Null Distance and Convergence of Lorentzian Length Spaces

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov–Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.


Introduction
Metric geometry [6,7] has led to identifying the 'metric core' of many results in Riemannian differential geometry, to clarifying the interdependence of various concepts, and to generalizations of central notions to lower regularity. In particular, Riemannian manifolds carry a natural metric structure and standard notions of convergence such as Gromov-Hausdorff (GH) and Sormani-Wenger intrinsic flat (SWIF) convergence [19] interact well with geometric quantities, in particular with curvature bounds.
Despite the increasing demand for a Lorentzian analog of this framework, particularly driven by general relativity (GR), see, e.g., [9], a comparable metric theory is still in its infancy. Only recently Sormani and Vega [18] put forward a solution to one prime obstacle, i.e., the fact that the Lorentzian distance does not induce a metric structure. 1 They constructed a 'null distance' capable of encoding both the topological and the causality structure of the manifold, and first convergence results built on the corresponding metric and integral current structures were established by Burtscher and Allen [4].
In a somewhat parallel approach more directly rooted in GR, Kunzinger and Sämann [14] introduced a notion of Lorentzian length spaces. Based on the time separation function, this construction provides a close Lorentzian analog of (metric) length spaces. In particular, it allows one to extend beyond the manifold level the synthetic approach to (sectional) curvature bounds, which was introduced for general semi-Riemannian manifolds by Alexander and Bishop [2].
In this work, we provide a natural next step toward a comprehensive notion of metric limits for Lorentzian manifolds by considering GH convergence based on the null distance in the class of Lorentzian length spaces, thereby extending the results in [4] beyond the manifold level. Furthermore, we show that in certain warped product Lorentzian length spaces GH convergence interacts well with synthetic curvature bounds.
This work is structured in the following way: We collect preliminaries on the null distance, the convergence results of [4], Lorentzian synthetic curvature bounds, and Lorentzian length spaces in Sect. 2. Then, in Sect. 3, we extend the null distance to the setting of Lorentzian (pre-)length spaces and, following [4,18], establish its fundamental properties. Finally, in Sect. 4, we study GH limits of warped product Lorentzian length spaces and prove or main results on their interaction with curvature bounds.
In the remainder of this introduction, we collect some basic notions and conventions. All manifolds are assumed to be smooth, connected, Hausdorff, second countable, of arbitrary dimension n ≥ 2, and without boundary. A spacetime (M, g) is a time-oriented Lorentzian manifold, where we use the signature (−, +, . . . , +). We will deal with metrics of various regularity, but generally assume them to be smooth unless explicitly stated otherwise. Causality notions will be based on locally Lipschitz curves, and we denote the chronological and the causal relation by I and J and write p q and p ≤ q if q ∈ I + (p) and p ∈ J + (q), respectively. A generalized time function τ : M → R is a function that is strictly increasing along all future-directed causal curves. It is called a time function if it is continuous. For points p, q ∈ M , the time separation function ρ(p, q) is the supremum of the length of all future-directed causal curve segments from p to q with the understanding that ρ(p, q) = 0 if there is no such curve, i.e., if q ∈ J + (p). In all matters of semi-Riemannian geometry and causality theory, we will adopt the conventions and notations of [17] and [16].
Finally, we will often deal with warped products M = B × f F , where f > 0 is the warping function and (B, g B ) and (F, g F ) are semi-Riemannian manifolds. The metric g on M = B × F is then given by g = g B + f 2 g F , where, as usual, we notationally suppress the projections. We will most of the time deal with the case that the base B is a real interval I and the fiber (F, g F ) is Riemannian. Then, the warped product metric takes the form g = −dt 2 +f 2 g F .

Preliminaries
The null distance of Sormani and Vega [18] provides a way of encoding the manifold topology as well as the causal structure of a spacetime. Given a generalized time function τ on (M, g), the null distance is defined bŷ Here, the null length of β : We say thatd τ encodes the causality of M if a property that is stronger than definiteness [18,Lem. 3.12]. For warped product spacetimes I × f S, it holds by [18,Thm. 3.25] that the null distance induced by any smooth time function (depending only on t ∈ I) is definite and encodes the manifold topology as well as the causality. Note that the completeness assumption on the fiber is not needed, a fact which also follows from our Theorem 4.11.
