The Hawking-Penrose singularity theorem for C^{1,1}-Lorentzian metrics
Melanie Graf, James D.E. Grant, Michael Kunzinger, Roland Steinbauer
aa r X i v : . [ m a t h - ph ] J u l The Hawking–Penrose singularity theorem for C , -Lorentzianmetrics Melanie Graf ∗ ,James D.E. Grant † ,Michael Kunzinger ∗ ,Roland Steinbauer ∗ ,July 13, 2018 Abstract
We show that the Hawking–Penrose singularity theorem, and the generalisation ofthis theorem due to Galloway and Senovilla, continue to hold for Lorentzian metricsthat are of C , -regularity. We formulate appropriate weak versions of the strong energycondition and genericity condition for C , -metrics, and of C -trapped submanifolds. Byregularisation, we show that, under these weak conditions, causal geodesics necessarilybecome non-maximising. This requires a detailed analysis of the matrix Riccati equationfor the approximating metrics, which may be of independent interest. Keywords:
Singularity theorems, low regularity, regularisation, causality theory
MSC2010:
The classical singularity theorems of General Relativity show that a Lorentzian manifold thatsatisfies physically “sensible” conditions cannot be geodesically complete. In particular, if oneattempts to “extend” such a manifold, then one cannot extend with a C -Lorentzian metric.It is then natural to ask whether one can extend with a lower regularity Lorentzian metric.In certain situations with a large amount of symmetry, one can show that even a low level ofregularity cannot be maintained. For example, in recent work, Sbierski [32] has shown thatthe Schwarzschild solution cannot be extended as a continuous Lorentzian metric.Generally speaking, the singularity theorems of Penrose [30], Hawking [10] and Hawking–Penrose [12] hold for C -Lorentzian metrics. In [17] and [16], it has been shown, however,that the theorems of Penrose and Hawking hold for metrics that are C , , i.e. metrics that aredifferentiable, with all derivatives locally Lipschitz. Such a level of regularity is of significanceto us for a variety of reasons. From a mathematical point of view, such metrics have thefollowing properties: ∗ University of Vienna, Faculty of Mathematics, [email protected], [email protected],[email protected] † Department of Mathematics, University of Surrey, [email protected] L ∞ loc . In particular, Rademacher’s theoremimplies that the curvature exists almost-everywhere.From the point of view of physics, the curvature of a metric being bounded but discontinuous,rather than blowing up, would, via the Einstein field equations, give rise to (or be generatedby) a finite jump in the energy-momentum tensor of the matter variables. This scenario isquite acceptable physically, and arises in the classical example of the Oppenheimer–Snydersolution [25] and the whole class of matched spacetimes (see e.g. [18, 19]). As such, there areboth physical and mathematical motivations for studying the class of C , -metrics.When one attempts to generalise the proof of the singularity theorems to the case of a C , -metric, however, the fact that the curvature tensor is only defined almost-everywhereposes significant problems. The standard proof of the singularity theorems relies on the existence of conjugate points(or focal points) along suitable classes of geodesics in the Lorentzian manifold. Such conjugatepoints are shown to exist by a study of Jacobi fields (or, equivalently, Riccati equations) alongthese geodesics. However, if the curvature tensor is only defined almost-everywhere, it is quitepossible that, since a geodesic curve has measure zero, the curvature may not be defined alongany given geodesic, so the Jacobi equation (and, hence, the notion of a conjugate point) isnot well-defined along said geodesic. In Riemannian geometry, a standard example of ametric that is C , but not C is the metric on a hemisphere joined at the equator to aflat cylinder [29, 27]. This metric has strictly positive curvature on the hemisphere and zerocurvature on the cylindrical part, which implies that the curvature is not well-defined on thegeodesic that traverses the join between the two regions. A similar phenomenon occurs inLorentzian geometry in the Oppenheimer–Snyder model, where the curvature tensor is notwell-defined along the geodesics that generate the boundary between the interior and exteriorregions of the solution. As such, the notion of a Jacobi field is not defined along such geodesics.The importance of conjugate points (or focal points) in the proof of the singularity theo-rems is the connection with maximising properties of causal geodesics. In particular, a causalgeodesic from a point stops being maximising if and only if either a) there exists a distinctcausal geodesic between the same endpoints of the same length or b) the geodesic encountersa conjugate point. Given suitable geometrical conditions on the Lorentzian metric (e.g. aRicci curvature bound, a “convergence condition” such as the existence of a trapped surface,and a completeness condition), one can use Riccati comparison techniques to show that allcausal geodesics of a suitable type will encounter conjugate points, and hence stop beingmaximising curves between their endpoints. It should perhaps be pointed out, however, thatthe cut-locus of a point in a Lorentzian manifold is necessarily a closed set, of which conjugatepoints form a subset of zero measure. Therefore, almost all geodesics stop maximising dueto their intersection with another geodesic with the same endpoint of the same length. Assuch, most causal geodesics will no longer be maximising even before they encounter their A number of technical obstacles for a proof in the C , -case are listed in Sect. 6.1 of the review article [33],see also [34, Sec. 8.1]. A similar statement holds for causal geodesics emanating from a submanifold of M . global geometry of the manifold, there is no way to estimate (in terms of, say, the curvature) thedistance that one must traverse along a given curve before one encounters such an intersec-tion. The power of conjugate points (and focal points) is the fact that they lead to geodesicsno longer being maximising and we can estimate when they occur.In this paper, we show that the Hawking–Penrose singularity theorem [12] can be gen-eralised to C , -Lorentzian metrics. The Hawking–Penrose theorem is, perhaps, the mostrefined of the classical singularity theorems, in the sense that it requires the most delicateanalysis of the effects of curvature. As a consequence, the technical issues that arise fromthe lack of a suitable concept of a “conjugate point” are considerably more pronounced whenone attempts to generalise the Hawking–Penrose theorem to the C , -setting, than they werewith the Penrose or Hawking theorems. The most general version of the Hawking–Penrosetheorem, which is stated in “causal” language, states the following: Theorem 1.1. [12, pp. 538] Let ( M, g ) be a spacetime with g a C -metric with the followingproperties:(C.i) M is chronological, i.e., contains no closed timelike curves;(C.ii) Every inextendible causal geodesic in M contains conjugate points;(C.iii) There is an achronal set S such that E + ( S ) or E − ( S ) is compact.Then ( M, g ) is causally geodesically incomplete. Hawking and Penrose also prove the following more “analytical” result: Theorem 1.2. [12, Sec. 3, Cor.] A spacetime ( M, g ) with C -metric that(A.1) is chronological;(A.2) satisfies the strong energy condition, Ric(
X, X ) ≥ ∀ causal X ∈ T M ; (1.1) (A.3) satisfies the genericity condition, i.e., along every causal geodesic γ there is a point atwhich ˙ γ c ˙ γ d ˙ γ [ a R b ] cd [ e ˙ γ f ] = 0; (1.2) (A.4) contains at least one of the following(i) a compact achronal set without edge,(ii) a closed trapped surface or(iii) a point p such that on every past (or every future) null geodesic from p the expan-sion θ of the null geodesics from p becomes negative,cannot be causally geodesically complete. In [12], Theorem 1.2 is proved as a Corollary of Theorem 1.1. Since the bulk of this paper is dedicatedto proving the analogue of Theorem 1.2, we will hereafter refer to Theorem 1.2 as the “Hawking–Penrosesingularity theorem”. C -metrics, Theorem 1.2 is proved as a corollary of Theorem 1.1. In particular, thegenericity condition (1.2) along with strong energy condition (1.1) are used, in conjunctionwith a matrix Riccati equation for the second fundamental form of a geodesic congruence,to show that any of the conditions (A.4) imply that every inextendible causal geodesic in M contains conjugate points, and that Condition (C.iii) of Theorem 1.1 holds. Therefore, theconditions of Theorem 1.2 imply those of Theorem 1.1.In the C , -case, which we study in this paper, the logical structure of the argument is verysimilar. We first prove an appropriate version of Theorem 1.1 for C , -metrics. To this end, wefirst note that Condition (C.ii) in Theorem 1.1 explicitly depends on the concept of a conjugatepoint, and so cannot be directly generalised to the case of C , -metrics. However, an inspectionof the proof of the Hawking–Penrose theorem shows that, rather than Condition (C.ii), theproperty that is actually required for their result is the following:(C.ii ′ ) Every inextendible causal geodesic in M stops being maximising;One of our fundamental results is, therefore, Theorem 7.4, which states that, with minormodifications, Theorem 1.1, with Condition (C.ii) replaced with Condition (C.ii ′ ) continuesto hold if the metric g is assumed to be C , . The web of causality results required in theproof of Theorem 1.1, generalised to the C , -setting, is summarised in Appendix A.In the C , -case, however, the step from Theorem 1.1 to Theorem 1.2 is considerably morecomplicated. We show that appropriate versions of the curvature conditions (1.1) and (1.2)lead to causal geodesics becoming non-maximising between their endpoints. We prove thisresult by studying appropriate smooth approximations g ε to the C , -metric g , where the g ε satisfy appropriate weakened versions of (1.1) and (1.2). By a refined analysis of the matrixRiccati equation along geodesics with respect to the g ε -metrics, we are able to show that g ε -causal geodesics develop conjugate points, and, hence, are non-maximising. From this, weargue that g -causal geodesics also become non-maximising. At this point, our main results,Theorem 2.5 and Theorem 2.6 follow from Theorem 7.4.The techniques that we develop in going from Theorem 7.4 to Theorem 2.5 and Theo-rem 2.6 are the main technical developments in this paper. In particular, the estimates thatwe develop in Sections 3 and 4 are new, and may well be of independent interest. We conclude this introduction by fixing our notation and conventions as well as introducingan improved version of the smooth Hawking–Penrose theorem that we will also deal withduring this work.All manifolds will be denoted by M and assumed to be smooth, Hausdorff, second count-able, n -dimensional (with n ≥ M we will consider Lorentzianmetrics g of regularity of at least C , and signature ( − , + . . . , +) with Levi-Civita connection ∇ and with a time orientation fixed by a continuous vector field. We say a curve γ : I → M from some interval I ⊆ R to M is timelike (causal, null, future or past directed) if it islocally Lipschitz and ˙ γ ( t ), which exists almost everywhere by Rademacher’s theorem, is time-like (causal, null, future or past directed) almost everywhere. Following standard notation,for p, q ∈ M we write p ≪ q if there exists a future directed timelike curve from p to q Note that the metrics g ε are smooth, so the classical notion of a conjugate point is well-defined. To the best of our knowledge. In particular, these are not estimates that follow from the standard Rauch comparison theorem for Jacobifields. p ≤ q if there exists a future directed causal curve from p to q or p = q ) and set I + ( A ) := { q ∈ M : p ≪ q for some p ∈ A } and J + ( A ) := { q ∈ M : p ≤ q for some p ∈ A } .We note that we require causal (timelike, . . . ) curves to be Lipschitz, whereas other standardsources use piecewise C curves instead (see, e.g., [11], [24]). However, as was shown in [21,Thm. 7], [15, Cor. 3.10], this has no impact on the relations ≪ and ≤ for C , -metrics. Wecall a C , -spacetime ( M, g ) globally hyperbolic if it is causal (i.e., contains no closed causalcurves) and J ( p, q ) := J + ( p ) ∩ J − ( q ) is compact for all p, q ∈ M . We further define theRiemann curvature tensor by R ( X, Y ) Z = [ ∇ X , ∇ Y ] Z − ∇ [ X,Y ] Z and the Ricci tensor byRic( X, Y ) = P ni =1 h E i , E i ih R ( E i , X ) Y, E i i , which in case of g being C , are L ∞ loc -tensor fields.Here and in the following ( E i ) ni =1 will denote (local) orthonormal frame fields and ( e i ) ni =1 willdenote orthonormal frames in individual tangent spaces T p M . Generally we will considerembedded submanifolds S of codimension m . We define the second fundamental form byII( V, W ) := nor( ∇ V W ) for all V, W tangent to S and the shape operator derived from anormal unit field ν by S ν ( X ) = ∇ X ν . For any tangent vector v ∈ T p M we denote by γ v thegeodesic with γ v (0) = p , ˙ γ v (0) = v . Throughout, a codimension 2 submanifold of M will bereferred to as a “surface”.Condition (A.4)(ii) of Theorem 1.2 has been generalized in [6] to include trapped sub-manifolds of arbitrary co-dimension m (1 < m < n ) by adding an additional curvature as-sumption, which in the classical case m = 2 automatically follows from the energy condition.For a precise formulation let S be a (smooth) spacelike ( n − m )-dimensional submanifoldand let e ( q ) , . . . , e n − m ( q ) be an orthonormal basis for T q S , smoothly varying with q in aneighbourhood (in S ) of p ∈ S . For a geodesic γ starting at p let E , . . . , E n − m denote theparallel translates of e ( p ) , . . . , e n − m ( p ) along γ . Let H S := n − m P n − mi =1 II( e i , e i ) denote themean curvature vector field of S , and let k S ( v ) := g ( H, v ) be the convergence of v ∈ T M | S .Now a closed spacelike submanifold S is called (future) trapped if for any future-directed nullvector ν ∈ T S ⊥ the convergence k S ( ν ) is positive. This is equivalent to the mean curvaturevector field H S being past pointing timelike on all of S . With this definition one has thefollowing extension of the classical Hawking–Penrose theorem ([6, Thm. 3]). Theorem 1.3.
