Aspects of $C^0$ causal theory

This paper serves as an introduction to $C^0$ causal theory. We focus on those parts of the theory which have proven useful for establishing spacetime inextendibility results in low regularity -- a question which is motivated by the strong cosmic censorship conjecture in general relativity. This paper is self-contained; prior knowledge of causal theory is not assumed.


Introduction
Recently, there has been an interest in low regularity aspects of Lorentzian geometry motivated in part by the strong cosmic censorship conjecture in general relativity. Roughly, the conjecture states that the maximal globally hyperbolic development of generic initial data for the Einstein equations is inextendible as a suitably regular Lorentzian manifold. Formulating a precise statement of the conjecture is itself a challenge because one needs to make precise the phrases 'generic initial data' and 'suitably regular Lorentzian manifold.' Understanding the latter is where general relativity in low regularity and inextendibility results become significant. The strongest form of the conjecture would prove inextendibility in the lowest regularity possible -continuity of the metric. Proving C 0 -inextendibility results is a nontrivial pursuit. The classical arguments of diverging curvature quantities only prove C 2inextendibility (since the curvature tensor requires two derivatives of the metric to be defined). The first example of a C 0 -inextendibility result is Sbierski's impressive proof of the C 0 -inextendibility of Schwarzschild [19,20]. Since then other inextendibility results have been found [4][5][6][7] and also within the context of Lorentzian length spaces [8] and Lorentz-Finsler spaces [14].
Understanding which spacetimes are C 0 -inextendible is a highly investigated research problem in Lorentzian geometry. Therefore an understanding of causal theory for C 0 spacetimes is necessary for anyone who wants to break into the field. This paper serves as an introduction to C 0 causal theory. We focus on those parts of the theory which have proven useful for inextendibility results. These are -I + (p) is open (Theorem 2.11).
The main difference between C 0 and smooth (at least C 2 ) causal theory is the existence of bubbling sets in C 0 spacetimes. This was shown in [3]. Bubbling sets are open sets of the form B + (p) = int J + (p) \ I + (p). In appendix A.1 we show that B + (p) = ∅ for all points in a C 2 spacetime. Hence bubbling sets are irrelevant in C 2 causal theory. But they play a prominent role in C 0 causal theory. Section 4.1 introduces them. In section 4.2 we offer a notion of a trapped set for C 0 spacetimes and prove a C 0 version of Penrose's theorem: if a C 0 spacetime has a noncompact Cauchy surface, then there are no trapped sets.
The treatment of C 0 causal theory in [3] and [18] uses a sequence of wider and narrower smooth metrics to approximate the C 0 metric. They then infer C 0 causal theory results from knowledge of smooth causal theory. Our approach is different. We obtain our results directly by using continuity to locally approximate the metric with wider and narrower metrics built from the Minkowski metric (Lemma 2.8). See also [13] which includes the results above and generalizes causal theory even further with notable applications. This paper is self-contained; prior knowledge of causal theory is not assumed. We only assume the Hopf-Rinow theorem and basic integration theory.
2 Preliminary causal theory for C 0 spacetimes 2.1 C 0 spacetimes Let k ≥ 0 be an integer. A C k metric on a smooth manifold M is a nondegenerate symmetric tensor g : T M × T M → R with constant signature whose components g µν = g(∂ µ , ∂ ν ) in any coordinate system are C k functions. Symmetric means g(X, Y ) = g(Y, X) for all X, Y ∈ T M . Nondegenerate means g(X, Y ) = 0 for all Y ∈ T M implies X = 0. With constant signature means there is an integer r such that at each point p ∈ M , there is a basis e 0 , . . . , e r , . . . , e n ∈ T p M such that g(e µ , e µ ) = −1 for 0 ≤ µ ≤ r and g(e µ , e µ ) = 1 for r + 1 ≤ µ ≤ n. If g(e 0 , e 0 ) = −1 and g(e i , e i ) = 1 for all i = 1, . . . , n, then g is called a Lorentzian metric and (M, g) is called a Lorentzian manifold. If g(e µ , e µ ) = 1 for all µ = 0, 1, . . . , n, then g is called a Riemannian metric and (M, g) is called a Riemannian manifold. When working with Lorentzian manifolds, our convention will be that Greek indices µ and ν will run through 0, 1, . . . , n and Latin indices i and j will run through 1, . . . , n.
If (M, g) is a Lorentzian manifold, then a nonzero vector X ∈ T p M is timelike, null, or spacelike if g(X, X) < 0, = 0, > 0, respectively. A nonzero vector is causal if it is either timelike or null. A Lorentzian manifold (M, g) is time-oriented provided there is a C 1 timelike vector field X on M . A causal vector Y ∈ T p M is future directed if g(X, Y ) < 0 and past directed if g(X, Y ) > 0. Note that −X defines an opposite time-orientation, and so any statement/theorem in a spacetime which is time-oriented by X has a time-dual statement/theorem with respect to the time-orientation given by −X.
Definition 2.1. Let k ≥ 0. A C k spacetime (M, g) is a time-oriented Lorentzian manifold with a C k metric such that M is connected, Hausdorff, and second-countable.
We now proceed to define timelike and causal curves. Our class of curves that we consider should be sufficiently regular so that we can integrate along them, but not too regular so that limit curves are not considered causal curves. The class of locally Lipschitz curves live in this Goldilocks zone.
Fix a C 0 spacetime (M, g) and a complete Riemannian metric h on M . Let I ⊂ R be an interval (i.e. any connected subset of R with nonempty interior). A locally Lipschitz curve γ : I → M is a continuous function such that for any compact K ⊂ I there is a constant C such that for any a, b ∈ K, we have where d h is the Riemannian distance function associated with h. Proposition 2.2 shows that we can integrate along locally Lipschitz curves. Proposition 2.2. If γ : I → M is locally Lipschitz, then the components γ µ = x µ • γ in any coordinate system are differentiable almost everywhere and these derivatives are in L ∞ loc .
We include a discussion of locally Lipschitz curves in appendix A.2 where we prove Proposition 2.2. But we mention here that Proposition 2.2 is an immediate consequence of Radaemacher's theorem which is a higher-dimensional generalization of the well-known fact that if f : [a, b] → R has Lipschitz constant C, then f is differentiable almost everywhere and |f ′ | ≤ C almost everywhere. A discussion of Rademacher's theorem in this setting along with references can be found in [2].
(1) A causal curve is a locally Lipschitz curve γ : I → M such that γ ′ is future-directed causal almost everywhere.
(2) A timelike almost everywhere curve is a causal curve γ : I → M such that γ ′ is futuredirected timelike almost everywhere.
-Our definition of a timelike curve is analogous to the locally uniform timelike curves which appear in [3].
-Note that 'future-directed' is implicit in our definition of causal and timelike curves. Hence all causal curves in this paper will be future-directed.
