Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

Abstract Within the synthetic‐geometric framework of Lorentzian (pre‐)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399–447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non‐branching result for spaces with timelike curvature bounded below.


Introduction
In the theory of (metric) length spaces, Alexandrov spaces and CAT(K)spaces angles play an important role. They are an indispensable tool and in particular allow one to characterize curvature bounds with a monotonicity condition on angles. Thus a fruitful way forward in the theory of Lorentzian (pre-)length spaces [KS18] is to introduce the notion of an hyperbolic angle into the Lorentzian synthetic setting. As compared to the metric case, several difficulties have to be overcome, foremost the issue of keeping track of the time orientation and causal relations of the points and curves considered. Thus several proofs become more involved than one would expect from the metric case.
Lorentzian (pre-)length spaces have been developed to allow one to define timelike or causal curvature bounds for spacetimes of low regularity or for situations where one does not even have a Lorentzian metric, like in some approaches to Quantum Gravity, e.g. the causal set approach [BLMS87] or causal Fermion systems [Fin18] (see also [KS18,Subsec. 5.3]). Triangle comparison theorems in Lorentzian geometry were pioneered by Harris in [Har82] and triangle comparison (to model spaces) is the key concept to define timelike and causal curvature bounds in [KS18]. The study of spacetimes (and cone structures) of low regularity is a very active field with connections to General Relativity and in particular to singularity theorems, the cosmic censorship conjecture and (in-)extendibility (cf. [CG12, FS12, Säm16, GL17, Sbi18, GGKS18, GLS18, Min19, GKSS20, Gra20, LLS21, KOSS22]). Questions of (in-)extendibility of spacetimes can be formulated in the setting of Lorentzian length spaces and this approach enabled [GKS19], for the first time, to relate inextendibility to a blow-up of (synthetic) curvature. Furthermore, as pioneered by Kronheimer and Penrose [KP67] and Busemann [Bus67], causality theory and (parts of) Lorentzian geometry can be studied decoupled from the manifold structure [KS18,Subsec. 3.5], [ACS20,BGH21]. A large class of Lorentzian length spaces have been investigated in [AGKS21], where warped products of a one-dimensional base with metric length space fibers are considered. These so-called generalized cones have particularly nice causal properties. Moreover, one can relate Alexandrov curvature bounds of the fibers to timelike curvature bounds of the entire space and vice versa, and a first synthetic singularity theorem was established in this setting. Recently, the gluing of Lorentzian pre-length spaces was developed in [BR22] and applied to gluing of spacetimes, thereby establishing an analogue of a result by Reshetnyak for CAT(K)-spaces, which roughly states that gluing is compatible with upper curvature bounds.
Cavalletti and Mondino [CM20] took the field even further by introducing a synthetic notion of Ricci curvature bounds by using techniques from optimal transport analogous to the Lott-Villani-Sturm theory of metric measure spaces with Ricci curvature bounded below [LV09,Stu06a,Stu06b]. They built on work by McCann [McC20] and Mondino-Suhr [MS22b], who characterize timelike Ricci curvature bounds in terms of synthetic notionsà la Lott-Villani-Sturm for smooth spacetimes. A related topic was then to find canonical measures in the Lorentzian setting akin to Hausdorff measures in the metric one. This has been achieved recently in [MS22a], where also a synthetic dimension for Lorentzian pre-length spaces is given, the compatibility of the Lorentzian measures with the volume measure of continuous spacetimes is shown and its relation to [CM20] is studied.
Finally, there is the related approach of Sormani and Vega [SV16] that introduces a metric on Lorentzian manifolds with a suitable time function, the so-called null distance. Its relation to spacetime convergence has been explored in [AB22], while in [KS22] the null distance is introduced on Lorentzian length spaces and a kind of compatibility between the two approaches is established.
In the final stages of preparing the article we have been made aware of the preprint [BMdOS22] by Barrera, Montes de Oca and Solis, where the authors also introduce angles in Lorentzian length spaces. Compared to our work they are investigating only curvature bounds from below but discuss an angle comparison condition and first variation formula, which we do not. On the other hand in our more comprehensive study we consider bounds from below and above, exponential and logarithmic map, triangle inequality of angles and equivalence of monotonicity comparison and timelike curvature bounds. Thus our two works nicely complement each other and their simultaneous appearance is a clear indication that this direction of research will prove fruitful.
In the following subsection we present the main results of our article and outline the structure of the paper.

Main results and outline of the article
The plan of the paper is as follows. In Subsection 1.2 we introduce the relevant background on Lorentzian geometry and fix some notations and conventions. Then, to conclude the introduction we give a brief review of the theory of Lorentzian length spaces in Subsection 1.3 and of curvature comparison in Subsection 1.4.
In Section 2 we introduce angles in Lorentzian pre-length spaces in timelike triangles and between timelike curves. A main tool is the Lorentzian version of the law of cosines (Lemma 2.4) and corollaries thereof. As a compatibility check we establish that in a smooth and strongly causal spacetime the angle between continuously differentiable timelike curves agrees with the synthetic angle in the sense of Lorentzian pre-length spaces. We also show that one can alternatively use K-comparison angles for the definition of angles (for any K ∈ R).
After these technical preparations we study in detail angles between timelike curves that have the same time orientation in Section 3. A central result is the triangle inequality.
Two timelike curves of the same time orientation and starting at the same point are defined to be equivalent if they have zero angle. Together with the triangle inequality this makes the space of (future/past) directed timelike directions at a point into a metric space. This allows one to define the timelike tangent cone as the Minkowski cone over the metric space of (future/past) directed timelike directions. Also, one can define an exponential and a logarithmic map and we establish foundational properties and relations of these objects.
Moreover, we introduce a monotonicity condition for angles at the past/ future endpoint of a timelike triangle, called future/past K-monotonicity comparison. This gives then a bound on the angle between timelike geodesics of the same time orientation (Corollary 3.8), and that angles exist, if finite (Lemma 3.9). Another central result is that timelike curvature bounds imply future/past K-monotonicity comparison as follows.
Theorem 3.10 (Triangle comparison implies future K-monotonicity comparison). Let (X, d, ≪, ≤, τ ) be a locally strictly timelike geodesically con-nected Lorentzian pre-length space and let K ∈ R. If X has timelike curvature bounded below (above) by K, it also satisfies future K-monotonicity comparison from below (above).
In Section 4 we establish the triangle inequality between timelike curves of arbitrary time orientation in Theorem 4.5. Moreover, we improve upon a geodesic non-branching result of [KS18], where we do not need any additional hypotheses, as follows.
Theorem 4.8 (Timelike non-branching). Let X be a strongly causal Lorentzian pre-length space with timelike curvature bounded below by some K ∈ R. Then timelike distance realizers cannot branch.
Finally, in Subsection 4.1, we define a general K-monotonicity comparison condition (Definition 4.9), show that it implies the existence of angles in Lemma 4.10 and, this being the main result of the article, that it characterizes timelike curvature bounds (Definition 1.20) as follows.
Theorem 4.13 (Equivalence of triangle and monotonicity comparison). Let (X, d, ≪, ≤, τ ) be a locally strictly timelike geodesically connected Lorentzian pre-length space and let K ∈ R. Then X has timelike curvature bounded below (above) by K if and only if it satisfies K-monotonicity comparison from below (above).
The proof of the Lorentzian law of cosines Lemma 2.4 and of a related lemma is outsourced to the Appendix A.

Notation and conventions
We fix some notation and conventions as follows. One of our main examples are spacetimes (M, g), where the Lorentzian metric g is of different regularity classes. Here M denotes a smooth, connected, second countable Hausdorff manifold and the Lorentzian metric g on M is continuous, C 2 or smooth. Note that our convention is that g is of signature (− + + + . . .). A vector v ∈ T M is called Moreover, we assume that (M, g) is time-oriented (i.e., there exists a continuous timelike vector field ξ, that is, g(ξ, ξ) < 0 everywhere). We call (M, g) a continuous/C 2 -/smooth spacetime. Furthermore, a causal vector v ∈ T M is future/past directed if g(v, ξ) < 0 (or g(v, ξ) > 0, respectively), where ξ is the global timelike vector field giving the time orientation of the spacetime (M, g). Analogously, one defines the causal character and time orientation of sufficiently smooth curves into M . The (Lorentzian) length L g (γ) of a causal curve γ : [a, b] → M is defined as L g (γ) := b a −g(γ,γ) and the time separation function is defined as follows. For x, y ∈ M set τ (x, y) := sup {L g (γ) : γ f.d. causal from x to y} ∪ {0}.
Two events x, y ∈ M are timelike related if there is a future directed timelike curve from x to y, denoted by x ≪ y. Analogously, x ≤ y if there is a future directed causal curve from x to y or x = y. The spacetime (M, g) is strongly causal if for every point p ∈ M and for every neighborhood U of p there is a neighborhood V of p such that V ⊆ U and all causal curves with endpoints in V are contained in U , (in which case V is called causally convex in U ).
Finally, let us recall the notion of sectional curvature on a smooth semi-Riemannian manifold.
Definition 1.1 (Sectional curvature). Let (M, g) be a semi-Riemannian manifold, p ∈ M a point. A non-degenerate subspace P ⊆ T p M is nondegenerate if (g| p )| P : P × P → R is a non-degenerate symmetric bilinear map. Let P ⊆ T p M be a non-degenerate 2-dimensional subspace (or plane). Let v, w ∈ P be linearly independent. Let R be the Riemann curvature tensor of g, then the sectional curvature of M at p on the plane P is g(R(v,w)w,v) g(v,v)g(w,w)−g(v,w) 2 .