A spacetime ( M, g ) with C -metric satisfying conditions (A.1)–(A.3) of The-orem 1.2 and(A.4) (iv) contains a spacelike (future) trapped submanifold S of co-dimension < m < n such that additionally n − m X i =1 h R ( E i , ˙ γ ) ˙ γ, E i i ≥ for any future directed null geodesic with ˙ γ (0) orthogonal to S ,cannot be causally geodesically complete. In Section 7, this result will also be shown to hold in the C , -setting.This paper is organised in the following way. In Section 2, we first define the appropriateweak notions of curvature conditions on Lorentzian metrics and convergence conditions on C -submanifolds that are required for our study of metrics that are C , . We then state our mainresults, Theorems 2.5 and 2.6, which are the analogues of the Hawking–Penrose Theorem 1.2 Note that we follow the convention of [11] for the curvature tensor, which is the opposite of that employedin [24, 16, 17]. C , -case. The remainder of the paper is concernedwith the proof of these results. In Section 3, we consider the regularisation of the C , -metricand, in particular, study the effect of smoothing on the curvature and genericity condition.In Section 4, we develop estimates for matrix Riccati equations that allow us to show thatgeodesics with respect to the smooth approximating metrics must develop conjugate (or focal)points. As mentioned previously, the estimates obtained in Sections 4 are, perhaps, the maintechnical advance in this paper, and may be of independent interest in their own right. Theresults of Section 4 are used in Section 5 to yield Theorems 5.1 and 5.3, which show that, underour curvature and genericity assumptions, causal geodesics will not remain maximising. InSection 6, we show that if S is a submanifold of M satisfying any one of the conditions (A.4) ofTheorems 2.5 and 2.6, then E + ( S ) is compact, i.e., the submanifold is a trapped set. Finally,in Section 7, we first show, using results summarised in Appendix A, that Theorem 7.4, theanalogue of the “causal” version of the Hawking–Penrose Theorem (Theorem 1.1), holds in the C , -setting. The results from Sections 3–6 then quickly yield the main result Theorems 2.5and 2.6, i.e. the “analytical” version of the Hawking–Penrose theorem. The aim of this paper is to generalise Theorems 1.2 and 1.3 to C , -metrics. Since not all ofthe conditions in these theorems are well-defined at this lower level of regularity, we begin bydiscussing the alternative formulations that we will use in the C , - case.By the strong energy condition or causal convergence condition , we shall mean thatRic( X, X ) ≥ X . (2.1)We will also speak of the timelike (or null) convergence condition if (2.1) is only supposed tohold for all Lipschitz continuous timelike (or null) local vector fields X . Remark . This condition is natural in the C , -context and has been successfully usedin the proofs of other singularity theorems in this regularity (cf. [16, Rem. 1.2(i)] and [17,Rem. 1.2(i)]). Note that the Lipschitz condition is only relevant in the null case. Contraryto the situation with a timelike vector, which can clearly be extended to a smooth timelikelocal vector field, it is, in general, not possible to extend a given null vector to a smooth nulllocal vector field. Indeed, parallel transporting a given null vector at a given point alongradial geodesics emanating from that point results in a null vector field that is only Lipschitzcontinuous. It is possible that, with a C , -Lorentzian metric, one can extend a given nullvector to a C , null local vector field, and the condition for our results may be weakenedto requiring (2.1) to hold for all C , causal local vector fields X . However, since we willexplicitly use a null vector field obtained by parallel transport (and, hence, Lipschitz) in theproof of Lemma 3.6, we have not investigated this possibility. For simplicity, we also refrainfrom refining condition (2.1) to apply to local smooth timelike and Lipschitz null vector fields,although this would be possible throughout.Looking at the classical proof of Theorem 1.2, one finds that it is not the genericitycondition itself that plays a role, but rather a derived condition on the tidal force operatoralong causal geodesics γ . The required condition is that there exists t such that the operator R : ( ˙ γ ( t )) ⊥ → ( ˙ γ ( t )) ⊥ , v R ( v, ˙ γ ) ˙ γ (2.2)6s not identically zero. (The fact that this condition follows from the genericity condition (1.2)can be found in, e.g., [13, Cor. 9.1.1].) Thus, we will henceforth refer to (2.2) as the genericitycondition, which we now formulate for C , -metrics, and which reproduces (2.2) in the smoothcase, as we shall see below (Lemma 3.5). Definition 2.2.
Let g ∈ C , be a Lorentzian metric on M , and let γ : I → M be a causalgeodesic for g . Then we say that the genericity condition holds along γ if there exists some t ∈ I and a neighbourhood U of γ ( t ), as well as continuous vector fields X and V on U such that X ( γ ( t )) = ˙ γ ( t ) and V ( γ ( t )) ∈ ( ˙ γ ( t )) ⊥ for all t ∈ I with γ ( t ) ∈ U , and there existssome c > h R ( V, X ) X, V i > c (2.3)in L ∞ ( U ). In this case, we say that the genericity condition is satisfied for γ at t ∈ I .Regarding the initial conditions (A.4), we first remark that the definition of an “achronalset without edge” and of a “smooth (or at least C -) future trapped submanifold” for C , -metrics can be carried across unchanged from the smooth case since the mean curvature isstill Lipschitz continuous. We will however wish to generalise the notion of a future trappedsubmanifold slightly to allow us to use C -submanifolds. We say that a(n at least C ) sub-manifold e S is a future support submanifold for a C -submanifold S at q ∈ S if dim( e S ) = dim S , q ∈ e S , and e S is locally to the future of S near q , i.e. there exists a neighbourhood U of q in M such that e S ∩ U ⊂ J + ( S, U ). Using such future support submanifolds we define past pointingtimelike mean curvature at q ∈ S by requiring the existence of a future support submanifoldwith past-pointing timelike mean curvature at q (see, for instance, [1]).This leads to the following definition of a future trapped submanifold of M (which reducesto the usual one if S is at least C ). Definition 2.3.
A closed ( C -) submanifold S of codimension m (1 ≤ m < n ) is called futuretrapped if, for any p ∈ S , there exists a neighbourhood U p of p such that S ∩ U p is achronalin U p and S has past-pointing timelike mean curvature at all of its points (in the sense ofsupport submanifolds).Similarly, to replace the point condition (A.4)(iii) in Theorem 1.2, we define a (future)trapped point as follows: Definition 2.4.
We say that a point p is future trapped if, for any future-pointing null vector ν ∈ T p M , there exists a t such that there exists a spacelike C -surface e S ⊂ J + ( p ) with γ ν ( t ) ∈ e S and k e S ( ˙ γ ν ( t )) > θ ( t ) along a geodesic γ defined in terms of Jacobi tensorclasses (cf. Lemma 4.1) and the shape operator S ˙ γ ( t ) derived from ˙ γ for the submanifold S t := exp p ( t V ), where V is the set of all (properly normalised) null vectors contained insome neighbourhood of ˙ γ (0) (see section 6.3 for details). Our definition then provides a C , -generalisation of the trace of such a shape operator becoming negative.With these definitions we will prove the following generalisation of Theorem 1.2: Theorem 2.5 (Hawking–Penrose for C , -metrics) . Let ( M, g ) be a spacetime with a C , -metric. If M A.1) is causal;(A.2) satisfies the strong energy condition (2.1) ;(A.3) satisfies the genericity condition along any inextendible causal geodesic (Definition 2.2);(A.4) contains at least one of the following(i) a compact achronal set without edge;(ii) a closed future trapped ( C -)surface (Definition 2.3);(iii) a future trapped point (Definition 2.4),then it cannot be causally geodesically complete. Note that the C , -version requires that ( M, g ) be causal rather than chronological since,contrary to the smooth case, the other conditions that we impose do not exclude the existenceof closed null curves. The problem will be evident in the proof of Theorem 5.3, where we willuse approximations to show that no inextendible null geodesic can be globally maximising,and our argument breaks down for closed null curves.Finally, we will also prove a C , -generalization of Theorem 1.3. Theorem 2.6.
Let ( M, g ) be a spacetime with a C , -metric that satisfies conditions (A.1)to (A.3) of Theorem 2.5 and(A.4) (iv) contains a (future) trapped C -submanifold (Definition 2.3) of co-dimension
0. Thusany g -causal vector is timelike for g . Then [4, Prop. 1.2] (cf. also [15, Prop. 2.5]) gives: Proposition 3.1.
Let ( M, g ) be a C -spacetime and let h be some smooth background Rie-mannian metric on M . Then for any ε > , there exist smooth Lorentzian metrics ˇ g ε and g ε on M such that for all < ε < ε ′ , ˇ g ε ′ ≺ ˇ g ε ≺ g ≺ ˆ g ε ≺ ˆ g ε ′ , and d h (ˇ g ε , g ) + d h (ˆ g ε , g ) < ε ,where d h ( g , g ) := sup p ∈ M, = X,Y ∈ T p M | g ( X, Y ) − g ( X, Y ) |k X k h k Y k h . (3.1) Moreover, ˆ g ε ( p ) and ˇ g ε ( p ) depend smoothly on ( ε, p ) ∈ R + × M , and if g ∈ C , then, letting g ε be either ˇ g ε or ˆ g ε , we additionally have(i) g ε converges to g in the C -topology as ε → , and(ii) the second derivatives of g ε are bounded, uniformly in ε , on compact sets. Curvature quantities for g ε -metrics will be denoted by a subscript, as in R ε or Ric ε .Next we recall the consequences of the strong energy condition (2.1) provided by [16,Lemma 3.2] and [17, Lemma 2.4] for nets ( g ε ) ε> (with g ε = ˇ g ε or g ε = ˆ g ε ) of approximatingsmooth metrics. Lemma 3.2.
Let M be a smooth manifold with a C , -Lorentzian metric g and smoothRiemannian background metrics h , ˜ h on M and T M , respectively. Let K ⋐ M and let C , δ > . Then we have:(i) If Ric(
Y, Y ) ≥ for every g -timelike smooth local vector field Y , then ∀ κ < ∃ ε > ∀ ε < ε ∀ X ∈ T M | K with g ( X, X ) ≤ κ and k X k h ≤ C : Ric ε ( X, X ) > − δ. (3.2) (ii) If Ric(
Y, Y ) ≥ for every Lipschitz-continuous g -null local vector field Y , then ∃ η > ∃ ε > ∀ ε < ε : if p ∈ K , X ∈ T p M with k X k h ≤ C and ∃ Y ∈ T M | K , g -null with d ˜ h ( X, Y ) ≤ η and k Y k h ≤ C : (3.3)Ric ε ( X, X ) > − δ. For later use, we also record the following result, cf. e.g. the proof of [16, Prop. 4.3]:
Lemma 3.3.
Let ( M, g ) be a globally hyperbolic C , -spacetime and let p , q ∈ M . Denote by d and d ˇ g ε the time-separation functions with respect to g and ˇ g ε , respectively. Then, we have d ˇ g ε ( p, q ) → d ( p, q ) ( ε → . The following basic Friedrichs-type Lemma collects some general convergence propertiesthat will be used repeatedly in subsequent sections.
Lemma 3.4.
Let a ∈ L ∞ loc ( R n ) , f ∈ C ( R n ) , b ε ∈ C ( R n ) ( ε > ), and b ε → b locallyuniformly for ε → . Let ρ ∈ D ( R n ) be a standard mollifier. Then(i) ( a · f · b ) ∗ ρ ε − ( a ∗ ρ ε ) · ( f ∗ ρ ε ) · b ε → ( ε → ) locally uniformly.(ii) If ρ is non-negative and a · f · b ≥ c ∈ R then ∀ ˜ c < c ∀ K ⋐ R n ∃ ε ∀ ε < ε : ( a ∗ ρ ε ) · ( f ∗ ρ ε ) · b ε > ˜ c on K. roof. (i) We have( a · f · b ) ∗ ρ ε − ( a ∗ ρ ε ) · ( f ∗ ρ ε ) · b ε = ( a · f · b ) ∗ ρ ε − ( a · f ) ∗ ρ ε · b ∗ ρ ε + ( a · f ) ∗ ρ ε · b ∗ ρ ε − ( a ∗ ρ ε ) · ( f ∗ ρ ε ) · b ε . Here, both the first and the second term on the right hand side go to 0 locally uniformly bya variant of the Friedrichs Lemma (cf. the proof of [16, Lemma 3.2]).(ii) Since ( a · f · b ) ∗ ρ ε ≥ c , the claim follows from (i).A convenient consequence of the previous Lemma concerns basic properties of curvaturequantities associated to a C , -metric g : Arguing in a local chart, Lemma 3.4 shows that if g ε is as in Proposition 3.1, then R ε − R ∗ ρ ε → R ∗ ρ ε → R in any L p loc (1 ≤ p < ∞ ), all the usual symmetry properties of theRiemann tensor for smooth metrics carry over to R pointwise a.e.Next we introduce some notation to deal with timelike and null geodesics simultaneously.Suppose that γ is a causal geodesic in a C , -spacetime ( M, g ). As is common in the smoothcase (see e.g. [13, Sec. 4.6.3]) we consider the quotient space [ ˙ γ ( t )] ⊥ := ( ˙ γ ( t )) ⊥ / R ˙ γ ( t ), i.e.vectors v, w ∈ ( ˙ γ ( t )) ⊥ are equivalent if there exists α ∈ R such that v = w + α ˙ γ ( t ). In thecase where γ is null, [ ˙ γ ( t )] ⊥ is an ( n − γ ( t )) ⊥ . When γ is timelike,[ ˙ γ ( t )] ⊥ coincides with ( ˙ γ ( t )) ⊥ . In order to enable a unified notation we will henceforth denotethe dimension of [ ˙ γ ( t )] ⊥ by d , i.e. d = n − d = n − γ ] ⊥ = S t [ ˙ γ ( t )] ⊥ . Every normal tensor field A along γ then induces aunique tensor class [ A ] along γ and the induced covariant derivative ∇ ˙ γ is well-defined fortensor classes and denoted by [ ˙ A ] = [ ∇ ˙ γ A ]. The metric g | [ ˙ γ ] ⊥ is positive definite in both thenull and the timelike case. Also recall that, for smooth metrics, the curvature (or tidal force)operator [ R ]( t ) : [ ˙ γ ( t )] ⊥ → [ ˙ γ ( t )] ⊥ , [ v ] [ R ( v, ˙ γ ( t )) ˙ γ ( t )] is well-defined since R ( ˙ γ, ˙ γ ) ˙ γ = 0.Before we proceed to construct suitable frames for the approximating curvature operators[ R ε ]( t ), we will show that for the case of a C -Lorentzian metric our definition of genericity(Definition 2.2) is equivalent to the classical one, i.e., (2.2) if the strong energy condition (2.1)holds. Clearly, (2.3) implies (2.2). For the converse, we have: Lemma 3.5.