The metric and integral current structure of Lorentzian manifolds based on the null distance was further studied in [4]. There, Allen and Burtscher showed that for any spacetime (M, g) with locally anti-Lipschitz time function τ , (M,d τ ) is a length space [4, Thm. 1.1]. They also proved first GH and SWIF convergence results for warped product spacetimes: Given a (connected, compact) Riemannian manifold (Σ, h), they consider sequences of warped product spacetimes (M = I × Σ, g j = −dt 2 + f 2 j (t)h) where I is a closed interval. If the (continuous) warping functions f j : I → (0, ∞) are uniformly bounded away from 0 and if they converge uniformly to a limit function f , then the corresponding null distancesd j converge uniformly tod f on M = I ×Σ and the metric spaces (M,d gj ) converge to (M,d g ) in the GH as well as in the SWIF sense [4,Thm. 1.4]. Here, g is the limiting warped product metric g = −dt 2 +f (t) 2 h.
The Lorentzian length spaces of [14] generalize the notion of length space to the Lorentzian world. Let (X, d) be a metric space and assume X is endowed with a preorder ≤ as well as a transitive relation contained in ≤, which we call the timelike and causal relation, respectively. If, in addition, we have a lower semicontinuous map 2 ρ : X × X → [0, ∞] that satisfies the reverse triangle inequality and ρ(x, y) > 0 ⇔ x y, then (X, d, , ≤, ρ) is called a Lorentzian pre-length space with time separation function ρ. Ann. Henri Poincaré A locally Lipschitz curve on an arbitrary interval γ : I → X that is nonconstant on any sub-interval is called (future-directed) causal (timelike) if for all t 1 < t 2 ∈ I we have γ(t 1 ) ≤ γ(t 2 ) (γ(t 1 ) γ(t 2 )). A causal γ is called null if no two points on the curve are timelike related. The length of a future-directed causal γ : [a, b] → X is defined via ρ by We call a future-directed causal curve γ : [a, b] → X maximal if it realizes the time separation, L ρ (γ) = ρ(γ(a), γ(b)). In analogy with metric length spaces, we call X a Lorentzian length space if, in addition to some technical assumptions (cf. [14,Def. 3.22]) ρ = T , where for any x, y ∈ X we set if there is a future-directed causal curve from x to y. Otherwise, we set T (x, y) := 0.
Causality theory in Lorentzian length spaces [1,11,14] extends standard causality theory [16] beyond the spacetime setting, to which it reduces for smooth strongly causal spacetimes. Hence, any smooth strongly causal spacetime is an example of a Lorentzian length space, but more generally, spacetimes with low regularity metrics and certain Lorentz-Finsler spaces [15] provide further examples [14,Sec. 5]. In particular, any continuous spacetime with strongly causal and causally plain metric (a condition that rules out causal pathologies, see [10,Def. 1.16]) is a (strongly localizable, for a definition see below) Lorentzian length space.
Based on pioneering work by Harris [12], Alexander and Bishop in [2] gave a characterization of sectional curvature bounds in terms of triangle comparison in smooth semi-Riemannian manifolds. We say that (M, g) has sectional curvature bounded below by some constant K, R ≥ K, if the sectional curvatures for all spacelike planes are bounded below by K and if for all timelike planes they are bounded above by K. Equivalently, we have Then, it holds [2, Thm. 1.1] that R ≥ K (R ≤ K) if and only if in any convex (totally normal) neighborhood the signed length of the geodesic between two points on a geodesic triangle is at least (at most) that of the corresponding points in the model triangle in L 2 (K). Here, the signed length of a geodesic in a convex neighborhood is defined as the signed length of the connecting vector |γ pq | ± = sign(γ pq ) | γ pq , γ pq |, with the sign of timelike vectors taken to be negative. Moreover, the Lorentzian model spaces L 2 (K) of constant sectional curvature K are Vol. 23 (2022) Null Distance and Convergence 4323 whereS 2 1 (r) is the simply connected covering manifold of the two-dimensional Lorentzian pseudosphere S 2 1 (r), R 2 1 is two-dimensional Minkowski space, and H 2 1 (r) is the simply connected covering manifold of the two-dimensional Lorentzian pseudohyperbolic space.
