  (B.1)

see Figure 2. Let us show that, under the future asymptotically predictable WASE conditions together with (B.1), for every compact set C ⊂ J^{+}(𝒮; ℳ) that meets I^{−}(ℐ^{+}; ℳ ) there exists a future inextendible (in ℳ) null geodesic η ⊂ ∂J^{+}(C; ℳ) starting on C and having future end point on ℐ^{+}. First, we claim that

ψ(J^{+}(C; ℳ) ∩ 𝒰 ) ⊂ J^{+}(K ∪ C′; ℳ′) ,  (B.2)

where C′ = ψ(C ∩ 𝒰̄ ). Indeed, let p ∈ ψ(J^{+}(C; ℳ) ∩ 𝒰 ), therefore there exists a future directed causal curve γ from C to ψ^{−1}(p) ∈ 𝒰 . If γ ⊂ 𝒰 , then p ∈ J^{+}(C′; ℳ′). If not, then γ exits 𝒰 when followed from ψ^{−1}(p) to the past at some point in J^{+}(𝒮; ℳ)∩∂𝒰 , and thus p ∈ ψ(J^{+}(𝒮; ℳ)∩∂𝒰 ) ⊂ J^{+}(K; ℳ′), which establishes (B.2). Since K ∪C′ is compact, by Lemma 4.5 and Proposition 4.13 in [54], each generator of ℐ_{ℳ′}^{+} meets ∂J^{+}(K ∪C′; ℳ ′) exactly once. It follows that, under the natural identification of ℐ_{ℳ}^{+} with ℐ_{ℳ′}^{+} , the criteria for ℋ-regularity discussed in Remark 4.5 are satisfied. Hence, we may apply Proposition 4.8 to obtain the desired null geodesic η.

authors of [37] had in mind some use of those conditions in the proof of Propostion 9.2.1. We have not investigated in detail whether or not the assertion made there can be justified if the supplementary hypothesis that (ℳ , g) is a regular predictable space is made, as the approach we advocate in Section 4.1 allows one to avoid the “WASE” framework altogether.

There exist several other proposals how to modify the WASE conditions of [37] to obtain better control of the space–times at hand [16, 50, 53], but we have not investigated in detail their suitability to the problems considered here.

C Some comments on the area theorems of Królak

Królak has previously extended the definition of a black hole to settings more general, in various ways, than the standard setting considered in Hawking and Ellis [37]. In each of the papers [47–49] Królak obtains an area theorem, under the implicit assumption of piecewise smoothness. It follows from the results presented here that the area theorems of Królak still hold without the supplementary hypothesis of piecewise smoothness, which can be seen as follows. First, in each of the papers [47–49] the event horizon ℋ is defined as the boundary of a certain past set, which implies by [37, Prop. 6.3.1 p. 187] that ℋ is an achronal closed embedded C^{0,1} hypersurface. Moreover, by arguments in [47–49] ℋ is ruled by future inextendible null geodesics and hence, in all the papers [47–49] ℋ is a future horizon as defined here. Now, because in [47, 48] the null generators of ℋ are assumed to be future complete, one can apply Theorem 1.1 to conclude that the area theorem holds, under the explicit assumptions of [47, 48], for the horizons considered there, with no additional regularity conditions.

On the other hand, in [49] completeness of generators is not assumed, instead a regularity condition on the horizon is imposed. Using the notation of [49], we shall say that a horizon H_{𝒯} (as defined in [49]) is weakly regular iff for any point p of H_{𝒯} there is an open neighborhood U of p such that for any compact set K contained in U ∩ \overline{W_{𝒯}} the set J^{+}(K) contains a N^{∞}-TIP from Ŵ_{𝒯}. (The set W_{𝒯} may be thought of as the region outside of the black hole, while Ŵ_{𝒯} represents null infinity.) This differs from the definition in [49, p. 370] in that Królak requires the compact set K to be in U ∩ W_{𝒯} rather than in the somewhat larger set U ∩ \overline{W_{𝒯}}. ^{26} Under this slightly modified regularity condition, the arguments of the proof of [49, Theorem 5.2] yield positivity of θ_{Al}: the deformation of the set 𝒯 needed in that proof in [49] is obtained using our sets S_{ϵ,η,δ} from Proposition 4.1. Our Theorem 6.1 then implies that area monotonicity holds for the horizons considered in [49], subject to the minor change of the notion of weak regularity discussed above, with no additional regularity conditions.