-Given a set S ⊂ M and a causal curve γ : I → M , we will write γ ⊂ S instead of γ(I) ⊂ S. Likewise with the intersection γ ∩ S. on the left is a timelike curve. The curve on the right is a timelike almost everywhere curve. It is not a timelike curve because it approaches a null vector at its break point. Remarks.
-The causal past J − (S, U ) and timelike past I − (S, U ) are defined time-dually. Any statement/theorem for J + has a corresponding time-dual statement/theorem for J − . Likewise with I + and I − . For example, the proof that I + is open (Theorem 2.11) has a corresponding time-dual proof that I − is open.
-If I + is defined via timelike almost everywhere curves, then it is not necessarily open (see [9]). This is the main distinction between timelike curves and timelike almost everywhere curves.
-If U = M , then we will write I + (S) instead. If S = {p}, then we will write I + (p, U ) instead. Likewise with J + . If we wish to emphasize the Lorentzian metric g being used, then we will write I + g and J + g .
-Given our convention, the constant curve γ : [0, 1] → M given by γ(t) = p for all t is not a causal curve. This is why we include the union with S in our definition of J + (S, U ).

Properties of timelike and causal curves
For this section fix a C 0 spacetime (M, g), and let h be a complete Riemannian metric on M . The goal of this section is to prove the following two important properties of timelike and causal curves: (1) I + (p) and I − (p) are open sets. This is Theorem 2.11.
(2) A causal curve is inextendible if and only if it has domain R when parameterized by h-arclength. This is Theorem 2.18.
To simplify arguments, we will often parameterize causal curves by x 0 within a coordinate neighborhood. This is possible because x 0 is a time function for a small enough neighborhood (see (3) in Lemma 2.8).
Proposition 2.6. Let τ : U → R be a time function and γ : [a, b] → U a causal curve. Then and γ has a reparameterization which is parameterized by τ .
Proof. Integrating gives The last inequality holds because ∇τ is a past-directed timelike vector field. Then τ • γ is a strictly increasing continuous function with a positive derivative almost everywhere; hence it's invertible with continuous inverse that's differentiable almost everywhere. The reparameterization we seek isγ = γ • (τ • γ) −1 .
To show (4), let X = X µ ∂ µ be any tangent vector. Then we have Using the fact that η = η ε − 2ε 1+ε (dx 0 ) 2 , we have By taking ε 0 > 0 small enough, we can ensure 2ε/(1 + ε) is larger than the bracket term. Then for this choice of ε 0 , there is a δ > 0 such that This proves the first implication in (4). The second implication is obvious. The third implication follows from a similar argument used to prove the first. Now we prove (5). If γ is an η ε -timelike curve parameterized by x 0 , then we have g(γ ′ , γ ′ ) < η ε (γ ′ , γ ′ ) − δ which shows that γ is a g-timelike curve (note that the property of being timelike is invariant under reparameterization by x 0 ). This shows the first inclusion in (5). The second inclusion is obvious. And the third inclusion follows from a similar argument used to prove the first. Definition 2.9. Let ·, · and | · | denote the standard inner product and norm on R n+1 with its standard global orthonormal basis {e 0 , e 1 , . . . , e n }. Given any open set U ⊂ R n+1 and any point p ∈ U , we define for −1 < ε < 1 Remark. C + ε is the usual interior of a cone in R n+1 which makes an angle θ with respect to the x 0 -axis where θ is given by cos θ = (1 + ε)/2. Note that C + 0 coincides with the forward lightcone in Minkowski space.
x 0 Figure 2: The coordinate system φ : U ε → R n+1 appearing in Lemma 2.8. The point p is located at the origin where the metric is exactly Minkowski: g µν (p) = η µν . Any causal curve γ ⊂ U ε will always be η −ε -timelike but it may be η ε -spacelike.
Proof. We only prove (1) as the proof of (2) is analogous. We first prove the left inclusion of (1). Let q ∈ C + ε (p, B). Let γ : [0, 1] → B be the straight line γ(t) = qt + (1 − t)p. Then γ ′ (t) = q − p. Put q − p = X = X µ e µ . By definition we have X 0 /|X| > (1 + ε)/2. Notice that |X| 2 = X, X = |X 0 | 2 + δ ij X i X j . Hence Since X is independent of t, there is an ε 0 > 0 such that η ε (γ ′ , γ ′ ) < −ε 0 . Therefore q ∈ I + η ε (p, B). Now we prove the right inclusion of (1). Suppose q ∈ I + η ε (p, B). Let γ ⊂ B be an η ε -timelike curve from p to q. To help visualize the proof consider ε = 15/17 and ε ′ = 3/5 which correspond to lightcones with 'slopes' 4 and 2, respectively. Consider the hyperplanes given by x 0 − 2x 1 = constant. Note that these hyperplanes are η ε ′ -null but η ε -spacelike. Let τ be the η ε -time function such that ∇τ is orthogonal to these hyperplanes. Apply Proposition 2.6 with g = η ε to conclude that γ lies above the particular hyperplane which intersects p. Now replace x 1 with any arbitrary direction orthogonal to ∂/∂x 0 , and apply Proposition 2.6 again to conclude that γ ⊂ C + ε ′ (p, B). Clearly this proof does not depend on the specific choices of ε and ε ′ . Proof. Fix q ∈ I + (p, U ) and let γ : [a, b] → U be a timelike curve from p to q. Let φ : U ε → R n+1 be a coordinate system about q from Lemma 2.8 with U ε ⊂ U . From continuity of the metric, for every ε 0 > 0, we can shrink U ε further so that |g µν (x) − η µν | < ε 0 for all x ∈ U ε . For the portion of γ within U ε , reparameterize γ by the time function x 0 . Put X = γ ′ so that within U ε we have X 0 = 1. By definition of a timelike curve, there is a δ > 0 such that g(X, X) < −δ almost everywhere. Then using η µν < g µν (x) + ε 0 and a similar calculation as in the proof of Lemma 2.8, we have Rearranging and using X 0 = 1, we arrive at Choose ε < 3/5. Then by the third implication in (4) of Lemma 2.8, we have |X i | < 2. Therefore the term in the bracket is bounded by 1 + 4n + 4n 2 . Thus we can choose ε and ε 0 sufficiently small so that With this choice of ε, the portion of γ within U ε is η ε -timelike. Let B ⊂ φ(U ε ) be an open Euclidean ball centered around φ(q). Choose t 0 < b such that φ • γ(t 0 ) ∈ B. Let q 0 = γ(t 0 ) and recall q = γ(b). Then we just showed q ∈ I + η ε q 0 , φ −1 (B) . Choose ε ′ ∈ (0, ε). Then (1) from Lemma 2.10 implies The last inclusion follows from (5) in Lemma 2.8.   Proof. By Proposition 2.2, the components of γ are differentiable almost everywhere and these derivatives are in L ∞ loc . Therefore the integral is well-defined and finite where t 0 , t ∈ I. Since γ is causal, we have γ ′ = 0 almost everywhere. Therefore s(t) is a strictly increasing continuous function; hence it's invertible. Moreover s is differentiable almost everywhere with s ′ > 0 wherever differentiable; hence s −1 is differentiable almost everywhere. The reparameterization we seek isγ = γ • s −1 . Then in Definition 2.14. Let γ : [a, b) → M be a causal curve. Suppose there exists a p ∈ M such that γ(t n ) → p for every sequence t n ր b. Then p is called the future endpoint of γ. Past endpoints are defined time-dually.
