Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
Stephanie B. Alexander, Melanie Graf, Michael Kunzinger, Clemens Sämann
aa r X i v : . [ m a t h . M G ] S e p Generalized cones as Lorentzian length spaces:Causality, curvature, and singularity theorems
Stephanie B. Alexander ∗ , Melanie Graf † , Michael Kunzinger ‡ , Clemens S¨amann § September 23, 2019
Abstract
We study generalizations of Lorentzian warped products with one-dimensional base of the form I × f X , where I is an interval, X is alength space and f is a positive continuous function. These general-ized cones furnish an important class of Lorentzian length spaces inthe sense of [KS18], displaying optimal causality properties that al-low for explicit descriptions of all underlying notions. In addition,synthetic sectional curvature bounds of generalized cones are directlyrelated to metric curvature bounds of the fiber X . The interest insuch spaces comes both from metric geometry and from General Rela-tivity, where warped products underlie important cosmological models(FLRW spacetimes). Moreover, we prove singularity theorems for thesespaces, showing that non-positive lower timelike curvature bounds im-ply the existence of incomplete timelike geodesics. Keywords:
Length spaces, Lorentzian length spaces, causality the-ory, synthetic curvature bounds, triangle comparison, metric geometry,warped products
MSC2010: ∗ [email protected] , Department of Mathematics, University of Illinois at Urbana-Champaign, USA. † [email protected] , Department of Mathematics, University of T¨ubingen,Germany. ‡ [email protected] , Faculty of Mathematics, University of Vienna,Austria. § [email protected] , Department of Mathematics, University of Toronto,Canada. ontents Warped products are of central importance to Riemannian geometry, in par-ticular in the study of constant curvature geometries and as a rich sourceof examples and counterexamples (cf., e.g., [Pet16]). Generalized cones andwarped products of metric spaces likewise play an important role in thetheory of length metric spaces with synthetic curvature bounds (
Alexandrovspaces ). These spaces, while including Riemannian manifolds with curva-ture bounds, allow singularities, being closed, for example, under Gromov-Hausdorff limits or gluing operations. Alexandrov spaces have yielded majorinsights into classical Riemannian geometry ([Per93], cf. [Kap07, Gro01]).Generalized cones and warped products provide examples and counterexam-ples in Alexandrov geometry (cf. [Che99, AB98, AB04, AB16]). Moreover,the first-order structure is captured by the tangent cone ; for instance, forAlexandrov spaces of curvature bounded below, the tangent cone at a pointis homeomorphic to a small metric ball centered at the point (cf. [BBI01]).In the smooth pseudo-Riemannian setting, Alexander and Bishop gavein [AB08] a characterization of sectional curvature bounds in terms of tri-angle comparison, including applications to Lorentzian and general semi-Riemannian warped products and so-called
Friedmann-Lemaitre-Robertson- alker (FLRW) -spacetimes. Lorentzian geometry enjoys a unique positionin the smooth pseudo-Riemannian world: Pseudo-Riemannian metrics ofsignature ( − , + , . . . , +) are central to the study of the physical theory ofGeneral Relativity and (apart from the Riemannian case) Lorentzian the-ory is the most mathematically well explored case, providing many strongtools and results which are absent in more general signatures. Addition-ally, the past decade has seen increasing interest from the mathematicalphysics community in the study of low regularity Lorentzian geometry andGeneral Relativity. An intensive study of causality theory (cf. [Min19b])for Lorentzian metrics of low regularity was initiated by P. Chru´sciel andJ.D.E. Grant in [CG12] and then pursued by various authors, see [Min15,KSS14, KSSV14, S¨am16]. In particular, Chru´sciel and Grant showed in[CG12] that for spacetimes with merely continuous metrics pathologies ofthe causal structure may occur, e.g. there are so-called causal bubbles , wherethe boundary of the lightcone is not a hypersurface but has positive measure(see also [GKSS19]). An important recent result for continuous Lorentzianmetrics is the C -inextendibility of the Schwarzschild solution to the Ein-stein equations, which was shown by J. Sbierski in [Sbi18] and has sparkedfurther research into low regularity (in-)extendibility and causality (e.g.[GLS18, GL17, DL17, GL18, GKS19]). The importance of such low regular-ity (in-)extendibility results is rooted in the strong cosmic censorship con-jecture (cf. e.g. [Ise15]), which, roughly, states that the maximal globally hy-perbolic development of generic initial data for the Einstein equations is in-extendible as a suitably regular Lorentzian manifold and which is intimatelyrelated to the question of determinism in General Relativity. Another areaof mathematical general relativity, where low regularity has recently come tothe forefront of current research, is the study of singularities and in particularof so-called singularity theorems, predicting causal geodesic incompletenessunder certain curvature and causality assumptions. The classical singularitytheorems of Hawking and Penrose have only recently been successfully ex-tended to the C , -setting ([KSSV15, KSV15, GGKS18]), which is a naturalregularity class to consider as curvature is still almost everywhere definedand locally bounded. With the final results in Section 6 this paper will di-rectly contribute to this line of research by proving singularity theorems forgeneralized cones. We also mention another natural generalization of smoothLorentzian geometry, namely cone structures on differentiable manifolds andLorentz-Finsler spacetimes, see [FS12, BS18, Min19a, MS19, LMO19], whichproved to be a significant extension of the field. Also, there currently isstrong interest in bringing techniques from Riemannian geometry and opti-mal transport into the Lorentzian setting, cf. [McC18, MS18]. For extend-ing these techniques further to a synthetic setting it might prove useful touse generalized cones as introduced and studied in the present paper as astarting point. Finally, a closely related direction of research is the recentapproach of Sormani and Vega [SV16] and its further development by Allen3nd Burtscher in [AB19] of defining a metric on a spacetime that is compat-ible with the causal structure in case the spacetime admits a time functionsatisfying an anti-Lipschitz condition.The importance of warped products, specifically, in General Relativity(cf., e.g., [O’N83, Wal84, Min07]) stems from the fact that the FLRW mod-els of the universe in cosmology are particular examples of warped productswith one-dimensional base, see e.g. [O’N83, Ch. 12]. Such spaces have a verysimple structure geometrically and provide a good starting point for tryingto generalize the smooth theory. Our main object of study will be gener-alizations of Lorentzian warped products with one-dimensional base to thecase where the fiber is merely a metric length space, but some of our resultsare new even if the fiber is a smooth Riemannian manifold. For example,we show that there is no causal bubbling in such spacetimes even if thewarping function f , and hence the Lorentzian metric, is merely continuousand that maximizing causal curves of positive length have to be timelike.Moreover, any globally hyperbolic smooth spacetime ( M, g ) splits isometri-cally as (
M, g ) ∼ = ( R × S, − β d t + h t ), where S is a Cauchy hypersurfacein M , β is a smooth positive function on R × S and h t is a t -dependentfamily of Riemannian metrics on each level set { t } × S (cf. [BS03, BS05]).Globally hyperbolic spacetimes can therefore be viewed as generalizations ofwarped products with one-dimensional base, so our methods may also findapplications to such spaces in future research.Both from the perspective of Lorentzian geometry and with a view tothe fundamental importance of warped products in General Relativity it istherefore of interest to study generalizations of such geometries beyond thesetting of smooth manifolds. A natural framework in which to carry out suchan extension is the theory of Lorentzian length spaces ([KS18, GKS19]), seeSubsection 1.2 below for a brief introduction. The plan of the paper is as follows. In the remainder of this introduction werecall the basic notions of the theory of Lorentzian length spaces. Section2 introduces Minkowski-cones, a Lorentzian analogue of cones over metricspaces, and a first instance of a generalized cone as defined in Section 3.The main result of Section 2 relates curvature bounds of the metric space X (the fiber) to timelike curvature bounds of the cone over X . In fact, weprove Theorem 2.5.
Let Y = Cone( X ) be the Minkowski cone over a geodesiclength space X . Then Y has timelike curvature bounded below (above) by if and only if X is an Alexandrov space of curvature bounded below (above)by − .
4n Section 3, we introduce the main object of this paper, a metric ana-logue of Lorentzian warped products with one-dimensional base and a lengthspace as fiber. We then study timelike and causal curves, introduce a time-separation function and establish the main features of causality theory forthese generalized cones . While there are a number of analogues to the met-ric theory of warped products (e.g., fiber independence of geodesics), thesecausality results require new methods. The main result in this section isthat generalized cones display no causal pathologies, a fact that is usedextensively later on. In more detail, we show that
Proposition 3.22. (Push-up and openness of I ± ) Every generalized cone Y = I × f X such that ( X, d ) is a length space has the property that p ≪ q ifand only if there exists a future directed causal curve from p to q of positivelength, i.e., push-up holds. Moreover, I ± ( p ) is open for any p ∈ Y . The above result is then used in Section 4 to establish that general-ized cones are examples of Lorentzian length spaces, without any additionalassumptions on the causality or on the warping function f : Theorem 4.8 & Corollary 4.9.
Any generalized cone I × f X , where ( X, d ) is a locally compact length space, is a strongly causal Lorentzian lengthspace. If X is a locally compact geodesic length space, then I × f X is a regularstrongly causal Lorentzian length space. In particular we prove that if the fiber X is a geodesic length space thatis proper then I × f X is globally hyperbolic. Section 5 is then devoted torelating synthetic curvature bounds (via triangle comparison) in generalizedcones to corresponding bounds in the fiber. Here the main results are asfollows: Theorem 5.3.
Let
K, K ′ ∈ R and let ( X, d ) be a geodesic length spacewith curvature bounded below/above by K . Then Y = I × f X has timelikecurvature bounded below/above by K ′ if I × f M ( K ) has timelike curvaturebounded below/above by K ′ . Theorem 5.7. If X is a geodesic length space, Y = I × f X has timelikecurvature bounded below (above) by K ′ and Y ′ = I × f M ( K ) has timelikecurvature bounded above (below) by K ′ then X has curvature bounded below(above) by K . Moreover, the first result above allows us to generate an abundance ofexamples of Lorentzian length spaces with timelike curvature bounds.We then apply our techniques in Section 6 to show that non-positivelower timelike curvature bounds imply the existence of incomplete timelikegeodesics. That is, we provide synthetic singularity theorems for generalizedcones. To be precise, we prove the following:5 orollary 6.4.
Let X be a geodesic length space, Y = I × f X with I =( a, b ) , f : I → (0 , ∞ ) smooth. Assume that Y has timelike curvature boundedbelow by K . Then:(i) If K < , then a > −∞ and b < ∞ and hence the time separationfunction τ Y of Y is bounded by b − a . Thus any such Y is timelikegeodesically incomplete.(ii) If K = 0 and f is non-constant, then a > −∞ or b < ∞ and hence Y is past or future timelike geodesically incomplete. In the present setup, these results are direct analogues of the LorentzianMyers’ theorem and of Hawking’s singularity theorem. Also, we relate time-like curvature bounds to big bang and big crunch singularities in Corollary6.7.In the appendix we describe a general approach to what we call
Lorentzianlength structures , analogous to the theory of length structures in metric ge-ometry (cf. [BBI01, Ch. 2]) based on which several basic results shown inSections 2 and 3 can be shown to hold in greater generality.
Here we give a very brief introduction to the theory of Lorentzian lengthspaces, as developed in [KS18], at the same time fixing some notations andterminology.Let X be a set endowed with a preorder ≤ and a transitive relation ≪ contained in ≤ . If x ≪ y or x ≤ y , we call x and y timelike or causallyrelated, respectively. If X is, in addition, equipped with a metric d anda lower semicontinuous map τ : X × X → [0 , ∞ ] that satisfies the reversetriangle inequality τ ( x, z ) ≥ τ ( x, y ) + τ ( y, z ) (for all x ≤ y ≤ z ), as well as τ ( x, y ) = 0 if x (cid:2) y and τ ( x, y ) > ⇔ x ≪ y , then ( X, d, ≪ , ≤ , τ ) is calleda Lorentzian pre-length space and τ is called the time separation function of X . A curve γ : I → X ( I an interval) that is non-constant on any sub-interval of I is called (future-directed) causal (timelike) if γ is locally Lips-chitz continuous and if for all t , t ∈ I with t < t we have γ ( t ) ≤ γ ( t )( γ ( t ) ≪ γ ( t )). It is called null if, in addition to being causal, no twopoints on the curve are related with respect to ≪ . For strongly causal con-tinuous Lorentzian metrics, this notion of causality coincides with the usualone ([KS18, Prop. 5.9]). In analogy to the theory of metric length spaces,the length of a causal curve is defined via the time separation function: For6 : [ a, b ] → X future-directed causal we set L τ ( γ ) := inf n N − X i =0 τ ( γ ( t i ) , γ ( t i +1 )) : a = t < t < . . . < t N = b, N ∈ N o . For smooth and strongly causal spacetimes (
M, g ) this notion of length co-incides with the usual one: L τ ( γ ) = L g ( γ ) ([KS18, Prop. 2.32]). A future-directed causal curve γ : [ a, b ] → X is maximal if it realizes the time sepa-ration, i.e., if L τ ( γ ) = τ ( γ ( a ) , γ ( b )). Standard causality conditions can alsobe imposed on Lorentzian pre-length spaces, and substantial parts of thecausal ladder ([MS08, Min19b]) continue to hold in this general setting, cf.[KS18, Subsec. 3.5].Lorentzian length spaces are close analogues of metric length spaces inthe sense that the time separation function can be calculated from the lengthof causal curves connecting causally related points. A Lorentzian pre-lengthspace that satisfies some additional technical assumptions (cf. [KS18, Def.3.22]) is called a Lorentzian length space if τ = T , where for any x, y ∈ X we set T ( x, y ) := sup { L τ ( γ ) : γ future-directed causal from x to y } , if the set of future-directed causal curves from x to y is not empty. Otherwiselet T ( x, y ) := 0.Any smooth strongly causal spacetime is an example of a Lorentzianlength space. More generally, spacetimes with low regularity metrics andcertain Lorentz-Finsler spaces [Min19a] provide further examples, cf. [KS18,Sec. 5].Finally, by a timelike geodesic triangle we mean a triple ( x, y, z ) ∈ X with x ≪ y ≪ z such that τ ( x, z ) < ∞ and such that the sides are realizedby future-directed causal curves (that is, there exist future directed causalcurves α, β, γ from x to y , from y to z and from x to z , respectively, with L ( α ) = τ ( x, y ), L ( β ) = τ ( y, z ) and L ( γ ) = τ ( x, z )). Curvature boundsare formulated by comparing such triangles with triangles of the same sidelengths in one of the Lorentzian model spaces L ( K ) of constant sectionalcurvature K . Here, L ( K ) = ˜ S ( r ) K = r R K = 0˜ H ( r ) K = − r , where ˜ S ( r ) is the simply connected covering manifold of the two-dimensionalLorentzian pseudosphere S ( r ), R is two-dimensional Minkowski space, and˜ H ( r ) is the simply connected covering manifold of the two-dimensionalLorentzian pseudohyperbolic space. In order to guarantee the existence of7omparison triangles in one of the model spaces, one needs to impose sizerestrictions on the sides, see [KS18, Lem. 4.6].Using this terminology, a Lorentzian pre-length space ( X, d, ≪ , ≤ , τ ) issaid to have timelike curvature bounded below (above) by K ∈ R if everypoint in X has a neighborhood U such that:(i) τ | U × U is finite and continuous.(ii) Whenever x, y ∈ U with x ≪ y , there exists a causal curve α in U with L τ ( α ) = τ ( x, y ).(iii) If ( x, y, z ) is a timelike geodesic triangle in U , realized by maximalcausal curves α, β, γ whose side lengths satisfy the appropriate sizerestrictions, and if ( x ′ , y ′ , z ′ ) is a comparison triangle of ( x, y, z ) in L ( K ) realized by timelike geodesics α ′ , β ′ , γ ′ , then whenever p , q arepoints on the sides of ( x, y, z ) and p ′ , q ′ are corresponding points of( x ′ , y ′ , z ′ ), we have τ ( p, q ) ≤ τ ′ ( p ′ , q ′ ) (respectively τ ( p, q ) ≥ τ ′ ( p ′ , q ′ )).We call such a U a comparison neighborhood. As a first explicit example we consider cones over metric spaces. Such spacesa very well-behaved and allow direct calculations even of spacelike distances.However, here we consider cones exclusively as Lorentzian length spaces,providing more details than in [Ale19], where such cones are considered inthe setting of
Lorentzian pseudometric spaces . In particular, they furnishinstances of generalized cones as defined in Section 3 (cf. Example 3.31).Proceeding by analogy with the metric geometry notion of cones overmetric spaces (cf. [BBI01, Subsec. 3.6.2]) we introduce the following: For X a geodesic length space, the Minkowski cone Y = Cone( X ) is defined as thequotient of [0 , ∞ ) × X resulting from identifying all points of the form (0 , p ).We equip Y with the cone metric d c as in [BBI01, Def. 3.6.16] (however thischoice is not important, as it suffices to pick some background metric on Y that induces the quotient topology on [0 , ∞ ) × X , to turn it into a Lorentzianpre-length space, see below). The equivalence class of { } × X in Y is calledthe vertex of Y and is denoted by 0 Y . Remark 2.1.