Let g ∈ C be a Lorentzian metric on M , and let γ : I → M be a causal geodesicfor g . Suppose that the genericity condition (2.2) is satisfied for γ at t ∈ I . If the strongenergy condition (2.1) holds then there exist a neighbourhood U of γ ( t ) , as well as Lipschitzvector fields X and V on U such that X ( γ ( t )) = ˙ γ ( t ) and V ( γ ( t )) ∈ ( ˙ γ ( t )) ⊥ for all t ∈ I with γ ( t ) ∈ U , and there exists some c > such that h R ( V, X ) X, V i > c on U .Proof. We assume that (2.2) holds at t . Let e , . . . , e n be orthonormal vectors at γ ( t ) (with e , . . . , e n − spacelike and e n timelike) such that ˙ γ ( t ) = ( e n − + e n ) if ˙ γ ( t ) is null or ˙ γ ( t ) = e n if ˙ γ ( t ) is timelike, respectively. Then Ric( ˙ γ ( t ) , ˙ γ ( t )) = P ki =1 h R ( e i , ˙ γ ( t )) ˙ γ ( t ) , e i i ≥ k = n − k = n − h R ( e j , ˙ γ ( t )) ˙ γ ( t ) , e j i has to be strictly positive. By continuity,extending e j and ˙ γ ( t ) to a neighbourhood U of γ ( t ) (e.g. by parallel transport) providesthe desired vector fields V and X such that (2.3) is satisfied.The next step is to use the C , -genericity condition to derive a lower bound on the tidalforce operator for approximating metrics along approximating causal geodesics.10 emma 3.6. Let g ∈ C , be a Lorentzian metric on M such that the strong energy conditionis satisfied, and let γ : I → M be a causal geodesic for g . Suppose that the genericity conditionis satisfied for γ at t ∈ I . Then there exist constants r > , c > , and C > such that thefollowing holds: Let g ε = ˇ g ε or g ε = ˆ g ε , and let γ ε be g ε -geodesics of the same causal characterw.r.t. g ε as that of γ w.r.t. g . Assume that γ ε converges to γ in C ( I ) and for each ε , let [ R ε ]( t ) := [ R ε ( . , ˙ γ ε ( t )) ˙ γ ε ( t )] : [ ˙ γ ε ( t )] ⊥ → [ ˙ γ ε ( t )] ⊥ . Then there exists ε > such that, for each ε ∈ (0 , ε ) there is a smooth parallel orthonormalframe [ E ε ]( t ) , . . . , [ E εd ]( t ) for [ ˙ γ ε ] ⊥ such that [ R ε ]( t ) > diag( c, − C, . . . , − C ) on [ t − r, t + r ] (3.4) in terms of this frame. Remark . As the proof will show, the conclusion of Lemma 3.6 remains valid if, for γ timelike resp. null, also the strong energy resp. genericity condition are assumed to hold onlyfor the timelike resp. null case.Moreover, since in all the following results the strong energy condition only enters viaLemmas 3.2 and 3.6, the claim in the final sentence of Remark 2.1 indeed holds. Proof of Lemma 3.6.
As the claim is local, we may assume that M = R n . We use thenotation of Definition 2.2, and may clearly set t = 0. Additionally we may assume that γ isparametrised to unit speed (if γ is timelike) or such that ˙ γ (0) = e n − + e n for two orthonormalvectors e n − , e n with e n timelike (if γ is null). Setting e n := ˙ γ (0) in the timelike case, byshrinking U and c we may assume that U is totally normal ([14, Sec. 4]) and relatively compactand replace X by the parallel transport (radially outward from γ (0)) of e n − + e n in the nullcase, respectively e n in the timelike case.We now briefly distinguish the timelike and the null case, first assuming that γ is null.We then replace V by the vector field obtained by transporting V ( γ ( t )) outwards from γ ( t ) along radial geodesics. Then by possibly shrinking U and c we still retain the genericityestimate (2.3) for X and V . By construction, the new V is either proportional to X nowhere oreverywhere, but the latter can’t occur by the symmetries of R and (2.3). Hence V is spacelikeand we normalise it. Thus we can choose an orthonormal Lipschitz frame E , . . . , E n on U such that E = V , E n is timelike and X = ( E n − + E n ).In the case where γ is timelike, by shrinking U and c further, we may replace V by V + h X, V i X and normalize it. Consequently, there exists a Lipschitz continuous orthonormalframe E = V, E , . . . , E n = X on U .After these preparations, we will now carry out the proof in several steps simultaneouslyin the timelike and the null case.To begin with, let 0 < c < c . We claim that there exists some C > R ij := h R ( E i , X ) X, E j i we have ( R ij ) di,j =1 > diag( c , − C , . . . , − C ) on U .To establish this, we need to find C > w =: ( w , ¯ w ) = 0 in R d , w ⊤ ( R ij − diag( c , − C , . . . , − C )) w >
0. Setting ¯ R := ( R ij ) di,j =2 , and denoting by λ min the Here and below, for d × d matrices A, B , we write
A > B if the matrix A − B is positive definite. R + C id, we have w ⊤ ( R ij − diag( c , − C , . . . , − C )) w = ( R − c ) w + 2 d X j =2 R j w j w + ¯ w ⊤ ( ¯ R + C id) ¯ w ≥ ( c − c ) w + 2 d X j =2 R j w j w + λ min k ¯ w k e ≥ ( c − c ) w − | w |k ( R j ) j k e k ¯ w k e + λ min k ¯ w k e , (3.5)where k . k e denotes the Euclidean norm. Setting C R := k ( R j ) j k e , we pick C > λ min ( x ) ≥ C R c − c for all x ∈ U . With this choice, the quadratic in the final line of (3.5) has noreal root, and therefore (3.5) is positive for all w ∈ R d \ { } .Since (component-wise) convolution with a non-negative mollifier as in Lemma 3.4(ii)preserves positive-definiteness, it follows that given 0 < c < c and C > C , we canachieve R ij ∗ ρ ε > diag( c , − C , . . . , − C ) for ε small. Furthermore, by the same argu-ment as in (5) in [16], R ε − R ∗ ρ ε → ε →
0) locally uniformly and, by Lemma 3.4(i), R εij − R ij ∗ ρ ε → R εij are defined as R εij =( h R ε ( E i , X ) X, E j i g ε ) di,j =1 . This implies that there exists an ε such that( R εij ) > diag( c , − C , . . . , − C ) (3.6)on U for all ε < ε .Next we note that by the explicit bounds derived in [14, Sec. 2] we may assume that U is g ε -totally normal for each ε < ε . Let p ε := γ ε (0). Since p ε → p , we can also achievethat p ε ∈ U for all ε < ε . Pick a g ε -orthonormal frame e ε , . . . , e εn at p ε such that, as above, e εn = ˙ γ ε (0) in the timelike case, whereas in the null case e εn is timelike and ˙ γ ε (0) ∝ e εn − + e εn .In addition, we may assume that e εi → E i ( p ) as ε →
0. Now denote by E ε , . . . , E εn the g ε -orthonormal frame on U that results from parallel transporting e ε , . . . , e εn out from p ε alongradial g ε -geodesics. Then, since E εi → E i uniformly on U , by further shrinking ε , we obtainfrom (3.6) that the matrix elements with respect to this frame satisfy( h R ε ( E εi , X ) X, E εj i g ε ) di,j =1 > diag( c , − C , . . . , − C ) (3.7)on U for ε < ε .Fix r > γ ([ − r, r ]) ⊆ U , so that, without loss of generality we have γ ε ([ − r, r ]) ⊆ U for all ε < ε . Then, by construction, E εi ( t ) := E εi ◦ γ ε ( t ) is a g ε -orthonormal smooth parallelframe along γ ε , and (3.7) implies that( h R ε ( E εi ( t ) , X ◦ γ ε ( t )) X ◦ γ ε ( t ) , E εj ( t ) i g ε ◦ γ ε ) di,j =1 > diag( c , − C , . . . , − C )on [ − r, r ] for ε ≤ ε . The claim now follows from the observation that h R ε ( . , X ) X, . i g ε ◦ γ ε −h R ε ( . , ˙ γ ε ) ˙ γ ε , . i g ε → − r, r ]. Given a causal geodesic γ without conjugate points, it is well known in the smooth casethat, under the strong energy condition, the initial expansion of the corresponding geodesic12ongruence must be bounded. In the following Lemma, we explicitly derive such boundsassuming only the weaker energy condition, Ric( ˙ γ, ˙ γ ) > − δ , that follows from the C , -version of the strong energy condition, cf. Lemma 3.2. We respect the conventions introducedin Section 3, so in particular d = n − γ timelike and d = n − γ null. Lemma 4.1.
Let g be a smooth Lorentzian metric on M . Then, for any T > , thereexists some δ = δ ( T ) > with the following property: Let γ be a future directed causalgeodesic without conjugate points on [ − T, T ] , and let [ A ] be the Jacobi tensor class along γ assuming the data [ A ]( − T ) = 0 and [ A ](0) = id . Then for any < r < T / the expansion θ = tr([ ˙ A ][ A ] − ) satisfies sup t ∈ [ − r,r ] | θ ( t ) | ≤ dT , (4.1) provided that Ric( ˙ γ, ˙ γ ) ≥ − δ on [ − T, T ] .Proof. Since [ A ]( − T ) = 0, [ B ] := [ ˙ A ][ A ] − is self-adjoint (cf., e.g., [13, Lemma 4.6.19]), so itsvorticity ω = ([ B ] − [ B ] t ) vanishes. By the Raychaudhuri equation we therefore have˙ θ = − Ric( ˙ γ, ˙ γ ) − tr( σ ) − d θ ≤ δ − d θ , (4.2)where the shear σ is given by σ = [ B ] − d θ · id. To estimate θ from below on [ − r, r ], assume thatthere exists t ∈ [ − r, r ] such that θ ( t ) < −√ dδ . Writing β = θ ( t ) < κ = − d δ < s + 1 d s + d κ = 0 , s (0) = β. (4.3)Denote by s κβ : [0 , b κβ ) → R the maximal solution of (4.3). Now if β ∈ ( −∞ , −√ dδ ), one has(cf. [35]) s κβ ( t ) = d p | κ | coth t p | κ | + arcoth βd p | κ | !! , (4.4) b κβ = − p | κ | arcoth βd p | κ | ! . (4.5)Since γ has no conjugate point before T , and since the maximal domain of definition of θ ( t + . ) must be contained in that of s κβ by Riccati comparison, we obtain T − t ≤ b κβ .Consequently, − d p | κ | coth (cid:16)p | κ | ( T − r ) (cid:17) ≤ − d p | κ | coth (cid:16)p | κ | ( T − t ) (cid:17) ≤ β. (4.6)The left hand side of (4.6) goes to − d/ ( T − r ) as κ →
0, so we may choose a κ < β ≥ − d/ ( T − r ). Translating back to δ and recalling that weassumed r ≤ T /
2, we see that we may choose δ > t asabove, β = θ ( t ) ≥ − d/T . So in total we have for sufficiently small δ thatinf t ∈ [ − r,r ] θ ( t ) ≥ min( − dT , −√ d δ ) = − dT . (4.7)13o obtain the analogous estimate from above, consider the Jacobi tensor t [ A ]( − t ) along t γ ( − t ). Then the corresponding past-directed expansion θ p ( t ) = − θ ( − t ) satisfies a Riccatiequation with the same bounds as θ , so the above arguments imply (4.7) also for θ p , yieldingthe claim.We may now prove the existence of conjugate points along causal geodesics in the smoothcase under the weakened version of the Ricci bounds derived in Lemma 3.2 from the strongenergy condition (2.1), as well as the bounds on the curvature operator derived in Lemma 3.6from the C , -genericity condition. Proposition 4.2.