Again, in parallel to the case of metric geometry, appropriate notions of synthetic (timelike or causal) curvature bounds based on triangle comparison have been introduced in Lorentzian length spaces. By a timelike geodesic triangle, we mean a triple (x, y, z) ∈ X 3 with x y z such that ρ(x, z) < ∞ and such that the sides are realized by future-directed causal curves. We then say, cf. [14,Def. 4.7], that a Lorentzian pre-length space (X, d, , ≤, ρ) has timelike curvature bounded below (above) by K ∈ R if every point in X has a so-called comparison neighborhood U such that: (i) ρ| U ×U is finite and continuous.
is a timelike geodesic triangle in U , realized by maximal causal curves α, β, γ whose side lengths satisfy the appropriate size restrictions (see [14,Lem. 4.6]), and if (x , y , z ) is a comparison triangle of (x, y, z) in L 2 (K) realized by timelike geodesics α , β , γ , then whenever p, q are points on the sides of (x, y, z) and p , q are corresponding points 3 of (x , y , z ), we have ρ(p, q) ≤ ρ (p , q ) (respectively, ρ(p, q) ≥ ρ (p , q )). We close this section by recalling further central notions in Lorentzian pre-length spaces. Generally, causality conditions such as strong causality and global hyperbolicity are translated in perfect analogy from the spacetime setting. (X, d, , ≤, ρ) is called causally path connected if for all x, y ∈ X with x y and all x, y with x ≤ y there is a future-directed timelike resp. causal curve from x to y. A localizing neighborhood Ω x of a point x ∈ X is a substitute for a convex neighborhood, and a Lorentzian pre-length space is called localizable if every x has such a neighborhood. It is defined by the conditions: If, in addition, the neighborhoods Ω x can be chosen such that (iv) Whenever p, q ∈ Ω x satisfy p q, then γ p,q is timelike and strictly longer than any future-directed causal curve in Ω x from p to q that contains a null segment, then (X, d, , ≤, ρ) is called regularly localizable. Finally, if every point x ∈ X has a neighborhood basis of open sets Ω x satisfying (i)-(iii), respectively, (i)-(iv), then (X, d, , ≤, ρ) is called strongly localizable, respectively, SRlocalizable.
In a strongly causal and localizable Lorentzian pre-length space, the length L ρ is upper semicontinuous, if it is regularly localizable, maximal causal curves have a causal character, and the push-up principle holds [14,Prop. 3.17,Thms. 3.18,3.20]. A Lorentzian pre-length space is called geodesic if for all x < y there is a future-directed causal curve γ from x to y with τ (x, y) = L τ (γ) (hence maximizing). Any globally hyperbolic Lorentzian pre-length space is geodesic [14,Thm. 3.30].

The Null Distance in Lorentzian Length Spaces
In this section, we extend the notion of null distance to the setting of Lorentzian (pre-)length spaces and establish its fundamental properties.
As we shall see in Theorem 3.13, existence of a 'reasonable' time function implies strong causality. While a significant part of the causal ladder for Lorentzian length spaces has been established in [1,14], the precise relationship between existence of time functions and stable causality in this general framework is still an open question. 4 Definition 3.2. A map β : [a, b] → X from a closed interval into a Lorentzian pre-length space X is called a piecewise causal curve if there exists a partition a = s 0 < s 1 < · · · < s k−1 < s k = b such that each β i := β| [si−1,si] is either trivial (i.e., constant) or future-directed causal or past-directed causal. Given, in addition, a generalized time function τ : X → R, the null length of β iŝ where Under this assumption, we indeed have the following fundamental existence result: be an scc Lorentzian pre-length space. Then, for any p, q ∈ X there exists a piecewise causal curve from p to q.
is open, and by causal path connectedness, the relation p q is always realized by the existence of a future-directed timelike curve from p to q. Moreover, since any p ∈ X lies on a timelike curve, we have X = x∈X I − (x) ∪ y∈X I + (y). Based on these observations, a straightforward adaptation of the proof of [18,Lemma 3.5] yields the claim. The only difference is that in the present situation the finite covering of any path from p to q will in general contain both timelike futures and timelike pasts of points in X. 5 We also have the following analog of [18, Lemma 3.6]: Lemma 3.6. Let τ be a generalized time function on a Lorentzian pre-length space and let β : [a, b] → X be piecewise causal from p to q. Then,  Proof. Symmetry and triangle inequality are immediate from the definition, and finiteness follows from Lemma 3.5. Thatd τ (p, p) = 0 is seen by considering the constant (hence piecewise causal) curve β ≡ p.
Since the previous proof relied on Lemma 3.5, we will henceforth usually assume the scc property in order to avoid degenerate situations. The next result corresponds to Lemmas 3.10-3.13 and 3.16-3.18 in [18] (with identical proofs): Proposition 3.8. Let τ be a generalized time function on an scc Lorentzian pre-length space X. Then, Ifτ is another generalized time function on X and λ ∈ (0, ∞), then dτ = λd τ ⇔τ = λτ + C for some constant C.
