# Comparison theorems for Lorentzian length spaces with lower timelike
curvature bounds

Waldemar Barrera, Luis Montes de Oca and Didier A. Solis

###### Abstract

In this article we introduce a notion of normalized angle for Lorentzian pre-
length spaces. This concept allows us to prove some equivalences to the
definition of timelike curvature bounds from below for Lorentzian pre-length
spaces. Specifically, we establish some comparison theorems known as the local
Lorentzian version of the Toponogov theorem and the Alexandrov convexity
property. Finally, as an application we obtain a first variation Formula for
non-negatively curved globally hyperbolic Lorentzian length spaces.

Keywords: Lorentzian length spaces, triangle comparison, spaces of bounded
curvature, first variation.

MSC Classification: 53C23, 53C50, 53C80.

##  1 Introduction

The beginning of the XXI century has brought a lot of excitement and whole new
perspectives both to Mathematical Relativity and Lorentzian geometry. In
particular, the first detection of gravitational waves [14] and the latest
observations of black holes [8] have boosted the interest of exploring
geometric tools that adapt well to non-smooth settings.

In the context of Lorentzian geometry, the search for an axiomatic approach to
the most remarkable aspects of relativity –such as causality— can be traced
back to the fundamental work of Kronheimer and Penrose [11]. In their
approach, the main features of causality theory could be established from a
small set of axioms rather than deduced from the smooth geometry structure of
spacetime. Thanks to Penrose’s insight, such idealizations have provided
effective foundations to deal with situations where smoothness is not
required, such as in the study of quantum gravity [17] and more recently in
the novel field of Lorentzian length spaces [13].

On the other hand, we have witnessed in the past decade a renewed interest for
synthetic geometric methods, arising from their extensive use in scenarios
where tools stemming from differential geometry are not available. Indeed,
several classical results from Riemannian geometry have been extended to the
more general scope of length spaces. Basically, a length space is a metric
space where the distance between any two points can be approached by the
length of curves joining them. Remarkably, for geodesic length spaces we are
able to define a synthetic notion of curvature by comparing geodesic triangles
with triangles in a suitable space form of constant curvature (see [6, 7, 15,
16] and references therein). The first attempts to establish a comparison
theory for Lorentzian manifolds dates back to the work of Harris in the proof
of Toponogov’s Splitting Theorem [10]. In their seminal work [13], Kunzinger
and Sämann developed a synthetic notion of (timelike) curvature bounds in a
Lorentzian non-smooth context and used it to explore the nature of
singularities. More recently, in [3] we find the first detailed study of
Lorentzian comparison theory, as well as further developments and techniques
in the context of Lorentzian pre-length spaces. Moreover, in [4] the main
analytic tools related to comparison theorems (such as properties of the
exponential map) are laid out, along with fundamental results pertaining
hyperbolic angles, most notably, the triangle inequality. In the present paper
we lay out the basic tools to deal with comparisons in Lorentzian length
spaces in a synthetic manner analog to the well established theory of
Alexandrov or CAT spaces, as well as some global results. In particular, we
discuss three equivalent notions of curvature bounds and a first variation
formula for Lorentzian length spaces bounded from below by $0$. We hope these
results will prove helpful in the endeavor of applying synthetic geometric
methods in Relativity.

The paper is organized as follows: in section 2 we set the main definitions
pertaining Lorentzian length spaces and fix the notation we will be using
throughout this work. In section 3 we recall the definition of bounded
timelike curvature. After revising in section 4 the notion of non-normalized
angle due to Alexander and Bishop [2], we establish the equivalence of
timelike curvature bounds and Alexandrov’s convexity property in section 5. On
section 6 we define the notion of angle in Lorentzian pre-length spaces and
use it to discuss the relation of timelike curvature bounds and the local
Lorentzian version of Toponogov’s property. Finally, in section 7 we prove a
global first variation formula for non-negative curvature bounded Lorentzian
length spaces.

##  2 Preliminaries

Throughout this section we will recall some basic notions in the context of
Lorentzian length spaces as defined in [13]. The first ingredient consist in
an axiomatic formulation of causality, close in spirit to the original
definition of causal spaces first proposed in [11].

######  Definition 1.

A _Lorentzian pre-length space_ is a quintuple $(X,d,\ll,\leq,\tau)$ where

  1. 1.

$(X,d)$ is a metric space,

  2. 2.

$\leq$ is a pre-order,

  3. 3.

$\ll$ is a transitive relation contained in $\leq$.

  4. 4.