Let g be a smooth Lorentzian metric on M . Then given c > , C > , and < r < π √ c there exist δ = δ ( c, C, r ) > , and T = T ( c, C, r ) > with the following property:If γ is a causal geodesic and t ∈ R is such that γ is defined at least on [ t − T, t + T ] and(i) Ric( ˙ γ, ˙ γ ) ≥ − δ on [ t − T, t + T ] , as well as(ii) there exists a smooth parallel orthonormal frame [ E ]( t ) , . . . , [ E d ]( t ) for [ ˙ γ ] ⊥ such that,in terms of this frame the tidal force operator satisfies [ R ]( t ) > diag( c, − C, . . . , − C ) on [ t − r, t + r ] ,then γ possesses a pair of conjugate points in [ t − T, t + T ] .Proof. Clearly we may assume that t = 0. Now suppose, to the contrary, that no matterhow small δ > T > γ satisfying (i) and (ii) withoutconjugate points in [ − T, T ]. Then for any such choice there is a unique Jacobi tensor class[ A ] along γ (depending on T and δ ) with [ A ]( − T ) = 0 and [ A ](0) = id. With [ E ]( t ) , . . . , [ E d ]( t ) as in (ii), henceforth we will consider all linear endomorphisms of [ ˙ γ ] ⊥ as matrices inthis basis. Set [ ˜ R ]( t ) := diag( c, − C, . . . , − C ). Then by (ii), [ ˜ R ]( t ) < [ R ]( t ) on [ − r, r ].Set [ B ] := [ ˙ A ] · [ A ] − . Then (cf., e.g., [2, ch. 12]) [ B ] is self-adjoint and satisfies the matrixRiccati equation [ ˙ B ] + [ B ] + [ R ] = 0 . (4.8)Denote by [ ˜ B ] the solution to (4.8), with [ R ] replaced by [ ˜ R ] and initial value prescribed atsome t ∈ [ − r, r ]. We will show that we can find a t ∈ [ − r, r ] and an initial value [ ˜ B ]( t )satisfying [ ˜ B ]( t ) ≥ [ B ]( t ). Once this is established then, since [ R ] > [ ˜ R ] on [ − r, r ], theRiccati comparison theorem of [5] implies that [ B ]( t ) ≤ [ ˜ B ]( t ) for all t ∈ [ t , r ].We will in fact seek t in [ − r,
0] and [ ˜ B ]( t ) in the form ˜ β ( t ) · id, where ˜ β ( t ) is greateror equal the largest eigenvalue of [ B ]( t ). Since we can without loss of generality assume that T > r and that δ < δ ( T ), our assumption on the absence of conjugate points in conjunctionwith Lemma 4.1 yields for the expansion θ = tr([ B ]):max t ∈ [ − r,r ] | θ ( t ) | ≤ ν ≡ ν ( T ) := 4 dT . (4.9)Also, θ satisfies the Raychaudhuri equation˙ θ + 1 d θ + tr( σ ) + tr([ R ]) = 0 , (4.10)14here, as before, σ = [ B ] − d θ · id. Denoting the eigenvalues of [ B ] by β i (1 ≤ i ≤ d ), σ haseigenvalues β i − θd , and since tr([ R ]) ≥ − δ by assumption we find˙ θ ≤ δ − d X i =1 (cid:16) β i − θd (cid:17) ≤ δ − (cid:16) β max ( t ) − θ ( t ) d (cid:17) =: − l. (4.11)Here, β max is the maximum eigenvalue of [ B ] and t ∈ [ − r,
0] is chosen such that (cid:12)(cid:12) β max − θd (cid:12)(cid:12) attains its minimum on [ − r,
0] in t . Using (4.9), we see − ν ≤ θ (0) ≤ − lr + θ ( − r ) ≤ − lr + ν, which implies l ≤ νr . Combining this with (4.11) gives β max ( t ) ≤ s(cid:18) νr + δ (cid:19) + θ ( t ) d ≤ s(cid:18) νr + δ (cid:19) + νd =: f ( ν, δ, r ) ≡ f. (4.12)Consequently, we may set ˜ β ( t ) := f ( ν, δ, r ) and [ ˜ B ]( t ) := f ( ν, δ, r ) · id to indeed achieve that[ B ]( t ) ≤ [ ˜ B ]( t ) on [ t , r ].Since both [ ˜ R ] and [ ˜ B ]( t ) are diagonal, the Riccati equation for [ ˜ B ] decouples and hasthe explicit solution [ ˜ B ]( t ) = 1 d diag( H c,f ( t ) , H − C,f ( t ) , . . . , H − C,f ( t )) . Here (cf. [35, 7]) H c,f ( t ) = d √ c cot( √ c ( t − t ) + arccot( f / √ c )) , and H − C,f ( t ) = d √ C tanh (cid:0) √ C ( t − t ) + artanh( f / √ C ) (cid:1) , and due to our assumption 0 < r < π √ c these functions are defined on [ t , r ] (for f sufficientlysmall). As was noted above, since [ ˜ R ]( t ) < [ R ]( t ) for all t ∈ [ − r, r ] and [ B ]( t ) ≤ [ ˜ B ]( t ),Riccati comparison implies [ B ]( t ) ≤ [ ˜ B ]( t ) for all t ∈ [ t , r ]. In particular, for the smallesteigenvalue β min of [ B ] we obtain β min ( t ) ≤ d H c,f ( t ) ( t ∈ [ t , r ]) . (4.13)We are now going to show that for δ small enough and T large enough, H c,f ( t ) < t ∈ [ r , r ]. In fact, since H c,f is monotonically decreasing, it suffices to secure that H c,f ( r ) < k := arccot( f / √ c ) < π . Then H c,f ( r ) < √ c (cid:16) r − t (cid:17) + k ∈ (cid:16) π , π (cid:17) . (4.14)To achieve this, first note that 3 r √ c < π , so that √ c ( r − t ) < π . Since k < π , (4.14) can besatisfied by choosing δ and ν so small that √ c ( r − t ) + k > π . Shrinking ν further, we canalso achieve that H c,f ( r ) < − ν , so altogether we obtain for t ∈ [ r , r ]: β min ( t ) ≤ d H c,f (cid:16) r (cid:17) < − νd ≤ θ ( t ) d .
15y (4.11) this gives ˙ θ ≤ − (cid:16) β min − θd (cid:17) + δ ≤ − (cid:16) d (cid:16) H c,f (cid:16) r (cid:17) + ν (cid:17)(cid:17) + δ on [ r , r ]. Consequently, − ν ≤ Z r r ˙ θ ( t ) dt ≤ − r h(cid:16) d (cid:16) H c,f (cid:16) r (cid:17) + ν (cid:17)(cid:17) − δ i , and thereby − d s(cid:18) νr + δ (cid:19) − ν ≤ H c,f (cid:16) r (cid:17) . However, as δ ց T → ∞ , the left hand side of this inequality tends to 0, while theright hand side has the limit d √ c cot (cid:16) √ c (cid:16) r − t (cid:17) + π (cid:17) <
0, a contradiction.
We will next prove that in the C , -case under suitable causality conditions complete causalgeodesics stop being maximising, provided the strong energy condition (2.1) and the genericitycondition (Definition 2.2) hold. We will do so separately in the timelike and in the nullcase with the respective causality conditions adapted to the later use of the correspondingstatements in the proof of the main theorem. Theorem 5.1.
Let g ∈ C , be a globally hyperbolic Lorentzian metric on M that satisfies thetimelike convergence condition. Moreover, suppose that the genericity condition holds alongany timelike geodesic. Then no complete timelike geodesic γ : R → M is globally maximising. Proof.
Let γ : R → M be a complete geodesic and suppose that γ were maximising betweenany two of its points. We approximate g from the inside by a net ˇ g ε , so each ˇ g ε is globallyhyperbolic as well. Without loss of generality assume that γ satisfies the genericity conditionat t = 0. Then by Lemma 3.6 there exist c > C > < r < π √ c such that, whenever γ ε is a net of ˇ g ε -geodesics that converge to γ in C , there exists some ε > ε < ε , condition (ii) of Proposition 4.2 is satisfied for R ε .Choose δ = δ ( c, C, r ) > T = T ( c, C, r ) > T > T . Sinceˇ g ε is globally hyperbolic, for any ε > g ε -geodesic γ ε from γ ( − ˜ T ) to γ ( ˜ T ) (cf. [4, Prop. 1.21 and Th. 1.20]). We choose the parametrisationsuch that γ ε ( − ˜ T ) = γ ( − ˜ T ) and v := ˙ γ ( − ˜ T ) and v ε := ˙ γ ε ( − ˜ T ) have the same h -norm fora fixed Riemannian background metric h . We define ˜ T ε by γ ε ( ˜ T ε ) = γ ( ˜ T ), so γ ε | [ − ˜ T , ˜ T ε ] ⊆ J − ( γ ( ˜ T )) ∩ J + ( γ ( − ˜ T )). Therefore there is a subsequence ε k such that v ε k converges to avector w with k w k h = k v k h and ˜ T ε k → b ∈ [ − ˜ T , ∞ ].Consequently, γ ε k converges in C to the (future) inextendible g -geodesic γ w : [ − ˜ T , b ) → M with γ w ( − ˜ T ) = γ ( − ˜ T ) and ˙ γ w ( − ˜ T ) = w . Since our spacetime is non-totally imprison-ing (which follows from global hyperbolicity by the same proof as for smooth metrics, [24,Lem. 14.13]), this geodesic must leave the compact set J − ( γ ( ˜ T )) ∩ J + ( γ ( − ˜ T )), hence b > b Recall that a timelike geodesic is globally maximising if it maximises between any two of its points. b = ∞ and γ w ( b ) = γ ( ˜ T ). Also, γ w | [0 ,b ] must be maximising since thedistances converge by Lemma 3.3. We now distinguish two cases:If w = v , then γ w is a maximising geodesic from γ ( − ˜ T ) to γ ( ˜ T ) different from γ , so γ can’t be maximising beyond ˜ T , contradicting our assumption.If, on the other hand, v = w , then γ w = γ and b = ˜ T . Let K be a compact neighbourhoodof γ ([ − T, T ]). Since γ ε k → γ in C ([ − T, T ]), there exist k ∈ N , ˜ C >
0, and κ < k ≥ k we have γ ε k ([ − T, T ]) ⊆ K , as well as k ˙ γ ε k ( t ) k h ≤ ˜ C and g ( ˙ γ ε k ( t ) , ˙ γ ε k ( t )) < κ for all t ∈ [ − T, T ]. Lemma 3.2(i) therefore implies that R ε k ( ˙ γ ε k ( t ) , ˙ γ ε k ( t )) ≥ − δ ( c, C, r ) on [ − T, T ]for k sufficiently large. This shows that γ ε k also satisfies condition (i) from Proposition 4.2 for k large. But then any such γ ε k incurs a pair of conjugate points within [ − T, T ], contradictingthe fact that it was supposed to be maximising even on [ − ˜ T , ˜ T ε k ] ⊃ [ − T, T ] since ˜ T ε k → ˜ T .The proof of the previous Theorem uses Proposition 4.2 to guarantee the existence ofconjugate points for ˇ g ε -geodesics close to γ , but the essence of the argument can be formulatedin a much more general way using cut functions. Let T ⊆
T M be the set of all future directedtimelike vectors, then one defines the timelike cut function s : T → R by s ( v ) := sup { t : L ( γ v | [0 ,t ] ) = d ( γ (0) , γ ( t )) } . (5.1)This function clearly depends on the metric and so a natural question is how, given a C , -metric g , the ˇ g ε k -cut functions s k relate to the g -cut function s . The following theorem showsthat at least for a globally hyperbolic spacetime a uniform upper bound on the s k must alsobe an upper bound for s . Theorem 5.2.
Let ( M, g ) be a spacetime with a globally hyperbolic C , -metric and let g k =ˇ g ε k . Let U ⊆ T be open such that U ⊆ T k for large k . If s k | U ≤ T then s | U ≤ T .Proof. The proof uses the same arguments as in Theorem 5.1: Let v ∈ U , ˜ T > T and assume,for the sake of contradiction, that s ( v ) > ˜ T . Then γ v maximises the distance between γ v (0)and γ v ( ˜ T ) and even remains maximising a bit further. Choosing γ k as in the previous proof,the same arguments give a sequence γ k that converges in C to γ (in particular, ˙ γ k (0) ∈ U forlarge k ) and is maximising on [0 , ˜ T k ] ⊃ [0 , T ] for large k , but this contradicts s k | U ≤ T .There is an analogous result to Theorem 5.1 for null instead of timelike curves. However,assuming global hyperbolicity in the null case renders such a statement mostly useless forthe proof of the Hawking–Penrose Theorem because inextendible yet maximising null curvesneed to be excluded everywhere in the spacetime and not just in some globally hyperbolicsubset (contrary to timelike curves, which will appear only briefly at the end of the proofwhen one already works in some Cauchy development). Fortunately in the null case thereis a sharper distinction between maximising and non-maximising geodesics because a nullgeodesic stops maximising if and only if it leaves the boundary of a lightcone, and one canexploit the structure of such boundaries to show that inextendible null geodesics which arenot closed cannot be maximizing. However, the methods of the following proof fail for closednull curves (which are not well behaved with respect to approximation), so these had to beexcluded in the statement of Theorem 2.5 by assuming that the spacetime is causal insteadof merely chronological in the classical theorem. Theorem 5.3.