A brief introduction to Lorentzian (pre-)length spaces
Kronheimer and Penrose [KP67] studied causality theory from an abstract point-of-view and their basic object is a so-called causal space. Here we use a slightly more general version of this.
Definition 1.2 (Causal space). A set X with two binary relations ≪, ≤, where both are transitive, ≤ is reflexive, and any timelike related pairs p ≪ q are also causally related p ≤ q (sometimes denoted as ≪ ⊆ ≤), is called a causal space.
We define the chronological and causal futures and pasts and the chronological and causal diamonds as follows.
A Lorentzian pre-length space generalizes spacetimes by taking the causal relations and the time separation function as its fundamental objects, thereby foregoing the smooth structure of the manifold completely. Definition 1.3 (Lorentzian pre-length space). A Lorentzian pre-length space (X, d, ≪, ≤, τ ) is a causal space (X, ≪, ≤) together with a metric d on X and a map τ : • τ is lower semicontinuous (with respect to d), • τ (p, r) ≥ τ (p, q) + τ (q, r) for p ≤ q ≤ r (reverse triangle inequality) and To introduce the notion of intrinsic time separation functions, we define the length of curves as follows.
For past directed causal and past directed timelike, we reverse these relations. Upon parameter reversal, they are future directed causal / timelike. Definition 1.5 (Length of curves). Let (X, ≪, ≤, d, τ ) be a Lorentzian prelength space. We can define the length of future directed causal curves as in metric spaces, replacing the supremum with an infimum in the definition of the variational length. We define the length L τ (γ) = inf{ i τ (γ(t i ), γ(t i+1 )) : (t i ) a partition of I}.
One of the main examples of an Lorentzian pre-length space is a spacetime (M, g) together with its timelike and causal relations ≪, ≤, its time separation function τ and a metric induced by a complete Riemannian background metric, cf. [KS18,Ex. 2.11]. Moreover, spacetimes of low regularity (i.e., with continuous metric and sufficiently well-behaved causality) and Lorentz-Finsler spaces are Lorentzian pre-length spaces as well, cf. [KS18, Prop. 5.8, Prop. 5.14].
Definition 1.6 (Intrinsic space). A Lorentzian pre-length space is strictly intrinsic or geodesic if for all p < q there exists a future directed causal curve γ from p to q of length L τ (γ) = τ (p, q). Such a curve is called a distance realizer.
A Lorentzian pre-length space is intrinsic if for all p < q and all ε > 0 there exists a future directed causal curve γ from p to q of length L τ (γ) > τ (p, q) − ε. Such a curve is called an ε-distance realizer.
A subset A ⊆ X in a Lorentzian pre-length space is called (causally) convex if for any two points p, q ∈ A their causal diamond is contained in A, i.e., J(p, q) ⊆ A.
A key technical tool in smooth semi-Riemannian geometry is the existence of geodesically convex neighborhoods, in which the causality is particularly simple and where one has a complete description of length-maximizing curves. The analogue of this notion in the present context is the following. A Lorentzian pre-length space X is called localizable if every x ∈ X has an open, so-called localizing neighborhood Ω x such that (i) The d-length of all causal curves contained in Ω x is uniformly bounded.
(ii) The localizing neighborhood Ω x can be turned into a Lorentzian prelength space by using the restrictions d| Ωx×Ωx , ≪| Ωx×Ωx and ≤| Ωx×Ωx , and define the local time separation function ω x : For p < q we require that there is an ω x -realizer γ p,q in Ω x from p to q, and that this makes Ω (iii) For every y ∈ Ω x we have I ± (y) ∩ Ω x = ∅.
If, in addition, the neighborhoods Ω x can be chosen such that (iv) Whenever p, q ∈ Ω x satisfy p ≪ q then γ p,q is timelike and strictly longer than any future-directed causal curve in Ω x from p to q that contains a null segment, then (X, d, ≪, ≤, τ ) is called regularly localizable.
Locally distance realizing curves can be thought of as an analog of geodesics.
Definition 1.7 (Geodesic). Let X be a Lorentzian pre-length space. A (say) future directed causal curve γ : I → X is a geodesic if for each t ∈ I there is a neighborhood [a, b] of t (i.e., a < t < b, but allowing for equality at the endpoints of I) such that γ| [a,b] is a distance realizer, i.e. τ (γ(a), γ(b)) = L τ (γ| [a,b] ).
In localizable spaces, the notion of a geodesic was previously defined using localizable neighborhoods as follows.
Definition 1.8 (Geodesicà la [GKS19, Def. 4.1] ). Let X be a localizable Lorentzian pre-length space. A (say) future directed causal curve γ : I → X is a geodesic if it is locally maximal, i.e., for every t 0 ∈ I there is a localizing neighborhood Ω γ(t 0 ) and a neighborhood [a, b] of t 0 in I such that γ| [a,b] is maximal from γ(a) to γ(b) in Ω γ(t 0 ) .
In this article we will use the first definition and the latter notion actually satisfies the first one when everything is considered in the Lorentzian prelength space Ω x . Thus, when developing the theory here, it is enough to use the first version, and not employ localizability unless explicitly stated. Definition 1.9. We call τ locally finite-valued if every point has a neighborhood U such that τ < ∞ on U ×U . Similarly, we call τ locally continuous if every point has a neighborhood U such that τ is continuous on U × U .
Moreover, X is locally causally closed if every point has a neighborhood U such that for all x n , y n ∈ U with x n → x ∈Ū , y n → y ∈Ū and x n ≤ y n for all n ∈ N, then x ≤ y.
Finally, X is causally path-connected if for all x, y ∈ X with x < y (or x ≪ y) there is a future directed causal (or timelike) curve from x to y.
The definition of local causal closedness is the original definition given in [KS18,Def. 3.4]. However, in [ACS20] an issue with non-strongly-causal Lorentzian pre-length spaces was pointed out and an alternative definition has been proposed in [ACS20, Def. 2.19]. We opted for the former one as we are only concerned with strongly causal spaces and for compatibility with previous results in [ KS18,AGKS21].
Any timelike distance realizer can be parametrized with respect to τarclength as long as τ is locally finite-valued and locally continuous, cf. [KS18,Subsec. 3.7].
Definition 1.11 (Lorentzian length space). A Lorentzian length space is a Lorentzian pre-length space which is locally causally closed, causally path connected, intrinsic and localizable.
12. An open set U ⊆ X is called timelike geodesically connected if whenever x, y ∈ U with x ≪ y, there exists a future-directed maximal geodesic in U from x to y. An open set U ⊆ X is called strictly timelike geodesically connected if whenever x, y ∈ U with x ≪ y, there exists a future-directed maximal geodesic in U from x to y, and that any futuredirected maximal geodesic in U from x to y is timelike. X is called locally strictly timelike geodesically connected if it is covered by strictly timelike geodesically connected neighborhoods.

Curvature comparison for Lorentzian pre-length spaces
Curvature bounds for metric spaces, generalizing sectional curvature bounds of Riemannian manifolds, are defined by comparing distances in triangles to distances in comparison triangles in two-dimensional (Riemannian) manifolds of constant curvature, see e.g. [BH99,BBI01,AKP22]. Timelike and causal curvature bounds were introduced analogously for Lorentzian prelength spaces in [KS18]. In this setting we measure distances with the time separation, so we restrict to causal triangles.
Definition 1.13 (Geodesic triangles). A timelike (geodesic) triangle ∆ = (p 1 , p 2 , p 3 ) in a Lorentzian pre-length space X consists of three points p 1 ≪ p 2 ≪ p 3 ∈ X (with τ (p i , p j ) < ∞ for i < j) and three future directed causal distance realizing curves α ij connecting p i to p j (for i < j). Analogously, we define a causal triangle.
An admissible causal (geodesic) triangle (p 1 , p 2 , p 3 ) in a Lorentzian prelength space X consists of three points p 1 ≪ p 2 ≤ p 3 or p 1 ≤ p 2 ≪ p 3 ∈ X (with τ (p i , p j ) < ∞ for i < j) and three possibly constant 1 future directed causal distance realizing curves α ij connecting p i to p j (for i < j). We call the sides between two vertices p i ≪ p j a timelike side (although it need not be realized via a timelike curve).
We call p 1 the past endpoint and p 3 the future endpoint of the triangle. A causal or timelike triangle is called non-degenerate if the reverse triangle inequality τ (p, r) ≥ τ (p, q) + τ (q, r) is strict, and it is called degenerate in the strict sense if the sides α 12 and α 23 are (reparametrized) parts of the longest side α 13 .

Comparison (model) spaces
Here we recall the two-dimensional Lorentzian manifolds of constant curvature K ∈ R, i.e., (scaled) (anti-)de Sitter spacetime and Minkowski spacetime.
Definition 1.14 (Model spaces). For integers 0 ≤ m ≤ n denote by R n m the vector space R n together with the inner product The Lorentzian K-planes or the comparison spaces of constant curvature K are the following.
Definition 1.15. Three numbers a 12 , a 23 , a 13 ≥ 0 satisfying the reverse triangle inequality a 12 +a 23 ≤ a 13 (making a 13 the largest) are said to satisfy the timelike size bounds for K if a 13 < D K . 3 Let M be a semi-Riemannian manifold. Let p ∈ M be a point and P ⊆ T p M be a plane. Then P is called spacelike if g| p is positive or negative definite on P , timelike if g| p is non-degenerate and indefinite on P .

Andersson-Howard
Definition 1.16 (Sectional curvature comparison). M satisfies sectional curvature comparison from below (or above) by K ∈ R if for all points p ∈ M and all spacelike planes P ⊆ T p M , the sectional curvature is ≥ K (or ≤ K) and for all timelike planes P ⊆ T p M , the sectional curvature is ≤ K (or ≥ K).
Remark 1.17. Equivalently, sectional curvature comparison from below can be written in compact form as g(R(v, w)w, v) ≥ K(g(v, v)g(w, w) − g(v, w) 2 ) (and with inequality reversed for sectional curvature comparison from above).
Remark 1.18. Sectional curvature comparison is not transitive in dimension larger than two: if M satisfies sectional curvature comparison from below / above by K ∈ R, it does not automatically satisfy it from below / above for anyK = K. But if M satisfies timelike sectional curvature comparison from below (or above) by K (i.e., the sectional curvature comparison just for timelike planes), it automatically satisfies timelike curvature comparison from below (or above) by anyK ≥ K (orK ≤ K) (but note the reversal of the inequality as compared to the Riemannian case). For two dimensional Lorentzian manifolds there are only timelike tangent planes, hence sectional curvature comparison is timelike sectional curvature comparison. We give a detailed account for this in the case of Lorentzian pre-length spaces (and the Lorentzian K-planes) and triangle comparison in Lemma 3.27.
Note that comparison triangles exist, when the side-lengths satisfy size bounds for K ∈ R. To be precise [AB08, Lem. 2.1] gives the following.
Proposition 1.19 (Comparison triangles exist). Given three non-negative reals a 12 , a 23 , a 13 satisfying the reverse triangle inequality a 12 +a 23 ≤ a 13 and timelike size bounds for K ∈ R, there exists a causal triangle ∆p 1 p 2 p 3 in a normal neighborhood in the K-plane such that τ (p i , p j ) = a ij (for i < j).
Any two such triangles ∆p 1 p 2 p 3 , ∆q 1 q 2 q 3 (for the same side-lengths a ij ) in the K-plane are related by an isometry ϕ mapping one to the other. The isometry ϕ is unique unless a 13 = 0 (making all a ij = 0) or the reverse triangle inequality is actually an equality.

Triangle comparison
In this subsection we introduce timelike curvature bounds as in [KS18,Subsec. 4.3].
Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space, L 2 (K) be the Kplane with time separation functionτ and p 1 ≪ p 2 ≪ p 3 be three timelike related points in X. A comparison triangle of ∆p 1 p 2 p 3 in the Lorentzian K-plane is a timelike triangle with vertices p 1 ≪ p 2 ≪ p 3 in the K-plane with agreeing side-lengths, i.e., τ (p i , p j ) =τ (p i , p j ) for i < j. We say the vertex p i corresponds to p i . Moreover, we say that the side α ij connecting p i to p j (i < j) corresponds to the side α ij connecting p i to p j .
For a point q on some side α ij of the triangle we define the corresponding point q on α ij by requiring equal distances to the endpoints of the curve it is on, i.e., we require τ (p i , q) =τ (p i , q) (and then automatically, τ (q, p j ) = τ (q, p j ) as α ij and α ij are distance realizing). Note that it might be necessary to specify which side q should be considered to be on (as two sides can partially overlap in X, but not in L 2 (K), unless the triangle is degenerate). If we take two such points q 1 , q 2 (usually on different sides), we can compare the time separation of q 1 to q 2 with the time separation of the corresponding points in the comparison situation. This sets the stage for timelike and causal curvature comparison.
Definition 1.20 (Timelike curvature bounds). Let X be a Lorentzian prelength space and K ∈ R. An open subset U is called a timelike ≥ Kcomparison neighborhood (or timelike ≤ K-comparison neighborhood) or just comparison neighborhood if • τ is finite and continuous on U × U , • U is strict timelike geodesically connected and • for all timelike triangles ∆p 1 p 2 p 3 in U satisfying timelike size bounds for K, q 1 , q 2 two points on some sides α and β and all (any) comparison situations ∆p 1 p 2 p 3 , q 1 , q 2 in the K-plane, the time separation satisfies τ (q 1 , q 2 ) ≤τ (q 1 , q 2 ) (or τ (q 1 , q 2 ) ≥τ (q 1 , q 2 )) .
We say X has timelike curvature bounded below by K if it is covered by timelike ≥ K-comparison neighborhoods. Likewise, it has timelike curvature is bounded above by K if it is covered by timelike ≤ K-comparison neighborhoods.
Remark 1.21. Note that in the case of timelike curvature bounded below, q 1 ≪ q 2 implies q 1 ≪ q 2 , and in the case of timelike curvature bounded above, q 1 ≪ q 2 implies q 1 ≪ q 2 . Furthermore, if U is also a locally causally closed neighborhood we also have for curvature bounded below that q 1 ≤ q 2 implies q 1 ≤ q 2 and in curvature bounded above the other implication holds.
In an analogous way we can define causal curvature bounds for admissible causal triangles where two vertices are not necessarily timelike related, cf. [KS18,Subsec. 4.5].
Remark 1.22 (Automatic size bounds). If X is strongly causal and has timelike curvature bounded above / below, we can restrict ourselves to comparison neighborhoods U where timelike size bounds are automatically satisfied and of the form of timelike diamonds, and there exists a basis of the topology of such neighborhoods, see [Ber20, Rem. 2.1.8].

Angles
In this section we introduce comparison angles, establish foundational properties and define the notion of an angle between timelike curves.