$\tau:X\times X\to[0,\infty]$ is lower semi-continuous function satisfying

     * •

$\tau(x,z)\geq\tau(x,y)+\tau(y,z)$ for all $x\leq y\leq z$

     * •

$\tau(x,y)>0$ if and only if $x\ll y$.

If a pair of points satisfy $x\ll y$, ($x\leq y$) we say that $x$ and $y$ are
_chronologically_ (_causally_) _related_ , respectively. _Chronological_
(_causal_) _future_ and _past sets_ $I^{+}(x)$, $I^{-}(x)$ ($(J^{+}(y)$,
$J^{-}(y)$) are thus defined in the standard way. The function $\tau$ is
called a _time separation_ function.

Lorentzian pre-length spaces have just enough structure in order to establish
some of the most basic facts pertaining causality. Most notably, the so called
_push-up property_ (if $x\ll y\leq z$ or $x\leq y\ll z$ then $x\ll z$) and the
openness of the chronological sets $I^{\pm}(x)$.

A curve $\gamma:[a,b]\to X$ that is not constant on any subinterval of $[a,b]$
is called _future-directed timelike_ (_causal_) if $\gamma$ is locally
Lipschitz continuous with respect to $d$ and whenever $s,t\in[a,b]$ with $s<t$
then $\gamma(s)\ll\gamma(t)$ ($\gamma(s)\leq\gamma(t)$). Such curve is
_future-directed null_ if it is causal and no pair of points on the curve are
timelike related. Past directed curves are defined similarly.

In order to have a sensible notion of length of causal curves, we rely on the
time separation function. Thus, we define the $\tau-$_length_ of a future-
directed causal curve $\gamma$ as

$L_{\tau}(\gamma)=\inf\left\\{\displaystyle\sum_{i=0}^{n-1}\tau(\gamma(t_{i}),\gamma(t_{i+1})):a=t_{0}<t_{1}<\cdots<t_{n}=b,n\in\mathbb{N}\right\\}.$

where the infimum is taken over all possible partitions of $\gamma$. A future-
directed causal curve $\gamma$ is _maximal_ if
$L_{\tau}(\gamma)=\tau(\gamma(a),\gamma(b))$. Maximal curves are cornestone to
synthetic geometry, as they are the closest analogs to geodesics.

Essentially, a _Lorentzian length space_ $(X,d,\ll,\leq,\tau)$ is a Lorentzian
pre-length space with a local structure that resemble the one provided by
normal neighborhoods and whose time separation function can be recovered from
the $\tau$-length of curves. The former is achieved through the notions of
_localizing neighborhoods_ : that is, neighborhoods $\Omega_{x}$ around each
point $x\in X$ furnished with relations $\leq_{\Omega_{x}}$,
$\ll_{\Omega_{x}}$ and continuous functions
$\omega_{x}:\Omega_{x}\times\Omega_{x}\to[0,\infty)$ such that

  1. 1.

$(\Omega_{x},d|_{\Omega_{x}\times\Omega_{x}},\ll_{\Omega_{x}},\leq_{\Omega_{x}},\omega_{x})$
is a Lorentzian pre-length space.

  2. 2.

$I^{\pm}(y)\cap\Omega_{x}\neq\emptyset$, for all $y\in\Omega_{x}$.

  3. 3.

All causal curves contained in $\Omega_{x}$ have uniformly bounded $d$-length.

  4. 4.

For all $p\neq q\in\Omega_{x}$ with $p\leq q$ there exists a future causal
curve $\gamma_{pq}$ contained in $\Omega_{x}$ such that
$L_{\tau}(\gamma_{pq})=\omega_{x}(p,q)$ and whose $\tau$-length is maximal
among all future causal curves from $p$ to $q$ lying in $\Omega_{x}$.

The precise definition reads as follows (refer to Definition 3.22 of [13]).

######  Definition 2.

A Lorentzian pre-length space for which:

  1. 1.

Every point $x$ has a localizing neighborhood $\Omega_{x}$.

  2. 2.

Every point has a neighborhood in which the $\ll$ is closed.

  3. 3.

If $x\leq y$ then there exists a future causal curve from $x$ to $y$.

  4. 4.

$\tau(x,y)=\mathcal{T}(x,y)$, for all $x,y\in X$, where

$\mathcal{T}(x,y)=\sup\\{L_{\tau}(\gamma):\gamma\mbox{ is future-directed
causal curve from $x$ to $y$}\\}.$

is called a _Lorentzian length space_.