As a preparation for the following definition of the time sep-aration function, consider n -dimensional Minkowski-space R n , with scalar This means that p ′ lies on the side corresponding to the side containing p at the sametime separation of the vertex (i.e., e.g. if p lies on the side xy then τ ( x, p ) = τ ′ ( x ′ , p ′ ),etc.). Similarly for q ′ . h x, y i = − x y + P n − i =1 x i y i . Then ( n − H n − is isometrically embedded into R n as { x ∈ R n | h x, x i = − , x > } = { x ∈ R n | τ R n (0 , x ) = 1 , x > } =: Σ, where τ R n is the time separation function on R n . Let us denote this embedding by ψ . The induced Riemannian distance function on H n − is uniquely deter-mined by cosh d H n − ( x, y ) = −h ψ ( x ) , ψ ( y ) i for x, y ∈ H n − . Suppose nowthat x, y ∈ H n − and let s, t >
0. Then ψ ( x ) = ( p | x ′ | , x ′ ) for some x ′ ∈ R n − , and analogously for y . Setting θ := d H n − ( x, y ) we calculate h tψ ( y ) − sψ ( x ) , tψ ( y ) − sψ ( x ) i = − t − s − st h ψ ( x ) , ψ ( y ) i = − ( s + t − st cosh θ ) . This shows that Y := [0 , ∞ ) × H n − can be identified with the cone I + (0) ∪ ⊆ R n via ( s, x ) sψ ( x ), and restricting this identification, we see that(0 , ∞ ) × H n − corresponds to I + (0) ⊆ R n . Pulling back the causal structureand time separation of R n we make the following definitions: Two points( s, x ) and ( t, y ) in Y are said to satisfy ( s, x ) ≤ Y ( t, y ) if and only if s ≤ t and s + t − st cosh θ ≥
0, which is equivalent to sx ≤ ty in R n . In addition, for( s, x ) ≤ Y ( t, y ) the Minkowski cone time separation function τ Y is defined by τ Y (( s, x ) , ( t, y )) = τ R n ( sψ ( x ) , tψ ( y )) = √ s + t − st cosh θ , and otherwise τ Y (( s, x ) , ( t, y )) = 0.For future reference, let us briefly remark that the Minkowski cone timeseparation τ Y defined above induces a time separation τ C on C := (0 , ∞ ) × H n − via restriction and this restriction equals the time separation τ W ofthe Lorentzian warped product manifold W := (0 , ∞ ) × id H n − = (cid:0) (0 , ∞ ) × H n − , g := − dt + t h ., . i H n − (cid:1) :It suffices to show that the map ϕ : W → I + (0) ⊆ R n , ( s, x ) sψ ( x )is an isometry, as this will imply τ W (( s, x ) , ( t, y )) = τ R n ( sψ ( x ) , tψ ( y )) = τ C (( s, x ) , ( t, y )). We have Dϕ | ( r,z ) = r Dψ | z ◦ pr T H n − + ψ ( z )pr T R + , hencefor vectors ( S, X ) , ( T, Y ) ∈ T r R + × T z H n − we get g (( S, X ) , ( T, Y )) = − ST + r h X, Y i H n − = − ST + r h DψX, DψY i R n = h Sψ ( z ) , T ψ ( z ) i R n + h rDψ | z X, rDψ | z Y i R n = h Dϕ ( S, X ) , Dϕ ( T, Y ) i R n , because h ψ ( z ) , Dψ | z X i R n = 0 for any X ∈ T z H n − . So ϕ is an isometry, asclaimed.To equip the cone Y = [0 , ∞ ) × X as defined above with a time separationfunction, we proceed analogously, with the metric d X of X taking over therole of θ = d H from Remark 2.1. Thus we say that ( s, p ) ≤ ( t, q ) (resp.( s, p ) ≪ ( t, q )) if s ≤ t and s + t − st cosh d X ( p, q ) ≥ > τ (( s, p ) , ( t, q )) := p s + t − st cosh d X ( p, q ) , (1)9nd τ (( s, p ) , ( t, q )) := 0 otherwise. Proposition 2.2. ( Y, d, ≪ , ≤ , τ ) is a Lorentzian pre-length space. More-over, τ is continuous.Proof. Since the causal and timelike relations are defined via τ , and since τ is clearly continuous with respect to d c , it only remains to check the reversetriangle inequality for τ . So let ( s, p ) , ( t, q ) , ( u, r ) ∈ Y and fix three com-parison points ˜ p, ˜ q, ˜ r in H ⊆ R with d H (˜ p, ˜ q ) = d X ( p, q ), and so on (notethat some of these distances might be zero). Then from Remark 2.1 and thedefinition of τ it follows that (denoting by ˜ τ the time separation functionin R ), ˜ τ ( s ˜ p, t ˜ q ) = τ (( s, p ) , ( t, q )), etc. The reverse triangle inequality for τ therefore is immediate from that of ˜ τ . Lemma 2.3.
Suppose that Y = ( s, p ) ≪ ( t, q ) ∈ Y .(i) Let γ : [0 , a ] → Y , γ ( λ ) = ( r ( λ ) , σ ( λ )) be a maximizing timelike curvefrom ( s, p ) to ( t, q ) . Then σ is a minimizing geodesic from p to q in X .(ii) Conversely, suppose that σ is a minimizing geodesic from p to q in X .Let ˜ y and ˜ y be points in I + (0) ⊆ R with distance r (0) := s resp. r ( a ) := t from and such that the hyperbolic angle between them is d X ( p, q ) . For λ ∈ [0 , a ] , let r ( λ ) be the distance of the intersectionof the straight line connecting ˜ y to ˜ y with the half-ray in I + (0) thathas hyperbolic angle σ ( λ ) with the half-ray through ˜ y . Then λ ( r ( λ ) , σ ( λ )) is a τ -realizing curve from ( s, p ) to ( t, q ) in Y .Proof. (i) Let λ < λ < λ ∈ [0 , a ], let y i := ( r ( λ i ) , σ ( λ i )) and pick points˜ y i ( i = 1 , ,
3) in I + (0) ⊆ R such that their distance from 0 is r ( λ i ), thehyperbolic angle between ˜ y and ˜ y is d X ( σ ( λ ) , σ ( λ )), and the hyperbolicangle between ˜ y and ˜ y is d X ( σ ( λ ) , σ ( λ )). This means that τ ( y , y ) =˜ τ (˜ y , ˜ y ), as well as τ ( y , y ) = ˜ τ (˜ y , ˜ y ). Now by assumption, τ ( y , y ) = τ ( y , y ) + τ ( y , y ), and the ensuing equality for ˜ τ implies that the ˜ y i mustlie on a straight line in R . Consequently, their hyperbolic angles must addup, i.e., d X ( σ ( λ ) , σ ( λ )) + d X ( σ ( λ ) , σ ( λ )) = d X ( σ ( λ ) , σ ( λ )). It followsthat σ is indeed distance-realizing.(ii) This is straightforward from the definition of τ and Remark 2.1.As an immediate consequence of Lemma 2.3 (ii) (and the obvious factthat s ( s, q ) is a realizing geodesic from 0 Y ≡ (0 , q ) to ( t, q ) for all t > , q ∈ X ) we obtain: Corollary 2.4.
Any two causally related points in Y can be connected by arealizing geodesic, i.e., Y is geodesic . X and timelike curvaturebounds in the Minkowski cone Y over X , foreshadowing analogous resultsfor generalized cones in Section 5. In particular, the following theorem is aspecial case of Theorems 5.7 and 5.3 below (and is analogous to the resultin the metric case, cf. [BBI01, Thm. 4.7.1]). Theorem 2.5.
Let Y = Cone( X ) be the Minkowski cone over a geodesiclength space X . Then Y has timelike curvature bounded below (above) by if and only if X is an Alexandrov space of curvature bounded below (above)by − .Proof. We observe that timelike comparison triangles for Y and comparisontriangles for X can be related in the following way: Let ( s, p ) ≪ ( t, q ) ≪ ( u, r ) ∈ Y be the vertices of a timelike triangle ∆ in Y . If ( s, p ) = 0 Y ,choose three comparison points ˜ p, ˜ q, ˜ r in H ⊆ R with d H (˜ p, ˜ q ) = d X ( p, q ),and so on (note that ˜ p, ˜ q, ˜ r need not be pairwise distinct). If ( s, p ) = 0 Y ,choose two points ˜ q, ˜ r ∈ H with d H (˜ p, ˜ q ) = d X ( p, q ). Then by definition of τ , the points s · ˜ p, t · ˜ q and u · ˜ r in R form a timelike comparison triangle˜∆ for (( s, p ) , ( t, q ) , ( u, r )) (note that s · ˜ p = 0 if ( s, p ) = 0 Y and that thepoints s · ˜ p, t · ˜ q, u · ˜ r will always be pairwise distinct if ( s, p ) , ( t, q ) , ( u, r )are, even if ˜ p, ˜ q, ˜ r are not). Indeed, their ˜ τ -sidelengths in R are exactly the τ -lengths of the original triangle in Y , and so equivalently we may use thetwo-dimensional Minkowski space M spanned by s · ˜ p, t · ˜ q, u · ˜ r as a (flat)comparison space for ∆: M is clearly totally geodesic in R , so its timeseparation function is precisely the restriction of ˜ τ to M × M . Thus ˜∆ canjust as well be viewed as a subset of M .Suppose now, first, that Y has timelike curvature bounded below by 0.Let p ∈ X and let V ⊆ Y be a neighborhood of (1 , p ) on which timelikecomparison holds. Then there exists ε > U ⊆ X of p such that (1 − ε, ε ) × U ⊆ V and any triangle ( p, q, r ) in U can be lifted toa timelike triangle ( s, p ) ≪ ( t, q ) ≪ ( u, r ) in V (the last requirement followsfrom (1) and Lemma 2.3 (ii) by shrinking U but keeping ε fixed). Let ( p, q, r )form a triangle in U . Also, let m, n be points on the sides of ( p, q, r ) and˜ m, ˜ n be corresponding points on the sides of a comparison triangle (˜ p, ˜ q, ˜ r )in H . W.l.o.g. (renaming points if necessary) m lies on the side from p to q and n on the side from q to r . Given realizing geodesics in U for the edgesof ( p, q, r ), by Lemma 2.3 (ii) we obtain corresponding realizing geodesicsfor the sides of the triangle ∆ = (( s, p ) , ( t, q ) , ( u, r )) in V . Note that thepoints M = ( r pq ( λ m ) , m ) and N = ( r qr ( λ n ) , n ) on these geodesics satisfy( s, p ) ≤ M ≤ ( t, q ) ≤ N ≤ ( u, r ) and are timelike related (or equal). Let˜ M = ℓ ˜ m · ˜ m , ˜ N = ℓ ˜ n · ˜ n be points in R on the sides of the comparison triangle˜∆ := ( s · ˜ p, t · ˜ q, u · ˜ r ) in R . From the construction of r ( λ ) in Lemma 2.3 (ii)we see that r pq ( λ m ) = ℓ ˜ m and r qr ( λ n ) = ℓ ˜ n . So M, N ∈ ∆ and ˜ M , ˜ N ∈ ˜∆11re corresponding points and by (1) and the monotonicity of cosh it thenfollows that d X ( m, n ) ≥ d H ( ˜ m, ˜ n ), because τ ( M, N ) ≤ ˜ τ ( ˜ M , ˜ N ).Conversely, if X has curvature bounded below by −
1, then a similar (infact, easier) argument, this time based on Lemma 2.3 (i), shows that Y hastimelike curvature bounded below by 0.Bounds from above can be treated analogously. In this section, we introduce a generalization of warped products of metricspaces to the Lorentzian setting.
Definition 3.1.
For ( X, d ) a metric space and I ⊆ R an open interval, set Y := I × X and put the the product metric on Y , i.e., D (( t, x ) , ( t ′ , x ′ )) = | t − t ′ | + d ( x, x ′ ) for ( t, x ) , ( t ′ , x ′ ) ∈ Y . Let f : I → (0 , ∞ ) be continuous.Then Y ≡ I × f X is called a generalized cone and f is called warpingfunction . Alternatively, generalized cones can also be called (Lorentzian) warpedproducts with one-dimensional base .We first turn to the question of introducing an appropriate Lorentzianstructure on a generalized cone. To this end, we have to define causal curves.
Definition 3.2.
Let Y = I × f X be a generalized cone and let γ : J → Y be an absolutely continuous curve (with respect to D ). Such a curve hascomponents γ = ( α, β ) , where α : J → I and β : J → X are both abso-lutely continuous, and the metric derivative of β , v β , exists almost every-where (cf. [AGS05, Thm. 1.1.2]). We additionally require that α is strictlymonotonous. The curve γ is called timelikenullcausal if − ˙ α + ( f ◦ α ) v β <
0= 0 ≤ , almost everywhere. It is called future/past directed causal if α is strictlymonotonically increasing/decreasing, i.e., ˙ α > or ˙ α < almost every-where. Remark 3.3.
So far in the development of the theory of Lorentzian lengthspaces, locally Lipschitz continuous curves were used as causal curves. How-ever, as we shall establish in Lemma 3.13 below, every absolutely continuouscausal curve has a parametrization as a Lipschitz curve. So using absolutelycontinuous curves is compatible with the previous works [KS18, GKS19].Moreover, parametrizing a timelike curve with respect to arclength anyways12nly gives an absolutely continuous curve in general — an issue that alsonecessitated a special treatment in [KS18, Subsec. 3.7]. Analogous ques-tions arise for spacetimes with continuous metrics, in which case we refer to[GKSS19].
Definition 3.4. (Length of a causal curve) Let γ = ( α, β ) : [ a, b ] → Y be acausal curve. Its length L ( γ ) is defined as L ( γ ) := Z ba q ˙ α − ( f ◦ α ) v β . Remark 3.5.
Note that q ˙ α − ( f ◦ α ) v β is integrable as α is absolutelycontinuous, f is bounded on the compact image of α and the metric deriva-tive is integrable by [AGS05, Thm. 1.1.2]. Moreover, from this it followsthat the map t L ( γ | [ a,t ] ) = R ta q ˙ α − ( f ◦ α ) v β is absolutely continuous. Lemma 3.6.