26In Królak’s definition K is not allowed to touch the event horizon H_{𝒯}. But then in the proof of [49, Theorem 5.2] when 𝒯 is deformed, it must be moved completely off of the horizon and into W_{𝒯}. So the deformed 𝒯 will, in general, have a boundary in W_{𝒯}. The generator γ in the proof of [49, Theorem 5.2] may then meet 𝒯 at a boundary point, which introduces difficulties in the focusing argument used in the proof. The definition used here avoids this problem.

D Proof of Theorem 5.6

For q ∈ S_{0} let Γ_{q} ⊂ ℋ denote the generator of ℋ passing through q. Throughout this proof all curves will be parameterized by signed σ–distance from S_{0}, with the distance being negative to the past of S_{0} and positive to the future. We will need the following Lemma:

Lemma D.1 S_{0} is a Borel subset of 𝒮, in particular S_{0} is ℌ_{σ}^{n−1} measurable.

Proof: For each δ > 0 let

A_{δ} : = {p ∈ S_{0} | the domain of definition of Γ_{p}
contains the interval [−δ, δ]} .  (D.1)
B_{δ} : = {p ∈ S_{0} | the domain of definition of the inextendible geodesic
γ_{p} containing Γ_{p} contains the interval [−δ, δ]} . (D.2)

Lower semi–continuity of existence time of geodesics shows that the B_{δ}’s are open subsets of 𝒮. Clearly A_{δ′} ⊂ B_{δ} for δ′ ≥ δ. We claim that for δ′ ≥ δ the sets A_{δ′} are closed subsets of the B_{δ}’s. Indeed, let q_{i} ∈ A_{δ′} ∩ B_{δ} be a sequence such that q_{i} → q_{∞} ∈ B_{δ}. Since the generators of ℋ never leave ℋ to the future, and since q_{∞} ∈ B_{δ}, it immediately follows that the domain of definition of Γ_{q_{∞}} contains the interval [0, δ]. Suppose, for contradiction, that Γ_{q_{∞}}(s_{−}) is an endpoint on ℋ with s_{−} ∈ (−δ, 0], hence there exists s′ ∈ (−δ, 0] such that γ_{q_{∞}}(s′) ∈ I^{−}(ℋ). As q_{∞} is an interior point of Γ_{q_{∞}}, the σ–unit tangents to the Γ_{q_{i}}’s at q_{i} converge to the σ–unit tangent to Γ_{q_{∞}} at q_{∞}. Now I^{−}(ℋ) is open, and continuous dependence of ODE’s upon initial data shows that γ_{q_{i}}(s′) ∈ I^{−}(ℋ) for i large enough, contradicting the fact that q_{i} ∈ A_{δ′} with δ′ ≥ δ. It follows that q_{∞} ∈ A_{δ′}, and that A_{δ′} is closed in B_{δ}. But a closed subset of an open set is a Borel set, hence A_{δ′} is Borel in 𝒮. Clearly

S_{0} = ∪_{i}A_{1/i} ,

which implies that S_{0} is a Borel subset of 𝒮. The ℌ_{σ}^{n−1}–measurability of S_{0} follows now from [25, p. 293] or [19, p. 147]^{11}. □

Returning to the proof of Theorem 5.6, set

Γ_{q}^{+} = Γ_{q} ∩ J^{+}(q) ,
ℋ_{sing} = ℋ \ ℋ_{Al} ,
Ω = ∪_{q∈S_{0}}Γ_{q}^{+} ,
Ω_{sing} = Ω ∩ ℋ_{sing} .

By definition we have Ω_{sing} ⊂ ℋ_{sing} and completeness of the Hausdorff measure^{27} together with ℌ_{σ}^{n}(ℋ_{sing}) = 0 implies that Ω_{sing} is n–Hausdorff measurable, with

ℌ_{σ}^{n}(Ω_{sing}) = 0 .  (D.3)

Let ϕ: Ω → S_{0} be the map which to a point p ∈ Γ_{q}^{+} assigns q ∈ S_{0}. The arguments of the proofs of Lemmata 6.9 and 6.11 show that ϕ is locally Lipschitz. This, together with Lemma D.1, allows us to use the co–area formula [23, Theorem 3.1] to infer from (D.3) that