The following is an analog of [18,Prop. 3.14], whose proof, however, needs to be adapted to the current setting. Proposition 3.9. Let τ be a generalized time function on an scc Lorentzian pre-length space X. Then, the following are equivalent: Proof. (i)⇒(ii): Let p, q ∈ X. Due to |d τ (p, q)−d τ (p , q )| ≤d τ (p, p )+d τ (q, q ), it suffices to show that we can find an open neighborhood of p on whichd τ (p, p ) becomes arbitrarily small-the same reasoning then applies to q, q . Since X is ssc, p lies on a timelike curve. Contrary to the manifold setting of [18], however, it might not be an interior point of any such curve, which prevents us from arguing with a timelike diamond as in the proof of [18,Prop. 3.14].
However, by symmetry we may without loss of generality assume that there is a future-directed timelike curve α p : [−δ 0 , 0] → X with α p (0) = p. Since timelike futures and pasts are open in any Lorentzian pre-length space ([14, Lemma 2.12]) and since we assume τ to be continuous, it follows that, for any 0 < δ < δ 0 , the set is an open neighborhood of p. Any p ∈ U δ can be connected via a timelike curve to α p (−δ) and hence can, via α p , be connected to p by a piecewise causal curve. By definition ofd τ and U δ , we therefore obtain Consequently, by choosing δ sufficiently small we can maked τ (p, p ) as small as desired. (ii)⇒(i): By time symmetry, this works exactly as in the proof of [18,Prop. 3.14], but we give the argument for the sake of completeness. Fixing q 0 ∈ X, by ssc there is either a p 0 ∈ I + (q 0 ) or a p 0 ∈ I − (q 0 ), and it will suffice to consider the first possibility. Then, by Proposition 3.8 (ii), for any p in the open neighborhood I − (p 0 ) of q 0 we have τ (p) = τ (p 0 ) −d τ (p 0 , p), which is continuous by assumption.
The following result is a direct generalization of [18,Prop. 3.15]. For the reader's convenience, we adapt its proof to the current, topologically more general, setup. Proposition 3.10. Let (X, d, , ≤, ρ) be an scc Lorentzian pre-length space with generalized time function τ and suppose that (X, d) is locally compact. Then, the following are equivalent: (i)d τ induces the same topology as d.
Proof. (i)⇒(ii): By Proposition 3.9, τ is continuous. Also, since d is definite, the topology on X is Hausdorff, implying thatd τ is definite as well.
(ii)⇒(i): Also in this case,d τ is continuous by Proposition 3.9, and so it only remains to show that thed τ -topology O τ is finer than the d-topology O d . Let x ∈ X and ε > 0 and denote by B d ε (x) the open ε-ball for d. Since X is locally compact, we may assume ε small enough so that ∂B d ε (x) is compact. We now distinguish two cases: First, if ∂B d ε (x) = ∅, then X being connected implies B d ε (x) = X, and so, we may pick any ε 0 > 0 to obtain Bd τ ε0 (x) ⊆ B d ε (x). Otherwise, ε 0 := min{d τ (x, y) | y ∈ ∂B d ε (x)} exists and is strictly positive sincê d τ is definite. Let y ∈ B d ε (x) and pick a piecewise causal curve β : [a, b] → M from x to y. With z the first intersection of β with ∂B d ε (x), let β 0 be the initial part of β from x to z. Then,L τ (β) ≥L τ (β 0 ) ≥ ε 0 . Thus, alsod τ (x, y) ≥ ε 0 . We conclude that also in this case  y). Then, for any p ∈ U and any q ∈ X \ {p},d τ (p, q) > 0. In particular,d τ is definite on U .

Proposition 3.12. Let τ be a generalized time function on a locally compact ssc Lorentzian pre-length space X. Then,d τ is definite if and only if τ is locally anti-Lipschitz.
In the above definition of the anti-Lipschitz property, there is no requirement on the compatibility of the local distance function d U with the topology on X induced by the metric d. To introduce such an assumption, we call f : X → R topologically anti-Lipschitz on the open set U ⊆ X if there exists a definite distance function d U on U that induces the d-topology on U and such that for all x, y ∈ U we have  U , then γ([a, b]) ⊆ U . In particular, if X is a Lorentzian length space, then X is strongly causal.