Let g ∈ C , be a Lorentzian metric on M such that ( M, g ) is causal. More-over, suppose that the null convergence condition holds and that the genericity condition is atisfied along any null geodesic. Then no complete null geodesic γ : R → M is globallymaximising.Proof. The general shape of the argument is similar to the timelike case, however, since wedo not assume global hyperbolicity we will have to choose the approximating ˇ g ε -geodesicsdifferently.Assume γ : R → M were a null geodesic that is maximizing between any of its points andthat without loss of generality satisfies the genericity condition at t = 0. Then by Lemma 3.6there exist c > C > < r < π √ c such that, whenever γ ε is a net of ˇ g ε -null geodesicsthat converge to γ in C , there exists some ε > ε < ε , condition (ii)of Proposition 4.2 is satisfied for R ε . Choose δ = δ ( c, C, r ) > T = T ( c, C, r ) > T > T in a such a way that p := γ ( − ˜ T ) is different from q := γ ( ˜ T ).Then, by assumption, q ∈ ∂J + ( p ). We will now find a sequence ε k → q k ∈ ∂J + k ( p ) := ∂J +ˇ g εk ( p ) with q k → q : Let U k be a sequence of neighbourhoods of q with U k +1 ⊆ U k and T k U k = { q } . Then for any U k there exist points q ek ∈ U k \ J + ( p ) and q ik ∈ U k ∩ I + ( p ). Let ε k be such that q ik ∈ I + k ( p ) and ε k ≤ k and let c k be a curve in U k connecting q ik and q ek ∈ U k \ J + ( p ) ⊆ U k \ J + k ( p ). Then this curve must intersect ∂J + k ( p ) andwe choose q k to be such an intersection point.Since q k ∈ ∂J + k ( p ) there exists a past directed ˇ g ε k -null geodesic starting at q k that iscontained in ∂J + k ( p ) and is either (past) inextendible or ends in p (cf. Proposition A.7). Let γ k : I k → M denote an inextendible future directed reparametrisation of such a geodesic with γ k ( ˜ T ) = q k and k ˙ γ k ( ˜ T ) k h = k ˙ γ ( ˜ T ) k h . Since the h -norms of ˙ γ k ( ˜ T ) ∈ T q k M are boundedand q k → q , we may without loss of generality assume that the sequence ˙ γ k ( ˜ T ) converges tosome vector w ∈ T q M . This vector w must be g -null since the ˙ γ k were ˇ g ε k -null. Hence thereexists a unique inextendible g -geodesic γ w : ( a w , b w ) → M with ˜ T ∈ ( a w , b w ), γ w ( ˜ T ) = q and˙ γ w ( ˜ T ) = w and the γ k converge to γ w in C .Due to our choice of the γ k , for each k there either exists t k < ˜ T such that γ k ( t k ) = p and γ k | [ t k , ˜ T ] ⊆ ∂J + k ( p ) or γ k ⊆ ∂J + k ( p ). By extracting a subsequence we may assume thatthe first or the second possibility applies in fact for each k . In the second case we pick some s ∈ ( a w , ˜ T ) and note that by C -convergence γ k is defined on [ s, ˜ T ] for k large.In the first case, if the sequence t k is unbounded (below) we may again pick some s ∈ ( a w , ˜ T ) such that γ k ([ s, ˜ T ]) ⊆ ∂J + k ( p ) for k large. Finally, if ( t k ) is bounded, we may withoutloss of generality assume that t k → ˜ t with γ w (˜ t ) = p . Since p = q (by our choice of ˜ T ), ˜ t < ˜ T ,so also in this case there exists max(˜ t, a w ) < s < ˜ T such that γ k ([ s, ˜ T ]) ⊆ ∂J + k ( p ) ⊆ J + ( p )for large k .Thus in any case γ w | [ s, ˜ T ] ⊆ J + ( p ). Therefore, if γ w were not (a reparametrisation of) γ , the concatenation γ w | [ s, ˜ T ] γ | [ ˜ T , ˜ T +1] would be a broken null curve from a point in J + ( p ) to γ ( ˜ T + 1), hence γ ( ˜ T + 1) ∈ I + ( p ), which contradicts γ being maximising between any of itspoints. This shows that (with our choice of parametrisations) γ w must actually be equal to γ . But then in particular γ (˜ t ) = γ w (˜ t ) = p (if ( t k ) is bounded) and thus since γ cannot beclosed by assumption of causality, we must have t = − ˜ T . Thereby in each of the above cases γ k | [ − T, ˜ T ] ⊆ ∂J + k ( p ) for k large. Consequently, any such segment must be maximising for themetric ˇ g ε k . Also, since γ k → γ in C ([ − T, T ]), there exist a compact neighbourhood K of γ ([ − T, T ]), k ∈ N , ˜ C >
0, and η > k ≥ k we have γ k ([ − T, T ]) ⊆ K , as wellas k ˙ γ k ( t ) k h ≤ ˜ C and d ˜ h ( ˙ γ k ( t ) , ˙ γ ( t )) < η and k ˙ γ ( t ) k h ≤ ˜ C for all t ∈ [ − T, T ]. Lemma 3.2(ii)18herefore implies that R ε k ( ˙ γ k ( t ) , ˙ γ k ( t )) ≥ − δ ( c, C, r ) on [ − T, T ] for k sufficiently large. Thisshows that γ k also satisfies condition (i) from Proposition 4.2 for k large. But then any such γ k incurs a pair of conjugate points within [ − T, T ], contradicting the fact that it was supposedto be maximising even on [ − T, ˜ T ].To conclude this section we want to briefly discuss the difference in causality conditions im-posed on M in the classical Theorem 1.2 ( M being chronological) and in the C , -Theorems 2.5and 2.6 ( M being causal). Causality assumptions (of any kind) on M were first required inthis section to prove Theorem 5.1 and Theorem 5.3. The results proven in previous sectionsdid not require any causality assumption (with the exception of Lemma 3.3, which is onlyused in the proof of Theorem 5.1). Contrary to our results the smooth versions of thesetwo theorems do not require any causality conditions. Regarding Theorem 5.1, we note thateven in the proof of the (classical) Hawking–Penrose theorem its smooth counterpart (despitebeing valid on all of M ) is actually only applied to an open globally hyperbolic subset of M . This is also true in the proof of our result (see Theorem 7.4). However, Theorem 5.3 isrequired in multiple places (e.g., any result requiring strong causality indirectly uses Theo-rem 5.3 by virtue of Lemma A.19). As such, we have found it necessary to assume that the C , -spacetime is causal.Nevertheless, the assumption of causality of M only enters in the proof of Theorem 5.3at a single point, namely where we argue that since γ cannot be closed the equality of γ (˜ t )and γ ( − ˜ T ) implies that ˜ t = − ˜ T . Moreover, this theorem is the only ingredient in the proof ofTheorems 2.5 and 2.6 where causality of M is required. For all other steps it is sufficient that M be chronological. This can be seen from the following argument: Both the classical proof ofthe Hawking–Penrose theorem and the proofs of Theorem 2.5 and Theorem 2.6 presented hereargue by contradiction, i.e., one assumes that M is a causal geodesically complete spacetime(satisfying the conditions of the theorem) and derives a contradiction. Hence if one couldshow that Theorem 5.3 remains true while only assuming M to be chronological (and notcausal), one could invoke Lemma A.19 to gain that M is even strongly causal and the rest ofour proof would go through.We expect that Theorem 2.5 and Theorem 2.6, in fact, even hold for chronological C , spacetimes, but anticipate that a proof will require new methods. In its classical version the Hawking–Penrose theorem comes with three distinct initial con-ditions: the existence of a compact achronal set without edge (or equivalently an achronalcompact topological hypersurface, [16, Cor. A.19]), the existence of a trapped surface, or theexistence of a point such that along any future (or past) directed null geodesic starting at thispoint the convergence becomes negative. An analogue of the trapped surface condition forsubmanifolds of arbitrary co-dimension was introduced in [6]. In this section we will studythese initial conditions and their consequences in the C , -case. We begin with the most straightforward case: the existence of a compact achronal set withoutedge. 19 roposition 6.1.
Let ( M, g ) be a C , -spacetime, and let A be a compact achronal set withoutedge. Then E + ( A ) = A , in particular it is compact.Proof. This follows immediately from the fact that for an achronal set A any future directednull geodesic starting in a point p / ∈ edge( A ) must immediately enter I + ( A ). This can beseen as in [16, Prop. A.18].One should note that as in the smooth case one may even relax the causality assumptionson A a little: By using a covering argument as in [16, Thm. A.34] it would be sufficient toassume the existence of a compact spacelike hypersurface A in the Hawking–Penrose theorem. < m < n In this section, we follow the approach of Galloway and Senovilla [6] and consider trappedsubmanifolds of arbitrary codimension of a C , -spacetime ( M, g ). To work in full generality(and because we will need this generality to deal with the codimension zero case later on) wewill now define C -trapped submanifolds of codimension 1 < m < n . Our definition is similarin spirit to the definition of lower mean curvature bounds for C spacelike hypersurfaces in[1]. As mentioned in section 2, we say that a submanifold ˜ S is a future support submanifoldfor a C -submanifold S at q ∈ S if dim( ˜ S ) = dim S , q ∈ ˜ S , and ˜ S is locally to the future of S near q , i.e. there exists a neighbourhood U of q in M such that ˜ S ∩ U ⊆ J + ( S, U ). We usethis to define ’past pointing timelike mean curvature’ for C -submanifolds. Definition 6.2.
Let S be a C -submanifold of codimension m (1 < m < n ) in a C , -spacetime ( M, g ). We say that S has past-pointing timelike mean curvature in q in the senseof support submanifolds if there exists a C spacelike future support submanifold ˜ S for S in q with H ˜ S ( q ) past-pointing timelike.This leads to the following definition of a future trapped C -submanifold of M (which isobviously satisfied for C -submanifolds that are future trapped in the classical sense definedin [6]). Definition 6.3. A C -submanifold S of codimension m (1 < m < n ) of a C , -spacetime( M, g ) is called future trapped if it is closed (i.e., compact without boundary) and for any p ∈ S there exists a neighbourhood U p of p such that S ∩ U p is achronal in U p and S haspast-pointing timelike mean curvature in all its points (in the sense of support submanifolds).Our aim is a generalisation of the main results of [6] to the C , -setting. In fact, wewill show that under some additional curvature assumptions any future directed null geodesicstarting at a point q of a trapped submanifold S in the above sense eventually stops maximisingthe distance to the future support submanifold ˜ S at q (provided it exists for long enoughtimes).Using the notation introduced in section 1 (i.e., letting E , . . . , E n − m denote the paralleltranslates of an orthonormal basis e ( γ (0)) , . . . , e n − m ( γ (0)) for T γ (0) S along γ ) we start byproving the following mild extension of [6, Prop. 1]: Lemma 6.4.
Let S be a C spacelike submanifold of codimension m (1 < m < n ) in a smoothspacetime ( M, g ) , and let γ be a geodesic such that ν := ˙ γ (0) ∈ T M | S is a future-pointing ull normal to S . Suppose that c := k S ( ν ) > and let b > /c . Then there exists some δ = δ ( b, c ) > such that, if n − m X i =1 h R ( E i , ˙ γ ) ˙ γ, E i i ≥ − δ (6.1) along γ , then γ | [0 ,b ] is not maximising to S , provided that γ exists up to t = b .Proof. We closely follow the proof of [6, Prop. 1]. For vector fields V , W along γ that areorthogonal to γ and vanish at t = b we consider the energy index form (with ˙ V etc. denotingthe induced covariant derivative along γ ) I ( V, W ) := Z b h h ˙ V , ˙ W i − h R V ˙ γ ˙ γ, W i i dt − h ˙ γ (0) , II( V (0) , W (0)) i . For 1 ≤ i ≤ n − m , let X i := (1 − t/b ) E i . Then I ( X i , X i ) = Z b h /b − (1 − t/b ) h R E i ˙ γ ˙ γ, E i i i dt − h ˙ γ (0) , II( e i , e i ) i . Hence n − m X i =1 I ( X i , X i ) = ( n − m ) (cid:16) b − c (cid:17) − Z b (cid:16) − tb (cid:17) n − m X i =1 h R ( E i , ˙ γ ) ˙ γ, E i i dt ≤ ( n − m ) (cid:16) b − c (cid:17) + bδ . Obviously this last expression can be made negative by choosing δ = δ ( b, c ) small enough.It then follows that the energy index form is not positive-semidefinite, so there must exist afocal point of S on γ within (0 , b ], giving the claim.We now turn to the case of a C , -metric g . Let ˜ S be a C spacelike submanifold ofco-dimension m , and let ν ∈ T p ˜ S be a future-pointing null vector normal to ˜ S . As in thesmooth setting above, assume that γ is a geodesic with affine parameter t with ˙ γ (0) = ν ,and let e , . . . , e n − m be a local orthonormal frame on ˜ S around p := γ (0) (of regularity C , ).Again, denote by E , . . . , E n − m the parallel translates of e ( p ) , . . . , e n − m ( p ) along γ (whichare Lipschitz continuous vector fields along γ ).In trying to formulate a natural analogue of (6.1) (with δ = 0) we again face the problemthat the curvature operator (being only defined almost everywhere) cannot be restricted tothe Lebesgue null set γ ([0 , b ]). Similar to the case of the genericity condition (Definition 2.2),we shall therefore require the existence of continuous extensions of E , . . . E n − m and ˙ γ to aneighbourhood of the geodesic γ . In fact, with the notation introduced above we have: Proposition 6.5.