A causality theory for Lorentzian length spaces can be developed in a way that
resembles the classical theory for spacetimes. In particular, a causal
hierarchy can be established with the notion of global hyperbolicity at the
top. Just as in the classical smooth case, a causal Lorentzian length space is
_globally hyperbolic_ if the causal diamonds $J^{+}(x)\cap J^{-}(z)$ are
compact. In this case, the time separation function $\tau:X\times
X\to[0,\infty]$ is continuous and finite. Moreover, $(X,d,\ll,\leq,\tau)$
satisfies the Avez-Seifert property: for any pair of causally related points
$x\leq y$ there exists a maximal future causal curve $\gamma$ from $x$ to $y$
[1, 13].

##  3 Triangle Comparison

At the core of synthetic geometry is the notion of triangle comparison. In a
nutshell, the main idea is that curvature bounds can be recovered locally from
comparisons of the most basic geometric objects (lengths and angles) with
respect to those found in a two dimensional model space. As in the Alexandrov
case, geodesic triangle comparison is the main key to describe the curvature
in the Lorentzian context. This is achieved by looking at timelike triangles
in the Lorentzian space forms of constant sectional curvature $k$. We denote
these models by

$\mathbb{M}_{k}^{L}=\left\\{\begin{array}[]{ll}\mathbb{S}_{1}^{2}(r)&k=\frac{1}{r^{2}}\\\
\mathbb{R}_{1}^{2}&k=0\\\
\mathbb{H}_{1}^{2}(r)&k=-\frac{1}{r^{2}}\end{array}\right.,$

where $\mathbb{S}_{1}^{2}(r)$ is the simply connected cover of two-dimensional
de Sitter space, $\mathbb{R}_{1}^{2}$ is the two-dimensional Minkowski space
and $\mathbb{H}_{1}^{2}(r)$ is the simple connected cover of the two-
dimensional anti de Sitter space.

Notice that in all these cases there exist restrictions on the lengths of the
sides of a triangle akin to the triangle inequality in Euclidean geometry. In
fact, since we will be dealing with triangles whose sides are unique
maximizing timelike segments, the occurrence of conjugate points along
geodesics hinders the possibility of having such kind of triangles with
arbitrary long side lengths. Those restrictions are described in the
Realizability Lemma (Lemma 4.6 of [13] or Lemma 2.1 in [2]), and if the side
lengths of a triangle satisfy them, we would say that such triangle obey
_timelike size bounds for_ $k$. Roughly speaking, for $k=0$ the restriction is
given by the reverse triangle inequality, while for $k<0$ in addition we have
that the greatest side should be less than $\pi/\sqrt{-k}$. A timelike
geodesic triangle in a model space ${M}_{k}^{L}$ whose vertices satisfy $x\ll
y\ll z$ will be denoted by $\triangle xyz$.

######  Definition 3.

A _timelike geodesic triangle_ $(x,y,z)$ in a Lorentzian length space
$(X,d,\ll,\leq,\tau)$ is a triple of points in $X$ satisfying $x\ll y\ll z$
such that $\tau(x,z)<\infty$. Its _sides_ are maximal future-directed causal
curves $\alpha$ from $x$ to $y$, $\beta$ from $y$ to $z$ and $\gamma$ from $x$
to $z$, hence

$L_{\tau}(\alpha)=\tau(x,y),\quad L_{\tau}(\beta)=\tau(y,z),\quad
L_{\tau}(\gamma)=\tau(x,z).$

A _comparison triangle_ for the geodesic triangle $(x,y,z)$ is a triangle
$\triangle\bar{x}\bar{y}\bar{z}$ in a model space $\mathbb{M}_{k}^{L}$ with
geodesic segments joining them $\bar{\alpha}$, $\bar{\beta}$, $\bar{\gamma}$
whose lengths equal those of $\alpha$, $\beta$, $\gamma$,
respectively222Notice that comparison triangles are unique up to an isometry..
In other words

$\bar{\tau}(\bar{x},\bar{y})=\tau(x,y),\quad\bar{\tau}(\bar{y},\bar{z})=\tau(y,z),\quad\bar{\tau}(\bar{x},\bar{z})=\tau(x,z).$

where $\bar{\tau}$ is the time separation function in the model space
$\mathbb{M}_{k}^{L}$. If a point $q$ lies on a side of a timelike geodesic
triangle, we denote by $\bar{q}$ the point lying on the corresponding side of
the comparison triangle such that $\bar{\tau}(\bar{p},\bar{q})=\tau(p,q)$,
where $p\in\\{x,y\\}$ is the initial point of the side.

Figure 1: A timelike geodesic triangle and its comparison triangle in
$\mathbb{M}_{k}^{L}$

We now state the original definition of timelike curvature bound for
Lorentzian length spaces as established in [13].