Let ( Z, ρ ) be a metric space, J, J ′ intervals, λ : J → Z anabsolutely continuous curve and φ : J ′ → J strictly monotonous and suchthat both φ and φ − are absolutely continuous. Then ξ := λ ◦ φ is absolutelycontinuous and v ξ = ( v λ ◦ φ ) | φ ′ | . Proof:
That ξ is absolutely continuous follows as in [Nat55, Thm. 3, Ch.IX, § λ exists almost everywhere (cf.[AGS05, Thm. 1.1.2]), so let t ∈ J be such a point. Then for any h ∈ R suchthat t + h ∈ J we have ρ ( λ ( t + h ) , λ ( t )) = | h | v λ ( t ) + r ( h ) , (2)where the remainder term satisfies r ( h ) = o ( | h | ). Next, let s ∈ J ′ suchthat both φ ′ ( s ) and λ ′ ( φ ( s )) exist. The set of all such s has full measurein J ′ because the absolutely continuous function φ − maps sets of measurezero to sets of measure zero (Lusin’s property, cf. e.g. [AT04, Thm. 3.4.3]).Furthermore, let h ∈ R with s + h ∈ J ′ . We conclude that ρ ( ξ ( s + h ) , ξ ( s ))= ρ ( λ ( φ ( s + h )) , λ ( φ ( s ))) (2) = | hφ ′ ( s ) + r ( h ) | v λ ( φ ( s )) + r ( hφ ′ ( s ) + r ( h ))= | h || φ ′ ( s ) | v λ ( φ ( s )) + ( | hφ ′ ( s ) + r ( h ) | − | h || φ ′ ( s ) | ) | {z } = o ( | h | ) v λ ( φ ( s )) + o ( | h | )= | h || φ ′ ( s ) | v λ ( φ ( s )) + o ( | h | ) , yielding the claim.A direct corollary of the above lemma is that the length of causal curvesis invariant under reparametrizations.13 orollary 3.7. The length L is reparametrization invariant, i.e., if γ =( α, β ) is a causal curve defined on some interval J and φ : J ′ → J is strictlyincreasing and such that ϕ and ϕ − are absolutely continuous, then γ ◦ φ isa causal curve of the same length and time orientation. Remark 3.8.
Note that this means that Y with these future/past di-rected causal/timelike curves and this length functional is an example ofa Lorentzian length structure as defined in the Appendix, see DefinitionA.2.To establish that the length functional L is upper semicontinuous (withrespect to pointwise convergence) we need to describe the length in a vari-ational way. As we show below, the variational length is the same as thelength defined above via the (metric) derivative of the curve. Definition 3.9.
Let γ = ( α, β ) : [ a, b ] → Y be a causal curve. For s, t ∈ I , s ≤ t , set m s,t := min r ∈ [ s,t ] f ( r ) > . Then the variational length of γ isdefined as L var ( γ ) :=inf a = t Let ( X, d ) be a metric space and let γ = ( α, β ) : [ a, b ] → Y be a causal curve. Then for any a ≤ s ≤ t ≤ b we have: ( α ( t ) − α ( s )) − m α ( s ) ,α ( t ) d ( β ( s ) , β ( t )) ≥ . Proof: Without loss of generality let γ be future directed, i.e., ˙ α > γ is causal we have ( f ◦ α ) v β ≤ ˙ α and, as allinvolved quantities are non-negative, in fact ( f ◦ α ) v β ≤ ˙ α . Denote by L d the length functional of ( X, d ), then m α ( s ) ,α ( t ) L d ( β | [ s,t ] ) = m α ( s ) ,α ( t ) Z ts v β ≤ Z ts ( f ◦ α ) v β ≤ Z ts ˙ α = α ( t ) − α ( s ) . Finally, as d ( β ( s ) , β ( t )) ≤ L d ( β | [ s,t ] ) we conclude that m α ( s ) ,α ( t ) d ( β ( s ) , β ( t )) ≤ ( α ( t ) − α ( s )) .Also note that the variational length is invariant under reparametriza-tions as it is defined via partitions, cf. e.g. the proof of [Pap14, Prop. 1.1.8].Additionally, L var is additive which is easily inferred from the inequality (ii)in the next Lemma. 14 emma 3.11. Let a, b ∈ I with a ≤ s ≤ t ≤ u ≤ b and x, y, z ∈ X such that ( t − s ) − m s,t d ( x, y ) ≥ , (3)( u − t ) − m t,u d ( y, z ) ≥ . (4) Then(i) ( u − s ) − m s,u d ( x, z ) ≥ , and(ii) q ( t − s ) − m s,t d ( x, y ) + q ( u − t ) − m t,u d ( y, z ) ≤ q ( u − s ) − m s,u d ( x, z ) . Proof: Clearly, m s,u = min( m s,t , m t,u ) and without loss of generality wemay assume that m s,u = m s,t . Let ( X, Y, Z ) be a comparison triangle of( x, y, z ) in the plane R , i.e., k X − Y k = d ( x, y ), k X − Z k = d ( x, z ) and k Y − Z k = d ( y, z ). For c > η c on R as η c := − ( dx ) + c (( dx ) + ( dx ) ). We claim that ( s, X ) ≤ ( t, Y ) ≤ ( u, Z ) in ( R , η m s,t ). That ( s, X ) ≤ ( t, Y ) follows directly from (3), and that( t, Y ) ≤ ( u, Z ) follows from (4) and m s,t ≤ m t,u . Thus ( t, X ) ≤ ( u, Z ) bythe transitivity of the causal relation ≤ in ( R , η m s,t ), giving (i).To show (ii), denote by P the time separation function of ( R , η m s,t ).Then P (( s, X ) , ( t, Y )) = q ( t − s ) − m s,t d ( x, y ) , and P (( t, Y ) , ( u, Z )) = q ( u − t ) − m s,t d ( y, z ) ≥ q ( u − t ) − m t,u d ( y, z ) . Consequently, by the reverse triangle inequality for P we obtain q ( t − s ) − m s,t d ( x, y ) + q ( u − t ) − m t,u d ( y, z ) ≤ P (( s, X ) , ( t, Y )) + P (( t, Y ) , ( u, Z )) ≤ P (( s, X ) , ( u, Z ))= q ( u − s ) − m s,u d ( x, z ) , as m s,u = m s,t . Remark 3.12. The above Lemma 3.11 shows that the function T (( t, x ) , ( s, y )) := ( t − s ) − m s,t d ( x, y ) , if non-negative, and T (( t, x ) , ( s, y )) := 0 otherwise, satisfies the reverse tri-angle inequality. So in principle it could also be used to define a Lorentzian(pre-)length space. However, as it only involves the minimum of f on theinterval [ s, t ] it does not contain the full information of f on this interval and15t is not compatible with the smooth case (i.e., if X is a smooth Rieman-nian manifold and f is smooth). Despite this, it proves very useful whenhandling the variational length because the Lorentzian (pre-)length spacedefinition of length in ( I × X, ≪ , ≤ , T ) coincides with the variational length L var defined above. We will show in Proposition 3.14 that the variationallength equals the length defined in Definition 3.4. Lemma 3.13. Every future directed causal curve has a reparametrizationthat is locally Lipschitz continuous. In particular, any future directed causalcurve defined on a compact interval γ : [ a, b ] → Y has a reparametrization ˜ γ such that ˜ γ ( t ) = ( t, ˜ β ( t )) . Similarly, γ has a reparametrization γ ′ = ( α ′ , β ′ ) such that β ′ : [0 , L d ( β ′ )] → X is parametrized with respect to arc length. Proof: As this is a local question we may restrict to the case of compactintervals. Let γ = ( α, β ) : [ a, b ] → Y be future directed causal, then ˙ α > α − is absolutely continuous, hence can serve as an admissible parametrizationfor γ . Then ˜ γ := γ ◦ α − satisfies ˜ γ ( t ) = ( t, ˜ β ( t )), where ˜ β := β ◦ α − .By Corollary 3.7, ˜ γ is future directed causal and so we have f v β ≤ v ˜ β ≤ C , where C := min r ∈ [ α ( a ) ,α ( b )] f ( r ) > 0. This implies that ˜ β isLipschitz continuous, due to d ( β ( s ) , β ( t )) ≤ L d ( β | [ s,t ] ) = Z ts v ˜ β ≤ C | t − s | , where α ( a ) ≤ s < t ≤ α ( b ). Proposition 3.14. Let ( X, d ) be a metric space. Then the variational lengthof any causal curve γ in Y = I × f X agrees with its length, i.e., L ( γ ) = L var ( γ ) . Proof: Let γ be a (without loss of generality) future directed causal curve.As L var and L are invariant under reparametrizations, using Lemma 3.13 wemay assume that γ : [ a, b ] → Y is parametrized as γ ( t ) = ( t, β ( t )) (where a, b ∈ I ). Let a ≤ s < t ≤ b , then d ( β ( s ) , β ( t )) ≤ L d ( β | [ s,t ] ) = R ts v β and so1( t − s ) d ( β ( s ) , β ( t )) ≤ t − s ) ( Z ts v β ) ≤ t − s Z ts v β , where in the last step we used Jensen’s inequality. Thus we obtain1 t − s d ( β ( s ) , β ( t )) ≤ Z ts v β . (5)16gain using Jensen’s inequality we estimate (cid:16) t − s Z ts q − f v β (cid:17) ≤ t − s Z ts (1 − f v β ) = 1 − Z ts f v β t − s ≤ − m s,t Z ts v β t − s (5) ≤ − m s,t d ( β ( s ) , β ( t )) ( t − s ) , which yields Z ts q − f v β ≤ ( t − s ) s − m s,t d ( β ( s ) , β ( t )) ( t − s ) = q ( t − s ) − m s,t d ( β ( s ) , β ( t )) . (6)Now let a = t < t < . . . < t N = b be a partition of [ a, b ], then L ( γ ) = Z ba q − f v β = N − X i =0 Z t i +1 t i q − f v β (6) ≤ N − X i =0 q ( t i +1 − t i ) − m t i ,t i +1 d ( β ( t i ) , β ( t i +1 )) , and taking the infimum over all partitions of [ a, b ] gives L ( γ ) ≤ L var ( γ ).For the reverse inequality, note that we have by definition of L var as theinfimum over all partitions of the interval that L var ( γ | [ s,t ] ) ≤ q ( t − s ) − m s,t d ( β ( s ) , β ( t )) , (7)for all a ≤ s < t ≤ b . Let 0 < ε < b − a , set ˜ b := b − ε > a , h := ˜ b − aN where N ∈ N is such that h ≤ ε and set t i := a + ih for i = 0 , . . . , N .Then for t ∈ [ a, ˜ b ] we have t + h ∈ [ a, b ] and hence by Lemma 3.10 that17 − m t,t + h d ( β ( t ) , β ( t + h )) ≥ h Z ˜ ba q h − m t,t + h d ( β ( t ) , β ( t + h )) d t = 1 h N − X i =0 Z t i +1 t i q h − m t,t + h d ( β ( t ) , β ( t + h )) d t ( ∗ ) = 1 h Z h N − X i =0 q h − m t i + t,t i +1 + t d ( β ( t i + t ) , β ( t i +1 + t )) d t ≥ h Z h L var ( γ | [ a + t, ˜ b + t ] ) d t = 1 h Z h (cid:16) L var ( γ ) − L var ( γ | [ a,a + t ] ) | {z } ≤ L var ( γ | [ a,a + h ] ) − L var ( γ | [˜ b + t,b ] ) | {z } ≤ L var ( γ | [˜ b,b ] ) (cid:17) d t ≥ L var ( γ ) − L var ( γ | [ a,a + h ] ) − L var ( γ | [˜ b,b ] ) (7) ≥ L var ( γ ) − q h − m a,a + h d ( β ( a ) , β ( a + h )) − q ε − m b − ε,b d ( β ( b − ε ) , β ( b )) , where we used additivity of L var (cf. Lemma 3.11) and, in ( ∗ ), the substi-tution t ′ = t + a + ih = t + t i . As 0 ≤ h q h − m t,t + h d ( β ( t ) , β ( t + h )) ≤ ∈ L ([ a, b ]) for all h ≥ t + h ∈ [ a, b ], we have by dominatedconvergence and the above inequality that Z ˜ ba q − f v β = lim h ց Z ˜ ba r − m t,t + h d ( β ( t ) , β ( t + h )) h d t ≥ L var ( γ ) − q ε − m b − ε,b d ( β ( b − ε ) , β ( b )) . Thus L ( γ ) = L ( γ | [ a,b − ε ] ) + L ( γ | [ b − ε,b ] ) ≥ L var ( γ ) − q ε − m b − ε,b d ( β ( b − ε ) , β ( b )) + L ( γ | [ b − ε,b ] ) , and letting ε ց Proposition 3.15. Let ( X, d ) be a metric space and let γ n , γ ( n ∈ N ) becausal curves defined on the interval [ a, b ] such that γ n → γ pointwise. Then lim sup n L ( γ n ) ≤ L ( γ ) , i.e., L is upper semicontinuous. roof: Let σ = ( a = t < t < . . . < t N = b ) be a partition of [ a, b ], thenthe map Φ σ defined on the space of causal curves λ = ( α, β ) : [ a, b ] → Y given byΦ σ ( λ ) := N − X i =0 q ( α ( t i +1 ) − α ( t i )) − m α ( t i ) ,α ( t i +1 ) d ( β ( t i ) , β ( t i +1 )) is clearly continuous with respect to pointwise convergence. Then L var ( λ ) =inf σ Φ σ ( λ ) and so L = L var (by Proposition 3.14) is upper semicontinuousas the infimum of continuous functions (cf., e.g., [AB06, Lem. 2.41]). Theorem 3.16. (Limit curve theorem) Let ( X, d ) be a metric space andlet γ n = ( α n , β n ) ( n ∈ N ), γ = ( α, β ) : [ a, b ] → Y be absolutely continuouscurves such that each γ n is future/past directed causal. Moreover, let ˙ α = 0 almost everywhere and let γ n → γ pointwise. Then γ is causal. Proof: Let a ≤ s < t ≤ b be such that ˙ α ( s ) and v β ( s ) exist. For every n ∈ N we have by Lemma 3.10( α n ( t ) − α n ( s )) − m α n ( s ) ,α n ( t ) d ( β n ( s ) , β n ( t )) ≥ . Taking the limit n → ∞ yields( α ( t ) − α ( s )) − m α ( s ) ,α ( t ) d ( β ( s ) , β ( t )) ≥ , so (cid:16) α ( t ) − α ( s ) t − s (cid:17) − m α ( s ) ,α ( t ) d ( β ( s ) , β ( t )) ( t − s ) ≥ . Now letting t ց s we get˙ α ( s ) − f ( α ( s )) v β ( s ) ≥ . Moreover, similarly one shows that ˙ α ≥ γ n is future directed) or˙ α ≤ γ n is past directed), which yields ˙ α > α < γ is a future or past directed causal curve.At this point we define a natural time separation function on Y , whichdirectly generalizes the spacetime case. Some of the results could have beenobtained in an even more general setting as they follow just from the exis-tence of a causal structure and a length functional — a fact that was alreadyindicated in [KS18, Rem. 5.11(i)]. For the interested reader we sketch thisapproach Appendix A, but it is not needed in the following.19 efinition 3.17. (Time separation function) The time separation function τ : Y × Y → [0 , ∞ ] is defined as τ ( y, y ′ ) := sup { L ( γ ) : γ future directed causal curve from y to y ′ } , if this set is non-empty, and τ ( y, y ′ ) := 0 otherwise. Definition 3.18. (Causal relations) Let y, y ′ ∈ Y , then y and y ′ are chrono-logically related, denoted by y ≪ y ′ , if there exists a future directed timelikecurve from y to y ′ . Moreover, y and y ′ are causally related, denoted by y ≤ y ′ if there exists a future directed causal curve from y to y ′ or y = y ′ .Moreover, we define the chronological and causal future and past of apoint as I + ( y ) := { y ′ ∈ Y : y ≪ y ′ } , I − ( y ) := { y ′ ∈ Y : y ′ ≪ y } ,J + ( y ) := { y ′ ∈ Y : y ≤ y ′ } , J − ( y ) := { y ′ ∈ Y : y ′ ≤ y } . Lemma 3.19. The relations ≪ and ≤ are transitive, ≤ is reflexive and ≪ ⊆ ≤ . Proof: Transitivity follows by concatenating curves. Reflexivity of thecausal relation ≤ as well as the fact that every timelike curve is causal holdby definition. Thus ≪ ⊆ ≤ .This can be summarized as: Remark 3.20. The time separation function τ has the following properties:(i) τ ( y, y ′ ) = 0 if y ′ y and(ii) τ ( y, y ′ ) > y ≪ y ′ . Lemma 3.21. (Reverse triangle inequality) Let y , y , y ∈ Y with y ≤ y ≤ y , then τ ( y , y ) + τ ( y , y ) ≤ τ ( y , y ) . Proof: This follows from the standard proof from Lorentzian geometry:Let y , y , y ∈ Y with y ≤ y ≤ y and assume first that there are futuredirected causal curves from y to y and from y to y . Then, given