0 = ∫_{Ωsing} J(ϕ)dℌ_{σ}^{n} = ∫_{S_{0}} ℌ_{σ}^{1}(Ω_{sing} ∩ ϕ^{−1}(q))dℌ_{σ}^{n−1}(q) ,

where J(ϕ) is the Jacobian of ϕ, cf. [23, p. 423]. Hence

ℌ_{σ}^{1}(Ω_{sing} ∩ ϕ^{−1}(q)) = 0  (D.4)

for almost all q’s in S_{0}. A chase through the definitions shows that (D.4) is equivalent to

ℌ_{σ}^{1}(Γ_{q}^{+} \ ℋ_{Al}) = 0 ,  (D.5)

for almost all q’s in S_{0}. Clearly the set of Alexandrov points of ℋ is dense in Γ_{q}^{+} when (D.5) holds. Theorem 5.1 shows, for such q’s, that all interior points of Γ_{q} are Alexandrov points of ℋ, hence points q satisfying (D.5) are in S_{1}. It follows that S_{1} is ℌ_{σ}^{n−1} measurable, and has full (n − 1)–Hausdorff measure in S_{0}. This establishes our claim about S_{1}. The claim about S_{2} follows now from the inclusion S_{1} ⊂ S_{2}. □

E Proof of Proposition 6.6

Because of the identities

(f(p) + ⟨x − p, a_{p}⟩ − \frac{C}{2} ∥x − p∥^{2}) + \frac{C}{2} ∥x∥^{2}

= (f(p) + \frac{C}{2} ∥p∥^{2}) + ⟨x − p, a_{p} + Cp⟩ ,

and

(f(q) + ⟨x − q, a_{q}⟩ + \frac{C}{2} ∥x − q∥^{2}) + \frac{C}{2} ∥x∥^{2}

= (f(q) + \frac{C}{2} ∥q∥^{2}) + ⟨x − q, a_{q} + Cq⟩ + C∥x − q∥^{2} ,

27A measure is complete iff all sets of outer measure zero are measurable. Hausdorff measure is constructed from an outer measure using Carathéodory’s definition of measurable sets [24, p. 54]. All such measures are complete [24, Theorem 2.1.3 pp. 54–55].

we can replace f by x ↦ f(x) + C∥x∥^{2}/2 and a_{p} by a_{p} + Cp and assume that for all p, x ∈ A we have

f(p) + ⟨x − p, a_{p}⟩ ≤ f(x) ≤ f(p) + ⟨x − p, a_{p}⟩ + C∥x − p∥^{2} ,  (E.1)

and for all p, q ∈ A and x ∈ ℝ^{n}

f(p) + ⟨x − p, a_{p}⟩ ≤ f(q) + ⟨x − q, a_{q}⟩ + C∥x − q∥^{2} .  (E.2)

These inequalities can be given a geometric form that is easier to work with. Let P := {(x, y) ∈ ℝ^{n} × ℝ : y > C∥x∥^{2}}. Then P is an open convex solid paraboloid of ℝ^{n+1}. We will denote the closure of P by \overline{P}. From the identity

f(q) + ⟨x − q, a_{q}⟩ + C∥x − q∥^{2} = (f(q) − (4C)^{−1}∥a_{q}∥^{2}) + C∥x − (q − (2C)^{−1}a_{q})∥^{2}

it follows that the solid open paraboloids {(x, y) ∈ ℝ^{n}×ℝ : y > f(q)+⟨x−q, a_{q}⟩+ C∥x−q∥^{2}} are translates in ℝ^{n+1} of P. Let G[f] := {(x, y) ∈ ℝ^{n}×ℝ : x ∈ A, y = f(x)} be the graph of f. The inequalities (E.1) and (E.2) imply that for each p ∈ A there is an affine hyperplane H_{p} = {(x, y) ∈ ℝ^{n}×ℝ : y = f(p)+⟨x−p, a_{p}⟩} of ℝ^{n+1} and a vector b_{p} ∈ ℝ^{n+1} so that

(p, f(p)) ∈ H_{p},  (E.3)
(P + b_{p}) ∩ G[f] = ∅ but (p, f(p)) ∈ b_{p} + \overline{P} ,  (E.4)

and for all p, q ∈ A

H_{p} ∩ (b_{q} + P) = ∅.  (E.5)

As the paraboloids open up, this last condition implies that each b_{q} +P lies above all the hyperplanes H_{p}.

Let

Q := Convex Hull ( ⋃_{p∈A}(b_{p} + P)).