######  Definition 4.

A Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is said to have _timelike
curvature bounded below by_ $k\in\mathbb{R}$ if around any $p\in X$ there
exists a neighborhood $U\ni p$ with the following properties:

  1. (i)

$\tau|_{U\times U}$ is finite and continuous.

  2. (ii)

For every $x\ll y$ there exists a causal curve $\alpha$ in $U$ with
$L_{\tau}(\alpha)=\tau(x,y)$.

  3. (iii)

For any timelike geodesic triangle $(x,y,z)$ in $U$, realized by maximal
curves $\alpha$, $\beta$, $\gamma$ whose side lengths satisfy timelike size
bounds for $k$ the following holds: if triangle
$\triangle\bar{x}\bar{y}\bar{z})$ is a comparison triangle for $(x,y,z)$ in
$\mathbb{M}_{k}^{L}$ realized by timelike geodesics $\bar{\alpha}$,
$\bar{\beta}$ and $\bar{\gamma}$, then whenever $p$, $q$ are points on the
sides of $(x,y,z)$ and $\bar{p},\bar{q}$ are the corresponding points in
$(\bar{x},\bar{y},\bar{z})$, we have

$\tau(p,q)\leq\bar{\tau}(\bar{p},\bar{q}).$

If under the same hypothesis the alternative inequality

$\tau(p,q)\geq\bar{\tau}(\bar{p},\bar{q}).$

holds, we will say that $(X,d,\ll,\leq,\tau)$ has _timelike curvature bounded
above by_ $k$. The neighborhood $U$ is called a _comparison neighborhood for_
$p$.

Figure 2: Timelike curvature bounded from below by $k$

In the interest of having a lighter presentation, we include the technical
aspects of Definition 4 as follows:

######  Definition 5.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space. We say that $x\in
X$ has a _compatible neighborhood_ $U$ if

  1. (i)

$\tau|_{U\times U}$ is finite and continuous.

  2. (ii)

For every $x\ll y$ there exists a causal curve $\alpha$ in $U$ with
$L_{\tau}(\alpha)=\tau(x,y)$.

######  Remark 1.

Let $U$ be a compatible neighborhood in a Lorentzian pre-length space
$(X,d,\ll,\leq,\tau)$.

  1. 1.

In virtue of Proposition 3.34, Remark 4.3 and Remark 4.8 in [13], any maximal
timelike curve in $U$ can parameterized by arc length. Furthermore, any
intermediate value of $\tau$ along $\alpha$, $\beta$ or $\gamma$ is attained.
In particular, all of these follow when $(X,d,\ll,\leq,\tau)$ is a globally
hyperbolic Lorentzian length space.

  2. 2.

Let $(x,y,z)$ be a timelike geodesic triangle in $U$, realized by maximal
causal curves $\alpha$, $\beta$, $\gamma$ whose side lengths satisfy timelike
size bounds for $k$. If we take points $p\in\alpha$, $q\in\beta$, $r\in\gamma$
then the sides of triangle $(p,y,q)$ also satisfy timelike size bounds for
$k$. Moreover, if $p\ll r$, then the sides of triangle $(x,p,r)$ satisfy
timelike size bounds for $k$ and so do the sides of triangle $(r,q,z)$
provided $r\ll q$.

As can be readily seen, Definition 4 formalizes the intuitive notion that in
the presence of positive (negative) timelike curvature, triangles look fatter
(thinner) than flat triangles. In fact, this formulation agrees with the well
known notions in metric geometry of Alexandrov (curvature bounded from below)
and CAT (curvature bounded from above) spaces. Moreover, it is consistent with
the definition of curvature bounds for semi-Riemannian manifolds proposed in
[2]. However, there is a catch. According to [2], a Lorentzian manifold having
curvature bounded from _below_ by $k$ while having timelike curvature bounded
from _below_ (as a Lorentzian length space), it has sectional curvature on
timelike planes bounded from _above_ by $k$. A similar statement holds if the
words below and above are interchanged.

Among the first applications of the synthetic notion of curvature in
Lorentzian length spaces described in [13] we find the description of
curvature singularities, a topic of great interest in the realm of Relativity.
A Lorentzian length space has _timelike curvature unbounded from below_
(_above_) if there exist a compatible neighborhood that fails to be a
comparison neighborhood for all $k\in\mathbb{R}$. While this definition can be
used to spot singularities (for instance, the interior region of Schwarszchild
spacetime is a Lorentzian length space with timelike curvature unbounded from
below), its use often requires comparisons on the large, which are at odds
with the intuitive local character of curvature. As an example, a timelike
funnel with timelike $\lambda$ has timelike curvature unbounded from below
(see Examples 3.19 and 4.21 in [13]), but both $I^{-}(p)$ and $I^{+}(q)$ are
flat open subsets of it, hence Lorentzian length spaces in their own right
with timelike curvature bounded —both from below and from above– by $0$.