ε > γ from y to y and γ from y to y such that L ( γ i ) > τ ( y i , y i +1 ) − ε for i = 1 , 2. Consequently, τ ( y , y ) + τ ( y , y ) < L ( γ ) + L ( γ ) + ε ≤ τ ( y , y ) + ε , as the concatenation of γ and γ is a future directed causal curve from y to y . Since ε > y to y , we have τ ( y , y ) = 0 and y = y , which implies the claim.20pacetimes of low regularity (below Lipschitz) can exhibit the unwantedphenomenon of causal bubbling , as shown in [CG12](cf. also [GKSS19]) forspacetimes with continuous metrics. However, the additional structure of ageneralized cone excludes such pathologies.For the formulation of the following result, we recall some terminologyfrom [CG12]: Y is said to possess the push-up property if the followingholds: Whenever γ : [ a, b ] → M is a future/past directed causal curve from p = γ ( a ) to q = γ ( b ) with L ( γ ) > 0, there exists a future/past directedtimelike curve connecting p and q . Proposition 3.22. (Push-up and openness of I ± ) Every generalized cone Y = I × f X such that ( X, d ) is a length space has the property that p ≪ q ifand only if there exists a future directed causal curve from p to q of positivelength, i.e., push-up holds. Moreover, I ± ( p ) is open for any p ∈ Y . Proof: For each p ∈ I ≡ ( a, b ) we define the function h p : ( a p , b p ) → ( a, b ) as the unique maximal solution of the ODE dds h p = f ◦ h p with h p (0) = p on I . Here a p = R ap f ( s ) ds and b p = R bp f ( s ) ds , and h p is theinverse of r R rp f ( s ) ds . The function h p is strictly increasing, bijectiveand C . We are going to show that I + (( p , ¯ p )) = { ( q , ¯ q ) ∈ Y : d (¯ p, ¯ q ) < b p and q > h p ( d (¯ p, ¯ q )) } , (8)which is clearly open. In proving this we will also see that q ∈ I + ( p ) if thereexists a causal curve γ from p to q with L ( γ ) > A ( p ) := { ( q , ¯ q ) ∈ Y : d (¯ p, ¯ q ) < b p and q >h p ( d (¯ p, ¯ q )) } ⊆ I + ( p ). Let q ∈ A ( p ) and pick an almost minimizing unit-speed curve β : [0 , d (¯ p, ¯ q ) + ε ] → X from ¯ p to ¯ q in X , as well as c > d (¯ p, ¯ q ) + ε + c < b p and q = h p ( d (¯ p, ¯ q ) + ε + c ). We define α ( s ) := h p ( s + cd (¯ p, ¯ q )+ ε s ). Then γ = ( α, β ) is a future directed timelikecurve from p to q , since˙ α ( s ) = (cid:16) cd (¯ p, ¯ q ) + ε (cid:17) ˙ h p (cid:16) s + cd (¯ p, ¯ q ) + ε s (cid:17) > ˙ h p (cid:16) s + cd (¯ p, ¯ q ) + ε s (cid:17) = f (cid:16) h p (cid:16) s + cd (¯ p, ¯ q ) + ε s (cid:17)(cid:17) = f ( α ( s )) . Now we show that if q / ∈ A ( p ), then there cannot exist any future directedcausal curve γ from p to q with L ( γ ) > 0. Assume to the contrary that such acurve exists and is parametrized so that γ : [ p , q ] → Y and γ ( s ) = ( s, ¯ γ ( s )).We start with the case where d (¯ p, ¯ q ) < b p but q ≤ h p ( d (¯ p, ¯ q )). Let β ε :[0 , d (¯ p, ¯ q ) + ε ] → X ( ε > 0) be an almost minimizing unit-speed curve in X from ¯ p to ¯ q and set ˜ β ε := β ε ◦ h − p | [ p ,h p ( d (¯ p, ¯ q )+ ε )] and n ε ( s ) := ( s, ˜ β ε ( s )).Then n ε : [ p , h p ( d (¯ p, ¯ q ) + ε )] → Y is a null curve, or equivalently, v ˜ β ε ( s ) =21 f ( s ) . Thus, we have L d ( ˜ β ε ) = Z h p ( d (¯ p, ¯ q )+ ε ) p v ˜ β ε = Z h p ( d (¯ p, ¯ q )+ ε ) p f . So letting ε → d (¯ p, ¯ q ) = R h p ( d (¯ p, ¯ q )) p f ≥ R q p f . Since γ is causal, wehave v ¯ γ ≤ f and furthermore since L ( γ ) > f on some subset of [ p , q ] having non-zero measure. So, d (¯ p, ¯ q ) = Z h p ( d (¯ p, ¯ q )) p f ≥ Z q p f > Z q p v ¯ γ = L d (¯ γ ) , a contradiction.Finally, we treat the case where d (¯ p, ¯ q ) ≥ b p . We again assume that γ : [ p , q ] → Y with parametrization γ ( s ) = ( s, ¯ γ ( s )) is a future directedcausal curve from p to q . Since q < b and h p ( s ) → b as s ր b p we canchoose ε > q < h p ( b p − ε ). Let further x := γ ( x ) = ( x , ¯ x ) bethe point on γ such that b p − ε = d (¯ p, ¯ x ). So γ | [0 ,x ] is a causal curve from p to ( x , ¯ x ) with d (¯ p, ¯ x ) < b p and x < q < h p ( b p − ε ) = h p ( d (¯ p, ¯ x )).Hence, as above, d (¯ p, ¯ x ) = Z h p ( d (¯ p, ¯ x )) p f > Z q p f ≥ Z q p v ¯ γ = L d (¯ γ )leads to a contradiction. Remark 3.23. The preceding result can be understood as establishing thatgeneralized cones are causally plain , i.e., there is no causal bubbling. Thisnotion of causal plainness is however not the same as the one in [CG12, Def.1.16] for spacetimes with continuous metrics. The reason is that we cannotspeak about approximating smooth metrics, and hence have no notion oftimelike curves for approximating metrics which have their lightcones insidethose of the original Lorentzian metric. However, as shown in [GKSS19]our notion of causal plainness (i.e., the condition that push-up holds) isequivalent to the absence of external bubbling , cf. [GKSS19, Thm. 2.12].Furthermore, if X is a Riemannian manifold with continuous metric h (i.e.,if Y is a Lorentzian manifold with continuous metric g = − d t + f ( t ) h )it can be seen from the description (8) of I + that I + = ˇ I + and locally ∂I + = ∂J + , so that in this case Y is indeed causally plain as defined in[CG12, Def. 1.16]. Moreover, the preceding result also sheds some light onthe causality of the so-called Colombini-Spagnolo metrics (cf. [CG12, Sec.2.1], [CS89]), i.e., metrics on R × S of the form − d t + f ( t, x )d x , where f ( t, x ) = F ( t ) F ( x ) , for a specific continuous positive function F .22 orollary 3.24. Let ( X, d ) be a length space. The following description,analogous to (8) , holds for J + : J + (( p , ¯ p )) = I + (( p , ¯ p )) ∪ { ( q , ¯ q ) ∈ Y : ∃ a minimizing curve in X from ¯ p to ¯ q and d (¯ p, ¯ q ) < b p and q = h p ( d (¯ p, ¯ q )) } . (9) Further, if X is geodesic, then J ± ( p ) is closed.Proof. That J ± ( p ) is closed if X is geodesic follows from (9): Let q k =( q k , ¯ q k ) be elements of the right hand side of (9) that converge to ( q , ¯ q ).To see that also q is an element of this set it suffices to exclude the casewhere d (¯ p, ¯ q k ) → b p = d (¯ p, ¯ q ). However, in this case we would have q k = h p ( d (¯ p, ¯ q k )) → b , resulting in q = b , a contradiction to q ∈ I = ( a, b ).It remains to show (9). First, let q = ( q , ¯ q ) be an element of the righthand side of (9), let β : [0 , d (¯ p, ¯ q )] → X be a minimizing unit speed curveand set γ : [0 , d (¯ p, ¯ q )] → Y , γ ( s ) := ( h p ( s ) , β ( s )). Then γ is a null curveconnecting p and q , so q ∈ J + ( p ).Conversely, we have to show that for any ( q , ¯ q ) ∈ J + ( p ) \ I + ( p ) theremust exist a minimizing curve in X from ¯ p to ¯ q with d (¯ p, ¯ q ) < b p and q = h p ( d (¯ p, ¯ q )). Let γ = ( α, β ) be a causal curve from p to q with β : [0 , d ] → X parametrized by arc-length. By Proposition 3.22, since q / ∈ I + ( p ), we musthave L ( γ ) = 0, i.e., ˙ α = f v β = f a.e., so ˙ α = f ◦ α . Since α (0) = p it follows that α = h p . If β were not minimizing, i.e., if d (¯ p, ¯ q ) < d , therewould exist a curve ¯ β : [0 , d ] → X from ¯ p to ¯ q , parametrized proportional toarclength, which is strictly shorter than β , hence satisfies v ¯ β < v β a.e.Then ¯ γ := ( α, ¯ β ) is a timelike curve from p to q , contradicting the fact that q / ∈ I + ( p ). Consequently, β must be minimizing. Thus d = d (¯ p, ¯ q ), so that q = α ( d (¯ p, ¯ q )) = h p ( d (¯ p, ¯ q )). Lemma 3.25. Let Y = I × f X be a generalized cone, where ( X, d ) is alength space. Then the time separation function τ is lower semi-continuous(with respect to D ). Proof: As the standard proof from Lorentzian geometry only uses open-ness of I + and the reverse triangle inequality it still works in our setting:Let y, y ′ ∈ Y and first assume that 0 < τ ( y, y ′ ) < ∞ (in the case τ ( y, y ′ ) = 0there is nothing to show). Let 0 < ε < τ ( y, y ′ ), then by definition of τ there exists a future directed causal curve γ : [ a, b ] → Y from y to y ′ with L ( γ ) ≥ τ ( y, y ′ ) − ε > 0. By Remark 3.5 there are 0 < t ≤ t < b such that0 < L ( γ | [ a,t ] ) < ε and 0 < L ( γ | [ t ,b ] ) < ε . Setting y := γ ( t ), y := γ ( t )and U := I − ( y ), V := I + ( y ) we obtain that τ ( y, y ) ≥ L ( γ | [ a,t ] ) > 0, thus y ≪ y and hence y ∈ U , which by Proposition 3.22 is an open neighbor-hood of y . Analogously we get that y ′ is in the open set V . At this point23et ( r, r ′ ) ∈ U × V , then r ≪ y ≤ y ≪ r ′ and thus by the reverse triangleinequality (Lemma 3.21) we obtain τ ( r, r ′ ) ≥ τ ( r, y ) | {z } ≥ + τ ( y , y ) + τ ( y , r ′ ) | {z } ≥ ≥ τ ( y , y ) ≥ L ( γ | [ t ,t ] )= L ( γ ) − L ( γ | [ a,t ] ) − L ( γ | [ t ,b ] ) ≥ τ ( y, y ′ ) − ε − ε − ε τ ( y, y ′ ) − ε , which finishes this case. For the case τ ( y, y ′ ) = ∞ the above constructionshows the existence of arbitrarily long future directed causal curves from r to r ′ , so τ attains arbitrarily large values on suitable neighborhoods of( y, y ′ ). Proposition 3.26. Let Y = I × f X be a generalized cone, where ( X, d ) isa length space. Then ( Y, D, ≪ , ≤ , τ ) is a Lorentzian pre-length space. Proof: By Lemma 3.19 ( Y, ≪ , ≤ ) is a causal space and by Lemma 3.25 thetime separation function is lower semi-continuous. Finally, Remark 3.20 andProposition 3.22 give the required properties of τ , cf. [KS18, Def. 2.8]. Example 3.27. Let f : I → (0 , ∞ ) be continuous and let ( X, h ) be aRiemannian manifold. Then considered as a Lorentzian pre-length space,the warped product I × f X , i.e., the product manifold I × X endowed withthe continuous Lorentzian metric − dt + f h coincides with the generalizedcone I × f X , as is immediate from Definitions 3.4 and 3.17. Therefore, thereis no ambiguity in our notation. Definition 3.28. Let Y = I × f X be a generalized cone and let γ =( α, β ) : [ a, b ] → Y be a causal curve. Then the energy of γ is defined as E ( γ ) := 12 Z ba ˙ α − ( f ◦ α ) v β . Contrary to the length, the energy of a curve depends on its parametriza-tion. Nevertheless it will turn out to be a useful tool.The following is an analogue of [AB98, Thm. 3.1] in the Riemanniancase. Theorem 3.29. Let ( X, d ) be a geodesic length space and let γ = ( α, β ) : [0 , b ] → Y = I × f X be future directed causal and maximal. Then:(i) The fiber component β is minimizing in X .(ii) Fiber independence holds, i.e., the base component α is independent of β , i.e., α depends only on the length of β . More precisely, let ( X ′ , d ′ )24 e another geodesic length space, β ′ minimizing in X ′ with L d ′ ( β ′ ) = L d ( β ) and the same speed as β , i.e, v β = v β ′ . Then γ ′ := ( α, β ′ ) is a future directed maximal causal curve in Y ′ := I × f X ′ , which istimelike if γ is timelike in Y .(iii) If γ is timelike, then it has an (absolutely continuous) parametriza-tion with respect to arclength, i.e., − ˙ α + ( f ◦ α ) v β = − almosteverywhere.(iv) If γ is timelike and parametrized with respect to arclength (so b = L ( γ ) ), then the energy of γ , E ( γ ) , is minimal under all reparametriza-tions of γ on [0 , b ] .(v) If γ is timelike and parametrized with respect to arclength, then v β isproportional to f ◦ α ) .(vi) If γ is timelike, it has an (absolutely continuous) parametrization pro-portional to arclength such that − ˙ α + f ◦ α ) is constant. Proof: (i) Assume that β is not minimal (and hence not constant). We maysuppose that β is parametrized with respect to arclength, i.e., v β =1 almost everywhere and γ : [0 , b ] → X , where b = L d ( β ). Since β is not minimal, there exists another curve ¯ β from β (0) to β ( b ),parametrized with respect to arclength, with 0 < L d ( ¯ β ) < L d ( β ) = b .We set T := L d ( ¯ β ) < b and define ¯ γ = ( ¯ α, ¯ β ) : [0 , T ] → Y by setting¯ α ( s ) := α ( bT s ) for s ∈ [0 , T ]. Then clearly ¯ γ is timelike and using thereparametrization ¯ s = bT s we get L (¯ γ ) = Z T q ( ˙¯ α ( s )) − f ( ¯ α ( s )) d s = Tb Z b r(cid:16) bT ˙ α (¯ s ) (cid:17) − f ( α (¯ s )) d¯ s = Z b r ˙ α (¯ s ) − (cid:16) Tb f ( α (¯ s )) (cid:17) d¯ s > L ( γ ) , a contradiction.(ii) Let β ′ be minimizing in X ′ , defined on [0 , b ] with L d ′ ( β ′ ) = L d ( β ) and v β = v β ′ . Set γ ′ := ( α, β ′ ) : [0 , b ] → Y ′ . Then γ ′ is future directedcausal and L ( γ ′ ) = Z b q ˙ α − ( f ◦ α ) v β ′ = Z b q ˙ α − ( f ◦ α ) v β = L ( γ ) . 25t remains to show that γ ′ is maximal in Y ′ from ( α (0) , β ′ (0)) =:( t , x ′ ) =: y ′ to ( α ( b ) , β ′ ( b )) =: ( t , x ′ ) =: y ′ . To this end assumeto the contrary that there is a ˜ γ = ( ˜ α, ˜ β ) : [0 , b ] → Y ′ that is futuredirected causal from y ′ to y ′ and longer than γ ′ , i.e., L (˜ γ ) > L ( γ ′ ).Without loss of generality we may assume that ˜ γ is parametrized suchthat ˜ β has speed v ˜ β proportional to v β ′ . By minimality of β ′ thisimplies that v ˜ β ≥ v β ′ . Set ¯ γ := ( ˜ α, β ′ ) : [0 , b ] → Y ′ , then ¯ γ is futuredirected causal from y ′ to y ′ and L (˜ γ ) ≤ L (¯ γ ). Furthermore, weobtain L (¯ γ ) = Z b q ˙˜ α − ( f ◦ ˜ α ) v β ′ = Z b q ˙˜ α − ( f ◦ ˜ α ) v β = L (( ˜ α, β )) . Consequently, L (( ˜ α, β )) = L (¯ γ ) ≥ L (˜ γ ) > L ( γ ′ ) = L ( γ ), contradictingthe maximality of γ , as ˜ α and α have the same endpoints.