Definition and basic properties
Definition 2.1 (K-comparison angles). Let K ∈ R, let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space and let x, y, z ∈ X be causally related, i.e.
x ≤ y ≤ z or a permutation of this, and all time separations between these points finite. If K < 0 we demand that the side-lengths of ∆xyz satisfy the timelike size bounds for K. Let x ≪ y and x ≪ z and let ∆xȳz be a comparison triangle of ∆xyz in L 2 (K). The K-comparison (hyperbolic) angle at x is defined as∡ Additionally, for notational simplicity we set∡ x (y, z) :=∡ 0 x (y, z). To make the notation more concise we define the signed angle: If the angle is at a future or past endpoint, we set σ = −1, if it is not, σ = 1. We define the signed angle∡ K,S x (y, z) := σ∡ K x (y, z).
Remark 2.2. Note that in case x ≪ y ≪ z we have∡ K,S x (y, z) = −∡ K x (y, z), so minus the "usual" angle, see Figure 1. Moreover, we will sometimes refer to the value of σ when we want to distinguish cases depending on the time orientation of the vertices. Note also that∡ K x (y, z) =∡ K x (z, y). Remark 2.3 (Angles in comparison spaces). Letx,ȳ,z ∈ L 2 (K), such that x ≪ȳ,x ≪z then ∡ L 2 (K) is defined as follows: where γ r,s is the unit speed timelike geodesic from r to s and ·, · is the metric on L 2 (K). We have an analogous equation at the other vertices. Note that arcosh γ ′ x,ȳ (0), γ ′ x,z (0) is defined as the argument is greater equal than one by the reverse Cauchy Schwarz inequality for timelike vectors: The sign σ of the signed angle∡ K,S x (y, z) is just the sign of γ ′ x,ȳ (0), γ ′ x,z (0) (which is the reason to define it this way).
Physically, the angle is a monotonously increasing function in the change of velocity of a particle switching from α to β. Moreover, note that in case σ = 1, the angles might seem to go against our (spatial) intuition: The angle ω between two parts of a straight line in Minkowski space is 0 here, in contrast to the Euclidean angle taking the maximum value π, as indicated in Figure 2.
A fundamental tool in (semi-)Riemannian and metric geometry is the law of cosines. For the convenience of the reader we include a proof in our setting in the appendix A. Note that it also could be derived from [AB08, Lem. 2.2. and Lem. 2.3]. The definition of the comparison angle leads to the hyperbolic law of cosines as follows.
Lemma 2.4 (Law of cosines). For p, q, r in the Lorentzian K-plane forming a finite causal triangle (not necessarily in this order), let a = max(τ (p, q), τ (q, p)), b = max(τ (q, r), τ (r, q)), c = max(τ (p, r), τ (r, p)) be the side-lengths (note that in each maximum, one of the two arguments is 0) with a, b > 0, i.e., q is timelike related to p and r but p and r need only be causally related.

|K|
. Then the left-resp. right-handsides join up to an analytic formula in c and K resp. a, b, ω and K.
In particular, fixing two sides and varying the third, ω is a strictly increasing function in the longest side and a strictly decreasing function in the other two sides.
In the case σ = −1, we can even make the side opposite the angle spacelike.
Proof. Let q, r be not causally related. Without loss of generality we assume τ (p, r) ≥ τ (p, q) and consider the timelike geodesic α from p to q. We consider the parameter t ′ = sup α −1 (I − (r)), then the point q ′ := α(t ′ ) is the point along α being null related to r, i.e., q ′ ≤ r, q ′ ≪ r. Note ω := ∡ r). We define the side-lengths: a ′ = τ (p, q ′ ) < a = τ (p, q), b = τ (p, r). We apply the law of cosines (Lemma 2.4 with σ = −1 and a null side) to this situation. We now form a new triangle ∆pqr with τ (p,q) = a, τ (p,r) = b andq ≤r null related. Letω := ∡ L 2 (K) p (q,r). The triangles ∆pqr and ∆pq ′ r only differ by the value of ω and the side-length a > a ′ . Now we apply the monotonicity statement of the law of cosines to a. Thus ω is a decreasing function of a, thus ω <ω, and the formula in the law of cosines holds with equality for a, b, c = 0,ω. Replacingω by ω, one easily sees whether the right-hand-side increases or decreases.
Remark 2.7. The extended law of cosines fits nicely with the monotonicity of the law of cosines, i.e., taking a timelike triangle and making the side opposite the angle null makes the angle larger, and similarly going from a null side to a spacelike side opposite the angle increases the angle.
The following is helpful for calculations and is an example how one can utilize the law of cosines. The proof is outsourced to the appendix A.
Lemma 2.8 (Calculating one-sided comparison situations). (1) Let K ∈ R and let a + b + c ≤ d, and d satisfy timelike size bounds for K. Then construct a timelike triangle ∆p 1 p 2 p 3 in L 2 (K) with sidelengths a, b + c, d.
Moreover, if lim (s,t)∈A 0 ; s,tց0∡x (α(s), β(t)) exists and is finite, then we call ∡ x (α, β) the angle between α and β at x, and say the angle between α and β at x exists. Similarly to the signed angle for three points, we define the signed angle between curves: If x is the future or past endpoint, i.e., α and β have the same time orientation, we set σ = −1 and if it is not, i.e., α and β have different time orientation, we set σ = 1. We define the signed angle Remark 2.10. The set A K is precisely the set of (s, t) for which∡ K x (α(s), β(t)) is defined. Also, note that the signed angle between α and β has the same sign as the signed angle of ∆α(s)xβ(t) at x for any s, t > 0.
Note that the definition of the upper angle of two future or past directed timelike curves α, β makes sense, i.e., there are s, t → 0 with (s, t) ∈ A K : In case α and β have a different time orientation, i.e., σ = 1, the condition α(s) ≤ β(t) or conversely is automatically satisfied. In the other case, i.e., σ = −1, say without loss of generality α, β future directed. Then Another useful concept is the one of a hinge: Definition 2.11 (K-comparison hinges). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space, where τ is locally finite-valued. We call two future or past directed timelike geodesics α : x such that the angle between them exists a hinge.
Let (α, β) be a hinge, and let K ∈ R. Then a comparison hinge is a hinge (ᾱ,β) in L 2 (K), where the sides have the same time orientation and Remark 2.12. Comparison hinges always exist and are unique up to (unique) isometry of L 2 (K), if L τ (α) and L τ (β) satisfy timelike size bounds for K, where the isometry is unique if none of L τ (α), L τ (β) and ∡ x (α, β) is zero.
As a first compatibility check we establish below that in a (C 2 , strongly causal) spacetime the angle between two timelike curves with respect to the Lorentzian metric and the angle as defined above agree.
By the law of cosines formula, we see that the comparison angle betweeñ α ε (s) andβ ε (t) converges to the comparison angle betweenα(s) As a consequence we obtain that the angle between timelike curves does not depend on the comparison angle used. This is an analog of [BH99, Prop. II.3.1, p. 184] in the metric case.
Proposition 2.14 (All∡ K converge to ∡). Let X be a strongly causal Lorentzian pre-length space with τ locally finite-valued, α, β timelike curves (each can be future or past directed) with α(0) = β(0) = x. Then for all and the limit superior on the left-hand-side is a limit and finite if and only if the angle between α and β exists.

Now we have
l 2 ab and are done with the denominator. Now we look at the enumerator of this fraction: We use the Taylor series of cosh, i.e., cosh(x) = 1 + n≥2, even x n n! . For the whole enumerator, we thus have where we abbreviated the sum of terms where a 2 b 2 appeared by o(l 2 ab). We now look at each n separately, together with the denominator: for n = 2, we get the desired l 2 l 2 · a 2 +b 2 −c 2 2ab . For n > 2, we get l n n! times a n + b n − c n now we use induction on n to get a n + b n − c n = o(a 2 + b 2 − c 2 ).
In total, we have Note that the sum in the enumerator is For a, b, c → 0, this converges if and only if converges, and they converge to the same value, and similarly for K < 0.