Thus, τ is topologically locally anti-Lipschitz, and Theorem 3.13 implies that any point in M has a neighborhood base consisting of causally convex sets. In both cases, this implies that (M, d h , , ≤, ρ) is a Lorentzian length space (cf. [14,Th. 5.12,5.16]). Of course, for smooth Lorentzian manifolds and even for the much more general Lorentz-Finsler setting this is well known. Indeed, for closed cone structures the existence of a time function implies stable causality, which in turn implies strong causality ([15, Th. 2.30]). It is not known whether an analogous result also holds for Lorentzian length spaces (cf. [1] for the definition of stable causality of Lorentzian length spaces).
As in [18,Cor. 3.19] it follows directly from Lemma 3.6 (ii) that any causal curve β from p to q ≥ p is minimal witĥ In other words, causal curves are null distance realizers. Next, we transfer [18,Lemma 3.20] to the Lorentzian length space setting: Proof. The proof of [18,Lemma 3.20] relying only on the properties that I ± (p) is open for any p ∈ X and that τ is strictly increasing along future-directed causal curves, shows that β has the following property: If a < s 0 < s 1 < · · · < s k = b denotes the breaks of β, then no two points of β| [si,si+1] are timelike related, giving the first claim (cf. [14,Def. 2.18]). Suppose now that X is localizable, let t 0 ∈ (s i , s i+1 ), x 0 := β(t 0 ) and let Ω x0 be a localizing neighborhood of were not maximizing for ρ, then there would exist a causal curve γ p,q from But then, in particular, 0 < L ρ (γ p,q ) and thereby p q, a contradiction. Consequently, β| [si,si+1] is a geodesic in the sense of [11,Def. 4.1].
Finally, we immediately conclude the analog of [18,Cor. 3.21]: Corollary 3.17. Let X be an scc Lorentzian pre-length space X with generalized time function τ and suppose that p, q ∈ X are not causally related. If β is a piecewise causal curve from p to q that contains a timelike subsegment, then there exists a strictly shorter piecewise causal curve α from p to q, i.e., witĥ L τ (α) <L τ (β).

Warped Products
Warped products are of fundamental importance in Riemannian and Lorentzian geometry. In the context of general relativity, they are generalizations of Friedmann-Lemaître-Robertson-Walker spacetimes, which serve as basic cosmological models of our universe. Warped products and generalized cones likewise play a fundamental role in length spaces with synthetic curvature bounds. In the context of Lorentzian length spaces, they have been studied in [3], and we refer to this work for further background information. We start with a closer look at the null distance on such spaces.

Null Distance on Generalized Cones
To begin with, we recall some basics of warped products with one-dimensional basis, the so-called generalized cones from [3,Sec. 3,4].
Let (X, d) be a metric space and I ⊆ R an interval. Observe that in almost everywhere. It is called future-/past-directed causal if α is strictly monotonically increasing/decreasing, i.e.,α > 0 orα < 0 almost everywhere. The length L(γ) of a causal curve γ is defined by The time separation function ρ : [3]) is defined as ρ(y, y ) := sup{L(γ) : γ future-directed causal curve from y to y } , (12) if this set is non-empty, and ρ(y, y ) := 0 otherwise. The causal and timelike relations p ≤ q resp. p q are defined as usual via the existence of a causal resp. timelike future-directed curve from p to q. When I × X is equipped with this time separation function, we write Y ≡ I × f X and call it a generalized cone (or warped product with one-dimensional basis) with warping function f . According to [3,Prop. 3.26 Generalized cones are automatically equipped with the continuous time function τ ≡ t : (t, x) → t (see [3,Lemma 4.2]). Also, if X is path connected, Y = I × f X is scc. For convenience, in what follows when we write that X is a length space we will always tacitly assume that X possesses only a single accessibility component (cf. [7]), i.e., that any two points in X can be connected by a path of finite length. In particular, X is always supposed to be path connected. Let us denote the null distance on Y corresponding to τ = t byd f . By Proposition 3.9,d f is continuous.
The following result shows thatd f is a metric (i.e., definite) if f is uniformly bounded below by a positive constant: Proposition 4.3. Let (X, d) be a length space and let Y = I × f X be the corresponding generalized cone. Suppose that for some f min ∈ R >0 we have f min ≤ f (t) for all t ∈ I. Then, for p = (t p , x p ), q = (t q , x q ) Consequently,d f is a metric on Y and the time function τ is locally anti-Lipschitz.