(iii) Let γ be timelike and define φ ( s ) := R s q ˙ α − ( f ◦ α ) v β for s ∈ [0 , b ].Then φ : [0 , b ] → [0 , L ( γ )] is absolutely continuous and strictly mono-tonically increasing. Moreover, φ − exists and is absolutely contin-uous as ˙ φ > γ := γ ◦ φ − ≡ ( ˜ α, ˜ β ) is absolutely continuous by [Nat55, Thm. 3, Ch.IX] and satisfies − ˙˜ α + ( f ◦ ˜ α ) v β = − γ on [0 , b ].(v) The claim follows as in the proof of [AB98, Thm. 3.1] by establishingthat R I ( f ◦ ˜ α ) v ˜ β = R J ( f ◦ ˜ α ) v ˜ β for all intervals I, J ⊆ [0 , b ] of thesame length, where one uses point (iv) above.(vi) By the previous two points we can assume that γ is parametrized withrespect to arclength, i.e., − ˙ α +( f ◦ α ) v β = − v β = c ( f ◦ α ) for some constant c . For c = 0, the reparametriza-tion ˜ γ ( s ) := γ ( sc ) does the job, and for c = 0 (i.e., v β = 0) thereparametrization ˜ γ = ( ˜ α, ˜ β ) := γ ◦ φ − , where φ ( t ) := R t f ◦ α yields − ˙˜ α + f ◦ ˜ α ) = 0.As a first consequence of fiber-independence we obtain: Corollary 3.30. Let X be a geodesic length space and let Y = I × f X .Then any maximizing causal curve γ = ( α, β ) : [ − b, b ] → Y has a causalcharacter, i.e., γ is either timelike or null. roof: Denote by Y ′ the Lorentzian warped product I × f R , i.e., themanifold I × R endowed with the continuous metric − dt + f dx . Let β be parametrized by arclength and set β ′ : [ − b, b ] → R , β ′ ( t ) := t . Then byTheorem 3.29,(ii) (and Example 3.27), γ ′ := ( α, β ′ ) is a causal maximizerin Y ′ . We now use the same basic ideas as in the proof of [GL18, Thm. 1.1](the difference being that the construction of the relevant curves is different,due to the metric being not locally Lipschitz but having a warped productstructure) to show that γ ′ is either timelike or null. The same must thereforebe true for γ .Since we exclusively work in Y ′ from now on, we will drop the ′ from ournotation. Assume γ is neither null nor timelike. Without loss of generality,we may assume γ (0) = ( α (0) , 0) = (0 , γ (0) exists and is timelike and N := { s ∈ [ − b, 0] : ˙ γ ( s ) is null } has non-zero measure. Let ε be positiveand define γ ε : [ − b, → I × R by γ ε ( s ) = ( α ( s ) , β ε ( s )) := ( α ( s ) , √ − ε s + b √ − ε − b ) . Then γ ε ( − b ) = γ ( − b ) = ( α ( − b ) , − b ) and γ ε (0) = ( α (0) , β ε (0)) = (0 , b √ − ε − b ). Note that for ε < , there exists C > | β ε (0) | = | b √ − ε − b | ≤ Cε . We estimate L ( γ ε ) = Z − b p ˙ α − ( f ◦ α ) (1 − ε ) = Z [ − b, \ N p ˙ α − ( f ◦ α ) (1 − ε )++ √ ε Z N f ◦ α | {z } =: c> ≥ Z [ − b, \ N p ˙ α − ( f ◦ α ) + c √ ε = L ( γ | [ − b, ) + c √ ε. Next, note that there exists η > C > f ( t ) < C < ˙ α (0) for t ∈ [ − η , η ] and since α ( s ) = ( ˙ α (0) + h ( s )) s with h ( s ) → s → 0, there also exist 0 < η and 0 < C < C < ˙ α (0) < C suchthat C s > α ( s ) > C s for s ∈ [0 , η ]. Fix k with C < k < C and set s ε := − kC − k β ε (0). Then 0 < s ε < CkC − k ε for ε < . Let ε small enough suchthat s ε + | β ε (0) | < min { η , η C } . Then the straight lines g : s ( C s, s ) and g k : s ( ks − kβ ε (0) , s ) are timelike on [ β ε (0) , s ε ] and intersect each otherin s ε > 0. Further, α ( s ) > C s on (0 , s ε ], so ( α (0) , 0) = (0 , 0) lies strictlybelow g k but ( α ( s ε ) , s ε ) lies strictly above g k (since it lies strictly above g ( s ε ) which is equal to g k ( s ε )) and hence s ( α ( s ) , s ) = γ ( s ) intersects g k in some 0 < ¯ s < s ε . Note that g k | [ β ε (0) , ¯ s ] is a future directed timelikecurve from γ ε (0) to γ (¯ s ). Now we estimate the length of the concatenation27s follows: L ( γ ε ∗ g k | [ β ε (0) , ¯ s ] ) > L ( γ ε ) ≥ c √ ε + L ( γ | [ − b, )= L ( γ | [ − b, ¯ s ] ) + c √ ε − L ( γ | [0 , ¯ s ] ) ≥ L ( γ | [ − b, ¯ s ] ) + c √ ε − α (¯ s ) > L ( γ | [ − b, ¯ s ] ) + c √ ε − C ¯ s> L ( γ | [ − b, ¯ s ] ) + c √ ε − C s ε > L ( γ | [ − b, ¯ s ] ) + c √ ε − C CkC − k ε > L ( γ | [ − b, ¯ s ] )for ε small. This contradicts the maximality of γ . Example 3.31. (Minkowski cones as generalized cones.) Here we showthat Minkowski cones as defined in Section 2 can equivalently be viewedas generalized cones. Let X be a geodesic length space, let Y := Cone( X )be the Minkowski cone over X , with relations ≪ Y , ≤ Y and time separationfunction τ Y . Let G := (0 , ∞ ) × id X be the generalized cone with warpingfunction f = id over X . Since we did not explicitly treat generalized conesof the form I × f X with a non-open interval I and a function f that mightbe zero at the endpoints (though, as indicated in Remark 3.32 below, thesecases could be included relatively straightforwardly), we will compare theLorentzian pre-length space ( G, D, ≪ G , ≤ G , τ G ) with the Lorentzian pre-length space Y ′ := Y \{ } = (0 , ∞ ) × X with metric D (which is equivalent tothe restriction of the cone metric d c ), relations ≪ Y ′ := ≪ Y | Y ′ × Y ′ , ≤ Y ′ := ≤ Y | Y ′ × Y ′ and time separation function τ Y ′ := τ Y | Y ′ × Y ′ .Let x = ( x , ¯ x ) , y = ( y , ¯ y ) ∈ G , then by the description of I + in (8) x ≪ G y if and only if for corresponding points x ′ = ( x , ¯ x ′ ) ∈ W :=(0 , ∞ ) × id H n − and y ′ = ( y , ¯ y ′ ) ∈ W with d X (¯ x, ¯ y ) = d H n − (¯ x ′ , ¯ y ′ ) onehas x ′ ≪ W y ′ . Similarly from (9) we see that, since both X and H n − are geodesic, the same holds for ≤ . Lastly, by fiber independence (Theorem3.29,(ii)) we also have that τ G ( x, y ) = τ W ( x ′ , y ′ ). By the last two paragraphsin Remark 2.1, we have ≤ W = ≤ C , ≪ W = ≪ C and τ W = τ C (where C :=(0 , ∞ ) × H n − with the Minkowski cone structure as in Remark 2.1). Nowsince by definition ≤ , ≪ , τ for Minkowski cones are clearly fiber independentas well, we have x ≪ Y ′ y if and only if x ′ ≪ C y ′ if and only if x ≪ G y .The same holds for ≤ . Also clearly τ Y ′ ( x, y ) = τ C ( x ′ , y ′ ) = τ G ( x, y ). So theLorentzian pre-length spaces ( G, D, ≤ G , ≪ G , τ G ) and ( Y ′ , D, ≤ Y ′ . ≪ Y ′ , τ Y ′ )can be identified. Remark 3.32. We have confined ourselves in this section to generalizedcones I × f X with I an open interval, but note that general intervals I couldbe treated in complete analogy. One could also consider the case where f has isolated zeros (either in the interior of the interval or at the intervalboundaries) with the additional assumption that the improper integrals of28 f coming from both sides of each zero diverge. If f ( t ) = 0 for some t ∈ I ,we identify ( t, x ) ∼ ( t, x ′ ) for x, x ′ ∈ X to a point denoted by t Y . Definingall concepts analogously to the case where f > 0, it is easy to see that forany ( x , x ) ∈ Y with x < t and x ∈ X arbitrary we have ( x , x ) ≪ t Y and γ : s ( x + s, x ) is a future directed timelike curve from ( x , x )to t Y with τ (( x , x ) , t Y ) = L ( γ ). Further, I + ( t Y ) = ( I ∩ ( t, ∞ )) × X and I − ( t Y ) = ( I ∩ ( −∞ , t )) × X . So any two points ( x , x ) , ( y , y ), x < y with f having a zero on [ x , y ] are trivially timelike related. Therefore considering f having zeros in the interior of I largely reduces to the problem of allowing f to vanish at the boundary. Divergence of the integral of f as one approachesthe zeroes of f ensures that I ± remains open (see Proposition 3.22) and thus,with some modifications, the main results in Sections 3 and 4 should remainvalid, but this would need to be investigated more carefully. We already established that every generalized cone Y = X × f X , where( X, d ) is a length space, is a Lorentzian pre-length space in Proposition 3.26.Here we will show that such spaces are in fact Lorentzian length spaces if X is locally compact. To this end we need the following auxiliary results. Lemma 4.1. Let ( X, d ) be a metric space and let p = ( t , ¯ p ) , q = ( t , ¯ q ) ∈ Y ,then J ( p, q ) ⊆ { ( t, ¯ r ) ∈ Y : t ≤ t ≤ t , ¯ r ∈ ¯ B d t − t mt ,t (¯ p ) ∩ ¯ B d t − tmt,t (¯ q ) } , where ¯ B dδ ( x ) = { x ′ ∈ X : d ( x, x ′ ) ≤ δ } denotes the closed ball of radius δ in X . Proof: Let r = ( t, ¯ r ) ∈ J ( p, q ), p < r < q , and let γ = ( α, β ) : [0 , b ] → Y be a future directed causal curve from p = γ (0) to r = γ ( t ∗ ) to q = γ ( b ).Then ˙ α > t = α (0) ≤ α ( t ∗ ) = t ≤ α ( b ) = t .From the proof of Lemma 3.10 we conclude that t − t = α ( t ∗ ) − α (0) ≥ m t ,t d (¯ p, ¯ r ) , and analogously t − t ≥ m t,t d (¯ r, ¯ q ). Lemma 4.2. Let ( X, d ) be a metric space. Then any p = ( p , ¯ p ) ∈ Y has abasis of open, causally convex neighborhoods, i.e., neighborhoods such thatany causal curve with endpoints in that neighborhood is contained in it. Thisalso shows that such a generalized cone is strongly causal . Moreover, themap Y → I : ( t, x ) t is a time function , i.e., t is continuous and strictlyincreasing along any future directed causal curve. roof. Using the same arguments as in the proof of the previous Lemmaone easily checks that the family (cid:26) ( t, ¯ r ) ∈ Y : p − ε < t < p + ε, ¯ r ∈ B d t − ( p − ε ) mp − ε,p ε (¯ p ) ∩ B d p ε − tmp − ε,p ε (¯ p ) (cid:27) ε> satisfies the claim. Lemma 4.3. Every generalized cone has the property that for every point y there is a neighborhood U of y and a constant C > such that the (metric) D -arclength of every causal curve which is contained in U is bounded by C ,i.e., L D ( γ ) ≤ C . Proof: Let y = ( t, p ) ∈ Y and let I ′ ⊆ I be a compact interval containing t . Set C ′ := diam( I ′ ) and C := min r ∈ I ′ f ( r ) > 0. Moreover, let γ =( α, β ) : [ a, b ] → Y be a (without loss of generality) future directed causalcurve that is contained in U := I ′ × X . Then since C v β ≤ ( f ◦ α ) v β ≤ ˙ α we get L D ( γ ) = Z ba q ˙ α + v β ≤ Z ba ˙ α r C ≤ r C C ′ , as required.We want to establish that generalized cones are Lorentzian length spaces.For this we first need to show that the different notions of causal curves andtheir length agree with the ones in the setting of Lorentzian length spaces.In the following result (and thereafter), when comparing the differentnotions of causal curves, it will always be understood that parametrizationsare chosen in which the respective curves are never locally constant (cf.[BBI01, Ex. 2.5.3]). Lemma 4.4. The notion of causal curves for a generalized cone agrees withthe notion of a causal curves with respect to the relation ≤ (cf. [KS18, Def.2.18]). Proof: Clearly, every future or past directed curve in the sense of Def-inition 3.2 is causal with respect to ≤ . For the converse, note that sincethis is a local question it suffices to consider segments of causal curves. Solet γ = ( α, β ) : [ a, b ] → Y be a (without loss of generality) future directedcausal curve with respect to ≤ , i.e., ∀ a ≤ s ≤ t ≤ b : γ ( s ) ≤ γ ( t ). Thus forany a ≤ s < t ≤ b there is a future directed causal curve (in the sense ofDefinition 3.2) γ s,t = ( α s,t , β s,t ) : [0 , → Y from γ ( s ) to γ ( t ). This impliesthat α is strictly monotonically increasing as t is a time function (cf. Lemma4.2). 30e now want to construct a sequence of future directed causal curves(in the sense of Definition 3.2) that converges pointwise to γ . For σ :=( a = t < t , . . . , t N = b ) a partition of [ a, b ], denote by γ σ the futuredirected causal curve γ σ := γ t ,t ∗ . . . ∗ γ t N − ,t N obtained by concatenatingthe curves γ t i ,t i +1 (0 ≤ i ≤ N − σ k be a sequence of such partitionswhose norms (maximal length of a subinterval) tend to zero as k → ∞ .We show that γ k := γ σ k converges pointwise to γ . Let t ∈ [ a, b ] and let U be a neighborhood of γ ( t ). By Lemma 4.2 there exists a causally convexneighborhood V of γ ( t ) such that V ⊆ U . As γ − ( V ) is a neighborhood of t in [ a, b ], for k large any sub-interval of σ k containing t lies entirely in V ,so in particular γ k ( t ) ∈ U . Consequently, γ k → γ pointwise and thus by thelimit curve theorem 3.16 γ is a (future directed) causal curve in the sense ofDefinition 3.2. Proposition 4.5 (Local existence of maximal causal curves) . Let ( X, d ) be a locally compact metric space. Then every point in Y = I × f X hasa neighborhood U such that any two causally related points in U can beconnected by a maximal causal curve. Proof: Let p ∈ Y , U ′ = I ′ × X and C > W ⊆ X be a compact neighborhood of ¯ p in X and set V := I ′ × W .Further, let U ⊆ V ⊆ U ′ be causally convex in V (cf. Lemma 4.2). Let y, z ∈ U with y < z , and note that any causal curve from y to z has tobe contained in U . So local maximality in U implies global maximality.Let γ n : [ a, b ] → Y be a sequence of future directed causal curves from y to z such that L ( γ n ) → τ ( y, z ). Then, by Lemma 4.3, L D ( γ n ) ≤ C andso reparametrizing each γ n proportional to D -arclength on [ a, b ] yields asequence of uniformly D -Lipschitz curves ˜ γ n each of which is future directedcausal. By the theorem of Arzela-Ascoli (the sequence γ n is contained in thecompact set V ) we obtain a subsequence (˜ γ n k ) k of (˜ γ n ) n that convergencesuniformly to a Lipschitz curve γ from y to z . As y < z this curve cannot beconstant and so by possibly reparametrizing γ such that it is never locallyconstant we obtain a future directed causal curve γ from y to z that iscontained in U . Moreover, by Proposition 3.15 we get that L ( γ ) ≤ τ ( y, z ) = lim sup k L ( γ n k ) ≤ L ( γ ) , so γ is maximal.A similar argument gives that Y is locally causally closed ([KS18, Def.3.4]): Lemma 4.6. Let ( X, d ) be a locally compact metric space. Then every pointin Y has a neighborhood U such that for any y n , z n ∈ Y with y n → y ∈ ¯ U , z n → z ∈ ¯ U and y n ≤ z n for all n ∈ N , it follows that y ≤ z . τ -length introduced in [KS18, Def. 2.24]. Recall that the τ -length, L τ ( γ ), is defined as L τ ( γ ) := inf { N − X i =0 τ ( γ ( t i ) , γ ( t i +1 )) : a = t < t < . . . < t N = b } , where γ is a future directed causal curve (and by Lemma 4.4 this is the sameas causal with respect to ≤ ). Proposition 4.7. Let ( X, d ) be a locally compact metric space. If γ : [ a, b ] → Y is a future directed causal curve, then L ( γ ) = L τ ( γ ) . Proof: Let a = t < t . . . < t N = b be a partition of [ a, b ], then N − X i =0 τ ( γ ( t i ) , γ ( t i +1 )) ≥ N − X i =0 L ( γ | [ t i ,t i +1 ] ) = L ( γ ) , as L is additive. Taking the infimum over all partitions of [ a, b ] yields L ( γ ) ≤ L τ ( γ ).For the reverse inequality we cover γ ([ a, b ]) by neighborhoods U , . . . , U N as in Proposition 4.5 and choose a partition σ := ( a = t < t < . . . 