Angles between timelike curves of the same time orientation
In this section we study the angle between curves and, in particular, geodesics that have the same time orientation, i.e., are all future or past directed. Some proofs of the results here are given later in Section 4, where these results are proven in full generality, i.e., where the curves can have a different time orientation. However, no logical issues arise from this.
We start by establishing that (upper) angles satisfy the (usual) triangle inequality if all the curves have the same time orientation.
We establish the case where α, β, γ are all future directed. The prospective inequality is only non-trivial if ∡ x (α, β) < ∞ and ∡ x (β, γ) < ∞ -so let that be the case. Moreover, assume to the contrary that the inequality does not hold, so that there is an ε > 0 such that By definition of the upper angles, there is a B ′ ∈ (0, B) small enough such that the initial segments (defined on [0, B ′ ]) of α, β, γ are contained in a causally closed neighborhood, where τ is finite and continuous, and for all r, t, s ∈ (0, B ′ ) we have the upper bound whenever the comparison angles exist, i.e., whenever α(r) ≤ β(s) and β(s) ≤ γ(t) or the other way round.
In particular α(r n ), γ(t n ) are causally related, without loss of generality we assume α(r n ) ≤ γ(t n ) for all n. For n ∈ N let 0 < r n <r n be so close tõ r n that by the (local) continuity of τ and the continuity in the law of cosines we get∡ We define For large enough n, we have s n − < B ′ and s n + < B ′ : I − (β( B ′ 2 )) contains an initial segment of α and γ, thus α(r n ), γ(r n ) ∈ I − (β( B ′ 2 )) for large enough n.
We pick a particular such n and will drop the subscript n from now on.
We distinguish the cases s − < s + and s − ≥ s + . Consider first the case s − < s + : It is easy to see that Set a := α(r), b(s) := β(s), c := γ(t) and note that by construction, our indirect assumption reads We now create a comparison situation where all side lengths and two of the angles are realized: Choose comparison triangles for ∆xab(s) and ∆xb(s)c in the Minkowski plane R 2 1 , which share the side [xb s ] and such that the pointsā s andc s are on different sides of the line [xb s ], i.e., we Figure 4. This realizes the angles∡ x (a, b(s)) and∡ x (b(s), c). Note thatā s ,b s ,c s may depend on s in general. This choice of points can be made continuously in s, even when one of the sides becomes null (i.e., when extending s to s = s − or s = s + ). We claim that there is an s ∈ (s − , s + ) such thatā s ,b s ,c s lie on a line (which then is timelike). For s ∈ (s − , s + ) let L s be the line connectinḡ a s withc s . Setting s = s − , we get that a and b(s − ) are null related, but we can still form the comparison situation (denoted as above). In this new comparison configuration, it is obvious thatb(s − ) =:b − lies below L s − =: L − as bothb − andc s lie on the same side of [xā s ], andb − is in the causal future ofā s andc s is in the timelike future ofā s as the family of lines (L s ) s are causal and hence cannot tilt too much, see Figure 5.
Similarly, setting s = s + , we get that b(s + ) and c are null related. Here, it is obvious thatb s + lies above L s + . Now, by continuity of all the points, there is a value s ∈ (s − , s + ) such thatb s lies on L s .
Now we use this s and drop the s-subscript, i.e.,b :=b s . Then we havē τ (ā,c) =τ (ā,b) +τ (b,c) as they lie on a straight line (a timelike Minkowskigeodesic). We choosec such that ∆xāc is a comparison triangle of ∆xac, withc andc on the same side of [xā]. In this situation, the angles under consideration are given by •∡ x (a, c) = ∡x(ā,c).
By the reverse triangle inequality, we know that τ (a, where the last equality is due to the additivity of hyperbolic angles in the plane. This is a contradiction to the assumption, i.e., Equation (1).
At this point we consider the case s − ≥ s + . Now we may not get a single point on β, but we may get two of them: Set b ± = β(s ± ). Then we have We can form the following comparison situation: For the triangle ∆xα(r)γ(t) we get a comparison triangle ∆xāc. In the same diagram, we add a comparison triangle for ∆xα(r)b − and call the additional pointb − , and add a comparison triangle for ∆xb + γ(t) and call the additional pointb + . We set it up so thatā is to the left ofc,b − is to the right ofā andb + is to the left ofc, see Figure 6. We again define L as the (future directed timelike) straight line fromā toc. For the angles we get that ω 2 ω 1 Figure 6: As the red lines are null and the green line is timelike, b − is below L and b + is above L.
By assumption, we obtain ω 3 > ω 1 + ω 2 + ε, thus the line [xb − ] is to the left of [xb + ] (this follows from angle additivity in the plane). As [āb − ] is future directed null and L = [āc] is future directed timelike and both go to the right,b − lies below L. As [b +c ] is null and L = [āc] is timelike and both go to the right,b + lies above L.
We now construct a future directed timelike curve fromb − tob + in the comparison space (that means b We begin atb − . Asb − lies below L, we use the extension of the line [xb − ] to reach the line L, then go along a part of L. Asb + lies above L, we use the end of the line [xb + ] to reachb + . In particular,τ (b − ,b + ) > 0, and by the reverse triangle inequality Finally, note that in case ∡ x (α, γ) = ∞, one obtains analogous contradictions in the cases s − < s + and s − ≥ s + , as the comparison angle∡ x (a, c) is finite but arbitrarily large.
For each t, we consider the set of parameters s such that α(s) ≪ β(t), i.e., Now by the definition of comparison angles we obtain Again by continuity of τ this also holds in the limit s ր s t , i.e., For simplicity, we set o := cosh(ω + ε) and get the quadratic inequality That is, for s < t exp(−ω − ε) we have s < s t and thus α(s) ≪ β(t) by construction.
At this point we introduce the notion of direction of a timelike geodesic.
Lemma 3.4 (Properties of angles and direction). Let (X, d, ≪, ≤, τ ) be a strongly causal and locally causally closed Lorentzian pre-length space with τ locally finite-valued.
) is the reversed curve, which is a past directed timelike geodesic.
(iii) Let x ∈ X, the relation ∼ of having the same direction at x is an equivalence relation on D ± x . Proof: is degenerate and the corresponding comparison triangle ∆xȳz in R 2 1 = L 2 (0) is a straight timelike segment, and sõ ∡ x (α(s), α(t)) = ∡ L 2 (0) x (ȳ,z) = 0. Thus, the angle between α and itself exists and is zero.
(iii) Symmetry of ∼ is clear from the definition and reflexivity follows from point (i) above. To show transitivity let α, β, γ ∈ D + x with α ∼ β and β ∼ γ. Then by the triangle inequality for upper angles (Theorem 3.1) we get that which shows that the angle between α and γ exists and equals zero. So we have α ∼ γ, as required.
Let us emphasize point (ii) above: Contrary to the metric case the hyperbolic angle between the incoming and outgoing segments of an interior point of a timelike geodesic is zero and not π, cf. Figure 2.
The notion of direction allows us to define the space of (future or past directed) timelike directions at a point.
Lemma 3.5. Let (X, d, ≪, ≤, τ ) be a strongly causal, locally causally closed Lorentzian pre-length space with τ locally finite-valued and so that angles between (future directed) timelike geodesics always exist, and x ∈ X. Then Definition 3.6 (Space of timelike directions). Let (X, d, ≪, ≤, τ ) be a strongly causal, locally causally closed Lorentzian pre-length space with τ locally finite-valued and so that angles between (future directed) timelike geodesics always exist. We call the metric space This definition makes sense as the angle ∡ x is symmetric and satisfies the triangle inequality by Theorem 3.1. Moreover, the angle does not depend on the choice of representative and then by construction ∡ x is positive definite.
For technical reasons it is useful to also introduce the completion Σ ± x of the space of timelike directions D ± x . However, we will later show in Proposition 3.15 that for a large class of Lorentzian pre-length spaces with a lower curvature bound the space of timelike directions D ± x is already complete.
At this point we use angles to introduce another notion of curvature bound by demanding that the comparison angles behave monotonically along the sides of a timelike geodesic triangle. The analogous concept in metric geometry is called monotonicity condition in [BBI01, Def. 4.3.1].
Definition 3.7 (Future K-monotonicity comparison). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space and let K ∈ R. We say that X satisfies future timelike K-monotonicity comparison from below (above) if every point in X possesses a neighborhood U such that (i) τ | U ×U is finite and continuous.
(iii) Let x ≪ y ≪ z be timelike related and forming a timelike geodesic triangle in U , whose side lengths satisfy timelike size bounds for K.
We can define past K-monotonicity comparison analogously by choosing x ′ ∈ [x, z] and y ′ ∈ [y, z] using the comparison angles at z instead. Note that y ′ is on a different side here as compared to the future case. Moreover, let us remark that x ′ , y ′ or z ′ are named as such because they seem to be in the same direction as x, y or z from the point we are measuring the angle at. For an illustration of K-monotonicity comparison see Figure 7.
Figure 7: When comparing with L 2 (K) the interior side is shorter than expected -the K-angle has to behave accordingly.
A direct consequence of K-monotonicity and Proposition 2.14 is the following corollary.
Corollary 3.8 (Monotonicity implies bound on angle of geodesics). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space that satisfies future/past Kmonotonicity comparison from below (above) for some K ∈ R. Then for any x ∈ X and α, β : [0, B] → X future/past directed timelike geodesics with for all s, t ∈ [0, B] small enough such that timelike size bounds for K are satisfied and chosen such that α(s) and β(t) are causally related.
Moreover, K-monotonicity yields that the limit superior in the definition of the angle between geodesics is actually a (not necessarily finite) limit, see e.g. [ Lemma 3.9 (Monotonicity implies existence of angles). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space. If it satisfies timelike K-monotonicity comparison from above or below, the angle between any two future directed timelike geodesics starting at the same point x exists if it is finite (the limit superior in the definition of upper angles is a limit, but we do not know if it is finite).
If it satisfies timelike K-monotonicity comparison from below, we know the limit is finite (in particular, the angle exists).
Proof. This will be proven later in more generality in Lemma 4.10.
The connection of future K-monotonicity comparison to timelike curvature bounds is as follows.
Theorem 3.10 (Triangle comparison implies future K-monotonicity comparison). Let (X, d, ≪, ≤, τ ) be a locally strictly timelike geodesically connected Lorentzian pre-length space and let K ∈ R. If X has timelike curvature bounded below (above) by K, it also satisfies future K-monotonicity comparison from below (above).
Proof. This will be proven later in more generality in Theorem 4.13.  To be precise, for x ∈ X and α, α n , β, β n : [0, b] → X all future or all past directed timelike geodesics starting at x with α n → α, β n → β pointwise, then Proof: This will be proven later in more generality in Proposition 4.14.
In case of timelike curvature bounded above, the angle between geodesics is in fact continuous.
Proposition 3.12 (Angle between geodesics is continuous). Let (X, d, ≪, ≤ , τ ) be a Lorentzian pre-length space with timelike curvature bounded above. Then angles are continuous for geodesics, i.e., for x ∈ X and α, α n , β, β n : [0, b] → X all future or all past directed timelike geodesics starting at x with α n → α, β n → β pointwise, then Proof: This will be proven later in more generality in Proposition 4.15.
If the curves considered are not geodesics, then there is no semi-continuity of angles, as is well-known in the smooth spacetime case. We include an example below for the convenience of the reader.
As there is no automatic local existence of timelike geodesics and, moreover, the time of existence need not be locally uniform, we have to exclude these situations when considering the question of whether the space of timelike directions is complete. For convenience we give a name to the following condition which will be used several times.
Definition 3.14 (Non-lingering). Let (X, d, ≪, ≤, τ ) be a Lorentzian prelength space. A point x ∈ X has the uniform future/past non-lingering property if there is an s > 0 and an ε > 0 such that all future/past directed timelike geodesics γ starting at x = γ(0), when parametrized by darclength and extended maximally, are defined at least up to parameter s, and d(x, γ(s)) > ε. Moreover, we say that (X, d, ≪, ≤, τ ) is future/past non-lingering if every point x ∈ X has the uniform future/past non-lingering property.
By using cylindrical neighborhoods and the fact that the time coordinate is a temporal function on such a neighborhood (cf. [CG12, Prop. 1.10]), it directly follows that any continuous spacetime is future and past nonlingering.
As pointed out before, given a lower curvature bound, we can give sufficient conditions so that the space of timelike directions is complete. We are now ready to provide a proof of this fact.
Proposition 3.15 (Space of timelike directions complete). Let (X, d, ≪, ≤ , τ ) be a locally compact, strongly causal, locally causally closed, d-compatible and regularly localizable Lorentzian length space with timelike curvature bounded below. If x ∈ X has the uniform future non-lingering property, then D + x is a complete metric space, i.e., Σ + x = D + x .
Proof. Note that, by strong causality and local compactness of (X, d), small causal diamonds are compact. Let γ n : [0, L d (γ n )] → X be future directed timelike geodesics, d-unitspeed-parametrized, with γ n (0) = x and [γ n ] ∈ D + x forming a Cauchy sequence. We can assume by the non-lingering property that all γ n are initially but not completely contained in a compact set J(x, y) (for some y ≫ x), which is at the same time a d-compatible neighborhood. Denoting the initial segments by γ n | [0,bn] , we can assume that γ n ([0, b n ]) ⊆ J(x, y), γ n (b n ) ∈ ∂J(x, y) ∩ ∂J − (y) for all n ∈ N. By being in a d-compatible neighborhood, we get that the b n s stay bounded. By the limit curve theorem [KS18,Thm. 3.7], we find a subsequence of (γ n ) n which converges uniformly to a future directed causal curve γ : [0, b] → X which also has γ(b) ∈ ∂J(x, y) ∩ ∂J − (y). Without loss of generality we denote this subsequence again by (γ n ) n and note that γ is non-constant. By [KS18,Prop. 3.17], the limit curve γ is maximizing. We claim it is timelike and [γ n ] → [γ]. Now let ε > 0, and n be large enough such that ∡ x (γ n , γ m ) < ε for all m ≥ n. Let s, t be such that γ(s) ≪ γ n (t) (these exist: for each t there is an s > 0 such that this holds as γ is future directed causal). As γ k (s) → γ(s), we get that for large enough k, also γ k (s) ≪ γ n (t) and we can calculate the comparison angle using the law of cosines (Lemma 2.4 with σ = −1) as follows (we only do the K = 0 case, the others are similar, just a bit more involved).
Lemma 3.9 implies that X satisfies future K-monotonicity (where K is the timelike curvature bound from below). Moreover, by Corollary 3.8 we conclude that ε > ∡ x (γ k , γ n ) ≥∡ K x (γ k (s), γ n (t)). When taking the limit as k → ∞, we have to be careful. Denoting the enumerator in the left-hand-side of Equation (2) by x k and the denominator by y k , we have 1 ≤ x k y k < cosh(ε) =: 1 +ε. We have that y k → 0 precisely when x k → 0. If they do not converge to 0, τ (x, γ(s)) > 0 and thus γ is timelike by regular localizability. Thus we have a valid timelike triangle. Taking k → ∞, and then s, t → 0, we get that the upper angle satisfies ∡ x (γ n , γ) ≤ ε. Letting ε → 0 yields a ∡ x -convergent subsequence of ([γ n ]) n and hence we are done in this case. Now we exclude the other cases: If y k → 0, we have τ (x, γ k (s)) → 0 and together with x k → 0 this implies τ (γ k (s), γ n (t)) → τ (x, γ n (t)). Note that this also works for K = 0 (e.g. we have a cosh(τ (x, γ k (s))) factor on one of the terms, which converges to 1). Denoting the points u := γ(s) and z := γ n (t), we have x ≤ u but x ≪ u and τ (u, z) = τ (x, z) > 0, as γ n is timelike. If x = u, we concatenate the geodesics from x to u and from u to z having length 0 + τ (u, z) = τ (x, z), i.e., a distance realizer containing a null segment (the segment from x to u), in contradiction to regular localizability. Thus x = u and hence s = 0 -a contradiction to s > 0.