Remark. Future and past endpoints are unique since M is Hausdorff.
When γ : [a, b) → M has a future endpoint p, one is tempted to define a new curvẽ γ : [a, b] → M such thatγ(t) = γ(t) for t < b andγ(b) = p. However a problem arises: the extended curveγ may not be locally Lipschitz. For example if one extends the curve t → √ t + 1, 0 in two-dimensional Minkowski space from (−1, 0] to [−1, 0] so that it includes the past endpoint (0, 0), then the new curve defined on [−1, 0] will not be locally Lipschitz because diverges as t and t ′ approach 0. However if we reparameterize causal curves with respect to h-arclength, then this problem goes away.
Proof. Since γ is parameterized with respect to h-arclength, we have Then B is compact by Hopf-Rinow. Let t n ր b be any sequence. Then γ(t n ) has an accumulation point p ∈ B. Therefore there is a subsequence (still denoted by t n ) such that t n ր b and γ(t n ) → p. We show that p is the future endpoint of γ. Let γ(s n ) be any other sequence with s n ր b.
Then d h γ(s n ), γ(s m ) ≤ |s n − s m | implies that γ(s n ) is a Cauchy sequence. Since (M, h) is complete, Hopf-Rinow implies γ(s n ) converges to some point q. Put p n = γ(t n ) and q n = γ(s n ). Then the triangle inequality gives Each of the three terms on the right hand side can be made arbitrarily small. Therefore d h (p, q) = 0 and so p = q. Hence p is the future endpoint of γ.
Henceγ is locally Lipschitz with Lipschitz constant 1. Thereforeγ is a causal curve.
The following technical proposition is needed for Theorem 2.18.
Remark. Proposition 2.15 shows that if p is a future endpoint for a causal curve γ parameterized by h-arclength, then γ is future extendible. (1) If b = ∞, then γ is future inextendible.
(2) If b < ∞, then γ can be extended to a future inextendible causal curve.
Proof. Suppose b = ∞ and γ is future extendible. Then there is a causal curveγ : [a, ∞] → M which extends γ. Let p =γ(∞). Then for any neighborhood U of p, we have L h γ| U = ∞ which contradicts Proposition 2.16. This proves (1). Now suppose b < ∞ and set By Proposition 2.15, we know that γ extends to a causal curveγ on [a, b]. Using a coordinate system from Lemma 2.8 centered aroundγ(b), we can extendγ even further (e.g. by concatenatingγ with the positive x 0 -axis). Therefore the set appearing in the above supremum is nonempty. Suppose c < ∞. Then there is a causal curve λ : [a, c) → M which extends γ. Proposition 2.15 implies λ extends to a causal curveλ : [a, c] → M . We can extendλ using a coordinate neighborhood via the same argument above. This contradicts the definition of c. Therefore c = ∞. Thus γ extends to a h-arclength parameterized causal curve on [a, ∞) which is future inextendible by (1). This proves (2).

Limit curves
Fix a C 0 spacetime (M, g) with a complete Riemannian metric h on M . The purpose of this section is to prove the limit curve theorem in the C 0 setting.
Definition 2.19. Let γ n : I → M be a sequence of causal curves. A causal curve γ : I → M is a limit curve of γ n if there is a subsequence of γ n which converges to γ uniformly on compact subsets of I.
Remark. Note that limit curves are not necessarily unique. For this reason limit curves are called 'accumulation curves' in [2,3].
Theorem 2.20 (Limit Curve Theorem). Let γ n : R → M be a sequence of causal curves parameterized by h-arclength. If p is an accumulation point of the γ n , then there is an inextendible limit curve γ : R → M which passes through p.
The proof of of the limit curve theorem requires three things: (1) The existence of the limit curve γ which will follow from the Arzelá-Ascoli theorem. (2) An argument to prove that γ is in fact a causal curve. (3) An argument showing that γ is indeed inextendible. This last part is necessary because, although each γ n is parameterized by h-arclength, there is no guarantee that the limit curve γ will be. Theorem 2.21 (Arzelá-Ascoli). Let (M, d) be a metric space. If the sequence γ n : R → M is equicontinuous and for each t 0 ∈ R the set n {γ n (t 0 )} is bounded, then there exists a continuous γ : R → M and a subsequence of γ n which converges to γ uniformly on compact subsets of R.
Proposition 2.22. Let γ n : R → M be a sequence of causal curves parameterized by harclength which accumulate at a point p. Then there is a locally Lipschitz curve γ : R → M and a subsequence of γ n which converges to γ uniformly on compact subsets of R.
Proof. We apply Arzelá-Ascoli to the metric space (M, d h ). Equicontinuity follows from the h-arclength parameterization: For ε > 0 choose δ = ε. Then for all n and all |a − b| < δ, By restricting to a subsequence and shifting parameterizations, we can assume γ n (0) → p.
A similar argument shows that γ ′ (t 0 ) cannot be past-directed causal. Since it's not zero by assumption, we have γ ′ (t 0 ) is future-directed causal. Proposition 2.24. Suppose γ n : I → M is a sequence of causal curves parameterized by h-arclength which converges uniformly to a locally Lipschitz curve γ : I → M on compact subsets of I. Then γ is a causal curve.
Proof. By Proposition 2.23 it suffices to show that γ ′ = 0 almost everywhere. Let t 0 ∈ I and p = γ(t 0 ). Consider a neighborhood U from Lemma 2.8 centered around p. Assume U has compact closure. Define C by Claim: C > 0. Suppose not. Then we can find a sequence X n ∈ T qn M with g(∇x 0 , X n ) > 0 and h(X n , X n ) = 1 such that g(∇x 0 , X n ) → 0. Since U has compact closure, the hunit bundle of U within T M has compact closure. Therefore there is a subsequence (still denoted by X n ) such that X n → X and continuity implies h(X, X) = 1 and g(∇x 0 , X) = 0. However, since ∇x 0 is timelike, g(∇x 0 , X) = 0 implies X is the zero vector. This contradicts h(X, X) = 1 which proves the claim.
Fix a < b in I such that γ(a), γ(b) ∈ U . For n sufficiently large, we have γ n (a), γ n (b) ∈ U . For these large n, we have .
Since a < b was arbitrary, Lebesgue's differentiation theorem implies g(∇x 0 , γ ′ ) ≥ C > 0 almost everywhere. Thus γ ′ = 0 almost everywhere for points in I of γ which lie in U . Since t 0 ∈ I was arbitrary, we have γ ′ = 0 almost everywhere in I. Proposition 2.25. Let γ n : R → M be a sequence of causal curves parameterized by harclength which converge to a causal curve γ : R → M uniformly on compact subsets of R. Then γ is inextendible.