1. Con-sequently, there are future directed maximal causal curves γ σi from γ ( t i ) to γ ( t i +1 ) for i = 0 , . . . , N . The future directed causal curve γ σ := γ σ ∗ . . . ∗ γ σN has length L ( γ σ ) = N X i =0 L ( γ σi ) = N X i =0 τ ( γ ( t i ) , γ ( t i +1 )) ≥ L τ ( γ ) . By shrinking the cover ( U i ) i and adapting the partition σ accordingly we geta sequence γ k of future directed causal curves which, by an argument as inthe proof of Lemma 4.4 converges pointwise to γ , and satisfies L ( γ k ) ≥ L τ ( γ )for all k ∈ N . As L is upper semicontinuous by Proposition 3.15 we get L ( γ ) ≥ L τ ( γ ) and this finishes the proof.Thus there is no need to distinguish between L and L τ and the dif-ferent notions of causal curves also agree, so when applying the theory ofLorentzian length spaces to generalized cones we will always use the notionsof the latter. Theorem 4.8. Any generalized cone I × f X , where ( X, d ) is a locally com-pact length space, is a strongly causal Lorentzian length space. roof: By Proposition 3.26 and Lemma 4.6 Y is a locally causally closedLorentzian pre-length space. Moreover, by definition of the causal relationsit is causally path connected .Since the different notions of causal curves agree by Lemma 4.4 and as L τ = L by Proposition 4.7, we directly obtain that τ = T , where T ( y, z ) = sup { L τ ( γ ) : γ future-directed causal from y to z } , if the set is non-empty and T ( y, z ) = 0 otherwise. It remains to show that Y is localizable ([KS18, Def. 3.16]). To this endwe apply the argument of the proof of [GKS19, Lem. 4.3] to see that we canuse ω := τ | U × U for a suitable neighborhood U of a point y = ( t , x ) ∈ Y .Such a suitable neighborhood can be chosen by taking U to be one of thecausally convex neighborhoods from Lemma 4.2 that is contained in theneighborhoods of Lemma 4.3 and Proposition 4.5. Thus ω is finite andlower semicontinuous. To see that ω is also upper semicontinuous note thatwe can adapt the proof of [KS18, Thm. 3.28] to the simpler local situationin U by using the local existence of maximal causal curves (Proposition 4.5)and the upper semi-continuity of L (Proposition 3.15). Moreover, since U is open, one has I ± (( t , x )) ∩ U = ∅ .This yields that Y is localizable and hence by the above is a Lorentzianlength space. It also implies that Y is strongly causal in the sense of [KS18,Def. 2.35(iv)] by using the result for Lorentzian length spaces [KS18, Thm.3.26(iv)] and Lemma 4.2.Further, the Lorentzian length space I × f X is regular , i.e., maximalcausal curves have a causal character (cf. [KS18, Def. 3.22]). Thus by Propo-sition 4.8 and Corollary 3.30 we immediately get the following: Corollary 4.9. Any generalized cone I × f X , where ( X, d ) is a geodesic lo-cally compact length space, is a strongly causal and regular Lorentzian lengthspace. Lemma 4.1 shows that if X is proper the causal diamonds J ( p, q ) are pre-compact. Moreover, by Lemma 4.2 any generalized cone is strongly causal.This is already close to the usual notion of global hyperbolicity. In the nextProposition we will show that generalized cones, where X is proper andgeodesic, are in fact globally hyperbolic (as defined for Lorentzian lengthspaces in [KS18, Def. 2.35(v)]). Proposition 4.10. Let I × f X be a generalized cone, where ( X, d ) is ageodesic length space that is a proper metric space. Then I × f X is globallyhyperbolic. Note that this could also be inferred from the more general Theorem A.10. roof: From Theorem 4.8 we know that Y is a strongly causal Lorentzianlength space and hence non-totally imprisoning by [KS18, Thm. 3.26(iii)].Moreover, from Corollary 3.24 we know that J ± ( p ) is closed for every p ∈ Y and Lemma 4.1 implies that for all p, q ∈ Y the causal diamond J ( p, q ) iscontained in a compact set. Thus J ( p, q ) is compact and so Y is globallyhyperbolic in the sense of [KS18, Def. 2.35(v)].As any complete and locally compact length space is proper and geodesic(by the Hopf-Rinow-Cohn-Vossen theorem) we obtain the following corol-lary. Corollary 4.11. Let I × f X be a generalized cone, where X is a locallycompact, complete length space. Then I × f X is globally hyperbolic. Recall that a Lorentzian pre-length space is called geodesic ([KS18, Def.3.27]) if any two causally related points can be joined by a maximal causalcurve. As any globally hyperbolic Lorentzian length space is geodesic (Avez-Seifert, cf. [KS18, Thm. 3.30]), we conclude by the above that every gen-eralized cone is geodesic if X is proper and geodesic (in the metric spacesense). This implies the following stronger result. Corollary 4.12. Let X be geodesic, then I × f X is geodesic. Furthermore,any two timelike related points can be connected by a timelike geodesic.Proof. Let ( x , ¯ x ) , ( y , ¯ y ) ∈ Y = I × f X . Because X is geodesic there existsa minimal curve β : [0 , d X (¯ x, ¯ y )] → X , parametrized by arc-length, from ¯ x to ¯ y . Let X ′ = [0 , d X (¯ x, ¯ y )] (with the standard metric) and Y ′ = I × f X ′ .Since X ′ is proper and geodesic, Y ′ is geodesic and there exists a maximizingcurve γ ′ = ( α ′ , β ′ ) : [0 , d X (¯ x, ¯ y )] → Y ′ , with β ′ parametrized by arc-length,from ( x , 0) to ( y , d X (¯ x, ¯ y )). Then γ := ( α ′ , β ) is maximizing from ( x , ¯ x ) to( y , ¯ y ) in Y by Theorem 3.29 (ii). The second claim follows from Corollary3.30. In this Section we generalize the results of Section 2 to generalized cones,i.e., we relate (metric) curvature bounds of the fiber X to timelike curvaturebounds of the generalized cone Y = I × f X , and vice versa. Lemma 5.1. Let ( X, d ) and ( X ′ , d ′ ) be two geodesic length spaces. Let Y := I × f X and Y ′ := I × f X ′ . Then for any two pairs of points x = ( x , ¯ x ) , y =( y , ¯ y ) ∈ Y and x ′ = ( x , ¯ x ′ ) , y ′ = ( y , ¯ y ′ ) ∈ Y ′ with d X (¯ x, ¯ y ) ≥ d ′ X ′ (¯ x ′ , ¯ y ′ ) one has τ ( x, y ) ≤ τ ′ ( x ′ , y ′ ) . roof. If τ ( x, y ) = 0 this obviously holds, so assume x ≪ y . Let β : [ a, b ] → X be a minimizing unit-speed geodesic from ¯ x to ¯ y in X . Then by Corollary4.12, there exists a timelike curve γ ≡ ( α, β ) : [ a, b ] → Y from x to y with L ( γ ) = τ ( x, y ). Further, let β ′ : [ a, b ] → X ′ be a curve from ¯ x ′ to ¯ y ′ in X ′ such that L X ′ ( β ′ ) = d ′ X ′ (¯ x ′ , ¯ y ′ ) and v β ′ is constant. Then L ( β ) = d (¯ x, ¯ y ) ≥ d ′ X ′ (¯ x ′ , ¯ y ′ ) = L ( β ′ ) implies v β ′ ≤ v β = 1, so the curve γ ′ := ( α, β ′ ) : [ a, b ] → Y ′ is timelike and τ ( x, y ) = L ( γ ) = Z ba p ˙ α − ( f ◦ α ) ≤ Z ba q ˙ α − ( f ◦ α ) v β ′ = L ( γ ′ ) ≤ τ ′ ( x ′ , y ′ ) . Moreover, for causally related points also strict inequalities are preservedin the above Lemma, i.e., if d X (¯ x, ¯ y ) < d ′ X ′ (¯ x ′ , ¯ y ′ ) then τ ( x, y ) < τ ′ ( x ′ , y ′ ).From this one obtains immediately the following converse: Lemma 5.2. Let ( X, d ) and ( X ′ , d ′ ) be two geodesic length spaces. Let Y := I × f X and Y ′ := I × f X ′ . Then for any two pairs of causally relatedpoints x = ( x , ¯ x ) , y = ( y , ¯ y ) ∈ Y and x ′ = ( x , ¯ x ′ ) , y ′ = ( y , ¯ y ′ ) ∈ Y ′ with τ ( x, y ) ≤ τ ′ ( x ′ , y ′ ) one has d X ′ (¯ x ′ , ¯ y ′ ) ≤ d X (¯ x, ¯ y ) . We now turn to the relation between (metric) curvature bounds in thefiber X (in the sense of [BBI01, Def. 4.6.2]) and timelike curvature boundsin the generalized cone I × f X as defined in [KS18, Def. 4.7].In what follows we use M ( K ) to denote the Riemannian model spaceof constant sectional curvature K , i.e., M ( K ) = S ( r ) K = r R K = 0 H ( r ) K = − r . Theorem 5.3. Let K, K ′ ∈ R and let ( X, d ) be a geodesic length spacewith curvature bounded below/above by K . Then Y = I × f X has timelikecurvature bounded below/above by K ′ if I × f M ( K ) has timelike curvaturebounded below/above by K ′ .Proof. As in the proof of Theorem 4.8, for any w ∈ Y we can choosea causally convex neighborhood U ⊆ Y according to Lemma 4.2 suchthat τ | U × U is continuous and any two points x, y ∈ U with x ≪ y canbe connected by a maximal future-directed timelike curve γ in U with L ( γ ) = τ ( x, y ). We may further assume that U was chosen small enough tosatisfy the following conditions: 35i) There is an open set V ⊆ X on which triangle comparison with M ( K )holds and such that for all ¯ x ∈ X for which there exists x ∈ R suchthat ( x , ¯ x ) ∈ U we have ¯ x ∈ V .(ii) U ⊆ [ u , u ] × B d X ε ( ¯ w ), where ε and | u − u | are so small that, for some(fixed) ¯ w ′ ∈ M ( K ) we have that [ u , u ] × B d M K ) ε ( ¯ w ′ ) ⊆ I × f M ( K ) iscontained in a neighborhood U ′ on which timelike triangle comparisonwith L ( K ′ ) holds.Let ( x, y, z ) be a timelike geodesic triangle in U , realized by maximaltimelike curves γ xy , γ yz , γ xz whose side lengths a, b, c satisfy timelike sizebounds for K ′ , i.e. c ≥ a + b and if c = a + b and K ′ > c > a + b and K ′ < c < π √ | K ′ | .To establish that Y has timelike curvature bounded below by K ′ we haveto show that if ( x ′′ , y ′′ , z ′′ ) is a comparison triangle of ( x, y, z ) in L ( K ′ ),then for all points p, q on the sides of ( x, y, z ) and corresponding points p ′′ , q ′′ on the sides of ( x ′′ , y ′′ , z ′′ ), we have τ ( p, q ) ≤ τ L ( K ′ ) ( p ′′ , q ′′ ). (To showthat Y has timelike curvature bounded above by K ′ we have to show that τ ( p, q ) ≥ τ L ( K ′ ) ( p ′′ , q ′′ ).)We do this in two steps. First, we construct a comparison triangle ∆ ′ in Y ′ := I × f M ( K ) with τ ( p, q ) ≤ τ Y ′ ( p ′ , q ′ ) (respectively ≥ in case ofcurvature bounded above).The projection (¯ x, ¯ y, ¯ z ) of ( x, y, z ) onto X is a geodesic triangle ¯∆ in X (which can be degenerate) and if γ xy = ( α xy , β xy ) , γ yz = ( α yz , β yz ) and γ xz = ( α xz , β xz ) are the sides of ( x, y, z ), then the sides of ¯∆ = (¯ x, ¯ y, ¯ z )are the minimizing curves β xy , β yz , β xz and are contained in V . Because V was a neighborhood in X on which triangle comparison with M ( K )holds, there exists a triangle ¯∆ ′ = (¯ x ′ , ¯ y ′ , ¯ z ′ ) in M ( K ) such that d X (¯ p, ¯ q ) ≥ d M ( K ) (¯ p ′ , ¯ q ′ ) (respectively ≤ in case of curvature bounded above). Thistriangle ¯∆ ′ in M ( K ) can be lifted to a triangle ∆ ′ in Y ′ = I × f M ( K )given by x ′ := ( x , ¯ x ′ ) , y ′ := ( y , ¯ y ′ ) , z ′ := ( z , ¯ z ′ ). By fiber independence(Theorem 3.29,(ii)) ∆ ′ is a triangle with the same sidelengths as ∆ and thepoints p ′ and q ′ corresponding to p and q are exactly ( p , ¯ p ′ ) and ( q , ¯ q ′ ).Thus, by Lemma 5.1 τ ( p, q ) ≤ τ Y ′ ( p ′ , q ′ ) (respectively ≥ in case of curvaturebounded above).Now because ∆ satisfies timelike size bounds, the same is true for ∆ ′ .Further, because of the symmetries of M ( K ) we may additionally sup-pose that our comparison triangle ∆ ′ was chosen such that ¯ x ′ = ¯ w ′ , hence∆ ′ ⊆ U ′ by our choice of U (cf. item (ii)). So, by the timelike curvaturebound of Y ′ = I × f M ( K ) there exists a timelike comparison triangle∆ ′′ = ( x ′′ , y ′′ , z ′′ ) of ( x ′ , y ′ , z ′ ) in L ( K ′ ) such that for the points p ′′ , q ′′ on∆ ′′ corresponding to p ′ , q ′ , we have τ Y ′ ( p ′ , q ′ ) ≤ τ L ( K ′ ) ( p ′′ , q ′′ ) (respectively, τ Y ′ ( p ′ , q ′ ) ≥ τ L ( K ′ ) ( p ′′ , q ′′ ) for a timelike curvature bound from above).36oreover, by construction and fiber independence (cf. Theorem 3.29) ∆ ′′ must be a comparison triangle for ∆.Together with the inequality τ ( p, q ) ≤ τ Y ′ ( p ′ , q ′ ) (respectively ≥ ) frombefore we get τ ( p, q ) ≤ τ L ( K ′ ) ( p ′′ , q ′′ ) (respectively ≥ ), concluding the proof.In the case of a smooth warping function f we can give sufficient condi-tions so that Y ′ = I × f M ( K ) has sectional curvature bounded by K ′ andthus timelike curvature bounded by K ′ . Corollary 5.4. Let f : I → (0 , ∞ ) be smooth and let Y = I × f X , Y ′ = I × f M ( K ) . If f is K ′ -concave (convex), i.e., f ′′ − K ′ f ≤ ( f ′′ − K ′ f ≥ )and K = inf( K ′ f − ( f ′ ) ) ( K = sup( K ′ f − ( f ′ ) ) ) and X has curvaturebounded below (above) by K , then Y has timelike curvature bounded below(above) by K ′ . Proof: This follows directly from Theorem 5.3 and [AB08, Prop. 7.1] inthe special case that the base is one-dimensional and from the fact thatsectional curvature bounds in the sense of [AB08] imply timelike curvaturebounds in the sense of Lorentzian length spaces (cf. [KS18, Ex. 4.9]).We apply the preceding corollary to specific spaces and warping functionsto obtain: Corollary 5.5. Let X be a geodesic length space with curvature boundedbelow/above by K (third column). With the interval I given in the firstcolumn and the warping function given in the second column, I × f X hastimelike curvature bounded below/above by K ′ (forth column): I f X CB b/a by K I × f X TLCB b/a by K ′ (0 , π ) sin − − − π , π ) cos − − , ∞ ) id − , ∞ ) sinh − R exp 0 1 R R cosh 1 1A result in the converse direction holds as well, showing that if Y and Y ′ satisfy timelike curvature bounds then X has a curvature bound. To showthis, we first need the following Lemma that establishes that we can lift ageodesic triangle ˜∆ in X to a timelike geodesic triangle in Y , provided ˜∆ issmall enough. 