Timelike tangent cones
In metric geometry the tangent cone is a generalization of tangent spaces of smooth manifolds (see e.g. [BBI01, p. 321]) and a valuable tool, especially in Alexandrov spaces with curvature bounded below. Using the space of timelike directions at a point we introduce in an analogous way the timelike tangent cone at a point in an Lorentzian pre-length space. To this end we use the concept of the Minkowski cone over a metric space, introduced in [AGKS21, Sec. 2].
Moreover, the causal relation ≤ and the time separation function τ are defined via τ ((s, p), (t, q)) = s 2 + t 2 − 2st cosh d(p, q) if (s, p) ≤ (t, q) , and ≪ induced by τ (i.e., (s, p) ≪ (t, q) ⇔ τ ((s, p), (t, q)) > 0  [AGKS21] establish that the Minkowski cone over a geodesic length space is a geodesic Lorentzian pre-length space and that the time separation τ is continuous. However, it cannot be localizable as the vertex 0 is isolated with respect to ≪, i.e., there is no point x such that x ≪ 0.
However, Minkowski cones have localizable neighborhoods at all points except the vertex 0, which we establish below but first we need the following lemma, whose proof is elementary. (i) If s > 0 one has that d(y, y ′ ) ≤ log(t) − log(s).
Proposition 3.19 (Minkowski cones nearly localizable). Let (Y, d) be a metric space which is strictly intrinsic. Then any 0 = x ∈ X := Cone(Y ) has a localizable neighborhood. Moreover, 0 ∈ X has a neighborhood Ω 0 that satisfies all the conditions of a localizable neighborhood except that I − (0) ∩ Ω 0 = ∅. We call such a neighborhood nearly localizable.
Proof: As (nearly) localizable neighborhoods, we choose the sets {(t, y) : t − < t < t + } and {(t, y) : 0 ≤ t < t + }, respectively, for any 0 < t − < t + . These neighborhoods are causally convex, and any two causally related points can be connected by a τ -distance realizer ([AGKS21, Cor. 2.4]). Furthermore, by [AGKS21, Prop. 2.2] τ is continuous, and except at 0 there are no ≪-isolated points. For d-compatibility, the d c -length of causal curves within such a neighborhood is at most 4t + . This can be shown by using Lemma 3.18: Parametrize a (future directed) causal curve γ from p to q as γ(r) = (r, β(r)), where r ∈ [s, t] and β : [s, t] → Y . We can without loss of generality assume that s = 0, hence p = 0 ∈ Cone(Y ) (otherwise extend γ to 0 -this is always possible, but not necessarily within the neighborhood). At this point, let 0 = s 0 < s 1 < . . . < s M = t be a partition of [0, t]. We choose a refinement 0 = s where we used the concavity of the logarithm. Taking the supremum over all partitions of [0, t] yields L dc (γ) ≤ 4t and t ≤ t + if q lies in such a neighborhood.
Localizability of the Minkowski cone away from the vertex also works if Y is only a locally strictly intrinsic metric space by using timelike diamonds I((t − , y), (t + , y)) for y ∈ Y and t − , t + close enough to each other so that pr Y (I((t − , y), (t + , y))) is contained in a strictly intrinsic neighborhood. Remark 3.21 (Cone metric does not induce Euclidean topology). This is a warning that usually, e.g., for spacetimes, the cone metric d c induces a coarser topology than one might at first expect. In any case we have d c (0, p) = τ (0, p) for all p ∈ Cone(Y ) and hence B dc 1 (0) = {p ∈ Cone(Y ) : τ (0, p) < 1} , i.e., the hyperboloid {p ∈ Cone(Y ) : τ (0, p) = r} of radius r > 0 is bounded, closed and not compact for non-compact Y .
For example, if Y is the n−1-dimensional hyperbolic space H n−1 , then the Minkowski cone Cone(Y ) can be identified with I + (0)∪{0} in n-dimensional Minkowski spacetime R n 1 , and clearly the topology does not agree with the Euclidean one of R n . This can be most easily seen (in without loss of generality n = 1 + 1 dimensions) by considering the sequence p k := ( 1 k , y k ), where y k ∈ H 1 ∼ = R 1 is given by y k = (k 2 , √ k 4 − 1) in the realization H 1 ⊆ R 2 1 . Then d c (0, p k ) = 1 k , hence p k → 0 ∈ Cone(Y ), but p k ∼ = 1 k ·y k = (k, k 2 − 1 k 2 ) → 0 in R 2 with respect to the Euclidean topology ( 1 k · y k even diverges). Here we used the identification of p = (t, y) with t · ψ(y) ∈ I + (0) ∪ {0} ⊆ R n 1 , given in [AGKS21, Rem. 2.1].
Despite Remark 3.21 we will use the cone metric d c on the Minkowski cone Cone(Y ) over a metric space (Y, d), as it is a canonical choice (in contrast to e.g. the usual metric on Minkowski space).
At this point we are in the position to introduce the notion of a timelike tangent cone.
Definition 3.23 (Timelike tangent cone). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space with τ locally finite-valued and so that angles between future/past directed timelike geodesics always exist and let x ∈ X. The future/past timelike tangent cone T ± x at x is the Minkowski cone Cone(Σ x ) over the metric space Σ ± x of the completion of future/past directed timelike directions D ± x .
The following is straightforward: Lemma 3.24. Let M be an n + 1-dimensional strongly causal spacetime with metric of C 2 -regularity, and consider it as a Lorentzian length space. Let p ∈ M , then the space of future directed timelike directions at p, D + p , is isometric to n-dimensional hyperbolic space H n and the tangent cone T + x is isometric to I + (0) ⊆ T x M .
Proof. In a C 2 -spacetime, geodesics are solutions to the geodesic equation. Thus, future directed timelike geodesics passing through x are in one-toone correspondence to future directed timelike vectors in T x M ∼ = R n+1 1 . As angles agree (Lemma 2.13), geodesics have the same angle as the vectors they correspond to, and we can restrict to unit future directed timelike vectors, which form H n . The usual metric on H n is just defined as d(p, q) = ∡ 0 (p, q), which is the metric on D + x . For the tangent cone, we just note that Cone(H n ) ∼ = I + (0) ⊆ R n+1 A natural question is that of whether Σ ± x is a geodesic length space. Moreover, in the metric case it is not hard to see that the completion of a length space is a length space ([BBI01, Exc. 2.4.18]). We give an example below, where D + x is not intrinsic, and then later in Proposition 3.26 give sufficient conditions for D ± x to be intrinsic.
Example 3.25. In essence, this example is the Minkowski cone over a nonintrinsic space. However, it is defined as a subset of Minkowski spacetime and not as a Minkowski cone over a metric space. Let X = {(t, x, y) ∈ R 3 1 : x 2 ≥ yt} ⊆ R 3 1 be the closed exterior of a tilted double cone, given by yt = 1 4 ((y + t) 2 − (y − t) 2 ) = x 2 , see Figure 8. It is a non-convex, conical (with respect to 0) subset of Minkowski spacetime, and we consider it as a Lorentzian pre-length space by restricting τ from Minkowski space. We only have to consider∡ as angles exist and are invariant under scaling. The timelike geodesics through 0 are just straight lines. Thus, the space of directions D + 0 at 0 is a non-convex subset of H n , and thus not intrinsic. Note X was not intrinsic as well.
If we intrinsify X (see [Ber20, Thm. 1.7.3]), i.e., considering X with the new time separation functionτ (p, q) := sup{L τ (γ) : γ future directed causal from p to q} ∪ {0}, the lengths of geodesics stay the same, but angles will change. We need the other geodesics as well: For p, q not both on the cone such that the straight line connection is not contained in X, the connecting geodesic will be straight outside the cone, will be tangential to the cone where it touches it, and at such points it might switch over to have a part contained in the cone (which is a geodesic in the part of the cone which can be considered as a 1+1-dimensional Lorentzian manifold). Now we have a distance realizer γ from p to q. It is intuitively clear that the straight lines from 0 to γ(t) form a distance realizer in D + 0 , making it strictly intrinsic. We have the following sufficient criterion for the completion of the space of timelike directions to be intrinsic.
Proposition 3.26 (Sufficient conditions for space of timelike directions to be intrinsic). Let (X, d, ≪, ≤, τ ) be a strongly causal, locally causally closed Lorentzian pre-length space with τ locally finite valued and timelike curvature bounded above by K + ∈ R and below by K − ∈ R. Then Σ ± x is intrinsic. If additionally X is locally compact, d-compatible, regularly localizable and future and past non-lingering, then already D ± x itself is intrinsic.
Proof. We only establish the future case, the past case is completely analogous.
We want to employ [BBI01, Theorem 2.4.16], which states that a complete metric space which has ε-midpoints is already intrinsic. Note that we only have to find ε-midpoints for a dense subset, for which we use D + x ⊆ Σ + x . So let ε > 0, and let [α], [β] be two points in D + x , with representatives α ∈ [α], β ∈ [β] parametrized by τ -arclength. Note that by Lemma 3.9, the angle between α and β exists and is finite. By strong causality and the curvature bound we find a neighborhood of x such that all τ -distances can be realized locally. Moreover, by the definition of the angle, Proposition 2.14 and Proposition 3.2, for small enough t > 0 we have that s := te −∡x(α,β)−1 satisfies α(s) ≪ β(t) and both |∡ Let γ t : [0, 1] → X be a distance realizer (i.e., a timelike geodesic) from α(s) to β(t). For each r ∈ [0, 1] let η r,t : [0, 1] → X be a distance realizer from x to γ t (r). We now claim that, for small enough t and some r, the direction [η r,t ] is a 4ε-midpoint of [α] and [β]. We denote the points by a = α(s), b = γ t (r), c = β(t). (See Figure  9 for a drawing.) Note that a ≪ b ≪ c.  We realize this in a comparison configuration as follows: Form a comparison triangle ∆x −ā−c− in L 2 (K − ), and analogously form a comparison triangle ∆x +ā+c+ in L 2 (K + ). As b ∈ [a, c] and denoting the corresponding points byb ± ∈ [ā ± ,c ± ], we conclude by using the curvature bounds, that τ (x, b) ≤τ (x − ,b − ) and τ (x, b) ≥τ (x + ,b + ), whereτ andτ denote the time separation functions on L 2 (K − ) and L 2 (K + ), respectively. By the law of cosines 2.4, we can translate this to statements on the angles and conclude that∡ Moreover, since the comparison situations are planar we have that We have by the triangle inequality for angles (Theorem 3.1) that and by (3), (4) that At this point we claim that ∡ To this end we compare the following four triangles and their angle at x: (iii) a comparison triangle ∆x +b+c+ in Minkowski space L 2 (0) of ∆x +b+c+ and the angle ∡ L 2 (0) x + (b + ,c + ) =: ω 3 , and (iv) the triangle ∆x +b+c+ in L 2 (K + ) with the angle ω 4 := ∡ Recall that all the points depend on r, t. Our aim is to have uniform comparison estimates on the differences of the angles ω 1 , . . . , ω 4 , while varying r and t. We claim that for all ε > 0 there is T > 0 such that for all 0 < t < T we have |ω 1 − ω 2 | < ε, |ω 2 − ω 3 | < ε and |ω 3 − ω 4 | < ε hold uniformly in r.
In total, we get |ω 1 −ω 4 | < 3ε for small t, so we have proven the claim for 4ε and can replace the ∡ (5), introducing only an error of 4ε. Then all the points are in the same comparison space and we have ∡ To sum up, we have where two ε come from the approximation of angles, and three ε comes from the difference of ∡ Note that this argument was independent of r. As τ is locally continuous, and we are considering it only in a comparison neighborhood, we get that ∡ x (α(s), γ t (r)) is a continuous function in r, with values 0 for r = 0 and∡ K + x (a, c) > ∡x(α,β) 2 for r = 1 (if ε is small enough). In particular, there is a value of r such that∡ . This then implies that As Σ + x is complete, [BBI01, Theorem 2.4.16] shows it is intrinsic. Finally, if additionally, X is locally compact, d-compatible, regularly localizable and future and past non-lingering, then the spaces of timelike directions D ± x are complete for every x ∈ X by Proposition 3.15 and hence intrinsic by the above.
As the previous Proposition 3.26 used two-sided curvature bounds we comment on the relationship of different curvature bounds, which was not spelled out so explicitly before. In particular, a non-trivial space cannot have arbitrary curvature bounds from below and above.
Lemma 3.27 (Relation of different curvature bounds). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space with timelike curvature bounded above by K + and below by K − . If there exists a non-degenerate timelike triangle satisfying timelike size bounds for K − and K + inside a neighborhood, which is simultaneously a comparison neighborhood for K − and for K + , then K + ≤ K − .
For (X, d, ≪, ≤, τ ) a Lorentzian pre-length space with timelike curvature bounded below by K − , then it also has timelike curvature bounded below by K for all K ≥ K − . And if (X, d, ≪, ≤, τ ) has timelike curvature bounded above by K + , it also has timelike curvature bounded above by K for all K ≤ K + .
In particular, for K + ≤ K − , L 2 (K − ) has timelike curvature bounded above by K + , and L 2 (K + ) has timelike curvature bounded below by K − .