Proof. Since γ is a causal curve, it has an h-arclength reparameterizationγ : (a, b) → M by Proposition 2.13. Seeking a contradiction, suppose γ is not future inextendible. Thenγ is not future inextendible and so b < ∞ by Theorem 2.18. Proposition 2.15 implies there is a future endpoint p ∈ M such thatγ extends continuously through p. By Proposition 2.16, there is an open set U around p such that L h (λ) < 1 for all causal curves λ ⊂ U . Since p is the future endpoint of γ, we have lim t→∞ γ(t) = p and hence there is some t 0 such that γ [t 0 , ∞) ⊂ U . Since the sequence γ n converges uniformly to γ on compact subsets, there exists an N such that γ n [t 0 , t 0 If γ is a causal curve from p to q such that its Lorentzian length satisfies L(γ) ≥ L(λ) for any other causal curve λ from p to q, then γ is called a causal maximizer from p to q. In this section we will show that globally hyperbolic spacetimes always contain causal maximizers between causally related points (Theorem 3.2). We first define global hyperbolicity. Remarks.
-A spacetime is causal if there are no closed causal curves. A strongly causal spacetime implies that it's causal.
-For C 2 spacetimes strong causality can be weakened to causality in the definition of global hyperbolicity [1], and if the spacetime dimension is greater than 2, then even causality is not needed [10] . For C 0 spacetimes strong causality can be weakened to non-totally imprisoning [18].
The goal of this section is to prove the following fundamental result which was first established by Sämann in [18] and later proved independently in [6].
Theorem 3.2. Let (M, g) be a globally hyperbolic spacetime. Then given any q ∈ J + (p), there is a causal maximizer γ from p to q. Moreover L(γ) < ∞. Remarks.
-Theorem 3.2 shows that globally hyperbolic spacetimes are analogous to complete Riemannian manifolds: there is always a length-minimizing curve between any two points in a complete Riemannian manifold.
To prove Theorem 3.2, we first establish some facts about strongly causal spacetimes. The following proposition shows that compact sets in a strongly causal spacetime cannot contain inextendible causal curves. This is often referred to as the 'no imprisonment' property.
Fix a causal curve γ : [a, b] → K. There exists some set from the finite cover which contains γ(a), say V 1 . Define s 1 via s 1 = sup{t | γ(t) ∈ V 1 }. If s 1 = b, then γ(s 1 ) / ∈ V 1 , and so there exists some set which contains γ(s 1 ), say V 2 . Define s 2 via s 2 = sup{t | γ(t) ∈ V 2 }. If s 2 = b, then γ(s 2 ) / ∈ V 2 , and so there exists some set which contains γ(s 2 ), say V 3 . And so on until Then γ i ⊂ V i and by the first paragraph of this proof, we have L h (γ i ) ≤ 1 for all i = 1, . . . , k. Therefore The limit curve theorem guarantees the existence of a limit curve when there is one accumulation point. The next proposition shows that for strongly causal spacetimes we can apply the limit curve theorem to two accumulation points within a compact set.
Proposition 3.4. Suppose (M, g) is strongly causal and K ⊂ M is a compact set. Let γ n : [a n , b n ] → K be a sequence of causal curves parameterized by h-arclength such that γ n (a n ) → p and γ n (b n ) → q. Then there exists a, b ∈ R and a limit curveγ : [a, b] → M from p to q ofγ n | [a,b] whereγ n : R → M are inextendible causal curve extensions of γ n .
Proof. Let γ n : [a n , b n ] → K be a sequence of causal curves parameterized by h-arclength. By Theorem 2.18, we can extend γ n : [a n , b n ] → K to inextendible causal curvesγ n : R → M . By the limit curve theorem, there is a subsequence (which we still denote byγ n ) which converges to an inextendible causal curveγ that passes through p. Therefore there is an a > −∞ such that a n → a andγ(a) = p. Since every sequence in R contains a monotone subsequence, we can assume b n is monotone by restricting to a further subsequence. Then either (1) b n → ∞ or (2) b n → b < ∞. The first scenario is ruled out by Proposition 3.3. Therefore the second scenario must hold. The triangle inequality gives Since γ n (b n ) → q and b n → b, the RHS limits to 0. Thusγ n (b) → q. Thereforeγ| [a, b] is a causal curve from p to q which is a limit curve ofγ n | [a,b] .
Proposition 3.5. If (M, g) is globally hyperbolic, then J + (p) is closed for all p.
Proof. Let q be an accumulation point of J + (p). Then there is a sequence of points q n → q and causal curves γ n from p to q n . Let r ∈ I + (q). Since I − (r) is open, there is an integer N such that n ≥ N implies q n ∈ I − (r). For these n, we have γ n ⊂ K where K is the compact set K = J + (p) ∩ J − (r). Proposition 3.4 implies there is a causal curve from p to q.
The following technical lemma is needed for the proof of Theorem 3.2.
Lemma 3.6. Given any p ∈ M and ε > 0, there is a neighborhood U such that L(γ) < ε for all causal curves γ ⊂ U .
The proof of Theorem 3.2 hinges on the upper semi-continuity of the Lorentzian length functional. This is Proposition 3.7. We first use it to prove Theorem 3.2. In section 3.3 we prove Proposition 3.7. We first show L < ∞. Let K = J + (p) ∩ J − (q). By Lemma 3.6 for each x ∈ K, there is a neighborhood U x such that L(γ) ≤ 1 for all γ ⊂ U x . By strong causality, there are neighborhoods V x ⊂ U x such that γ ⊂ U x whenever γ : [a, b] → M is a causal curve with endpoints in V x . Since K is compact and covered by {V x } x∈K , there is a finite subcover V 1 , . . . , V N . Therefore L is bounded by N via the same proof used in Proposition 3.3. By definition of L there is a sequence of causal curves γ n : [a n , b n ] → M from p to q satisfying L ≤ L(γ n ) + 1/n. Let γ n be parameterized by h-arclength. Proposition 3.4 guarantees a limit curveγ : [a, b] → M from p to q ofγ n | [a,b] whereγ n : R → M are inextendible extensions of γ n . By restricting to a subsequence, we can assumeγ n | [a,b] converges uniformly toγ. By upper semi-continuity of the length functional, given any ε > 0 there exists an N such that n ≥ N implies Since this is true for all n ≥ N , we have L(γ) + ε ≥ L (note we used Proposition 2.2 here).
Since ε was arbitrary we have L(γ) ≥ L. Thusγ is a causal maximizer from p to q.

Cauchy surfaces imply global hyperbolicity
Fix a C 0 spacetime (M, g). In this section we show that global hyperbolicity is implied by the more familiar notion of a Cauchy surface: Theorem 3.8. If (M, g) has a Cauchy surface, then (M, g) is globally hyperbolic.