Here we present a novel example of a globally hyperbolic Lorentzian length
space with arbitrary small $k$-compatible neighborhoods in which no comparison
is possible. Hence, in spite of being at the top of the causal ladder, any
open subset of it has unbounded timelike curvature both from above and from
below.

######  Example 1.

Let us consider $(X,d)=(\mathbb{R}^{2},d_{T})$ where $d_{T}$ is the taxicab
metric

$d_{T}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{1}-x_{2}|+|y_{1}-y_{2}|,$

and let the relations $\ll_{T}$, $\leq_{T}$ be the usual chronological and
causal relations in Minkowski space $\mathbb{R}^{2}_{1}$. Furthermore, for
$(x_{1},y_{1}),(x_{2},y_{2})\in\mathbb{R}^{2}$ we define

$\tau_{T}((x_{1},y_{1}),(x_{2},y_{2})=\left\\{\begin{array}[]{ll}y_{2}-y_{1}-|x_{2}-x_{1}|&\mbox{if
$(x_{1},y_{1})\leq(x_{2},y_{2})$}\\\ 0&\mbox{otherwise}\end{array}\right.$

We first show now that
$\mathbb{R}^{2,1}_{T}=(\mathbb{R}^{2},d_{T},\ll_{T},\leq_{T},\tau_{T})$
—dubbed _Lorentzian taxicab space_ — is a globally hyperbolic Lorentzian
length space.

A straightforward computation shows that $\tau_{T}$ satisfies the causal
properties of a time separation as described in Definition 1. Since $d_{T}$ is
equivalent to the standard Euclidean metric, the topology induced by $d_{T}$
is Euclidean, and as a consequence $d_{T}$ is continuous. Moreover, the class
of (Lipschitz) causal curves in both $\mathbb{R}^{2,1}_{T}$ and
$\mathbb{R}^{2}_{1}$ also coincide, which in turns implies that the causal
diamonds of $\mathbb{R}^{2,1}_{T}$ are just the standard causal diamonds of
$\mathbb{R}^{2}_{1}$. Compacity of the causal diamonds follows, and thus
$\mathbb{R}^{2,1}_{T}$ is a globally hyperbolic pre-length space.

Now we focus on the requirements of Definition 2. Global hyperbolicity implies
causal connectivity and causal closedness. Moreover, given any point
$(x,y)\in\mathbb{R}^{2,1}_{T}$ consider the $d_{T}$ open ball centered at
$(x,y)$

$\Omega_{(x,y)}=\\{(p,q)\in\mathbb{R}^{2}:|p-x|+|q-y|<1\\},$

and $\omega_{(x,y)}=\tau_{T}|_{\Omega_{(x,y)}}$. Conditions (1) and (2) of the
definition of a localizing neighborhood are immediate. In order to show (3),
let $\gamma:[a,b]\to\mathbb{R}^{1,1}_{T}$,
$\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t))$, be a future causal curve in
$\Omega_{(x,y)}$ and take a partition $a=t_{1}<t_{2}<\cdots<t_{N}=b$. Since
$\gamma_{2}(t_{i+1})-\gamma_{2}(t_{i})\geq|\gamma_{1}(t_{i})-\gamma_{1}(t_{i+1})|$
we have

$\begin{array}[]{rcl}\displaystyle\sum_{i=1}^{N-1}d_{T}(\gamma(t_{i}),\gamma(t_{i+1}))&=&\displaystyle\sum_{i=1}^{N-1}|\gamma_{1}(t_{i})-\gamma_{1}(t_{i+1})|+|\gamma_{2}(t_{i})-\gamma_{2}(t_{i+1})|\\\
&\leq&\displaystyle\sum_{i=1}^{N-1}2(\gamma_{2}(t_{i+1})-\gamma_{2}(t_{i}))\leq
4.\end{array}$

Thus, the $d_{T}$ arc-length of curves is bounded.