37 emma 5.6. Let Y = I × f X be a generalized cone, where X is a geodesiclength space. Let ( t , ¯ p ) ∈ Y , then for all neighborhoods V ⊆ Y of ( t , ¯ p ) there is a constant C > (depending on f and V ) such that any convexneighborhood U of ¯ p in X with diam( U ) ≤ C has the property that anygeodesic triangle in U can be lifted to a timelike geodesic triangle in V . Proof: Let ( t , ¯ p ) ∈ Y and V ⊆ Y a neighborhood of ( t , ¯ p ), and set f ( t ) =: m > 0. Then there is an ε > t ∈ [ t − ε, t + ε ] ⊆ I we have f ( t ) ≤ m and [ t − ε, t + ε ] × B ε √ m (¯ p ) ⊆ V . Moreover, set α ( t ) := t , α : [ t − ε, t + ε ] → I and t ± := α ( t ± ε ) ∈ I . Let U be a convexneighborhood in X with diam( U ) ≤ ε √ m =: C . Then U ⊆ B ε √ m (¯ p ).Let ¯∆ = (¯ x, ¯ y, ¯ z ) be a geodesic triangle in U . Let ˜ β : [0 , d (¯ x, ¯ y )] → U be aminimizing unit-speed geodesic connecting ¯ x and ¯ y . We now reparametrize˜ β as follows: β ( s ) := ˜ β ( s − t + εε d (¯ x, ¯ y )) for s ∈ [ t − ε, t ]. Clearly, β : [ t − ε, t ] → U is minimizing and v β = d (¯ x, ¯ y ) ε almost everywhere. We establishthat γ := ( α, β ) : [ t − ε, t ] → V is future directed timelike from x := ( t − , ¯ x )to y := ( t , ¯ y ): − ˙ α + ( f ◦ α ) v β = − f ( t ) d (¯ x, ¯ y ) ε ≤ − m U ) ε ≤ − . Thus we have ( t − , ¯ x ) ≪ ( t , ¯ y ) and analogously ( t , ¯ y ) ≪ ( t + , ¯ z ) =: z . As Y is geodesic by Corollary 4.12, there is a maximizing future directed timelikecurve from x to y , whose projection to X is a minimizing curve in X byTheorem 3.29 (i). Note that this projection is in general different from β .However, one can now proceed as in the proof of Corollary 4.12 and obtaina maximal timelike curve in Y whose projection is β or ˜ β , respectively.Analogously, one can argue the existence of maximizing causal curves from y to z and from x to z for which the X -components are the minimizinggeodesics from ¯ y to ¯ z and from ¯ x to ¯ z . Thus ∆ := ( x, y, z ) is a lift of ¯∆ andby construction ∆ and all its sides lie in V .With this preparation we can show the following: Theorem 5.7. If X is a geodesic length space, Y = I × f X has timelikecurvature bounded below (above) by K ′ and Y ′ = I × f M ( K ) has timelikecurvature bounded above (below) by K ′ then X has curvature bounded below(above) by K . Proof: Fix ¯ p ∈ X and t ∈ I . Let V ⊆ Y be a comparison neighborhoodfor ( t , ¯ p ) in Y such that all timelike triangles in V satisfy the size bounds.Let C > U be a convex neighborhoodof ¯ p in X with diam( U ) ≤ C . Let ¯∆ = (¯ x, ¯ y, ¯ z ) be a geodesic trianglein U satisfying the appropriate size bounds with its sides realized by unitspeed geodesics β ¯ r, ¯ s ( r, s ∈ { x, y, z } ). Let ¯ p, ¯ q be points on ¯∆, say ¯ p ∈ β ¯ x, ¯ y q ∈ β ¯ y, ¯ z . Then by Lemma 5.6 we can lift ¯∆ to a timelike geodesictriangle ∆ = ( x, y, z ) in I × f X , where x = ( x , ¯ x ), y = ( y , ¯ y ), z = ( z , ¯ z ),with x ≪ y ≪ z , and whose sides are realized by future directed timelikemaximal curves γ r,s = ( α r ,s , β ¯ r, ¯ s ) ( r, s ∈ { x, y, z } ). Let ¯∆ ′ = (¯ x ′ , ¯ y ′ , ¯ z ′ ) bea comparison triangle in M ( K ) of ¯∆, whose sides are realized by geodesics β ′ ¯ r ′ , ¯ s ′ in M ( K ). Again we lift ¯∆ ′ to a timelike geodesic triangle ∆ ′ in Y ′ of the same side lengths as ∆. Moreover, let ∆ ′′ be a comparison trianglein L ( K ′ ) of ∆, which therefore is also a comparison triangle of ∆ ′ . Let¯ p ′ , ¯ q ′ be points corresponding to ¯ p, ¯ q on ¯∆ ′ . We can lift these points topoints on p, q ∈ ∆ and p ′ , q ′ ∈ ∆ ′ as follows: Let t ¯ p ∈ [0 , d X (¯ r, ¯ s )], then p := γ r,s ( t p ) = ( α r ,s ( t ¯ p ) , ¯ p ), and analogously for ¯ q and ¯ p ′ , ¯ q ′ . Then let p ′′ , q ′′ be corresponding points on ∆ ′′ with respect to p, q , which are alsocorresponding to p ′ , q ′ . At this point we can use the curvature bounds toobtain τ ( p, q ) ≤ τ L ( K ′ ) ( p ′′ , q ′′ ) ≤ τ ′ ( p ′ , q ′ ) . Since by construction p ≤ q and p ′ ≤ q ′ , Lemma 5.2 now gives d M ( K ) (¯ p ′ , ¯ q ′ ) ≤ d (¯ p, ¯ q ), concluding the proof. Lemma 6.1. Let K ′ > K (resp. K ′ < K ). Then for small enough corre-sponding timelike triangles ∆ = ( x, y, z ) ∈ L ( K ) , x ≪ y ≪ z and ∆ ′ =( x ′ , y ′ , z ′ ) ∈ L ( K ′ ) , x ′ ≪ y ′ ≪ z ′ and corresponding points q ∈ yz ⊆ I + ( x ) , q ′ ∈ y ′ z ′ ⊆ I + ( x ′ ) with q = y, z (and consequently q ′ = y ′ , z ′ ) we have τ L ( K ′ ) ( x ′ , q ′ ) > τ L ( K ) ( x, q ) (resp. < )Proof. We use the notation of [AB08], in particular the modified distancefunction h K,x on L ( K ) is given as h K,x = ( (1 − cos √ KE x ) /K = P ∞ n =1 ( − K ) n − ( E x ) n (2 n )! , K = 0 E x / , K = 0 . Here, it is understood that if the argument of cosine is imaginary, thencos( iϕ ) = cosh ϕ . Moreover, E x is defined as E x := inf {h γ ′ (0) , γ ′ (0) i : γ : [0 , → L is a geodesic joining p and q } .Along timelike geodesics in the model space L ( K ) the modified distancefunction h K,x satisfies( h K,x ◦ γ )¨ + h ˙ γ, ˙ γ i Kh K,x ◦ γ = h ˙ γ, ˙ γ i , γ : [0 , → L ( K ) be the geodesic from y to z in L ( K ) and γ ′ : [0 , → L ( K ′ ) the corresponding geodesic from y ′ to z ′ in L ( K ′ ). Defining f := h K,x ◦ γ − h K ′ ,x ′ ◦ γ ′ and using K < K ′ as well asstrict positivity of h K,x , we get ¨ f = σKh K,x ◦ γ − σK ′ h K ′ ,x ′ ◦ γ ′ < σK ′ f ,where σ := −h ˙ γ, ˙ γ i = −h ˙ γ ′ , ˙ γ ′ i > 0. Now, as in [AB08, Prop. 5.2], we have f (0) = f (1) = 0 and get f ≥ f ≤ σK ′ f . We now argue that f > , s ∈ (0 , 1) with f ( s ) = 0.Then f has a local minimum at s , so ¨ f ( s ) ≥ 0, contradicting the strictinequality ¨ f ( s ) < σK ′ f ( s ) = 0.Thus, h K,x ◦ γ ( s ) > h K ′ ,x ′ ◦ γ ′ ( s ) for all s ∈ (0 , h K,x ( q ) > h K ′ ,x ′ ( q ′ ). Since q h K,x ( q ) and q ′ h K ′ ,x ′ ( q ′ ) are strictly in-creasing functions of the signed distance from x to q and the signed distancefrom x ′ to q ′ , respectively, this implies τ L ( K ) ( x, q ) < τ L ( K ′ ) ( x ′ , q ′ ) . The proof of the K ′ < K case is analogous. Theorem 6.2. Let X be a geodesic length space, Y = I × f X with f : I → (0 , ∞ ) smooth. Assume that Y has timelike curvature bounded below (above)by K , then f is K -concave (convex), i.e., f ′′ − Kf ≤ ( f ′′ − Kf ≥ ).Proof. We only treat the case where Y has timelike curvature bounded belowby K . The proof for a bound from above is analogous.Assume to the contrary that f is not K -concave, i.e., there exists t ∈ I such that f ′′ ( t ) > Kf ( t ). Then there exists an interval J ⊆ I , t ∈ J and K ′ > K such that f ′′ > K ′ f on J . Define Y ′ := J × f R , then Y ′ is a smooth two-dimensional Lorentzian manifold with (timelike) sectionalcurvature R = f ′′ f > K ′ . Since sectional curvature bounds in the sense of[AB08] imply timelike curvature bounds in the sense of Lorentzian lengthspaces (cf. [KS18, Ex. 4.9], based on [AB08, Thm. 1.1]), we get that Y ′ has timelike curvature bounded above by K ′ > K . Let x ′ = ( x , , y ′ =( y , ¯ y ′ ) , z ′ = ( z , 0) be points in Y ′ forming a small timelike geodesic triangle.Let p ′ = x ′ and q ′ = ( q , ¯ q ′ ) ∈ y ′ z ′ and let p ′′ , q ′′ and p ′′′ , q ′′′ be correspondingpoints for p ′ , q ′ on the sides of the comparison triangle for ∆( x ′ , y ′ , z ′ ) in L ( K ′ ) and L ( K ), respectively. Then τ Y ′ ( p ′ , q ′ ) ≥ τ L ( K ′ ) ( p ′′ , q ′′ ) > τ L ( K ) ( p ′′′ , q ′′′ ) , where we used Lemma 6.1 to get the last strict inequality.Now let ¯ x ∈ X be arbitrary and set x := ( x , ¯ x ) , y := ( y , ¯ y ) , z :=( z , ¯ x ), where ¯ y ∈ X is chosen such that d X (¯ x, ¯ y ) = d R (0 , ¯ y ′ ) = | ¯ y ′ | . Thenby fiber independence (Theorem 3.29,(ii)), ∆ = ( x, y, z ) is a triangle in Y corresponding to ( x ′ , y ′ , z ′ ) in Y ′ . Let p = x and let q be the point40orresponding to q ′ on the side yz of ∆. Then, again by fiber independence, τ Y ( p, q ) = τ Y ′ ( p ′ , q ′ ) and using the assumption that Y has timelike curvaturebounded below by K we obtain the contradiction τ L ( K ) ( p ′′′ , q ′′′ ) ≥ τ Y ( p, q ) = τ Y ′ ( p ′ , q ′ ) > τ L ( K ) ( p ′′′ , q ′′′ ) . Remark 6.3. In the future it would also be interesting to examine if onemight still obtain K -concavity ( K -convexity) in the barrier sense for non-smooth warping functions f .We now relate non-positive lower timelike curvature bounds to the exis-tence of singularities, i.e., incomplete causal geodesics. To this end we firstrecall some notions and results from [GKS19]. A geodesic in a Lorentzianlength space X is a causal curve that is locally maximizing ([GKS19, Def.4.1]) and for a smooth strongly causal spacetime ( M, g ) one has that causalpregeodesics of ( M, g ) are geodesics in the synthetic sense above and viceversa (of the same causal character) ([GKS19, Thm. 4.4]). Moreover, ageodesic γ : [ a, b ) → X is extendible if there is a geodesic ¯ γ : [ a, b ] → X suchthat ¯ γ | [ a,b ) = γ . Otherwise, γ is called inextendible . Also, we have an anal-ogous notion of timelike geodesic completeness in the synthetic setting: ALorentzian length space X is said to have property (TC) if all inextendibletimelike geodesics have infinite τ -length ([GKS19, Def. 5.1]). This notion isagain compatible with the smooth spacetime case as a smooth and stronglycausal spacetime is timelike geodesically complete if and only if it has prop-erty (TC) ([GKS19, Lem. 5.2]). Consequently, we call a timelike geodesic incomplete if it has finite τ -length and the space X timelike geodesicallyincomplete if there is a timelike geodesic which is incomplete. Analogousnotions are defined for past and future incompleteness. Corollary 6.4. Let X be a geodesic length space, Y = I × f X with I =( a, b ) , f : I → (0 , ∞ ) smooth. Assume that Y has timelike curvature boundedbelow by K . Then:(i) If K < , then a > −∞ and b < ∞ and hence the time separationfunction τ Y of Y is bounded by b − a . Thus any such Y is timelikegeodesically incomplete.(ii) If K = 0 and f is non-constant, then a > −∞ or b < ∞ and hence Y is past or future timelike geodesically incomplete.Proof. We first show (i): Assume first that b = ∞ . Set u := f ′ f . Theorem6.2 implies f ′′ ≤ Kf , so u satisfies the differential inequality u ′ ≤ − u + K ≤ min {− u , K } . Since K < s ∈ ( a, ∞ )with u ( s ) < 0. Let s max be the maximal s > s such that −∞ < u | [ s ,s ) < s := s − u ( s ) > s . Then integrating u ′ ≤ − u shows that41 ≤ u ( s + s − s < u ( s ) < s , min { s max , s } ). But from this we seethat s max ≤ s < ∞ and u → −∞ as s → s max . Since u = f ′ f this implies f → s → s max , contradicting f > a, ∞ ).To show that a > −∞ we simply reverse the time orientation, i.e., weconsider ˜ f ( s ) := f ( − s ) instead of f .For (ii), since f is non-constant we have f ′ ( s ) = 0 for some s . If f ′ ( s ) < 0, then u ( s ) < b = ∞ as in (i). If f ′ ( s ) > 0, we can again just reverse the time orientation to get ˜ f ′ ( − s ) < a = −∞ . Remark 6.5. If ( X, h ) is a smooth n -dimensional Riemannian manifold,then Y = I × f X with metric g = − dt + f ( t ) h is a smooth Lorentzianmanifold and we may compare Corollary 6.4 with the Hawking singularitytheorem and the Lorentzian Myers’ theorem ([O’N83, Thm. 55A and 55B],[BEE96, Thm. 11.9]) applied to Y . A key assumption in both these theoremsis a bound on the timelike Ricci curvature, which is implied by certainsectional curvature bounds: In any smooth ( n + 1)-dimensional Lorentzianmanifold a bound on timelike sectional curvatures from below/above by K implies a bound on the timelike Ricci curvature from above/below by − n K . However, even in the smooth setting it is not known, to the bestof our knowledge, if a bound on merely the timelike sectional curvatures,which is strictly weaker than assuming a sectional curvature bound in thesense of [AB08], will imply triangle comparison for timelike triangles, i.e.,a timelike curvature bound as in [KS18], or vice versa. Thus, a timelikecurvature bound as in [KS18] might in general not imply the correspondingtimelike Ricci curvature bounds. However, in the specific warped productsetting we are considering, the following simple relationship holds: For any V ∈ T X we have Ric( ∂ t , ∂ t ) = − n