Remark. In fact global hyperbolicity is equivalent to the existence of a Cauchy surface [18].  Proof. If I + (S) ∩ S = ∅, then there would be a timelike curve γ : [a, b] → M with endpoints on S. We can extend γ to an inextendible causal curveγ via Theorem 2.18. But theñ γ intersects S twice which contradicts the definition of a Cauchy surface. Therefore S is achronal.
Seeking a contradiction, suppose p ∈ S is an edge point of S. Let U be a neighborhood of p. Then there is a timelike curve γ : [a, b] → U such that γ(a) ∈ I − (p, U ), γ(b) ∈ I + (p, U ), and γ ∩ S = ∅. We can extend γ to an inextendible causal curveγ : R → M . Since S is a Cauchy surface, there exists some t 0 such thatγ(t 0 ) ∈ S. By assumption we have t 0 / ∈ [a, b]. Suppose t 0 < a. Since p ∈ S, there is a sequence of points p n ∈ S such that p n → p. Therefore for all sufficiently large n, we have p n ∈ I + γ(a), U . Then there is a causal curve fromγ(t 0 ) to γ(a) to p n . This contradicts the definition of a Cauchy surface. Likewise supposing t 0 > b yields a contradiction.
Corollary 3.11. If S is a Cauchy surface, then S is a C 0 hypersurface.
Proof. This follows from Theorem A.6. Proof. Fix p ∈ M . Since M is time-oriented, there is a C 1 timelike vector field X on M . Let γ p denote the maximal integral curve of X through p. Since γ p is maximal and hence inextendible as a continuous curve, it is an inextendible causal curve. Let γ p : R → M be parameterized by h-arclength with γ p (0) = p. Since γ p is an inextendible causal curve, it must intersect S at some point t 0 . If t 0 = 0, then p ∈ S. If t 0 > 0, then γ p | [0,t 0 ] is a C 1 timelike curve, so p ∈ I − (S). Likewise, if t 0 < 0, then p ∈ I + (S). The disjointness follows from S being achronal.  Proof. Let γ : R → M be an inextendible causal curve. Seeking a contradiction, suppose γ does not intersect I + (S). Since S is a Cacuhy surface, there exists some t 0 such that γ(t 0 ) ∈ S. Let t 1 > t 0 . By Lemma 3.12, we have γ(t 1 ) ∈ S ∪ I − (S). If γ(t 1 ) ∈ S, then there is a causal curve from γ(t 0 ) to γ(t 1 ) which is a contradiction. If γ(t 1 ) ∈ I − (S), then there is a causal from γ(t 0 ) to γ(t 1 ) to a point on S -again a contradiction. Therefore γ must intersect I + (S). Likewise γ must intersect I − (S). Proof. Extend γ to an inextendible causal curveγ : R → M via Theorem 2.18. If the conclusion did not hold, then Proposition 3.14 implies there exists a t 0 < 0 such that γ(t 0 ) ∈ I + (S). By Lemma 3.12, we must have γ(0) ∈ S ∪ I − (S). But this implies we can find a causal curve which intersects S twice. Lemma 3.16. Let S be a Cauchy surface. If p ∈ I + (S), then sup {L h (γ) | γ is a causal curve from S to p} < ∞.
Proof. Suppose this is not true. Then we can find a sequence of h-arclength parameterized causal curves γ n : [a n , 0] → M from S to p such that a n → −∞. By Theorem 2.18 we can extend these curves to inextendible causal curvesγ n : R → M . By the limit curve theorem there is a subsequence (still denoted byγ n ) which converges to an inextendible causal curve γ : R → M passing through p = γ(0).
Recall that a causal curve γ :  Proof. Suppose strong causality failed at the point p. Then there is a neighborhood U and a sequence of causal curves γ n : [0, b n ] → M parameterized by h-arclength such that γ n (0) → p and γ n (b n ) → p but each γ n leaves U . Note this implies there is a c > 0 such that b n > c for all n. Extend each γ n to inextendible causal curvesγ n : R → M via Theorem 2.18. Since γ n (0) → p, the limit curve theorem yields a limit curve γ : R → M of theγ n such that γ(0) = p. Therefore there is a subsequence (still denoted byγ n ) such thatγ n converges to γ uniformly on compact subsets. By restricting to a further subsequence, we either have (1) b n → ∞ or (2) b n → b < ∞. Suppose the second case. Then the triangle inequality gives Each of the terms on the RHS limits to 0. Therefore γ(b) = p. Since b ≥ c > 0, we have a closed causal curve through p. This contradicts Lemma 3.17.
Therefore we must have b n → ∞. Proposition 3.15 implies that there exists a t 0 ≥ 0 such that γ(t 0 ) ∈ I + (S). By passing to a further subsequence, we can assume b n ≥ t 0 . Thereforeγ n (t 0 ) = γ n (t 0 ). Fix q ∈ I + (p). There exists an N such that n ≥ N implies γ n (b n ) ∈ I − (q) and γ n (t 0 ) ∈ I + (S) since these are open sets. Therefore for these n, there is a causal curve λ n from S to γ n (t 0 ) to γ n (b n ) to q. But L h (λ n ) → ∞ since b n → ∞. This contradicts Lemma 3.16 provided we show q ∈ I + (S). Indeed since q ∈ J + (S), we have q / ∈ I − (S). Therefore q ∈ S ∪ I + (S) by Lemma 3.12. But q / ∈ S otherwise λ n would be a causal curve which intersects S twice. Hence q ∈ I + (S).
Lemma 3.19. Let S be a Cauchy surface. Then for all p ∈ I − (S) and q ∈ I + (S), we have sup {L h (γ) | γ is a causal curve from p to q} < ∞.
Proof. By Lemma 3.16, we have sup{L h (γ) | γ is causal from S to q} = b < ∞. Likewise the time-dual of Lemma 3.16 gives sup{L h (γ) | γ is causal from p to S} = a < ∞. Then any causal curve from p to q has h-length bounded by a + b. Proof. Seeking a contradiction, suppose the supremum was infinite. By Proposition 3.15, there is a causal curve from q to q ′ ∈ I + (S). Likewise the time-dual of Proposition 3.15 guarantees a point p ′ ∈ I − (S) and a causal curve from p ′ to p. Since the supremum is infinite, we have sup{L h (γ) | γ is causal from p ′ to q ′ } = ∞. But this contradicts Lemma 3.19. Proof. Let q be an accumulation point of J + (p). Then there is a sequence h-arclength parameterized causal curves γ n : [0, b n ] → M from p to q n where q n → q. Extend these to inextendible causal curvesγ n : R → M via Theorem 2.18. By the limit curve theorem, there is a subsequence (still denoted byγ n ) which converges to an inextendible causal curve γ : R → M passing through p. By restricting to a further subsequence, we either have (1) b n → ∞ or (2) b n → b < ∞. It suffices to show that only (2) can hold. For then the same triangle inequality argument used in the proof of Proposition 3.18 implies γ : [0, b] → M is a causal curve from p to q.