Even though $\mathbb{R}^{2}_{1}$ and $\mathbb{R}^{2,1}_{T}$ share topology and
casuality, their geodesic structures are rather different. As opposed to the
Minkowski case, the Lorentzian taxicab admits infinitely many maximal curves
joining any pair of causally related points. Indeed, given
$(x_{1},y_{2})\leq_{T}(x_{2},y_{2})$ take any future causal curve and a
partition as above. Further assume $\gamma_{1}$ is a monotone function. Thus

$\begin{array}[]{rcl}\displaystyle\sum_{i=1}^{N-1}\tau_{T}(\gamma(t_{i}),\gamma(t_{i+1}))&=&\displaystyle\sum_{i=1}^{N-1}\gamma_{2}(t_{i+1})-\gamma_{2}(t_{i})-|\gamma_{1}(t_{i+1})-\gamma_{1}(t_{i})|\\\
&=&\displaystyle\sum_{i=1}^{N-1}\gamma_{2}(t_{i+1})-\gamma_{2}(t_{i})-\displaystyle\sum_{i=1}^{N-1}\gamma_{1}(t_{i+1})-\gamma_{1}(t_{i})\\\
&=&\gamma_{2}(b)-\gamma_{2}(a)-(|\gamma_{1}(b)-\gamma_{1}(a)|)\\\
,&=&\tau_{T}((x_{1},y_{1}),(x_{2},y_{2}))\end{array}$

Hence $L_{\tau_{T}}(\gamma)=\tau_{T}((x_{1},y_{1}),(x_{2},y_{2}))$, and

$\tau_{T}((x_{1},y_{1}),(x_{2},y_{2}))=L_{\tau_{T}}(\gamma)\leq\mathcal{T}((x_{1},y_{1}),(x_{2},y_{2}))\leq\tau_{T}((x_{1},y_{1}),(x_{2},y_{2})).$

which shows that $\gamma$ is maximizing. As an immediate consequence,
condition (4) of localizing neighborhoods holds and also $\mathcal{T}=\tau$.
Thus $\mathbb{R}^{2,1}_{T}$ is a globally hyperbolic Lorentzian length space.

Figure 3: Infinitely many maximal causal curves joining causally related
points in $\mathbb{R}^{2,1}_{T}$.

We now apply directly the definition of timelike curvature bounds to show that
there exists arbitrary small neighborhoods of $\mathbb{R}^{2,1}_{T}$ with no
curvature bounds. We focus first at the case $k=0$. Let $\varepsilon>0$ and
consider the triangle $(x,y,z)$ in $\mathbb{R}^{2,1}_{T}$ whose vertices are
$x=(0,0)$, $y=(-2\varepsilon,3\varepsilon)$, $z=(\varepsilon,7\varepsilon)$,
and whose sides are the linear segments connecting them. Moreover, set the
comparison triangle $\triangle\overline{x}\overline{y}\overline{z}$ in
Minkowski space given by $\overline{x}=(0,0)$,
$\overline{y}=(\sqrt{10}\varepsilon,3\varepsilon)$ and
$\overline{z}=(0,6\varepsilon)$.

Further let $q=({\varepsilon}/{4},{13\varepsilon}/{2})$ and notice it belongs
to the segment joining $y$ with $z$. Its corresponding point in the comparison
triangle $\triangle\overline{x}\overline{y}\overline{z})$ is
$\overline{q}=(\sqrt{10}\varepsilon/4,{21\varepsilon}/{4})$. Then

$\tau_{T}(x,q)=\displaystyle\frac{25\varepsilon}{4}>\displaystyle\frac{\sqrt{431}\varepsilon}{4}=\overline{\tau}(\overline{x},\overline{q})$

On the other hand, set $q=(-{5\varepsilon}/{4},4\varepsilon)$ on the segment
from $y$ to $z$ and its corresponding point
$\overline{q}=({3\sqrt{10}\varepsilon}/{4},{15\varepsilon}/{4})$. Thus

$\tau_{T}(x,q)=\frac{11\varepsilon}{4}<\frac{\sqrt{135}\varepsilon}{4}=\overline{\tau}(\overline{x},\overline{q})$

Therefore, none of the curvature conditions of Definition 4 hold in a
neighborhood $U$ containing a triangle isometric to $(x,y,z)$.

We can use the same choice of triangle $(x,y,z)$ and points $p$, $q$ in order
to obtain similar inequalities in the model spaces $\mathbb{M}_{k}^{L}$ with
$k\neq 0$ provided that $\varepsilon$ is small enough so that $(x,y,z)$
satisfies timelike size bounds.

######  Remark 2.

The above example admits an straightforward generalization: consider a metric
space $(X,d_{X})$ and a Lorentzian pre-length space
$(Y,d_{Y},\ll_{Y},\leq_{Y},\tau_{Y})$ let us take the metric $d:(X\times
Y)\times(X\times Y)\to\mathbb{R}_{\geq 0}$ defined as

$d((a,b),(x,y))=d_{X}(a,x)+d_{Y}(b,y).$

Let us set $\ll$, $\leq$ and $\tau:(X\times Y)\times(X\times Y)\to[0,\infty]$
defined as follows:

  * •

$(a,b)\leq(x,y)$ if and only if $\tau_{Y}(b,y)\geq d_{X}(a,x)$ and $b\leq y$.