Let ( M, g ) be a strongly causal C , -spacetime, ˜ S ⊆ M a C spacelikesubmanifold and suppose that k ˜ S ( ν ) > c > and let b > /c . If there exists a neighbourhood U of γ | [0 ,b ] and continuous extensions ¯ E , . . . ¯ E n − m and ¯ N of E , . . . E n − m and ˙ γ , respectively,to U such that n − m X i =1 h R ( ¯ E i , ¯ N ) ¯ N , ¯ E i i ≥ , (6.2) then γ | [0 ,b ] is not maximising to ˜ S . roof. We again proceed by regularisation. Let g ε = ˇ g ε , then as in the proof of Lemma 3.6we may without loss of generality suppose that M = R n , and that R ε = R ∗ ρ ε . Since ˜ S is a C -submanifold, k ˜ S is continuous on ˜ S and k ˜ S,ε → k ˜ S uniformly on compact subsets. Thus,there exists a neighbourhood V in T M | ˜ S of ν and an ε such that for all ε ≤ ε one has k ˜ S,ε ( v ) > c for all v ∈ V . Shrinking U , we may assume that there exists ε such that for all g ε with ε ≤ ε the submanifold U ∩ ˜ S is g ε -spacelike and, shrinking V if necessary, we havethat the projection W := π ( V ) of V onto ˜ S is contained in U ∩ ˜ S .Further shrinking ε and V if necessary, for each ε < ε let e ε , . . . , e εn − m be a g ε -orthonormal frame for ˜ S on W such that e εi → e i uniformly on W for ε →
0. For each v ∈ V , denote by E εi ( t ) the parallel transport of e εi ( π ( v )) along the g ε -geodesic γ εv with˙ γ εv (0) = v .By (6.2) we have n − m X i =1 g ( R ( ¯ E i , ¯ N ) ¯ N , ¯ E i ) ∗ ρ ε ≥ . Since without loss of generality U is relatively compact and γ εv ([0 , b ]) ⊆ U for all v ∈ V and all ε ≤ ε , Lemma 3.4 (i) implies that g ( R ( ¯ E i , ¯ N ) ¯ N , ¯ E i ) ∗ ρ ε − g ε ( R ε ( ¯ E i , ¯ N ) ¯ N , ¯ E i ) → U , as well as g ε ( R ε ( ¯ E i , ¯ N ) ¯ N , ¯ E i ) ◦ γ εv − g ε ( R ε ( E εi , ˙ γ εv ) ˙ γ εv , E εi ) → , b ] as ( ε, v ) → (0 , ν ), for 1 ≤ i ≤ n − m .Now let 1 /c < b ′ < b , and pick δ := δ ( b ′ , c ) as in Lemma 6.4. Then by the above wemay shrink V and ε in such a way that condition (6.1) is satisfied along γ εv on [0 , b ′ ] for each v ∈ V and each ε ≤ ε .Consequently, any γ εv with v being g ε -null stops maximising the g ε -distance to ˜ S at pa-rameter t = b ′ the latest (if v is not a g ε -normal to ˜ S it must stop maximising the distanceimmediately (cf. Remark 6.6 (ii) below), if it is a null normal Lemma 6.4 applies).Now assume that the g -null geodesic γ maximises the distance to ¯ U ∩ ˜ S until the parametervalue b . We then proceed in parallel to the final part of the proof of Theorem 5.3: Let b ′′ besuch that b ′ < b ′′ < b and set q := γ ( b ′′ ). There exist points q k ∈ ∂J + k ( ¯ U ∩ ˜ S ) with q k → q . ByProposition A.7, since q k ∈ ∂J + k ( ¯ U ∩ ˜ S ) there exists a past directed ˇ g ε k -null geodesic startingat q k that is contained in ∂J + k ( ¯ U ∩ ˜ S ) and is either past inextendible or ends in ¯ U ∩ ˜ S . Againlet γ k : I k → M denote an inextendible future directed reparametrisation of such a geodesic,this time with γ k ( b ′′ ) = q k and k ˙ γ k ( b ′′ ) k h = k ˙ γ ( b ′′ ) k h . As in Theorem 5.3 we may assume that˙ γ k ( b ′′ ) converges to a g -null vector v and that the γ k converge to the corresponding geodesic γ v in C .For each k there either exists some 0 < t k < b ′′ with γ k ( t k ) ∈ ¯ U ∩ ˜ S and γ k | [ t k ,b ′′ ] ⊆ ∂J + k ( ¯ U ∩ ˜ S ), or γ k | [0 ,b ′′ ] ⊆ ∂J + k ( ¯ U ∩ ˜ S ). In the second case we set t k = 0, to obtain a sequencethat without loss of generality converges to some t ′ and t ′ = 0 < b ′′ or γ v ( t ′ ) ∈ ¯ U ∩ ˜ S .Since q ¯ U ∩ ˜ S the second case also gives t ′ < b ′′ and there exists t ′ < t ′′ < b ′′ such that γ k | [ t ′′ ,b ′′ ] ⊆ ∂J + k ( ¯ U ∩ ˜ S ) ⊆ J + ( ¯ U ∩ ˜ S ) for large k . Consequently, γ v | [ t ′′ ,b ′′ ] ⊆ J + ( ¯ U ∩ ˜ S ), andas in Theorem 5.3 this implies that γ = γ v .We now note that by shrinking U we may assume that γ can only intersect ¯ U ∩ ˜ S once: infact, we may locally view ˜ S ∩ ¯ U as a submanifold of some spacelike hypersurface ˆ S . By [16,Lemma A.25], there exists an open set W in M such that W ∩ ˆ S is a Cauchy hypersurface in W . Also, since M is strongly causal, W can be chosen in such a way that γ can only intersectit once by Lemma A.18. 22onsequently, we must have t ′ = 0. Since γ k | [ t k ,b ′′ ] ⊆ ∂J + k ( ¯ U ∩ ˜ S ), any such segment mustbe maximising for ˇ g ε k . For k large we have ˙ γ k ( t k ) ∈ V since γ k → γ . Therefore, by what wasshown above, γ k must stop maximising the distance to ¯ U ∩ ˜ S already at t = t k + b ′ < b ′′ , acontradiction. Remark . (i) In case m = 2 (i.e., the traditional trapped surface case) a slightly perturbedversion of (6.2) (namely with right hand side − δ for any given δ >
0) is automaticallysatisfied if the null convergence condition holds: Choose e n − , e n such that e n is timelike,˙ γ (0) = e n − + e n and e , . . . , e n is an orthonormal basis and denote the parallel translates of e , . . . , e n along γ by E , . . . , E n . Now let ¯ E , . . . , ¯ E n be arbitrary continuous extensions of E , . . . , E n to a neighbourhood U of γ and set ¯ N = ¯ E n − + ¯ E n .Cover γ by finitely many totally normal neighbourhoods. Then in each such neighbour-hood V we may parallelly transport E , . . . , E n from some point of γ in V radially outward toobtain local orthonormal fields ˜ E , . . . , ˜ E n , and ˜ N = ˜ E n − + ˜ E n . Then P n − i =1 h R ( ˜ E i , ˜ N ) ˜ N , ˜ E i i =Ric( ˜ N , ˜ N ) ≥ V . Now, as in section 3, shrinking U produces (6.2) with right hand sidenegative but arbitrarily close to 0. The proof of Proposition 6.5 then still gives the desiredresult.(ii) If v ∈ T M | ˜ S is future directed causal, but not a null normal to ˜ S , then γ v enters I + ( ˜ S ) immediately: This is well known for smooth metrics ([24, Lem. 10.50]). If g is only C , one cannot use the exponential map to construct a C -variation with a given variationalvector field, but since this is a local question (and clearly true if v is timelike) we may assumethat M = R n , γ v (0) = 0 and v is null. We now construct suitable variations as follows: Since v / ∈ T ˜ S ⊥ there exists y ∈ T ˜ S such that h y, v i g >
0. Let α : [0 , b ] → ˜ S be a C -curve with˙ α (0) = y (and α (0) = 0). We define a C -variation σ : [0 , t ] × [0 , s ] → R n by σ ( t, s ) := γ v ( t ) + (1 − tt ) α ( s ). Now let t , s > h y, ˙ γ v ( t ) i g ( σ ( t,s )) > c > t ≤ t and s ≤ s . We will show that σ ( ., s ) is a timelike curve for small s and t , proving the claim. Expanding α ( s ) and g ( σ ( t, s )) in a Taylor series around s = 0 gives α ( s ) = sy + O ( s ) and | g ( σ ( t, s )) − g ( γ v ( t )) | ≤ Cs (1 − tt ) + O ( s ) (where C > s, t ) as s → h ∂ t σ ( t, s ) , ∂ t σ ( t, s ) i g ( σ ( t,s )) = h ˙ γ v ( t ) , ˙ γ v ( t ) i g ( σ ( t,s )) − st h ˙ γ v ( t ) , y i g ( σ ( t,s )) + O ( s ) ≤ s (cid:18) ˜ C (cid:18) − tt (cid:19) − c t (cid:19) + O ( s ) . The bracketed term evidently is negative for small t and thus for such t the curve t σ ( t, s )will be a timelike curve from 0 to γ v ( t ) for small s . Proposition 6.7.
Let ( M, g ) be a strongly causal C , -spacetime and let S be a ( C ) trappedsubmanifold of co-dimension < m < n such that, if m = 2 , the support submanifolds ˜ S fromDefinition 6.2 satisfy (6.2) for all null normals and, if m = 2 , the null convergence conditionis satisfied. Then E + ( S ) is compact or M is null geodesically incomplete.Proof. Assume M is null geodesically complete and fix a Riemannian metric h on M andlet K := { v ∈ T M | S : v future directed , null , k v k h = 1 } . Clearly K is compact and byProposition 6.5 and Remark 6.6 for any v ∈ K there exists a time t v such that exp( t v v ) ∈ I + ( ˜ S ) ⊆ I + ( S ). Since ( v, t ) exp( tv ) is continuous there even exists a neighbourhood U v such that exp( t v w ) ∈ I + ( S ) for all w ∈ U v . By compactness we may cover K by finitely23any of these U v and thus there exists T such that E + ( S ) ⊆ exp([0 , T ] · K ). This shows that E + ( S ) is relatively compact.It remains to show that E + ( S ) is closed. Let p i = exp( t i v i ) ∈ E + ( S ) be a sequence with p i → p for some p ∈ M . Clearly p / ∈ I + ( S ), so it remains to show that p ∈ J + ( S ). Since t i ≤ T and v i ∈ K we may assume that t i → t and v i → v ∈ K . But then since p i ∈ E + ( S )we must have t i ≤ t v for i large, hence p = exp( tv ) ∈ J + ( S ) and we are done. Corollary 6.8.
Let ( M, g ) and S be as in the previous proposition. Then E + ( S ) ∩ S is anachronal set and E + ( E + ( S ) ∩ S ) is compact or M is null geodesically incomplete.Proof. This follows verbatim as in the smooth case, see [6, Prop. 4] or [33, Prop. 4.3], usingthat by definition for any p ∈ S there exists a neighbourhood U p such that S ∩ U p is achronalin U p . In the classical smooth version of the Hawking–Penrose theorem there is a third initial con-dition concerning a ‘trapped point’ p , which is a point p such that the expansion becomesnegative for any future directed null geodesic starting in p . This condition can again be formu-lated in a precise way in the language of Jacobi tensors, see e.g. [2, Prop. 12.46], by demandingthat for any future directed null geodesic γ starting in p the expansion θ ( t ) associated to theunique Jacobi tensor class [ A ] along γ with [ A ](0) = 0 and [ ˙ A ](0) = id becomes negative forsome t >
0. This formulation unfortunately does not generalise to a C , -metric (one of thereasons for this being that there is no sensible way to formulate the Jacobi equation). Thereis, however, an equivalent formulation for smooth metrics using a shape operator of spacelikeslices of the lightcone of p (which is similar to the use of co-spacelike distance functions andtheir level sets in the timelike or Riemannian case, cf. [2, Appendix B.3]):Let γ be a null geodesic and assume that the expansion of the Jacobi tensor class [ A ] along γ with [ A ](0) = 0 and [ ˙ A ](0) = id becomes negative for some t >
0. We set t := inf { t >η : θ ( t ) < } , where η > , η ] · ˙ γ (0) is contained in a neighbourhoodwhere exp p is a diffeomorphism. This ensures that γ ( t ) must come before the first conjugatepoint of p and so there exists t > t such that γ | [0 ,t ] does not contain points conjugate to p along γ . Thus, there exists a neighbourhood U ⊆ T p M of [0 , t ] · ˙ γ (0) such that exp p | U is adiffeomorphism onto its image: It clearly is a local diffeomorphism and if it were not injectiveon any such neighbourhood there would exist vectors X k , Y k ∈ T p M , X k = Y k , convergingto X, Y ∈ [0 , t ] · ˙ γ (0) with exp p ( X k ) = exp p ( Y k ), hence exp p ( X ) = exp p ( Y ). Since exp p islocally injective X = Y but this contradicts exp p being injective on [0 , t ] · ˙ γ (0) by causalityof M .Now, one can look at the level sets S t := exp p ( t ˜ U ), where ˜ U := { v ∈ U : v null , g ( T, v ) = g ( ˙ γ (0) , T ) } for some fixed timelike vector T ∈ T p M , and their shape operators S ˙ γ ( t ) ( t ) : T γ ( t ) S t → T γ ( t ) S t derived from the normal ˙ γ ( t ). Proceeding as in [9, Prop. 3.4] one gets that this shapeoperator satisfies a Riccati equation along γ and lim t ց t S ˙ γ ( t ) ( t ) = id. Identifying T γ ( t ) S t with[ ˙ γ ( t )] ⊥ , a quick calculation shows that the tensor class [ B ] along γ defined by [ ˙ B ] = S ˙ γ [ B ]on (0 , t ) and [ B ]( t ) = [ A ]( t ) also satisfies the Jacobi equation and hence can uniquely beextended to ( −∞ , ∞ ). From the limiting behaviour of S ˙ γ ( t ) ( t ) as t ց B ](0) = 0and thus by uniqueness of Jacobi tensors [ B ] = [ A ] on [0 , t ), so S ˙ γ ( t ) ( t ) = [ ˙ A ]( t )[ A ] − ( t )and θ ( t ) = tr S ˙ γ ( t ) ( t ) for t < t . Consequently, a negative θ ( t ) corresponds to a negative24race of the shape operator of the spacelike surface S t with respect to the normal ˙ γ . Since k S t ( ˙ γ ( t )) = − tr S ˙ γ ( t ) ( t ) this is equivalent to k S t ( ˙ γ ( t )) being positive.This condition can now be generalised to C , -metrics and, as introduced in section 2, wegive the following definition of a (future) trapped point. Note that this can very roughly beseen as a condition on the mean curvature of the level set S t (which is now at best Lipschitz)in the sense of support submanifolds and hence bears some similarities to our definition ofpast-pointing timelike mean curvature for C -submanifolds. Definition 6.9.
We say that a point p is future trapped if for any future-pointing null vector v ∈ T p M there exists a t such that there exists a spacelike C -surface ˜ S ⊆ J + ( p ) with γ v ( t ) ∈ ˜ S and k ˜ S ( ˙ γ v ( t )) > E + ( p ) is compact for a trapped point p . Proposition 6.10.
Let ( M, g ) be a strongly causal C , -spacetime and assume that the nullconvergence condition holds. If p ∈ M is a future trapped point and M is null geodesicallycomplete then E + ( p ) is compact.Proof. The proof is completely analogous to the one of Proposition 6.7, using that ˜ S is asurface and thus condition (6.2) is not required if the null convergence condition holds (cf.Remark 6.6). As in the smooth case we will first prove a C , -version of Theorem 1.1. To do so, we willroughly follow the original proof in [12]. However, we will split the argument into smallerpieces to better highlight the places where the reduced regularity of the metric has to betaken into account. In an attempt to keep our presentation concise we start only with theproof of [12, Lemma 2.12] (which will be Corollary 7.2 here), but for completeness all neces-sary preliminary results are collected in the appendix. Our notation in this section follows,e.g., [24], but is also explicitly defined in the introduction or the appendix. In what followswe always assume S to be non-empty. Lemma 7.1.