Exponential and logarithmic map
The notion of a timelike tangent cone allows us to introduce an exponential and logarithmic map into our setting, which are indispensable geometric tools. For exponential and logarithmic maps in the metric case see [BBI01, p. 321ff.]. We first define the logarithmic map, which (locally) assigns points to elements in the timelike tangent cone encoding the time separation and the equivalence class of the connecting geodesic, i.e., the timelike direction.
Definition 3.28 (Logarithmic map). Let (X, d, ≪, ≤, τ ) be a locally uniquely timelike geodesic Lorentzian pre-length space, i.e., every point has a neighborhood U such that in U two timelike related points can be joined by a unique timelike geodesic contained in U . Let x ∈ X and let U be such a neighborhood of x. Then, for all y ∈ U with x ≪ y (y ≪ x) there exists an unique future (past) directed timelike geodesic γ from x to y contained in U . Then set log + x (y) := (τ (x, y), [γ]) ∈ T + x and log − x (y) := (τ (y, x), [γ]) ∈ T − x , respectively. So, the logarithmic map log ± x : U ∩ I ± (x) → T ± x maps points to elements of the tangent cone.
Note that the image log ± x (U ∩ I ± (x)) ⊆ T ± x is star-shaped with respect to (0, ⋆) ∈ T ± x in the sense that for an element (t, [γ]) ∈ log ± x (U ∩ I ± (x)) and λ ∈ [0, 1], also the scaled element (λt, At this point we introduce the exponential map, that projects elements of the timelike tangent cone down to points in the space. Moreover, it is the inverse of the logarithmic map. Definition 3.29 (Exponential map). Let (X, d, ≪, ≤, τ ) be a locally uniquely timelike geodesic Lorentzian pre-length space with timelike curvature bounded below, let x ∈ X and U as in Definition 3.28. Then the future/past exponential map exp ± x :

3.35]).
Remark 3.30. The exponential map is well-defined: Let α, β be two timelike geodesics starting at x, defined on [0, r], parametrized with respect to τ -arclength, and ending at different points α(r) = p = q = β(r). Then, α, β agree on a closed set and so there is a point α(s) ∈ I − (q) which is not on β, and by uniquely timelike geodesicness the concatenation of α| [0,s] and the unique timelike geodesic from α(s) to q is strictly shorter than β. Thus we get a non-degenerate triangle ∆xα(s)β(r). Then we conclude ∡ x (α, β) ≥∡ x (α(s), β(r)) > 0 by using the timelike curvature bound from below, Corollary 3.8 and Theorem 3.10. Thus [α] and [β] are different points in D + x . Moreover, as we restrict to the image of U under log ± x we haveγ(r) ∈ U .
As we assume that the space is locally uniquely timelike geodesic and has timelike curvature bounded from below the exponential map is well-defined. In principle, one could define an exponential map also for general Lorentzian pre-length spaces as set-valued maps, cf. the discussion in the metric case [BBI01, Rem. 9.1.43].
Remark 3.31. Let X ⊆ Y be two Lorentzian pre-length spaces included in each other (i.e. the relations and time separation function on X is just the restriction of those of Y ) with logarithm and exponential map defined, and let x ∈ X. We can consider the space of directions with respect to X and with respect to Y . Then (D + x ) X ⊆ (D + x ) Y and in particular, (T + x ) X is a subset of (T + x ) Y . We then have that the logarithmic map and exponential map with respect to Y are extensions of the logarithmic map and exponential map with respect to X.
Example 3.32. The image of the logarithm, hence the domain of the exponential map, need not be open, even in well-behaved spaces (as happens in metric geometry). Let (see Figure 10). As a subset of a spacetime, X is a Lorentzian pre-length space. It is a quite well-behaved space: It is geodesically convex and thus strictly intrinsic, τ is continuous, there exist no ≪-isolated points, it is a d-compatible set and distance realizing curves do not change their causal character, so the whole space is a regularly localizable neighborhood, and is even strongly regularly localizable. It is strongly causal and even causally simple (i.e., the relation ≤ is a closed subset of X × X), but not globally hyperbolic. It has timelike curvature bounded above and below by K = 0. However, the space of directions D + 0 at p = 0 is isometric to (0, ∞) and thus not complete, and the image log + 0 (I + (0)), which is contained in x n ∈ T + 0 corresponding to the points ( 1 − (1 − 1 n ) 2 , 1 n ) ∈ I + (0) in R 2 1 ) (on the considered circle, transferred to T + 0 by the logarithm on Minkowski space) converges to 0, but is not contained in log + 0 (I + (0)).
On the one hand, it is necessary that all curves are timelike when considering their angles. On the other hand, the limit curve theorem gives only causal limit curves in general. However, when the sequence of angles is bounded the limit curve has to be timelike, given that one has curvature bounded below.
Proof: First, by [KS18,Prop. 3.17] the limit curve γ is maximal and by [KS18,Thm. 3.18] it is either timelike or null. Indirectly assume that γ is null and without loss of generality we can assume that all curve (segments) are in one regularly localizable neighborhood, which is a comparison neighborhood as well. Let C > 0 be such that ∡ x (γ n , γ 0 ) ≤ C for all n ∈ N and let K ∈ R be the timelike curvature bound from below. Let t > 0 and s > 0 such that γ(s) ≪ γ 0 (t) and γ 0 is a distance realizer on [0, t]. Then for n large enough we have that γ n (s) ≪ γ 0 (t) and hence x (γ n (s), γ 0 (t)) =: ω n , by using Corollary 3.8 and Theorem 3.10. At this point we consider the triangle ∆ n := ∆xγ n (s)γ 0 (t), which has side-lengths a n := τ (x, γ n (s)), b := τ (x, γ 0 (t)) and c n := τ (γ n (s), γ 0 (t)). Note that a n → 0 and c n → τ (γ(s), γ 0 (t)) =: c ≤ b, by the reverse triangle inequality. For simplicity we assume that K = 0 -the other cases being analogous. By the law of cosines Lemma 2.4 for K = 0 and σ = −1 we conclude that If c < b, the enumerator stays positive in the limit, and the denominator converges to 0, yielding ω n → ∞, contradicting ω n < C for all n ∈ N. If c = b, this is a contradiction to X being regular (cf. [KS18,Thm. 3.18]): We get that the concatenation of γ from x to γ(s) with the null segment γ(s) to γ 0 (t), of overall length 0 + c has the same length as the distance realizer γ 0 | [0,t] , which goes from x to γ 0 (t) and which has the length b = c. Note that a maximal causal curve from γ(s) to γ 0 (t) exists by localizability and strong causality.
With the above result in hand we are able to establish that the exponential map is continuous.
Lemma 3.34 (Exponential map continuous). Let (X, d, ≪, ≤, τ ) be a locally uniquely timelike geodesic, globally hyperbolic, locally causally closed, regularly localizable and future or past non-lingering, respectively, Lorentzian pre-length space with timelike curvature bounded below. Then exp ± x is continuous away from 0 ∈ T ± x for all x ∈ X.
Proof: We only establish the future directed case, i.e., for exp + x . The past directed case is completely analogous. Let x ∈ X and U as in definition 3.28. By global hyperbolicity and local timelike geodesic connectedness, we can shrink U to be with compact closure. Let (t n , [γ n ]) ∈ log + x (I + (x) ∩ U ) converge to (t, [γ]), and let γ n and γ be defined up to parameter t n and t. Then by the limit curve theorem [KS18,Thm. 3.7], we can assume (without loss of generality) that (γ n ) n converges to a future directed causal curveγ. Note thatγ is non-constant by the non-lingering property (Definition 3.14). We have to show that [γ] = [γ] and to this end we need to establish thatγ is timelike. As [γ n ] → [γ] the sequence (∡ x (γ n , γ 0 )) n is bounded and because γ n →γ uniformly we obtain thatγ is timelike by Lemma 3.33. Now, by semicontinuity of angles, Proposition 3.11, we get ∡ x (γ,γ) ≤ lim n ∡ x (γ, γ n ) = 0, so [γ] = [γ] and we are done.
Similarly, we can show that the logarithm is continuous.
Lemma 3.35 (Logarithmic map continuous). Let (X, d, ≪, ≤, τ ) be a locally uniquely timelike geodesic, globally hyperbolic, locally causally closed, regularly localizable Lorentzian pre-length space with timelike curvature bounded above. Then log ± x is continuous for all x ∈ X.
Proof: We only establish the future directed case, i.e., for log + x . The past directed case is completely analogous. Let x ∈ X and U as in Definition 3.28, and by global hyperbolicity and local timelike geodesic connectedness we can assume without loss of generality that U is relatively compact and by shrinking U further that U is a regularly localizing neighborhood. Indirectly, assume that log + x is not continuous, so there is a sequence (y n ) n in U ∩ I + (x) with y n → y ∈ U ∩ I + (x) and log + x (y n ) → log + x (y). Thus, by definition, we have log(y n ) = (t n , [γ n ]) and log + x (y) = (t, [γ]), where t n = τ (x, y n ), t = τ (x, y), γ n are future directed timelike geodesics from x to y n and γ is a future directed timelike geodesic from x to y. As t n = τ (x, y n ) and τ is continuous (on U ), t n → t = τ (x, y). By assumption we know that γ n (t n ) = y n → y. By the limit curve theorem, we get a limit curveγ of a subsequence of (γ n ) n from x to y and we indirectly assume . As x ≪ y and X is chronological,γ is non-constant. By [KS18,Prop. 3.17]γ is also maximizing and still contained inŪ . From regular localizability and [KS18, Thm. 3.18] we conclude thatγ is timelike, hence by uniqueness of geodesics inŪ , γ andγ are reparametrizations of each other. Finally, by Proposition 3.11 we estimate Combining the last two results, we obtain Corollary 3.36. The exponential map exp ± x is a homeomorphism away from 0 for locally uniquely timelike geodesic, globally hyperbolic, locally causally closed, regularly localizable, future or past non-lingering Lorentzian prelength spaces with timelike curvature bounded above and below.

Angles between timelike curves of arbitrary time orientation
In this final section we consider angles between curves of different time orientation. This is needed to complete the characterization of (timelike) curvature bounds via K-angle monotonicity and for the triangle inequality of angles.
In fact, for a special case of the triangle inequality of angles, we need the following corollary of the straightening lemma for shoulder angles [AB08, Lemma 2.4] (see also [Kir18,Lemma 3
Then we have that ∡ (q, r) holds if and only if τ (m, q) < τ (m ′ , q ′ ) holds. Moreover, the same equivalence holds for the reversed inequalities and equality on one side holds if and only if equality holds on the other.
To this end let γ p ′ ,r ′ : [0, 1] → L 2 (K) be the timelike geodesic from p ′ to r ′ , then there is a parameter λ such that m ′ = γ p ′ ,r ′ (λ), which we calculate as τ (p,m)+τ (m,r) . Then the required inequality (8) translates to where we used the formula of the non-normalized angle including the two side-lengths, the cosh of the angle and the sign of the angle. This reduces to and similarly for the inequality reversed. Under this condition, the straightening Lemma yields that which in our terminology means that Now we note that ∆p ′ m ′ q ′ and ∆pmq have two equal sides (all but the [mq] side) and use the monotonicity statement in the law of cosines 2.5 (σ = −1, varying a short side) to get τ (m ′ , q ′ ) ≥ τ (m, q).
Finally, we establish the general case where q ≤ r. Let ε > 0 be small enough. We considerq ∈ [mq] with τ (q, q) = ε. Thenq ≪ r, so we can apply the above: Let the causal triangle∆ ′ = ∆p ′q′r′ in L 2 (K) with side-lengths τ (p ′ ,q ′ ) = τ (p,q), τ (q ′ ,r ′ ) = τ (q, r), τ (p ′ ,r ′ ) = τ (p, m)+ τ (m, r) and a point m ′ on the side [p ′r′ ] with τ (p ′ ,m ′ ) = τ (p, m). Thus we get that the same for flipped inequalities and equality on one side holds if and only if equality holds on the other. By continuity of τ and ∡ L 2 (K) we get as ε → 0 that or with reversed inequalities, so we still need to show strictness in either of these. For this, we consider what happens in case of equality.
If τ (m, q) = τ (m ′ , q ′ ), we get that ∆ 1 and ∆p ′ m ′ q ′ have the same sidelengths and ∆ 2 and ∆m ′ q ′ r ′ have the same side-lengths. Thus, angle additivity in L 2 (K) gives us that ∡ (q, r) we use angle additivity in L 2 (K) to get ∡ L 2 (K) m (p, r) = 0 and thus p, m, r lie on a geodesic. Consequently, p ′ = p, q ′ = q, r ′ = r, m ′ = m is actually a valid choice for ∆ ′ and m ′ and thus τ (m, q) = τ (m ′ , q ′ ).   This definition is complementary to [GKS19,Def. 4.5], where extendibility of geodesics was introduced. Thus we opted for calling it prolongation instead of extendibility to easily distinguish them.
Smooth spacetimes satisfy geodesic prolongation. However, smooth spacetimes with boundary do not satisfy geodesic prolongation: There, timelike geodesics which pass non-tangentially into the boundary have (half-)closed domains and cannot be extended. For details see the following example. has very nice properties. In fact, it has timelike curvature bounded below and above by 0, it is globally hyperbolic, causally path connected, strongly regularly localizable and it is non-lingering at each point. However, the distance realizer α(t) = (t, t 0 −t 2 ) is only defined on (−∞, t 0 ], has α(t 0 ) = (t 0 , 0) and cannot be extended as a geodesic. Thus H fails to satisfy geodesic prolongation at each point on the boundary, see Figure 12. Finally, we are in a position to establish the triangle inequality for angles for arbitrarily time oriented timelike curves. This complements Theorem 3.1.
(i) If α, β are future directed, γ is past directed and the angle ∡ x (α, γ) exists, we get the triangle inequality of angles (ii) If we assume that X is locally causally closed, has timelike curvature bounded below and satisfies geodesic prolongation, the triangle inequality of angles also holds for α, γ being future directed curves and β being a past directed geodesic.
See Figure 13 for an illustration of the two cases.
By time reversal and switching the roles of α and γ, this statement and that of Theorem 3.1 cover all possible combinations of time orientation.
Proof: (i): We restrict to a neighborhood of x such that τ is finite and continuous and all curve segments are contained in it. Let α, β be future directed, γ past directed and such that the angle ∡ x (α, γ) exists. By definition (and existence) of the angle ∡ x (α, γ) we have that for all ε > 0 there is a δ > 0 such that ∀r, s, t ∈ (0, δ) we have the upper bound whenever α(r) ≤ β(s) or the other way around (γ(t) ≪ β(s) and γ(t) ≪ α(r) are always satisfied, as γ is past directed).
We denote a = α(r), b = β(s) and c = γ(t). We let a ≤ b (fixing s > 0, there is an r such that this is satisfied).
We form a comparison situation in L 2 (0) as follows: Letx,ā,b,c containing comparison triangles for ∆cxb and ∆xab, where we letā andc on the same side of (the line given by the segment) [xb]. We consider the segment L = [cb] and its relation toā. We vary r and will get two cases eitherā lying on L is possible or if not, we approximate appropriately. For this, we use continuity ofā when varying r. For small r, it is clearā will lie below L (asx is below L). As r → sup α −1 (J − (b)) =: r + we have τ (a, b) → 0 and thusā has to approach ∂I − (b). As the part of ∂I − (b) lying on the same side of [xb] asc lies entirely above or on L except forb itself. Thus eitherā is eventually above L orā →b (which is on L).
First, we discuss the case whereā is eventually on or above L, so we have a value for r whereā is on L. Now we have the situation wherē c ≪ā ≤b all lie on a straight line, i.e.,τ (c,b) =τ (c,ā) +τ (ā,b). As we have τ (c, b) ≥ τ (c, a)+τ (a, b) and the side-lengths agree with the ones in the comparison triangles ∆cxb, ∆xāb except τ (c, a), we get thatτ (c,ā) ≥ τ (c, a). This yields for the angles that where the last inequality holds by the monotonicity of the law of cosines (Remark 2.5, varying the longest side,τ (c,ā) ≥ τ (c, a)). In the comparison configuration, i.e., in Minkowski spacetime we can use additivity of angles, and thus we get that which is what we need. We have proven that the triangle inequality of angles is not violated by more than 3ε. Now letā be under L for all r 6 , thenā →b as r → r + . We adapt the above proof to this case, and can assume that we have the above construction except for the location ofā. Now, we can take r close enough to r + such thatτ (c,ā) ≥ τ (c, a) −ε for some constantε > 0 to be specified later. This is possible asā →b, τ (c, b) ≥ τ (c, a) and τ is continuous. Using the law of cosines (2.4), we get where we used the equality of all distances except τ (c, a) andτ (c,ā) in the comparison situation. The difference is We have that τ (c, a),τ (c,ā) ≤ τ (c, b) stay bounded as r → r + and τ (x, a) is increasing and thus bounded away from 0. Thus the right-hand-side goes to 0 as r → r + and thusε → 0. As the inverse function of cosh is uniformly continuous, we find a small enoughε > 0 for each ε > 0 such that