  * •

$(a,b)\ll(x,y)$ if and only if $\tau_{Y}(b,y)>d_{X}(a,x)$.

  * •

For $(a,b),(x,y)\in X\times Y$ we have

$\tau((a,b),(x,y))=\left\\{\begin{array}[]{ll}\tau_{Y}(b,y)-d_{X}(a,x)&\mbox{if
$(a,b)\leq(x,y)$}\\\ 0&\mbox{otherwise}\end{array}\right.$

As can be checked, the _taxicab Lorentzian product_ $(X\times
Y,d,\ll,\leq,\tau)$ is a Lorentzian pre-length space.

##  4 Non-normalized angles

According to Euclidean geometry, side lengths and angle measure are the
fundamental quantities associated to a triangle. Since the notion of timelike
curvature bounds involves length comparison, it is natural to ask if there are
alternative formulations involving angle measurements. This is indeed the case
for Alexandrov and CAT spaces. Moreover, in the context of semi-Riemannian
geometry, an affirmative answer is given in Proposition 2.1 of [2]. Refer to
[12] for a thorough analysis on the properties of non-normalized angles.

######  Definition 6.

Let $(M,\langle\cdot,\cdot\rangle)$ be a semi-Riemannian model space of
curvature $k$. For a geodesic triangle $\triangle xyz$ in $M$ with
$\alpha,\gamma:[0,1]\to M$ geodesics connecting $x$ with $y$, and $x$ with
$z$, respectively, we denote $\measuredangle
yxz=\langle\alpha^{\prime}(0),\gamma^{\prime}(0)\rangle$ and call it the _non-
normalized angle_ at $p$.

######  Remark 3.

In the scenario depicted above, choose for instance a point
$p=\alpha(\lambda)$, other than $x$ on the side $\alpha$. Then it follows

$\angle pxz=\lambda\angle yxz.$

Hence, $\angle pxz$ and $\angle yxz$ though not equal, only differ by the
scaling factor $\lambda$. Hence the use of the term non-normalized angles is
fully accurate.

In view of the above remark, we can relate the non-normalized angles when the
endpoints vary along the sides of a geodesic triangle. We state this relation
in the form of a lemma, which will be used often in the following results.

######  Lemma 1.

Let $\triangle xyz$ be a timelike geodesic triangle in a Lorentzian model
space $\mathbb{M}_{k}^{L}$ realized by maximal timelike curves
$\alpha:[0,a]\to\mathbb{M}_{k}^{L}$, $\beta:[0,b]\to\mathbb{M}_{k}^{L}$,
$\gamma:[0,c]\to\mathbb{M}_{k}^{L}$ whose side lengths satisfy timelike size
bounds for $k$. Then, for every $a_{0}\in[0,a]$, $b_{0}\in[0,b]$ and
$c_{0}\in[0,c]$ we have

  1. (a)

$\measuredangle\bar{\alpha}(a_{0})\bar{x}\bar{z}=\frac{a_{0}}{a}\measuredangle\bar{y}\bar{x}\bar{z}$
and
$\measuredangle\bar{y}\bar{x}\bar{\gamma}(c_{0})=\frac{c_{0}}{c}\measuredangle\bar{y}\bar{x}\bar{z}$.

  2. (b)

$\measuredangle\bar{\beta}(b_{0})\bar{y}\bar{x}=\frac{b_{0}}{b}\measuredangle\bar{z}\bar{y}\bar{x}$
and
$\measuredangle\bar{z}\bar{y}\bar{\alpha}(a_{0})=\frac{a-a_{0}}{a}\measuredangle\bar{z}\bar{y}\bar{x}$.

  3. (c)

$\measuredangle\bar{\gamma}(c_{0})\bar{z}\bar{y}=\frac{c-c_{0}}{c}\measuredangle\bar{x}\bar{z}\bar{y}$
and
$\measuredangle\bar{x}\bar{z}\bar{\beta}(b_{0})=\frac{b-b_{0}}{b}\measuredangle\bar{x}\bar{z}\bar{y}$.

Two of the main results pertaining the above notion are the Hinge Lemma (Lemma
2.2 in [2]) and the Straightening Lemma (Lemma 2.4 in [2]). We present these
lemmas in a context adapted to our ends. Notice that in its original
formulation, these results are stated using signed distances.