Let ( M, g ) be a spacetime with a C , -metric g , let S be an achronal and closedsubset of M and suppose that strong causality holds on M . Then H + ( E + ( S )) is non-compactor empty.Proof. The proof is completely analogous to the smooth one found in, e.g., [13, Lemma 9.3.2].Note that Lemma 9.3.1 and Lemma 8.3.8 from that reference still hold (see Corollary A.16and Proposition A.10) and that the curve β , which starts outside of D + ( E + ( S )) and endsin S , must intersect H + ( E + ( S )) by Lemma A.12. Corollary 7.2. Let ( M, g ) be a spacetime with a C , -metric g that is strongly causal.Let S ⊆ M be an achronal set and assume that E + ( S ) is compact. Then there exists afuture-inextendible timelike curve γ contained in D + ( E + ( S )) ◦ . cf. [12, Lemma 2.12] roof. The proof is completely analogous to the smooth case, [12, Lemma 2.12]. By Lemma A.8we may assume that S is closed. The idea is that, if every timelike curve that meets E + ( S ) alsomeets H + ( E + ( S )) (or equivalently leaves D + ( E + ( S )) ◦ ), then, using that H + ( ∂J + ( S )) isa topological hypersurface by Lemma A.15, one can define a continuous map from E + ( S )to H + ( E + ( S )) via the flow of a smooth timelike vector field. This gives a contradic-tion since E + ( S ) is non-empty and compact but H + ( E + ( S )) is empty or non-compact byLemma 7.1.The next Lemma will extract the part of the proof of Theorem 7.4, where the originalproof (and also the one in [33]) argues using the continuous dependence of conjugate pointson the geodesic, which is evidently a problem for C , -metrics. There are, however, smoothproofs that avoid this, see e.g. [13, Lemma 9.3.4]. While that proof should also work in C , (and we will refer to parts of it), we will still present a different argument of the crucial stepmore in line with the original proof. Lemma 7.3. Let ( M, g ) be a spacetime with a C , -metric g that is strongly causal andassume that no inextendible null geodesic in M is globally maximising. Let S be achronal andassume that E + ( S ) is compact, and let γ be a future inextendible timelike curve contained in D + ( E + ( S )) ◦ . Then F := E + ( S ) ∩ J − ( γ ) is achronal and E − ( F ) is compact.Proof. By Lemma A.8 we may without loss of generality assume that S is closed. Since F ⊆ E + ( S ) and E + ( S ) is achronal, it follows that F is achronal. Moreover, E + ( S ) is, byassumption, compact and J − ( γ ) is closed, therefore F is compact. We need to show that E − ( F ) is compact. To do so, first note that the same arguments as in [13, Lemma 9.3.4] showthat E − ( F ) ⊆ F ∪ ∂J − ( γ ) . (7.1)Now let v ∈ T M | F be past pointing causal. Then, by the definition of F , the pastinextendible geodesic c v : [0 , b ) → M with initial velocity ˙ c v (0) = v must be contained in J − ( γ ). We show that c v ∩ I − ( γ ) = ∅ : If c v never met I − ( γ ) it would have to be a null geodesicand lie entirely in ∂J − ( γ ) \ E − ( γ ) (since E − ( γ ) = ∅ because γ is future inextendible timelike).In particular c v (0) ∈ ∂J − ( γ ) \ E − ( γ ), so by Proposition A.7 (note that the image of γ is aclosed set by Lemma A.20), there exists a future directed, future inextendible null geodesic λ that starts at c v (0) and is contained in ∂J − ( γ ). But then c v λ either is an inextendiblebroken null geodesic, hence not maximizing by Lemma A.3, or it is an inextendible unbrokennull geodesic, hence not maximizing by assumption. Hence by Lemma A.2, c v λ cannot lieentirely in ∂J − ( γ ), giving a contradiction. Consequently, for all v ∈ T M | F , there exists a t v with c v ( t v ) ∈ I − ( γ ). Since I − ( γ ) is open there exists a neighbourhood U v ⊆ T M of v such that c w is defined on [0 , t v ) and c w ( t v ) ∈ I − ( γ ) for all w ∈ U v . By compactness of F one can cover the set of all h -unit, past pointing causal vectors in T M | F by finitely many ofthese neighbourhoods, which shows that E − ( F ) ∩ ∂J − ( γ ) is relatively compact. In fact, itis actually compact as can easily be seen using a limit argument as in the final part of theproof of Proposition 6.7 (which does not use null completeness). This shows that E − ( F ) iscompact by (7.1) and compactness of F .Combining these preliminary results allows us to prove the low-regularity version of Theo-rem 1.1. Again the argument proceeds very similarly to the smooth case, but we neverthelessgive a complete proof. cf. [12, pp. 545]. heorem 7.4. Let ( M, g ) be a spacetime with a C , -metric g . Then the following fourconditions cannot all hold:(C.i) M contains no closed timelike curves;(C.ii) Every inextendible timelike geodesic contained in an open globally hyperbolic subset stopsbeing maximizing;(C.iii) Every inextendible null geodesic stops being maximizing;(C.iv) There is an achronal set S such that E + ( S ) or E − ( S ) is compact.Proof. We assume, to the contrary, that all four conditions hold. From conditions (C.iii)and (C.i), Lemma A.19 implies that M is strongly causal. In condition (C.iv) we assume,without loss of generality, that E + ( S ) is compact.Let γ be a future inextendible timelike curve contained in D + ( E + ( S )) ◦ given by Corol-lary 7.2, and let F := E + ( S ) ∩ J − ( γ ) as in Lemma 7.3. Then, by Lemma 7.3, the set F isachronal and E − ( F ) is compact. Therefore, by Corollary 7.2, there exists a past-inextendibletimelike curve λ contained in the set D − ( E − ( F )) ◦ .Next we show γ ⊆ D + ( E − ( F )) ◦ : We have γ ⊆ D + ( E + ( S )) ◦ , so every past inextendiblecausal curve starting at γ must meet E + ( S ). This meeting point is obviously in J − ( γ ) ⊆ J − ( γ ), so every past inextendible causal curve starting at γ meets E + ( S ) ∩ J − ( γ ) = F ,which gives γ ⊆ D + ( F ). Also γ cannot meet ∂D + ( F ), which is equal to F ∪ H + ( F ) byProposition A.10, since F ⊆ E + ( S ) and γ ⊆ D + ( E + ( S )) ◦ and if γ met H + ( F ) it would alsomeet I + ( H + ( F )) by being timelike, hence leave D + ( F ) (by Lemma A.13). This means that γ ⊆ D + ( F ) ◦ ⊆ D + ( E − ( F )) ◦ (by achronality of F ).So both γ and λ are contained in D ( E − ( F )) ◦ . By [16, Thm. A.22], D ( E − ( F )) ◦ is globallyhyperbolic. Now choose sequences { p k } ⊆ λ and { q k } ⊆ γ with the following properties:(i) p k +1 ∈ I − ( p k ) and q k +1 ∈ I + ( q k ),(ii) both { p k } and { q k } leave every compact subset of M , and(iii) q ∈ I + ( p ). To see that this is possible, note that λ ⊆ J − ( E − ( F )) ⊆ J − ( F ) ⊆ J − ( J − ( γ )) and since λ is timelike Lemma A.2 gives that λ ⊆ I − ( J − ( γ )) = I − ( γ ).By [31, Prop. 6.4] there exist maximizing causal curves γ k : [ a k , b k ] → M from p k to q k . Each γ k must intersect E − ( F ) (because it connects D − ( E − ( F )) ◦ with D + ( E − ( F )) ◦ , cf. the remarkpreceding [24], Lemma 14.37) in some point r k . By compactness of E − ( F ) (see Lemma 7.3)we may assume that r k → r after passing to a subsequence if necessary, so there exists acausal limit curve ˜ γ by Theorem A.6.Now because every γ k is maximising the sequence { γ k } is limit maximising in the senseof [20, Def. 2.11] and thus ˜ γ has to be maximising (again by Theorem A.6). Also, since { p k } and { q k } leave every compact set, ˜ γ is inextendible. Because ˜ γ is maximising it has to be ageodesic (cf. [21, Thm. 1.23]).If ˜ γ is null this immediately contradicts the third assumption and we are done. Since D ( E − ( F )) ◦ is globally hyperbolic, to establish a contradiction to condition C.ii it only remainsto show that ˜ γ ⊆ D ( E − ( F )) ◦ if it is timelike. Since it is the limit of the γ k ’s we certainlyhave ˜ γ ⊆ D ( E − ( F )). Now, Proposition A.10 implies ∂D ( E − ( F )) ⊆ H + ( E − ( F )) ∪ E − ( F ) ∪ H − ( E − ( F )) . I + ( H + ( E − ( F ))) = I + ( E − ( F )) \ D + ( E − ( F )) (see Lemma A.13) it follows that ˜ γ ∩ H + ( E − ( F )) = ∅ and, analogously, ˜ γ ∩ H − ( E − ( F )) = ∅ . Now assume there exists t suchthat ˜ γ ( t ) ∈ E − ( F ). We show that then ˜ γ ( t ) ∈ D ( E − ( F )) ◦ . By achronality of E − ( F ) andthe above we get˜ γ ( t + 1) ∈ D + ( E − ( F )) \ ( E − ( F ) ∪ H + ( E − ( F ))) = D + ( E − ( F )) ◦ ⊆ D ( E − ( F )) ◦ and by the same argument also ˜ γ ( t − ∈ D ( E − ( F )) ◦ . But then since D ( E − ( F )) ◦ is globallyhyperbolic we have that the causal diamond J (˜ γ ( t − , γ v ( t + 1)) ⊆ D ( E − ( F )) ◦ and hence˜ γ ( t ) ∈ D ( E − ( F )) ◦ .Collecting this and the results established in the previous sections, we are now in theposition to prove Theorem 2.5 and Theorem 2.6. Proof of the Hawking–Penrose Theorem for C , -metrics Proof.
We show that, for a causally geodesically complete spacetime (
M, g ), assumptions (A.1)to (A.4) in Theorem 2.5 and Theorem 2.6 imply that conditions (C.i) to (C.iv) of Theorem 7.4are satisfied.Clearly, causality is a stronger assumption than being chronological, so (A.1) implies (C.i).Theorem 5.1 shows that the strong energy and the genericity conditions (i.e. assumptions (A.2)and (A.3) of Theorem 2.5) imply that condition (C.ii) of Theorem 7.4 is satisfied. Similarly,Theorem 5.3 shows that assumptions (A.1), (A.2) and (A.3) of Theorem 2.5 imply thatcondition (C.iii) of Theorem 7.4 holds.Finally, Proposition 6.1 shows that assumption (A.4.i) implies condition (C.iv). Since wehave already established that conditions (C.i) and (C.iii) of Theorem 7.4 hold, Lemma A.19in the appendix implies that (
M, g ) is strongly causal. Therefore, one can apply Propo-sition 6.7 (with Corollary 6.8) and Proposition 6.10 to show that any one of the assump-tions (A.4.ii), (A.4.iii) or (A.4.iv) (together with assumptions (A.1)–(A.3)), implies that con-dition (C.iv) of Theorem 7.4 is satisfied.
A Causality results in C , Standard expositions of causality theory ([11, 33, 8, 3, 23]) usually assume the metric to be atleast C . Most results, however, remain true for C , -metrics, see [4, 21, 15] and the appendixof [16]. In this appendix we will collect further results that are not included in these previousworks, but are necessary for the proof of Theorem 7.4.In the following we will always assume that ( M, g ) is a spacetime with a C , -metric unlessexplicitly stated otherwise. We also fix a smooth Riemannian background metric h . A.1 Limit curves and the structure of ∂J + ( S ) Two important results from [4] are that I ± ( S ) is open ([4, Prop. 1.21]) and that the push-upprinciple remains true ([4, Lem. 1.22]) for causally plain spacetimes. As these include theclass of spacetimes with Lipschitz continuous metrics ([4, Cor. 1.17]), we have Lemma A.1.
Let S ⊆ M . Then I ± ( S ) is open. Lemma A.2.
Let p, q, r ∈ M be such that p ≤ q ≪ r or p ≪ q ≤ r . Then p ≪ r .
28e will also repeatedly be making use of the following result, see [21, Lem. 2]:
Lemma A.3.
Let p, q ∈ M such that there exists a future directed causal curve c from p to q . Then either q ∈ I + ( p ) or c is (can be reparametrised to) a maximising null geodesic from p to q . Using the usual notation, we set E + ( S ) := J + ( S ) \ I + ( S ). It is easily checked that (asfor smooth metrics) we have: Lemma A.4.
Let S ⊆ M . Then both E + ( S ) and ∂J + ( S ) are achronal sets, ∂J + ( S ) isclosed, but E + ( S ) need not be. Lemma A.5.
Let S ⊆ M . Then ∂J + ( S ) is an achronal, closed topological hypersurface.Proof. Clearly J + ( J + ( S )) = J + ( S ), so [24, Corollary 14.27], which is easily verified to holdfor C , -metrics as well, gives the desired result.To proceed further we are going to need some results on limits of causal curves. Thus wewill now state that what is essentially Theorem 3.1.(1) from [20] remains true for C , -metrics. Theorem A.6.
Let y be an accumulation point of a sequence of (future directed) causalcurves. There is a subsequence parametrized with respect to h -length, γ k : [ a k , b k ] → M ( a k and b k may be infinite), ∈ [ a k , b k ] such that γ k (0) → y and such that the following propertieshold. There are a ≤ and b ≥ , such that a k → a and b k → b . If there is a neighbourhood U of y such that only a finite number of γ k is entirely contained in U then there is a causalcurve γ : [ a, b ] → M , such that γ k converges h -uniformly on compact subsets to γ . This limitcurve is past, respectively future, inextendible if and only if a = −∞ , respectively b = ∞ .Further, if γ k is limit maximising (in the sense of [20, Def. 2.11]) then γ is maximising.Proof. The existence of such a limit curve follows from the smooth version [20, Thm. 3.1.(1)]in the same way as in the proof of [31, Thm. 1.5]. This also immediately gives the statementabout inextendibility. That the limit of a limit maximising sequence is maximising follows asin the smooth case (see [20, Thm. 2.13]), using that for C , -metrics the Lorentzian distancefunction is still lower semi-continuous (see [16, Lemma A.16]) and that the length functionalis still upper semi-continuous (see [31, Thm. 6.3] and note that it does not require the samestart and end points but only a uniform bound on the Lipschitz constants).We now use this to show that as in the smooth case the boundary of the causal future ∂J + ( S ), is ruled by null geodesics that are either past inextendible or end in ¯ S . This resultis needed for the proof of both Theorem 5.3 and Proposition 6.5. Proposition A.7.