Proof. We adapt arguments from [4,Lemma 4.9]. If q ∈ J ± (p) and β is a causal curve from p to q then by (10) Now let q ∈ J ± (p) and take (according to Lemma 3.5) a piecewise causal Therefore, for α : J → X the concatenation of the α i we obtain that Taking the infimum over all curves β we obtain that d(x p , x q ) ≤ 1 fmind f (p, q). It is evident from (13,14) thatd f is definite. The final claim then follows from Proposition 3.12. Proof. Let p = (t p , x p ) = (t q , x q ) = q ∈ I × f X. If t p = t q , thend f (p, q) ≥ |t p − t q | > 0 by Proposition 3.8 (i). On the other hand, if t p = t q and x p = x q , then there exists some δ > 0 such that (at least) a one-sided interval, say [t p , t p + δ] is contained in I. Let β be a piecewise causal curve connecting p and q. If β leaves the δ-strip [t p , t p + δ], then its null length is at least δ. On the Lemma 4.7. Let Y = I × f X be a generalized cone, where (X, d) is a length space. Then, for any p, q ∈ Y there exists a piecewise null curve connecting p and q (i.e., a piecewise causal curve whose every segment is null).
Proof. Let I = a, b and write p = (t p , x p ), q = (t q , x q ), where without loss of generality we assume t p ≤ t q . We first consider the case where in fact t p = t q ∈ (a, b) and pick δ > 0 such that [t p , t p + δ] ⊆ I. If x p = x q , there is nothing to do. Otherwise, let β : [0, L] → X be a unit speed curve connecting x p to x q . Consider the initial value problem α 0 (s) = f (α 0 (s)) α 0 (0) = t p .
The maximal solution to (15) is given by the inverse of the map r → The solution α 1 of (16) has an analogous explicit description as α 0 . Combining these formulae for α 0 and α 1 with the fact that f is bounded below and above on [t p , t p + δ] by positive constants, we conclude that for any s 1 sufficiently small the following conditions are satisfied: connects the point (t p , x p ) to the point (t p , β(s 1 )). Starting from this new point, we can iterate the procedure, and by choosing for s 1 a suitably small fraction of [0, L], we obtain the desired broken null curve connecting p and q in a finite number of steps. Suppose now that t p < t q . By introducing, if necessary, an intermediate point r with t p < t r < t q and considering p, r and r, q separately, we may assume that at most one of t p , t q is a boundary point of I, say t p (and the following argument works just as well if both t p and t q lie in the interior of I).
In this case, the solution to (15) attains all t-values between t p and b, hence in particular the value t q , say at s = s . Then, following the null curve γ = (α 0 , β) until s we obtain a point (t q , z), which according to the first part of the proof can itself be connected to q by a piecewise null curve.
Based on Lemma 4.7, we can also prove the following dual to Proposition 4.3: Proposition 4.8. Let (X, d) be a length space and let Y = I × f X be the corresponding generalized cone. Suppose that for some f max ∈ R + we have f (t) ≤ f max for all t ∈ I. Then, for p = (t p , x p ), q = (t q , x q ) Proof. Equation (17) was already established in Proposition 4.3. To show (18), let ε > 0 and choose a unit speed path β : [0, L] → X connecting x p to x q such that L d (β) = L < d(x p , x q ) + ε. Let us suppose, without loss of generality, that t p ≤ t q . As in the proof of Lemma 4.7, we then first construct a null curve γ = (α, β) emanating from x p and reaching an endpoint r with t r = t q . Since q ∈ J + (p), the explicit description of J + (p) in [3,Cor. 3.24] shows that α is defined on [0, L ] for some L < L, and again as in the proof of Lemma 4.7, we can then extend α to all of [0, L] such that γ = (α, β) : [0, L] → Y is a piecewise null curve connecting p and q. Denoting the break points of γ by s 0 , . . . , s k , we then haveL Here, since γ is null and v β ≡ 1, we have |α( giving the claim for ε → 0. Remark 4.9. Combining Propositions 4.3 and 4.8, it follows that if the warping function f is bounded from above and below by positive constants, then the null distanced f induces on any fiber {t 0 } × X of the generalized cone I × f X a metric that is (bi-Lipschitz) equivalent to the one induced by the original metric d (by ((t 0 , x 1 ), (t 0 , x 2 )) → d(x 1 , x 2 )). In the special case of pure