Now we have a chain of inequalities as abovẽ
which gives analogous to the above that the triangle inequality of angles is not violated by more than 4ε.
(ii): At this point we assume that X is locally causally closed, has timelike curvature bounded below by K ∈ R and satisfies geodesic prolongation. Let α, γ be future directed timelike curves and β a past directed geodesic.
First, we reduce this to the case where β and γ join up to a geodesic. By geodesic prolongation, we can extend β to the future of x by a future directed geodesicβ. By Lemma 3.4.(ii), we have that ∡ x (β,β) = 0. We claim that the following triangle inequalities of angles hold Note these claimed triangle inequalities are just like the desired triangle inequality, i.e., a future-past-future case. Assuming this, we get the desired triangle inequality where the first inequality is by the future-only triangle inequality of angles, i.e., Theorem 3.1.
So we are left with (without loss of generality) establishing ∡ x (α, β) ≥ ∡ x (α,β). Let U be a comparison neighborhood of x, and by strong causality we can assume that all the points and geodesic segments are contained in U . Let ε > 0 and choose some parameters r, s, t > 0 small enough such that where we used Proposition 2.14 to replace the Minkowski comparison angle with the K-comparison angle and without loss of generality choose the time orientation in the third line (the other case being analogous). We denote a := α(r), b := β(s) and c :=β(t). We now construct the situation of the straightening lemma 4.1: We form comparison triangles ∆xāc for ∆xac and ∆bxā for ∆bxa in L 2 (K) such that they share the side [xā] andb,c lie on different sides of (the line extending) x) +τ (x,c), as required by the straightening lemma. These triangles realize the K-comparison angles ∡ a). Let ∆ā ′b′c′ be a comparison triangle for ∆abc and At this point we use the lower curvature bound to obtain τ (x, a) = τ (x,ā) ≤τ (x ′ ,ā ′ ). Thus by the straightening lemma 4.1, we get that As this holds for all ε > 0 we conclude that ∡ x (α,β) ≤ ∡ x (α, β) and we are done. Example 4.6 (Necessity of the assumptions in the triangle inequality with mixed time orientations). If α, γ are future directed and β is past directed, the triangle inequality may fail as the following example shows. We consider a causal funnel (cf. [KS18,Ex. 3.19]) with a (Minkowski-)timelike curve β, i.e., X = β((−∞, 0]) ∪ J + (β(0)) in n-dimensional Minkowski spacetime R n 1 , and set x := β(0), see Figure 14. It exhibits timelike branching, say β branches into future directed timelike geodesics α and γ at x. Then α and γ have ∡ x (α, γ) > 0. However, as both the concatenation of β and α and the concatenation of β and γ is a geodesic, we have that ∡ x (α, β) = ∡ x (β, γ) = 0 by Lemma 3.4,(ii). Thus the triangle inequality of angles is not satisfied: 0 = ∡ x (α, β) + ∡ x (β, γ) ≥ ∡ x (α, γ). As a causal funnel has timelike curvature unbounded below, this shows the necessity of the timelike curvature bound below in Theorem 4.5.
At this point, we use the triangle inequality of angles to get an analogue of the metric fact that the angles along a geodesic add to π.
In [KS18,Thm. 4.12] we established that given a timelike lower curvature bound timelike distance realizers do not branch using either one of two additional hypotheses. The first being that the space is timelike locally uniquely geodesic and the second being that locally in X maximal future directed timelike curves are strictly longer than any future directed causal curve with the same endpoints that contains a null segment. These two hypotheses are only used to establish the existence of a non-degenerate timelike triangle. Using angles we were able to remove these two hypotheses and therefore establish that K-monotonicity comparison from below implies timelike non-branching. However, after the fact we were able to avoid the use of angles completely, and therefore make the result even stronger because we do not need local timelike geodesicness for applying Theorem 3.10, i.e., that a lower timelike curvature bound implies future K-monotonicity comparison from below. Thus the result obtained is now completely analogous to the metric case, see e.g. [Shi93,Lem. 2.4] where a lower bound on curvature (in the Alexandrov sense) implies non-branching of distance realizers.
Theorem 4.8 (Timelike non-branching). Let X be a strongly causal Lorentzian pre-length space with timelike curvature bounded below by some K ∈ R. Then timelike distance realizers cannot branch.
In the second case, there are s, t ∈ (0, B) such that, without loss of generality, x ≪ α(s) ≪ β(t) is a non-degenerate timelike triangle. Thus also for every r > 0 the timelike triangle p := α(−r) = β(−r) ≪ α(s) =: q ≪ β(t) =: z is non-degenerate. At this point we can argue as in the proof of [KS18,Thm. 4.12]. For the sake of completeness we include the argument here. Let∆ = ∆pqz be a comparison triangle of ∆pqz in L 2 (K). Letx 1 ∈ [p,q] correspond to x andx 2 ∈ [p,z] correspond to x as well, i.e., τ (p,x i ) = τ (p, x) = r for i = 1, 2. As∆ is non-degenerate we havex 1 ∈ [p,z]. Consequently,τ (x 1 ,z) <τ (x 2 ,z) as otherwise the broken geodesic going fromp tox 1 toz is as long as the unbroken geodesic [p,z]. Finally, we arrive at where we used the timelike curvature bound from below in the last inequality -a contradiction.

Monotonicity comparison
In this final subsection we introduce a version of K-monotonicity which holds for an arbitrary angle in a timelike geodesic triangle by using the signed angle ∡ K,S . The main result is that K-monotonicity is equivalent to curvature bounded by K.
Definition 4.9 (K-monotonicity). Let (X, d, ≪, ≤, τ ) be a Lorentzian prelength space and let K ∈ R. We say that X satisfies timelike K-monotonicity comparison from below (above) if every point in X possesses a neighborhood U such that (i) τ | U ×U is finite and continuous.
(ii) Whenever x, y ∈ U with x ≪ y, there exists a future-directed maximal timelike geodesic α in U from x to y. where (s, t) ∈ D precisely when α(s), β(t) are causally related. We require this to be monotonically increasing (decreasing) in s and t for timelike K-monotonicity comparison from below (above).
A direct consequence of K-monotonicity comparison is that it implies the existence of angles between timelike geodesics depending on monotonicity comparison from below or above and the time orientation of the geodesics. Lemma 4.10 (Monotonicity implies existence of angles between geodesics of arbitrary time orientation). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space. If X satisfies timelike K-monotonicity comparison from above or below, the angle between any two timelike geodesics starting at the same point x (both future or past directed or one future, one past directed) exists if it is finite 7 .
Moreover, in case X satisfies timelike K-monotonicity comparison from below and one of the two timelike geodesics is future directed and one is past directed (i.e., the σ = 1 case), the angle between them exists.
Similarly, in case X satisfies timelike K-monotonicity comparison from above and the two timelike geodesics are both future directed or both past directed (i.e., the σ = −1 case), the angle between them exists.