Seeking a contradiction, suppose (1) holds. Let q 0 ∈ I + (q). Then for all sufficiently large n, we have q n ∈ I − (q 0 ). Therefore there are causal curves λ n from p to q 0 with L h (λ n ) → ∞. But this contradicts Lemma 3.20.

Upper semi-continuity of the Lorentzian length functional
This section is solely devoted to proving the upper semi-continuity of the Lorentzian length functional. This is Proposition 3.7 which played a chief role in the proof of Theorem 3.2.
Remark. The proof of upper semi-continuity in [6] used approximating smooth metrics and the fact that the Lorentzian length of a causal curve can be found by taking the length of a limit of interpolating geodesics. This last fact was somewhat of a folklore theorem until Minguzzi proved it in [12,Theorem 2.37]. The proof in this section relies on similar ideas but instead of approximating via geodesics, we approximate via η −ε -maximizers (see Definition 3.23).
The following proof is inspired by [12,Theorem 2.37].
Proof. The proof is an application of Lebesgue's dominated convergence theorem. Define c via 1/c 2 = inf {h(X, X) | x ∈ B ε , X ∈ T x M, δ µν X µ X ν = 1}. Since B ε has compact closure, we have 1/c 2 > 0. Then for all x ∈ B ε and X ∈ T x M , we have δ µν X µ X ν ≤ c 2 h(X, X).
for each k. Thus by the dominated convergence theorem, it suffices to show λ ′ k → γ ′ almost everywhere. Let A denote the set of points in [a, b] where γ is not differentiable. Likewise, let A k denote the set of points in [a, b] where λ k is not differentiable. Then D = [a, b]\(A ∪ k A k ) has full measure and represents the set of differentiable points which belong to γ and each λ k .
Fix t * ∈ D. We will show λ ′ k (t * ) → γ ′ (t * ). Letγ = φ • γ andλ k = φ • λ k where φ is the coordinate map for B ε . Let a k be the greatest point on the partition P k such that a k < t * and b k be the least point on the partition such that t * < b k . Then γ(a k ) = λ k (a k ) and γ(b k ) = λ k (b k ). Therefore the triangle inequality gives We get zero for the second term on the RHS becauseλ k is composed of straight lines. Now we use the definition of the derivative to bound the first term on the RHS. First notice the identityγ Fix ε 0 > 0. By definition of the derivative, there exists a δ > 0 such that |t − t * | < δ implies Choose N large enough so that k ≥ N implies t * − a k < δ and b k − t * < δ. Then using the identity above, we have Lemma 3.25. For any p ∈ M there exists a neighborhood U and a constant C such that for any p 0 ∈ U and any 0 < ε < 3/5, there is a neighborhood Proof. Fix p ∈ M . Let U 3/5 be a neighborhood from Lemma 2.8 with compact closure.
By the first paragraph of this proof, we havẽ Define c via 1/c 2 = inf {h(X, X) | x ∈ U 3/5 , X ∈ T x M, δ µν X µ X ν = 1}. Since U 3/5 has compact closure, we have 1/c 2 > 0. Then for all x ∈ U 3/5 and X ∈ T x M , we have δ µν X µ X ν ≤ c 2 h(X, X). In particular, if h(X, X) = 1, then |X 0 | 2 ≤ δ µν X µ X ν ≤ c 2 . Since h(σ ′ , σ ′ ) = 1, we have |X 0 (t)| < c 5/2. Therefore −η −ε (X, X) < −g(X, X) + εC 2 where C 2 = c 2 (5/2)(6 + 4n + 4n 2 ). This establishes L η −ε (σ) < L(σ) + CL h (σ) √ ε. The other inequality is analogous. Proof. Fix p ∈ M and a U 3/5 neighborhood with compact closure. Let γ n : [a, b] → U 3/5 be a sequence of causal curves parameterized by h-arclength which converge uniformly to a causal curve γ : [a, b] → U 3/5 . For any p 0 ∈ γ [a, b] there is a B ε ⊂ U 3/5 around p 0 from Lemma 3.25. Since γ [a, b] is compact, we cover it by finitely many neighborhoods B 1 ε . . . , B l ε . We order these sets in the following way: There exists some set which contains γ(a), say B 1 ε . Define s 1 via in which case we take l = 1 and stop. Otherwise γ(s 1 ) / ∈ B 1 ε , and so there exists some set which contains γ(s 1 ), say B 2 ε . Define ε }. Either s 2 = b in which case we take l = 2 and stop. Otherwise γ(s 2 ) / ∈ B 2 ε , and so there exists some set which contains γ(s 2 ), say B 3 ε . And so on. Repeat this process until and γ 2 = γ| [t 1 ,t 2 ] and so on until γ l = γ| [t l−1 ,b] . Then γ i ⊂ B i ε for all i = 1, . . . , l. Let t 0 = a and t l = b. For each i = 1, . . . , l, we partition the domain [t i−1 , t i ] into k equal subintervals (i.e. via P k ) and let λ i k denote the interpolating η −ε -maximizer of γ i with respect to P k . By Proposition 3.24, we have L η −ε (γ i ) = lim k→∞ L η −ε (λ i k ). Fix k large enough so that L η −ε (γ i ) ≥ L η −ε (λ i k ) − ε/l for all i. Then using Lemma 3.25, we have Note that λ i k is the concatenation of the curves λ i k,j for j = 1, . . . , k where λ i k,j is the straight line in B i ε joining p i k,j to q i k,j where We extend λ i k,j just slightly to a new curveλ i k,j such thatλ i k,j is still the straight line between its end pointsp i k,j andq i k,j . Moreover, we choose the pointsp i k,j andq i k,j sufficiently close to p i k,j and q i k,j , respectively, such that By uniform convergence of the γ n , there is an integer N such that n ≥ N implies γ n |C i k,j is contained in D i k,j for all i = 1, . . . , l and all j = 1, . . . , k. Sinceλ i k,j is an η −ε -maximizer, (2) from Proposition 3.22 implies Using Lemma 3.25, we have Thus, for these n ≥ N , we have Since ε was arbitrary, the result follows.
Proof of Proposition 3.7: Since γ [a, b] is compact, we can cover it by finitely many neighborhoods U 1 , U 2 . . . , U l given from Lemma 3.26. We order these sets in the following way: There exists some set which contains γ(a), say U 1 . Define s 1 via s 1 = sup{t | γ [a, t) ⊂ U 1 }. Either s 1 = b in which case we take l = 1 and stop. Otherwise γ(s 1 ) / ∈ U 1 , and so there exists some set which contains γ(s 1 ), say U 2 . Define s 2 via s 2 = sup{t | γ [s 1 , t) ⊂ U 2 }. Either s 2 = b in which case we take l = 2 and stop. Otherwise γ(s 2 ) / ∈ U 2 , and so there exists some set which contains γ(s 2 ), say U 3 . And so on until s l = b ∈ U l . For i = 1, . . . , l − 1, choose 4 Bubbling spacetimes

Bubbling sets and causally plain spacetimes
We begin with a motivating example. Consider the spacetime (M, g) where , the metric components a(t) and b(t) are not C 1 . In fact they are not even Lipschitz. If γ is a curve beginning at the origin, then γ(x) = t(x), x will be null when t ′ (x) = |t| 1/2 .