f ′′ f = − n K ( ∂ t , V ), where K ( ∂ t , V ) denotesthe sectional curvature of the timelike plane spanned by ∂ t and V . SoTheorem 6.2 shows that triangle comparison for timelike triangles impliesboth a bound on the sectional curvatures of all timelike planes orthogonal to X and on Ric( ∂ t , ∂ t ). More specifically, timelike curvature bounded belowby K implies that Ric( ∂ t , ∂ t ) ≥ − nK .Now the comparison with Hawking’s theorem is straightforward: It iswell known that the key assumptions for Hawking’s theorem to hold boildown to Ric( ˙ γ, ˙ γ ) ≥ γ starting or-thogonally to an initial Cauchy surface Σ with mean curvature H Σ < β < { t } × X . Then themean curvature H Σ equals n f ′ ( t ) f ( t ) and ˙ γ ( t ) = ∂ t | γ ( t ) , so Hawking’s singu-larity theorem corresponds exactly to the K = 0 case of Corollary 6.4 (andwhether one has a past or a future singularity is determined by the sign of H Σ = n f ′ ( t ) f ( t ) , which equals the sign of f ′ ( t ) f ( t ) ).42n the K < I × f X is bounded by b − a < ∞ , so this case corresponds to the LorentzianMyers’ theorem. Note that while the standard formulation of the LorentzianMyers’ theorem (as in [BEE96]) requires Ric( W, W ) ≥ n | K | > W , one can use the techniques of the proof of the Hawkingsingularity theorem to show that to get a bound on the timelike diameterit is sufficient to assume Ric( ˙ γ, ˙ γ ) ≥ n | K | > γ starting orthogonally to a Cauchy surface Σ (cf. [Gra16, Thm. 4.2and Rem. 4.3]). However, the bound obtained in this way may be larger than π √ | K | . With this in mind, we see that also the K < big bang and big crunch singu-larities as follows: Definition 6.6. Let Y = ( a, b ) × f X be a generalized cone, where f issmooth. Then(i) the generalized cone Y has a big bang singularity at a if f ( t ) → and f ′ ( t ) → ∞ as t ց a , and(ii) the generalized cone Y has a big crunch singularity at b if f ( t ) → and f ′ ( t ) → −∞ as t ր b . Corollary 6.7. If Y has a big bang or a big crunch singularity, then Y cannot have timelike curvature bounded above by any K ∈ R .Proof. Assume that f ( s ) → f ′ ( s ) → −∞ as s → b and that Y hastimelike curvature bounded above by K . First, if K ≥ f ′′ ≥ Kf ≥ f ′ → −∞ . So let K < s be such that f ≤ s , b ). Then on this interval f ′′ ≥ Kf ≥ K , so f ′ ( s ) ≥ K ( s − s ) + f ′ ( s ),contradicting f ′ ( s ) → −∞ . The big bang case follows by reversing the timeorientation. By transporting the notion of a (generalized) cone into the synthetic Lorentziansetting we achieved the following objectives: • We established that the causality of generalized cones is optimal, evenfor fibers that are not necessarily manifolds. • Generalized cones are instances of strongly causal Lorentzian lengthspaces. 43 Timelike curvature bounds of a generalized cone can be related to the(metric) curvature bounds of its fiber and vice versa. • Our results allow one to generate an abundance of examples of Lorentzianlength spaces with timelike curvature bounds. • We proved singularity theorems for generalized cones that are directanalogues of the Hawking singularity theorem and the Lorentzian My-ers’ theorem. That is, we showed how non-positive lower timelike cur-vature bounds imply the existence of incomplete timelike geodesics.Our methods are also expected to be applicable to more general spaces withone-dimensional fiber, like the Colombini-Spagnolo metrics discussed in Re-mark 3.23. More generally, topics of further investigation are the following: • Generalize our methods and results to spaces of the form I × X , whereon each fiber { t } × X one has a t -dependent metric. This would gen-eralize the metric splitting for globally hyperbolic spacetimes. • Extend the results of Lemma 5.4 to non-smooth warping functions f that are K ′ -convex or K ′ -concave in the support sense (cf. [AB03]). • Put a Lorentzian structure on general warped products B × f F , wherethe base B is a Lorentzian length space and the fiber F is a metric(length) space. • Extend the singularity theorems of Section 6 to lower regularity (i.e.,lower the regularity of the warping function f ). • Relate our approach to the theory of Sormani and Vega [SV16], inparticular to the results on warped products of Allen and Burtscher[AB19]. Acknowledgments This work was supported by research grants P28770and J4305 of the Austrian Science Fund FWF. A Appendix: Lorentzian length structures In this appendix we want to outline that in the situation where one merelyhas • a notion of causal curves and • a notion of length of such curves,44ne can already construct a Lorentzian pre-length space without push-up ortopology, hence without lower semicontinuity of τ , but such that neverthelessthe time separation function is intrinsic. So in a certain sense the situationis even better than starting out with a Lorentzian pre-length space. In doingso we can reproduce some of the results of Section 3 but in greater generality.Also, this fact was already mentioned in [KS18], so here we expand on [KS18,Rem. 5.11(ii)] and provide some arguments to illustrate the matter.In analogy with the metric case (cf. [BBI01, Sec. 2.1]) we first define a Lorentzian length structure . To this end we also need to introduce notionsof admissible curves and length functionals. Definition A.1. Let ( X, d ) be a metric space and denote by A the class ofabsolutely continuous curves from an interval into X . Let I ± ⊆ C ± ⊆ A be four families of of absolutely continuous curves. Then ( I ± , C ± ) is called admissible if it satisfies the following axioms. (C1) Every curve in C ± (hence in I ± ) is never constant, i.e., restrictionsto non-trivial subintervals are non-constant. (C2) The classes I ± and C ± are closed under (non-trivial) restrictions,e.g., if γ : [ a, b ] → X is in I ± or in C ± , and a ≤ c < d ≤ b then therestriction γ | [ c,d ] of γ to [ c, d ] is in I ± or C ± , respectively. (C3) The classes I ± and C ± are closed under concatenations, that is if γ : [ a, b ] → X is a curve such that the (non-trivial) restrictions γ | [ a,c ] and γ | [ c,d ] are in I ± or in C ± for some a ≤ c ≤ b then so is theirconcatenation γ . (C4) The classes I ± and C ± are closed under reparametrizations: Let γ : J ′ → X be in I ± or in C ± and let φ : J → J ′ be a strictly increasing con-tinuous map defined on an interval J ⊆ R such that it and its inverseare absolutely continuous. Then γ ◦ φ ∈ I ± or ∈ C ± , respectively.Moreover, if φ is as above but orientation reversing, i.e., strictly de-creasing, then γ ◦ φ ∈ I ∓ or ∈ C ∓ , respectively.We call the curves in I ± future/past directed timelike curves and the onesin C ± future/past directed causal curves. Moreover, set I := I + ∪ I − and C := C + ∪ C − . Definition A.2. Lorentzian length structure (LLStr) A Lorentzian lengthstructure on a metric space ( X, d ) is an admissible tuple ( I ± , C ± ) of subsetsof A together with a function L : C → [0 , ∞ ] , called the Lorentzian length functional , which satisfy the following list ofproperties: L1) L is additive : If γ : [ a, b ] → X is in C then L ( γ ) = L ( γ | [ a,c ] ) + L ( γ | [ c,d ] ) for any c ∈ ( a, b ) . (L2) L is invariant under reparametrizations , i.e., for γ and φ as in (C4)we require L ( γ ◦ φ ) = L ( γ ) . (L3) For every γ ∈ I we have L ( γ ) > . (L4) The length structure respects the topology of X in the followingsense: L depends continuously on the parameter of the curve, e.g., if γ : [ a, b ] → X is in C we set L ( γ, a, t ) := L ( γ | [ a,t ] ) . Then t L ( γ, a, t ) is continuous.We write ( X, d, I ± , C ± , L ) for a Lorentzian length structure. Recall that a causal space ( X, ≪ , ≤ ) on a set X is constituted by twotransitive relations ≪ and ≤ on X , where ≤ is additionally reflexive andcontains ≪ , cf. [KS18, Def. 2.1] (this is a slightly stronger notion than theone introduced in [KP67]). Definition A.3. Let ( X, d, I ± , C ± , L ) be a Lorentzian length structure. For x, y ∈ X define x ≤ y if x = y or there exists a γ ∈ C + from x to y .Moreover, define x < y if x ≤ y and x = y . Also, define x ≪ y if thereexists a γ ∈ I + from x to y . Lemma A.4. Let ( X, d, I ± , C ± , L ) be a Lorentzian length structure and letthe relations ≪ , ≤ be given by Definition A.3. Then ( X, ≪ , ≤ ) is a causalspace. Proof: This follows from axiom (C3) and I + ⊆ C + . Definition A.5. Let ( X, d, I ± , C ± , L ) be a Lorentzian length structure. For x, y ∈ X define the time separation of x and y by τ ( x, y ) := sup { L ( γ ) : γ ∈ C + connects x and y } , if the set of connecting future directed causal curves is non-empty, otherwiseset τ ( x, y ) := 0 . The time separation function τ has the following properties. Lemma A.6. Let ( X, d, I ± , C ± , L ) be a Lorentzian length structure and letthe time separation function τ and the causal relations ≪ , ≤ be as definedabove. Then(i) τ ( x, y ) = 0 if x y and(ii) τ ( x, y ) > if x ≪ y . roof: (i) This is immediate from the definitions.(ii) Let x ≪ y , then there is a timelike curve γ ∈ I + from x to y and byaxiom (L3) we have L ( γ ) > 0. Thus 0 < L ( γ ) ≤ τ ( x, y ).The reverse triangle inequality holds: Lemma A.7. Let ( X, d, I ± , C ± , L ) be a Lorentzian length structure withtime separation function τ and causal relations ≪ , ≤ as defined above. Thenfor all x, y, z ∈ X with x ≤ y ≤ zτ ( x, y ) + τ ( y, z ) ≤ τ ( x, z ) . Proof: If x = y then either τ ( x, x ) = 0 or there exists a causal curve in C + from x to x . In the first case we are done. Analogously, if y = z either τ ( y, y ) = 0 or there exists a causal curve in C + from y to y . Again in the firstcase we are done. So we can without loss of generality assume that there arecausal curves connecting x to y and connecting y to z . Let γ, λ ∈ C + with γ from x to y and λ from y to z . Then by axiom (C3) the concatenation γ ∗ λ is in C + and connects x to z . Thus L ( γ ) + L ( λ ) = L ( γ ∗ λ ) ≤ τ ( x, z ),by (L1) . Taking now the supremum over all future directed causal curvesconnecting x and y and the ones connecting y to z gives the claim.To investigate only the algebraic properties and further consequencesof having a set with a Lorentzian length structure we relax the notion ofLorentzian pre-length space as introduced in [KS18, Def. 2.8] and call theresulting generalization a bare Lorentzian pre-length space. In particular,we do not want to assume semi-continuity properties of the time separationfunction from the beginning. Definition A.8. Let ( X, ≪ , ≤ ) be a causal space together with a function τ : X × X → [0 , ∞ ] that satisfy the following properties:(i) If x ≪ y then τ ( x, y ) > .(ii) The reverse triangle inequality holds: for all x, y, z ∈ X with x ≤ y ≤ z one has τ ( x, y ) + τ ( y, z ) ≤ τ ( x, z ) . Then ( X, ≪ , ≤ , τ ) is called a bare Lorentzian pre-length space . X comes with a metric (topology), push-upto hold or that τ is lower semicontinuous with respect to this topology .Given a bare Lorentzian pre-length space and a metric d on X one canstill define causal curves, their length etc., and get the same basic properties(following the steps as in [KS18, Sec. 2]). For example, given now the causalrelations defined in Definition A.3 a ≤ -causal curve is a locally Lipschitzcurve γ such that γ ( s ) ≤ γ ( t ) for any two parameter values with s ≤ t (cf.[KS18, Def. 2.18]). Note that any curve in C + is ≤ -causal, and analogouslyfor timelike curves.Finally, we define T ( x, y ) := sup { L τ ( γ ) : γ future directed ≤ -causal from x to y } , if the set of connecting future directed ≤ -causal curves from x to y is non-empty. Otherwise, we set T ( x, y ) := 0. Definition A.9. A bare Lorentzian pre-length space ( X, ≪ , ≤ , τ ) is called bare Lorentzian length space if T = τ , i.e., if the time separation function τ is intrinsic (for some background metric d on X ). At this point we are able to establish that a Lorentzian length structuregives rise to a bare Lorentzian length space. Theorem A.10. Let ( X, d, I ± , C ± , L ) be a Lorentzian length structure anddefine the causal relations ≤ , ≪ and the time separation function τ as above.Then ( X, ≪ , ≤ , τ ) is a bare Lorentzian length space. Proof: That ( X, ≪ , ≤ , τ ) is a bare Lorentzian pre-length space followsfrom Lemmata A.4, A.6 and A.7.As noted, any C + -causal curve is ≤ -causal and analogously for I + and ≪ . Moreover, for any ≤ -causal curve γ (hence also for any γ ∈ C + ) from x to y one has L τ ( γ ) ≤ τ ( x, y ) by the definition of the τ -length. Let ( γ : [ a, b ] → X ) ∈ C + and let a = t < t < · · · < t N = b be a partition of [ a, b ], then by (L1) L ( γ ) = N − X i =0 L ( γ | [ t i ,t i +1 ] ) ≤ N − X i =0 τ ( γ ( t i ) , γ ( t i +1 )) . Of course, by endowing X with the discrete metric, any function from X to anothertopological space is continuous. Thus any such bare Lorentzian pre-length space withouttopology can be viewed as a Lorentzian pre-length space in the sense of [KS18, Def. 2.8](except that push-up need not hold). However, note that then there are no non-constantcurves, hence no causal curves. a, b ] yields that L ( γ ) ≤ L τ ( γ ) forany γ ∈ C + (defined on a compact interval). 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