Proof:
Let U be a comparison neighborhood given by Definition 4.9 above and let α, β : [0, ε) → U be two future or past directed timelike geodesics with x := α(0) = β(0). Without loss of generality we can assume that either both are future directed or α is past directed and β is future directed. Let s > 0 and t > 0 such that α(s) and β(t) are causally related. Then by K-monotonicity comparison from below (or above) we get that θ(s, t) =∡ K,S x (α(s), β(t)) is monotone (decreasing if we have Kmonotonicity comparison from below, increasing if we have it from above), and so this sequence has a limit. By Corollary 2.14 this limit is ∡ S x (α, β), but we do not yet know it is finite. Now if σ = 1, we have 0 as a lower bound for all signed comparison angles, and if σ = −1, we have 0 as an upper bound. So the angles are finite (and thus exist) in two cases: In case of K-monotonicity comparison from below and σ = 1, and in case of K-monotonicity comparison from above and σ = −1.
Analogous to Corollary 3.8 a direct consequence of K-monotonicity comparison and Proposition 2.14 is that, given a curvature bound, K-comparison angles bound the angle between geodesics.
Corollary 4.11 (Monotonicity implies bound on signed angle of geodesics). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space that satisfies K-monotonicity comparison from below (above) for some K ∈ R. Then for any x ∈ X and α, β : [0, B] → X timelike geodesics with α(0) = β(0) = x one has that for all s, t ∈ [0, B] small enough (with respect to timelike size bounds for K) such that α(s) and β(t) are causally related.
Monotonicity comparison also directly influences the length of the third side in a comparison hinge.
Corollary 4.12 (Monotonicity comparison implies hinge comparison). Let (X, d, ≪, ≤, τ ) be a Lorentzian pre-length space that satisfies K-monotonicity comparison from below (above) for some K ∈ R. Let (α, β) be a hinge, i.e., x ∈ X and α : [0, A] → X, β : [0, B] → X timelike geodesics of arbitrary time orientation with α(0) = β(0) = x, which satisfies the timelike size bounds for K. Then one can form a comparison hinge (ᾱ,β) in L 2 (K) and we have Proof. We only establish the case of K-monotonicity comparison from below, in the other case the previous Corollary 4.11 yields a reversed inequality and so gives an analogous proof. We denote the points by x, a = α(A) and b = β(B). We first assume that they are causally related, say a ≤ b. We consider the comparison hinge (ᾱ,β) in L 2 (K), and consider it as a triangle∆ = ∆xāb, wherex :=ᾱ(0) =β(0), a :=ᾱ(A) andb :=β(B). We assumeā ≤b. Furthermore, we consider a comparison triangle∆ ′ = ∆x ′ā′b′ for ∆xab. These two triangles have two agreeing side-lengths, i.e., For the angles we have By the previous Corollary 4.11, we know that ∡ S x (α, β) ≤∡ K,S x (a, b). Now we distinguish the cases where the angle is at. If σ = 1, we have that ∡ , and the monotonicity statement of the law of cosines (Remark 2.5, varying the longest side) gives thatτ , and the monotonicity statement of the law of cosines (Remark 2.5, varying a non-longest side) gives that Now for the case that a and b are not causally related (thus σ = −1). In case of monotonicity comparison from above there is nothing to do, so let X satisfy monotonicity comparison from below. We need to establish that τ (ā,b) = 0. Let ε > 0 small enough, then s ε = sup{s ∈ [0, A] : τ (α(s), b) > ε} < A, so a ε := α(s ε ) is the last point on α with τ (a ε , b) = ε. Then as above we get a pointā ε in the comparison hinge, and the above argument yields that ε = τ (a ε , b) ≥τ (ā ε ,b) ≥τ (ā,b). Now let ε → 0 to getτ (ā,b) = 0. Now for the case thatā andb are not causally related (thus σ = −1). In case of monotonicity comparison from below there is nothing to do, so let X satisfy monotonicity comparison from above. We need to establish that a, b are not timelike related, so indirectly assume a ≪ b. We have two triangles with two equal side-lengths:∆ with a non-timelike side and∆ ′ which is timelike. Now the monotonicity statement of the extended law of cosines (Corollary 2.6) yields that the angle in the non-causal triangle is bigger, i.e., ∡ . But by monotonicity comparison from above we get that ∡ x (α, β) ≤∡ K x (a, b), i.e., ∡ We are now in a position to prove the main result of this final section. We establish that timelike curvature bounds via triangle comparison (Definition 1.20) are equivalent to monotonicity comparison (Definition 4.9).
Theorem 4.13 (Equivalence of triangle and monotonicity comparison). Let (X, d, ≪, ≤, τ ) be a locally strictly timelike geodesically connected Lorentzian pre-length space and let K ∈ R. Then X has timelike curvature bounded below (above) by K if and only if it satisfies K-monotonicity comparison from below (above).
Proof: The first two conditions to be satisfied in timelike triangle comparison and K-monotonicity comparison are the same under the condition of local timelike geodesic connectedness when shrinking U to be a strictly timelike geodesically connected neighborhood. Moreover, let U be a comparison neighborhood given either by the timelike triangle comparison or by K-monotonicity comparison.
However, in case of timelike curvature bounded above there is an issue if y ′ is not causally related toz ′ , so σ = −1. Then the monotonicity statement of the extended law of cosines (Corollary 2.6) gives α = ∡ , which is what we wanted to prove. In total, X satisfies K-monotonicity comparison from below (or above, respectively).
Second, let X satisfy K-monotonicity comparison from below (above), let ∆ = ∆xyz be a timelike geodesic triangle in U satisfying timelike size bounds for K and let∆ = ∆xȳz be a comparison triangle of ∆ in L 2 (K). Let p ∈ [x, y], q ∈ [x, z], the other cases work nearly analogously by considering the angles at the other vertices. Moreover, if p = x or q = x there is nothing to do so let p = x = q. First we assume that p and q are causally related, say p < q. Letp,q be corresponding points of p, q on∆, and we assumep ≤q. By assumption we have∡ K,S x (p, q) ≤∡ K,S x (y, z) (or ∡ K,S x (p, q) ≥∡ K,S x (y, z)). Let (x,p ′ ,q ′ ) be a comparison triangle of ∆xpq.
(p,q). Then the law of cosines (Remark 2.5) implies that τ (p, q) =τ (p ′ ,q ′ ) ≤τ (p,q) (or τ (p, q) ≥τ (p,q), respectively). Now we consider the case wherep andq not causally related, which can only happen if σ = −1. We need to exclude p ≪ q, so let indirectly p ≪ q. For the timelike curvature bounded above case, we need τ (p, q) ≥ τ (p,q) = 0 which is automatically satisfied. For the timelike curvature bounded below case, we need τ (p, q) ≤τ (p,q) = 0, i.e. we need to exclude this case. We compare the triangles ∆xpq and ∆xp ′q′ . Two of the sides are equal, and in the first triangle the other side is spacelike. Thus the monotonicity statement of the extended law of cosines (Corollary 2.6) gives us that∡ K x (y, z) = ∡ , in contradiction to K-monotonicity comparison from below (note σ = −1).
In case p and q are not causally related (σ = −1) we cannot form timelike triangles, so the main argument does not work. For the timelike curvature bounded below case, we need 0 = τ (p, q) ≤τ (p,q) which is automatically satisfied. For the timelike curvature bounded above case, we need 0 = τ (p, q) ≥τ (p,q), i.e.p ≪q. First assume U is a causally closed neighbourhood and we indirectly assume the weakerp ≤q. We get that there exists a pointp ∈ [x, y] withp ≤ q null related. We compare the triangles ∆(x,p, q) and ∆(x,p,q): we have τ (x,p) < τ (x, p), τ (p, q) = 0 ≤τ (p,q) and the longest side-length is equal. Thus the strict law of cosines monotonicity (Remark 2.5) gives that∡ K x (p, q) > ∡ L 2 (K) x (p,q), which contradicts the inequality given by K-monotonicity comparison from above, i.e., (p,q). Without U being a causally closed neighbourhood, one does not necessarily get a null related pointp but a sequencep n with τ (p n , q) ց 0. Indirectly assumingp ≪q, a limiting argument gives the claim.
In total, X has timelike curvature bounded below (above) by K.
We now give a proof of the fact that signed angles are semi-continuous, given a curvature bound by using the equivalence of timelike curvature bounds and K-monotonicity comparison established in Theorem 4.13. In particular, this implies the case of geodesics with the same time orientation, i.e., Proposition 3.11. In the metric setting the situation is of course easier as one Proposition 4.14 (Semi-continuity of angles). Let (X, d, ≪, ≤, τ ) be a locally strictly timelike geodesically connected Lorentzian pre-length space with timelike curvature bounded below (above). Then signed angles of timelike geodesics are upper (lower) semicontinuous, i.e., α, α n , β, β n : [0, B] → X future or past directed timelike geodesics at x with α n → α, β n → β point-  Figure 15: When, comparing with L 2 (K), the interior side is shorter than expected, the K-angle has to behave accordingly.
Proof: First, we use the fact that the timelike curvature bound implies the corresponding K-monotonicity comparison by Theorem 4.13. We will prove the case of timelike curvature bounded below by K ∈ R -the other case is completely analogous. For s, t > 0 such that α n (s) and β n (t) are causally related set θ n (s, t) := ∡ K,S x (α n (s), β n (t)) and analogously for s, t > 0 such that α(s) and β(t) are causally related we define θ(s, t) :=∡ K,S x (α(s), β(t)). As the side lengths of the (comparison) triangles converge we get θ n → θ pointwise if both are defined.
Without loss of generality we assume that β is future directed. To see that the comparison angles of the approximating sequence exist pick sequences s k , t k ց 0 with α(s k ) ≪ β(t k ). By the openness of the timelike relation ≪ (cf. [KS18,Prop. 2.13]) we get that α n (s k ) ≪ β n (t k ) for large enough n ∈ N (depending on k), so both θ and θ n are defined at (s k , t k ).
Thus Corollary 4.11 gives for n, k large enough that ∡ S x (α n , β n ) ≤ θ n (s k , t k ) .
Under weak additional assumptions we even get continuity of angles.
Proposition 4.15 (Continuity of angles). Let (X, d, ≪, ≤, τ ) be a locally strictly timelike geodesically connected and locally causally closed Lorentzian pre-length space. Then angles are continuous between geodesics in the following cases. Let x ∈ X and α, α n , β, β n : [0, B] → X future or past directed timelike geodesics starting at x with α n → α, β n → β pointwise (in particular, α n is future directed if and only if α is and similarly for β).
(i) The space X has timelike curvature bounded above.
(a) The geodesics α, β have the same time orientation or (b) if α, β have different time orientations we assume that X is strongly causal and the angles ∡ x (α, β), ∡ x (α, β n ) and ∡ x (α n , β n ) are finite for all n ∈ N.
(ii) The space X has timelike curvature bounded below, is strongly causal and satisfies geodesic prolongation.
Proof: In both cases the timelike curvature bound implies K-monotonicity comparison by Theorem 4.13 and all geodesic segments are assumed to lie in a K-monotonicity comparison neighborhood.
At this point we use the triangle inequality of angles (Theorem 4.5).

Conclusion
In this article we introduced hyperbolic angles into the setting of Lorentzian length spaces and characterized timelike curvature bounds by an angle monotonicity property. We also introduced a diversity of metric tools, and thereby furthering the development of the theory considerably. Moreover, these tools and methods have further applications in Lorentzian geometry and General Relativity: The most striking example is the recently established splitting theorem for Lorentzian length spaces with non-negative curvature in [BORS22]. The splitting theorem can be understood as a rigidity statement of the classical Hawking-Penrose singularity theorem [HP70] and as such is of fundamental importance in Einstein's theory of gravity.
Finally, let us point out several directions of future research. First, hyperbolic angles could be used to define coordinates in Lorentzian (pre-) length spaces with timelike curvature bounded below or above. Second, a natural continuation is to establish the variation formula and an angle comparison condition as in [BMdOS22] also for curvature bounded from above (and for curvature bounded from below by a non-zero K ∈ R for the former). Further, we expect angles to play a significant role in globalization theorems for curvature bounds and in a Lorentzian version of billiards (cf. [BFK98]) with possible applications to particle collisions.
where the is in the denominator get canceled by the is appearing due to sin(ix) = i sinh(x).
At this point, we establish the case where c = 0, i.e., σ = −1 and without loss of generality q ≪ p ≤ r with p, r null related. We consider the timelike geodesic α from q to p. For each ε > 0 we consider the point p ε := α(1 − ε), then τ (p ε , r) > 0 and τ (p ε , r) → 0 as ε → 0. We also have ∡ L 2 (K) q (p ε , r) = ∡ L 2 (K) q (p, r). We now apply the timelike case above to the timelike triangle q ≪ p ε ≪ r. We note that the side-lengths converge as follows: a ε = τ (q, p ε ) → a = τ (q, p), b = τ (q, r) is constant and c ε = τ (p ε , r) → 0. As the term containing ω stays finite, the law of cosines continues to hold in the limit ε → 0, giving the claimed equality.
For the monotonicities, we without loss of generality assume that K ∈ {−1, 0, 1}. We look at each side varying separately. In each case, we say that ω and one of the side-lengths depend on some additional variable t (i.e., we insert ω(t) and a(t) or b(t) or c(t), respectively, into the law of cosines equations) and take the derivative of the law of cosines equations. Then sgn(a ′ (t)) = sgn(ω ′ (t)) (or sgn(c ′ (t))) makes ω monotonically increasing in a (or c) and sgn(a ′ (t)) = − sgn(ω ′ (t)) monotonically decreasing.
If c is the longest side, σ = 1 and the signs of the factors are clear, we get sgn(a ′ ) = − sgn(ω ′ ). Otherwise σ = −1. If b is the longest side, it is clear that 2b cosh(ω) ≥ 2a and so sgn(a ′ ) = sgn(ω ′ ). If a is the longest side, we have to be more careful. We use the law of cosines formula and transform it to a 2 + b 2 = c 2 + 2ab cosh(ω) , = ω ′ (σ sinh(ω) sinh(a) sinh(b)) .
If c is the longest side, σ = 1 and the signs of the factors are clear, giving sgn(a ′ ) = − sgn(ω ′ ). Otherwise σ = −1. If b is the longest side, cosh(a) sinh(b) − sinh(a) cosh(b) = sinh(b − a) ≥ 0, thus sgn(a ′ ) = sgn(ω ′ ). If a is the longest side, we have to be more careful. We multiply the law of So the sign of the factor of a ′ is minus the sign of the left-hand-side here (0 < a < π, so − sin(a) < 0). Now we use the reverse triangle inequality b ≤ a − c and monotonicity of cos to get cos(a) cos(c) − cos(b) ≤ cos(a) cos(c) − cos(a−c) = − sin(a) sin(c) ≤ 0, thus the factor of a ′ is positive and sgn(a ′ ) = sgn(ω ′ ).
Proof. (of Lemma 2.8) The analyticity and order of convergence statements directly follow from inserting the power series of cos, sin, cosh and sinh. For (1), we consider the triangles ∆p 1 p 2 p 3 and ∆p 1 p 2 q: They share one side-length and the angle ω = ∡ L 2 (K) p 2 (p 1 , p 3 ) = ∡ L 2 (K) p 2 (p 1 , q). We only do the K = 0 case since the other cases work analogously. By the law of cosines 2.4, we get the two equations a 2 + (b + c) 2 = d 2 − 2a(b + c) cosh(ω) , a 2 + b 2 = x 2 − 2ab cosh(ω) .