Since |t| 1/2 is continuous, we are guaranteed the existence of solutions. However it's not Lipschitz, so we are not guaranteed uniqueness. Indeed the solutions for the initial condition t(0) = 0 are given by the bifurcating family If we let p = (0, 0) ∈ M denote the origin, then this example demonstrates the proper inclusion:  Remark. Past bubbling sets are defined time-dually. When U = M , we will simply write B + (S). When S = {p} we will simply write B + (p, U ).
Since bubbling sets are unfamiliar (and hence undesirable), we set out to establish sufficient conditions which will guarantee they are empty.  Proof. Fix q ∈ int J + (p) . Then there is a neighborhood U ⊂ int J + (p) about q. Therefore there is a causal curve from p to a point q ′ ∈ I − (q, U ). Thus q ∈ I + (p) by the push-up property. Hence B + (p) = ∅.
In Appendix A.1 we demonstrate that normal neighborhoods can be used to show that C 2 spacetimes satisfy the push-up property. Therefore they are causally plain. The motivating C 0 spacetime from the beginning of this section demonstrates that spacetimes with regularity less than C 1 are not causally plain. Therefore a natural question to ask is: are C 1 spacetimes causally plain? The answer is yes. In fact Lipschitz is sufficient.  Proof. Let (M, g) be a Lipschitz spacetime. Let γ : [a, b] → M be a causal curve. We will construct a timelike curve λ : [a, b] → M such that λ(a) = γ(a), and λ(b) can be made arbitrarily close to γ(b) such that λ(b) ∈ I + γ(b) . This will prove the push-up property.
where A(t) = −g γ ′ (t), ∂ 0 is a continuous function of t. Consider the initial value problem Since f and f ′ are bounded on [0, b], we can now choose ε small enough so that Hence λ is a timelike curve, and by choosing ε sufficiently close to 0, we can make λ(b) as sufficiently close to γ(b) as we desire.
For the general case, we can cover any causal curve γ : [a, b] → M by finitely many such coordinate neighborhoods U 1 , . . . U k . Then we apply the technique above to each coordinate neighborhood resulting in timelike curves λ 1 , . . . , λ k . Then we concatenate each of these timelike curves yielding the desired timelike curve λ.

Trapped sets in C 0 spacetimes
Trapped sets play a prominent role in C 2 causal theory where they are used to prove the existence of singularities in a spacetime (i.e incomplete geodesics). The most notable example of this is Penrose's original singularity theorem [16,21]. In this section we offer a definition for trapped sets in C 0 spacetimes and prove a C 0 version of Penrose's theorem: if (M, g) has a noncompact Cauchy surface, then there are no trapped sets in M . Definition 4.6. Let (M, g) be a C 0 spacetime.
-Σ is future trapped if there is a nonempty future set F ⊂ J + (Σ) such that ∂F is compact.
Examples of future sets are I + (Σ) and J + (Σ). In bubbling spacetimes, the boundaries of these future sets may not be equal: ∂I + (Σ) = ∂J + (Σ), and so there can be a future set with boundary ∂F that lies between them. See Figure 4.  Proof. We first show ∂F is achronal. Fix p, q ∈ ∂F and suppose γ : [0, 1] → M is a timelike curve from p to q. Then I − γ(1/2) is an open set containing the point p and hence contains a point r ∈ F . Therefore q ∈ I + (r). But the definition of a future set implies I + (r) ⊂ F . Hence q ∈ int[F ] which contradicts the assumption q ∈ ∂F . Now we show ∂F has no edge points. Fix p 0 ∈ ∂F . Let p ∈ I − (p 0 ) and q ∈ I + (p 0 ) and let γ : [a, b] → M be a timelike curve from p to q. Using similar arguments as in the above paragraph, we have q ∈ int[F ] and p ∈ int[M \ F ]. Define Since int [F ] is open, we have t * < b. Likewise p ∈ int[M \ F ] implies t * > a. Since γ(t * ) is an accumulation point of int[F ], we have γ intersects ∂F : Corollary 4.8. If F is a nonempty future set, then ∂F is a C 0 hypersurface.
Proof. This follows from Theorem A.6.
The following proof is a direct analogue of Penrose's original proof [15]. We include it for the sake of (in)completeness. Proof. Let S be the Cauchy surface. Claim: S is connected. Since M is time-oriented there is a C 1 timelike vector fieldX on M . Let X =X/h(X, X). Since maximal integral curves are inextendible as continuous curves, the integral curves of X are inextendible causal curves and are parameterized by h-arclength. Let γ p : R → M denote the integral curve of X through p. Let φ : M × R → M denote the flow of X given by φ(p, t) = γ p (t). Let φ S : S × R → M denote the restriction of φ to S × R. Then φ S is one-to-one because integral curves don't intersect, and φ S is onto because S is a Cauchy surface. Since S is a C 0 hypersurface by Corollary 3.11, Brouwer's invariance of domain theorem implies φ S is a homeomorphism. Let π : S × R → S denote the natural projection. Put r = π • φ −1 S . Then r : M → S is a retraction of M onto S. Since M is connected, S = r(M ) is connected. This proves the claim.
Seeking a contradiction, suppose Σ is future trapped with future set F . Let r ∂F : ∂F → S denote the restriction of r to ∂F . Since r ∂F is one-to-one, Brouwer's invariance of domain theorem implies r ∂F is a homeomorphism of ∂F onto an open subset of S. Since ∂F is compact, r ∂F (∂F ) is closed in S. Therefore r ∂F (∂F ) = S since S is connected. But this contradicts S being noncompact.
Remark. It would be interesting to see what conditions on a C 0 spacetime would force a future trapped set. For instance in a C 2 spacetime we have [15,Proposition 14.60]: Trapped surface + null energy condition + null completeness =⇒ future trapped set

A Appendices
A.1 Differences between C 0 and smooth (at least C 2 ) causal theory In this appendix we highlight the main difference between causal theory in smooth (at least C 2 ) spacetimes and causal theory in C 0 spacetimes. The goal is to see how the twicedifferentiability of the metric is used in C 2 causal theory and the difference that arises with C 0 metrics. For references on C 2 causal theory one can look at classical sources such as [15,22] or more recent sources such as [2,12].