Let S ⊆ M . Any x ∈ ∂J + ( S ) \ ¯ S is the future end point of a causal curve γ ⊆ ∂J + ( S ) that either is past inextendible (and never meets ¯ S ) or has a past endpoint in ¯ S .This γ is (can be reparametrised to) a maximising null geodesic. If S is closed and x / ∈ J + ( S ) ,then this curve is past inextendible and contained in ∂J + ( S ) \ J + ( S ) .Proof. Let x ∈ ∂J + ( S ) \ ¯ S . Then there exists a sequence { x k } ⊂ I + ( S ) with x k → x andpast directed timelike curves γ k : [0 , b k ] → M from γ k (0) = x k to γ k ( b k ) ∈ S . Since x / ∈ ¯ S the γ k ’s leave a fixed neighbourhood of x and so by Theorem A.6 there exists (a subsequencewith) a limit curve γ with γ (0) = x that is either past inextendible or b k → b < ∞ and29 ( b ) = lim γ k ( b k ) ∈ ¯ S . Clearly, γ ⊆ J + ( S ). If γ were ever in I + ( S ), then x ∈ I + ( S ) byLemma A.2, a contradiction.That γ is (can be reparametrised to) a maximizing null geodesic follows immediately fromLemma A.3. Finally, if S is closed and x / ∈ J + ( S ) there can be no causal curve from x to S = ¯ S , so γ must be inextendible and γ ⊆ ∂J + ( S ) \ J + ( S ). A.2 Cauchy development and Cauchy horizon
Next, we are interested in the Cauchy developments and Cauchy horizons of both E + ( S ) and ∂J + ( S ) (and their relationship with each other). From now on we will generally require S tobe an achronal (non-empty) set. Note that this implies in particular S ⊆ J + ( S ) \ I + ( S ) = E + ( S ) . (A.1)From this one also immediately obtains the following Lemma: Lemma A.8.
Let S be achronal. Then ¯ S is also achronal. Further, if E + ( S ) is compact,then E + ( S ) = E + ( S ) .Proof. The first claim follows from the fact that I + ( ¯ S ) = I + ( S ) and openness of I + ( S ). Thesame equality also immediately gives E + ( S ) ⊆ E + ( ¯ S ). Now if E + ( S ) is compact, then (A.1)implies ¯ S ⊆ E + ( S ). This gives E + ( ¯ S ) = J + ( ¯ S ) \ I + ( ¯ S ) ⊆ J + ( E + ( S )) \ I + ( ¯ S ). Since J + ( E + ( S )) = J + ( S ) and I + ( ¯ S ) = I + ( S ), this shows the other inclusion. Definition A.9.
Let A be achronal. The future Cauchy development D + ( A ) of A is definedby D + ( A ) := { x ∈ M : every past inextendible causal curve through x meets A } (A.2)and its future Cauchy horizon H + ( A ) is defined by H + ( A ) := D + ( A ) \ I − (cid:0) D + ( A ) (cid:1) = n x ∈ D + ( A ) : I + ( x ) ∩ D + ( A ) = ∅ o . (A.3)Two important properties of D + ( A ) for closed achronal sets A are given in the followingproposition. Proposition A.10.
Let A be closed and achronal. Then D + ( A ) = { x ∈ M : every past inextendible timelike curve through x meets A } . (A.4) Furthermore ∂D + ( A ) = A ∪ H + ( A ) . (A.5) Proof.
The proofs can be found in [16, Lemma A.13] and [16, Lemma A.14].
Lemma A.11.
Let A be closed and achronal and let x ∈ D + ( A ) \ H + ( A ) . Then every pastinextendible causal curve through x must meet I − ( A ) .Proof. Any x ∈ D + ( A ) \ H + ( A ) is either in A or in D + ( A ) ◦ , so the result follows from [13,Lemma 8.3.6], which still holds for C , -metrics. We follow the convention of [10, 11, 24], rather than that of [26, 12, 28]. emma A.12. Let A be closed and achronal and x ∈ J + ( A ) \ D + ( A ) or x ∈ I + ( A ) \ D + ( A ) ◦ .Then every causal curve from x to A must also meet H + ( A ) .Proof. Let x ∈ J + ( A ) \ D + ( A ) or x ∈ I + ( A ) \ D + ( A ) ◦ . If x ∈ D + ( A ), then x ∈ ∂D + ( A ) = A ∪ H + ( A ) (see Proposition A.10). Thus x ∈ H + ( A ) since in either case x cannot be in A because A ⊆ D + ( A ) and I + ( A ) ∩ A = ∅ by achronality, so we are done.Now assume x / ∈ D + ( A ) and let λ be a causal curve from x to A ⊆ D + ( A ). Then thereexists t > λ ( t ) ∈ ∂D + ( A ) but λ ( t ) / ∈ D + ( A ) for all t < t . We have to showthat λ ( t ) ∈ H + ( A ). Assume to the contrary that λ ( t ) ∈ A \ H + ( A ) (cf. (A.5)). Then I + ( λ ( t )) ∩ D + ( A ) = ∅ by definition of H + . Now let p ∈ I + ( λ ( t )) ∩ D + ( A ), then I − ( p ) is anopen neighbourhood of λ ( t ) so there exists a t < t such that λ ( t ) is still in I − ( p ). Since t < t we have λ ( t ) / ∈ D + ( A ), so, by (A.4), there exists a timelike past inextendible curve γ starting at λ ( t ) that does not meet A . Concatenating any timelike curve from p to λ ( t )with γ shows that this timelike curve from p to λ ( t ) must meet A in a point that cannot be λ ( t ) itself (since λ ( t ) / ∈ D + ( A )). But this means that λ ( t ) ∈ I − ( A ), giving a contradictionto λ ( t ) ≥ λ ( t ) ∈ A and achronality of A .We use this to give a proof of [12, Equation (2.4)] in the C , -setting. Lemma A.13.
Let A be closed and achronal. Then I + ( H + ( A )) = I + ( A ) \ D + ( A ) .Proof. By Proposition A.10 we have H + ( A ) ⊆ D + ( A ) ⊆ I + ( A ) ∪ A , so I + ( H + ( A )) ⊆ I + ( A ).Let x ∈ I + ( H + ( A )) and assume x ∈ D + ( A ), then there exists a neighbourhood U of x suchthat U ∩ D + ( A ) = ∅ and U ⊆ I + ( H + ( A )), contradicting I + ( H + ( A )) ∩ D + ( A ) = ∅ (cf. (A.3)).So I + ( H + ( A )) ⊆ I + ( A ) \ D + ( A ).Now let x ∈ I + ( A ) \ D + ( A ). Then by Lemma A.12 any timelike curve from x to A mustmeet H + ( A ) in some point p so, since x / ∈ D + ( A ) ⊇ H + ( A ) we have p = x , and thus x mustbe in I + ( H + ( A )). Lemma A.14.
Let S be closed and achronal. Then edge( H + ( S )) ⊆ edge(S) . Proof.
We basically follow the proof of [11, Prop. 6.5.2]. Let q ∈ edge( H + ( S )) and let U k bea sequence of neighbourhoods of q with U k → { q } . By definition of edge (cf. [24, 14.23]), foreach n there exist points p k ∈ I − ( q, U k ) and r k ∈ I + ( q, U k ) connected by a future directedtimelike curve λ k that does not intersect H + ( S ). It then follows that λ k does not intersect D + ( S ) ⊇ S .In particular, r k ∈ I + ( q, U k ) ⊆ I + ( q ), so q ∈ I − ( r k ). Hence, I − ( r k ) is a neighbourhood of q , so I − ( r k ) ∩ H + ( S ) = ∅ , so r k ∈ I + ( H + ( S )). Therefore, by Lemma A.13, r k ∈ I + ( S ), but r k D + ( S ). Thus, if λ k would intersect D + ( S ), it would also have to intersect the boundaryof that set, i.e., S ∪ H + ( S ) (by (A.5)), and thereby S . But then Lemma A.12, applied to x = r k would imply that λ k intersects H + ( S ), a contradiction.It remains to show that q ∈ ¯ S . Since q ∈ D + ( S ) we have I − ( q ) ⊆ I − ( D + ( S )) ⊆ I − ( S ) ∪ D + ( S ). It follows that p k ∈ I − ( q ) \ D + ( S ) ⊆ I − ( S ). Let α k be a timelike curve from q to p k contained in U k and extend it to the past to become past inextendible. As q ∈ edge( H + ( S )) ⊆ H + ( S ) ⊆ D + ( S ), this curve must, by Proposition A.10, intersect S in apoint z k . Since p k ∈ I − ( S ) and S is achronal any such z k must lie between q and p k , hence z k ∈ U k . Thus z k → q , and therefore q ∈ S . 31 emma A.15. Let S be achronal. Then the Cauchy horizon H + ( ∂J + ( S )) of ∂J + ( S ) is aclosed, achronal topological hypersurface.Proof. Clearly H + ( ∂J + ( S )) is closed and achronality follows from Lemma A.13. By Lemma A.14(and Lemma A.4), edge ( H + ( ∂J + ( S ))) ⊆ edge( ∂J + ( S )) = ∅ (see Lemma A.5 and [16,Prop. A.18]), so the claim follows from [16, Prop. A.18]. Lemma A.16.
Let S be closed and achronal. Then H + ( E + ( S )) ⊆ H + ( ∂J + ( S )) .Proof. We roughly follow the proof of [13, Lemma 9.3.1]. Assume to the contrary that thereexists p ∈ H + ( E + ( S )) \ H + ( ∂J + ( S )). Since E + ( S ) ⊆ ∂J + ( S ) we have D + ( E + ( S )) ⊆ D + ( ∂J + ( S )), so p ∈ I − ( D + ( ∂J + ( S ))). Thus there exists q in I + ( p ) ∩ D + ( ∂J + ( S )) andbecause p H + ( ∂J + ( S )) and H + ( ∂J + ( S )) is closed, we may additionally assume that q / ∈ H + ( ∂J + ( S )). This q is in I + ( H + ( E + ( S ))) so by Lemma A.13 q / ∈ D + ( E + ( S )). Thusby Proposition A.10 there exists a past inextendible timelike curve λ starting in q that nevermeets E + ( S ). However, as any such curve must meet ∂J + ( S ) there exists z ∈ λ with z ∈ ∂J + ( S ) \ E + ( S ). By Proposition A.7 there exists a past inextendible null curve µ ⊆ ∂J + ( S ) \ E + ( S ) starting in z . Finally by Lemma A.11 the concatenation of λ and µ must enter I − ( ∂J + ( S )), contradicting the achronality of ∂J + ( S ). A.3 Strong causality
Finally we are going to collect some results concerning strong causality.
Definition A.17.
Strong causality holds at a point p ∈ M if for every neighbourhood U of p there exists a neighbourhood V of p with V ⊆ U such that every causal curve in M thatstarts and ends in V is entirely contained in U .As in the smooth case there is the following alternative definition. Lemma A.18.
Strong causality holds at p if and only if for every neighbourhood U of p thereexists a neighbourhood V of p with V ⊆ U such that no causal curve in M intersects V morethan once.Proof. See [22, Lem. 3.21].
Lemma A.19. If M is chronological and every inextendible null geodesic is not maximising,then strong causality holds throughout M .Proof. The proof is similar to the smooth case, see, e.g., [2, Prop. 12.39] or [13, Lem. 8.3.7].Assume to the contrary that strong causality does not hold at some point p ∈ M . Thenthere exists a neighbourhood U of p and neighbourhoods V k of p with T k ∈ N V k = { p } andfuture directed causal curves γ + k parametrised with respect to h -arclength that start at p k = γ + k (0) ∈ V k and end at q k = γ + k ( b k ) ∈ V k but leave U . Hence by Theorem A.6, thereexists a causal limit curve γ + starting at p . We may assume that this limit curve is futureinextendible: Otherwise b k → b < ∞ and p = lim k →∞ γ + k ( b k ) = γ + ( b ), so γ + is a closedcausal curve. But then Lemma A.2 and Lemma A.3 show that two points on γ + couldbe connected by a timelike curve because no inextendible null geodesic is maximising byassumption, contradicting chronology. 32y the same argument, only using the (also future directed) curves γ − k : [ − b k , → M defined by γ − k ( t ) := γ + k ( b k + t ), one obtains a past inextendible causal limit curve γ − startingat p . Together these two limit curves form an inextendible causal curve γ .Since γ is inextendible there are points x = γ ( t − ) and y = γ ( t + ) on γ that can beconnected by a timelike curve. We may assume y ∈ J + ( p ) and x ∈ J − ( p ) by Lemma A.2 and γ − k ( t − ) → γ ( t − ) and γ + k ( t + ) → γ ( t + ). Since the relation ≪ is open (see [21, Sec. 1.4] or [15,Cor. 3.12]) this implies γ − k ( t − ) ≪ γ + k ( t + ) for k large. Then γ − k ( t − ) = γ + k ( t − + b k ) ≪ γ + k ( t + )and by b k → ∞ we get t − + b k > t + for large enough k , but this yields γ + k ( t + ) ≤ γ + k ( t − + b k ) ≪ γ + k ( t + ), hence there exists a closed timelike curve through γ + k ( t + ), contradicting chronologyof M .As already remarked in [31, Def. 2.6], the proof of [24, Lem. 14.13] remains true even forcontinuous metrics and so strong causality implies that the spacetime is both non-totally and non-partially imprisoning , meaning that no future (or past) inextendible causal curve canremain in a compact set or return to it infinitely often. This gives Lemma A.20.
Let M be strongly causal and let γ be an inextendible causal curve in M .Then (the image of ) γ is a closed subset of M .Acknowledgements. We are greatly indebted to James Vickers for several discussions thathave importantly contributed to this work. We also thank Clemens S¨amann for valuableinput. The work of JG was partially supported by STFC Consolidated Grant ST/L000490/1.MG is the recipient of a DOC Fellowship of the Austrian Academy of Sciences. This workwas supported by project P28770 of the Austrian Science Fund FWF. Finally, we gratefullyacknowledge the kind hospitality of the Erwin Schr¨odinger Institute ESI during the thematicprogramme “Geometry and Relativity”.
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