Because P is convex if α_{1}, …, α_{m} ≥ 0 satisfy ∑_{i=1}^{m} α_{i} = 1 then for any v_{1}, …, v_{m} ∈ ℝ^{n+1}

α_{1}(v_{1} + P) + α_{2}(v_{2} + P) + ⋯ + α_{m}(v_{m} + P) = (α_{1}v_{1} + ⋯ + α_{m}v_{m}) + P.

Therefore if

B := Convex Hull {b_{p} : p ∈ A}

then

Q = ⋃_{v∈B}(v + P)

so that Q is a union of translates of P. Thus Q is open. Because P is open we have that if lim_{ℓ→∞} v_{ℓ} = v then v + P ⊆ ⋃_{ℓ}(v_{ℓ} + P). So if \overline{B} is the closure of B in ℝ^{n+1} then we also have

Q = ⋃_{v∈\overline{B}}(v + P).  (E.6)

For each p the open half space above H_{p} is an open convex set and Q is the convex hull of a subset of this half space. Therefore Q is contained in this half space. Therefore Q ∩ H_{p} = ∅ for all p.

We now claim that for each point z ∈ ∂Q there is a supporting paraboloid for z in the sense that there is a vector v ∈ \overline{B} with v + P ⊂ Q and z ∈ v + \overline{P}. To see this note that as z ∈ ∂Q there is a sequence {b_{ℓ}}_{ℓ=1}^{∞} ⊂ B and {w_{ℓ}}_{ℓ=1}^{∞} ⊂ P so that lim_{ℓ→∞}(b_{ℓ} + w_{ℓ}) = z. Fixing a p_{0} ∈ A and using that all the sets b_{ℓ} + P are above the hyperplane H_{p_{0}} we see that both the sequences b_{ℓ} and w_{ℓ} are bounded subsets of ℝ^{n+1} and by going to a subsequence we can assume that v := lim_{ℓ→∞} b_{ℓ} and w := lim_{ℓ→∞} w_{ℓ} exist. Then z = v + w, v ∈ \overline{B} and w ∈ \overline{P}. Then (E.6) implies v + P ⊂ Q and w ∈ \overline{P} implies z ∈ v + \overline{P}. Thus we have the desired supporting paraboloid.

We also claim that the graph G[f] satisfies G[f] ⊂ ∂Q. This is because for p ∈ A the point (p, f(p)) ∈ H_{p} and H_{p} is disjoint from Q. Thus (p, f(p)) ∉ Q. But from (E.4) (p, f(p)) ∈ b_{p} + \overline{P} and as b_{p} + \overline{P} ⊂ \overline{Q} this implies (p, f(p)) ∈ \overline{Q}. Therefore (p, f(p)) ∈ ∂Q as claimed.

Let F: ℝ^{n} → ℝ be the function that defines ∂Q. Explicitly

F (x) = inf{y : (x, y) ∈ Q}.

This is a function defined on all of ℝ^{n} and G[f] ⊂ ∂Q implies that this function extends f. For any x_{0} ∈ ℝ^{n} the convexity of Q implies there is a supporting hyperplane H for Q at its boundary point (x_{0}, F (x_{0})) and we have seen there also is a supporting paraboloid v + P for ∂Q = G[F] at (x_{0}, F (x_{0})). Expressing H and ∂(v + P) as graphs over ℝ^{n} these geometric facts yield that there is a vector a_{x_{0}} so that the inequalities

F (x_{0}) + ⟨x − x_{0}, a_{x_{0}}⟩ ≤ F (x) ≤ F (x_{0}) + ⟨x − x_{0}, a_{x_{0}}⟩ + C∥x − x_{0}∥^{2}

hold for all x ∈ ℝ^{n}. Because the function F is defined on all of ℝ^{n} (rather than just the subset A), it follows from [9, Prop. 1.1 p. 7] that F is of class C^{1,1}. □

Acknowledgments P.T.C. acknowledges useful discussions with or comments from Guy Barles, Piotr Hajłasz, Tom Ilmanen, Bernd Kirchheim, Andrzej Królak, Olivier Ley and Laurent Véron. R.H. benefited from discussions and/or correspondence with Joe Fu, Sergei Konyagin, and Ron DeVore. We are grateful to Lars Andersson for bibliographical advice. E.D. is grateful fo the KTH, Stockholm, for hospitality during the final stage of work on this project.

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