Let (M, g) be a C 2 spacetime. Then there is a unique affine connection ∇ such that ∇g = 0. A curve γ is a geodesic if ∇ γ ′ γ ′ = 0. A consequence of ∇g = 0 is that a geodesic must be either timelike, null, or spacelike. The equation ∇ γ ′ γ ′ = 0 is a second order differential equation. Introducing a coordinate system x µ and putting γ µ = x µ • γ, this differential equation is d 2 γ µ dt 2 + Γ µ αβ dγ α dt dγ β dt = 0. Since the metric is C 2 , the Christoffel symbols are C 1 . Thus the fundamental existence and uniqueness theorem for differential equations implies a map exp If γ is a causal curve from p to q and λ is a timelike curve from q to r, then using a finite number of normal neighborhoods and the properties above, we can deform the concatenation of γ and λ into a timelike curve from p to r [2,15]. This proves the push-up property.
Proposition A.1 (Push-up property). Let (M, g) be a C 2 spacetime. Then The push-up property implies: Proof. Fix q ∈ int J + (p) . Then there is a normal neighborhood U ⊂ int J + (p) about q. Therefore there is a causal curve from p to a point q ′ ∈ I − (q, U ). Thus q ∈ I + (p) by the push-up property.
It was shown in [3] that Proposition A.2 need not hold for C 0 spacetimes. See the example in the beginning of section 4.1. There can be nonempty bubbling sets in C 0 spacetimes. These are the open sets For C 2 spacetimes B + (p) = ∅ for all p by Proposition A.2. This highlights the main difference between C 2 and C 0 causal theory.

A.2 Properties of locally Lipschitz curves
In Definition 2.3 we defined causal and timelike curves via locally Lipschitz curves. In this section we establish the properties of locally Lipschitz curves. These curves are defined via a complete Riemannian metric h. Therefore we first show that if (M, g) is a C 0 spacetime, then there is a complete Riemannian metric h on M .
Proposition A.3. Let M be a smooth manifold which is connected, Hausdorff, and secondcountable. Then there is smooth a complete Riemannian metric h on M .
Proof. We could construct h via a partition of unity as in [17,Lemma 11.1], but there's another argument using Hopf-Rinow and the Whitney embedding theorem which is also pointed out in [17].
Since M is smooth, Hausdorff, and second-countable, we can apply the Whitney embedding theorem [11] to obtain a smooth proper embedding f : M → R N . By pulling back the Euclidean metric onto M , we have a Riemannian manifold (M, h). Let d h be the distance function on M induced by h. Since f is proper, any closed set in M maps to a closed subset of R N . Therefore any closed and bounded subset of (M, d h ) will be a closed and bounded subset within f (M ) ⊂ R N which is compact by the Heine-Borel theorem. Since M is connected, (M, h) is complete by Hopf-Rinow.
Fix a C 0 spacetime (M, g) and a complete Riemannian metric h on M . Let I ⊂ R be an interval (i.e. any connected subset of R with nonempty interior). A locally Lipschitz curve γ : I → M is a continuous function such that for any compact K ⊂ I there is a constant C such that for any a, b ∈ K, we have The next proposition is shows that the definition of locally Lipschitz does not depend on the choice of complete Riemannian metric h. See also [2]. Proof. Fix a compact set K ⊂ I. Let L denote the h 1 -length of γ| K . Set D = t∈K B h 1 γ(t), L . Here B h 1 denotes the closed geodesic ball with respect to h 1 . D is closed because the compactness of K implies its complement is open, and D is bounded by 3L. Therefore D is compact by Hopf-Rinow. Define C by 1/C 2 = inf {h 1 (X, X) | p ∈ D, X ∈ T p M, h 2 (X, X) = 1} Compactness of D implies 1/C 2 > 0. Then for all p ∈ D and X ∈ T p M , we have h 2 (X, X) ≤ C 2 h 1 (X, X).
Fix a, b ∈ K. Let σ denote a minimizing h 1 -geodesic between γ(a) and γ(b). Note that the definition of D implies σ ⊂ D. Therefore Thus, if γ is locally Lipschitz with respect to h 1 , then it is locally Lipschitz with respect to h 2 . Reversing the roles of h 1 and h 2 gives the reverse implication.
Of course one normally works with smooth or piecewise smooth curves in a spacetime. That these curves are locally Lipschitz follows from the next proposition. Define C by 1/C 2 = inf {δ µν X µ X ν | p ∈ B, X ∈ T p M, h(X, X) = 1}.
This proves the proposition when γ(K) ⊂ B. In the general case, we can cover γ(K) by finitely many such balls and then apply the triangle inequality to obtain the result.
Proposition 2.2 is a partial converse to the previous proposition.

Proof of Proposition 2.2:
Fix t 0 ∈ I. Let K ⊂ I be a compact interval containing t 0 such that γ(K) ⊂ U where φ : B → R n+1 be a coordinate system such that φ(B) is an open Euclidean ball with finite radius. Hence B has compact closure. Fix a, b ∈ K. Since γ is locally Lipschitz, there is a constant C > 0 such that d h γ(a), γ(b) ≤ C|b − a|. Define c via 1/c 2 = inf {h(X, X) | p ∈ B, X ∈ T p M, δ µν X µ X ν = 1}.
Since B has compact closure, we have 1/c 2 > 0. Then for all p ∈ B and X ∈ T p M , we have δ µν X µ X ν ≤ c 2 h(X, X).
Fix a, b ∈ K. Writeγ = φ • γ. Let σ : [0, 1] → B denote a minimizing h-geodesic joining γ(a) to γ(b). Let X = σ ′ . Then The last inequality follows because the shortest distance betweenγ(a) andγ(b) with respect to the Euclidean metric δ µν is just the straight line. For any µ, we trivially have γ µ (b) − γ µ (a) ≤ γ(a) −γ(b) . Thus Hence the components γ µ are Lipschitz functions on K. Therefore they are absolutely continuous and hence differentiable almost everywhere with derivative bounded by Cc almost everywhere on K.

A.3 Achronal and edgeless subsets in a spacetime
Fix a C 0 spacetime (M, g). A subset S ⊂ M is achronal if I + (S) ∩ S = ∅. We say S is achronal in U if I + (S, U ) ∩ S = ∅. We say S is locally achronal if for every p ∈ M , there is an open set U around p such that S is achronal in U . The edge of an achronal set S is the set of points p ∈ S such that for every neighborhood U of p, there is a timelike curve γ : [a, b] → U such that γ(a) ∈ I − (p, U ), γ(b) ∈ I + (p, U ), and γ ∩ S = ∅. We say S is edgeless if S is disjoint from its edge. A subset S ⊂ M is a C 0 hypersurface provided for each p ∈ S there is a neighborhood U ⊂ M and a homeomorphism φ : U → φ(U ) ⊂ R n+1 such that φ(U ∩ S) = φ(U ) ∩ P where P is a hyperplane in R n+1 .
Remark. The following theorem says that a locally achronal and edgeless set is a C 0 hypersurface, but the proof actually shows that this can be strengthened to a locally Lipschitz hypersurface.