First, notice that the result below agrees completely with its basic Euclidean
counterpart.

######  Lemma 2 (Hinge Lemma for included angles).

Let $\triangle x_{1}y_{1}z_{1}$ be two timelike geodesic triangles in
$\mathbb{M}^{L}_{k}$ satisfying timelike curvature bounds for $k$.

  * •

Suppose $\bar{\tau}(x_{1},y_{1})=\bar{\tau}(x_{2},y_{2})$ and
$\bar{\tau}(y_{1},z_{1})=\bar{\tau}(y_{2},z_{2})$. Then
$\bar{\tau}(y_{1},z_{1})\leq\bar{\tau}(y_{2},z_{2})$ if and only if
$\measuredangle y_{1}x_{1}z_{1}\leq\measuredangle y_{2}x_{2}z_{2}$.

  * •

Suppose $\bar{\tau}(x_{1},y_{1})=\bar{\tau}(x_{2},y_{2})$ and
$\bar{\tau}(y_{1},z_{1})=\bar{\tau}(y_{2},z_{2})$. Then
$\bar{\tau}(x_{1},z_{1})\leq\bar{\tau}(x_{2},z_{2})$ if and only if
$\measuredangle x_{1}y_{1}z_{1}\leq\measuredangle x_{2}y_{2}z_{2}$.

  * •

Suppose $\bar{\tau}(x_{1},z_{1})=\bar{\tau}(x_{2},z_{2})$ and
$\bar{\tau}(y_{1},z_{1})=\bar{\tau}(y_{2},z_{2})$. Then
$\bar{\tau}(x_{1},y_{1})\leq\bar{\tau}(x_{2},y_{2})$ if and only if
$\measuredangle y_{1}z_{1}x_{1}\leq\measuredangle y_{2}z_{2}x_{2}$.

######  Lemma 3 (Hinge Lemma for shoulder angles).

Let $\triangle x_{1}y_{1}z_{1}$ be two timelike geodesic triangles in
$\mathbb{M}^{L}_{k}$ satisfying timelike curvature bounds for $k$.

  * •

Suppose $\overline{\tau}(x_{1},y_{1})=\overline{\tau}(x_{2},y_{2})$ and
$\overline{\tau}(x_{1},z_{1})=\overline{\tau}(x_{2},z_{2})$. If
$\overline{\tau}(y_{1},z_{1})\leq\overline{\tau}(y_{2},z_{2})$ then
$\measuredangle x_{2}y_{2}z_{2}\leq\measuredangle x_{1}y_{1}z_{1}$ or
$\measuredangle x_{2}z_{2}y_{2}\leq\measuredangle x_{1}z_{1}y_{1}$.

  * •

Suppose $\overline{\tau}(x_{1},y_{1})=\overline{\tau}(x_{2},y_{2})$ and
$\overline{\tau}(y_{1},z_{1})=\overline{\tau}(y_{2},z_{2})$. If
$\overline{\tau}(x_{1},z_{1})\leq\overline{\tau}(x_{2},z_{2})$ then
$\measuredangle y_{2}x_{2}z_{2}\leq\measuredangle y_{1}x_{1}z_{1}$ or
$\measuredangle y_{2}z_{2}x_{2}\leq\measuredangle y_{1}z_{1}x_{1}$.

  * •

Suppose $\overline{\tau}(x_{1},z_{1})=\overline{\tau}(x_{2},z_{2})$ and
$\overline{\tau}(y_{1},z_{1})=\overline{\tau}(y_{2},z_{2})$. If
$\overline{\tau}(x_{1},y_{1})\leq\overline{\tau}(x_{2},y_{2})$ then
$\measuredangle x_{2}y_{2}z_{2}\leq\measuredangle x_{1}y_{1}z_{1}$ or
$\measuredangle y_{2}x_{2}z_{2}\leq y_{1}x_{1}z_{1}$.

As a first application of the notion of non-normalized angles we prove an
improved version of timelike curvature bounds more suited for applications.
Namely, triangle comparison can be more easily performed when one of the
points $p$ or $q$ in Definition 4 agree with a vertex, while the other point
is chosen on its opposite side.

######  Proposition 4.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space and suppose that
for every point $X$ there exists a compatible neighborhood $U$ with the
following property: for any timelike geodesic triangle $(x,y,z)$ in $U$,
realized by maximal timelike curves $\alpha$, $\beta$, $\gamma$ whose lengths
satisfy timelike size bounds for $k$, whenever $p$ is a point on one side of
$(x,y,z)$ and $v\in\\{x,y,z\\}$ is the vertex opposite to it, then we have

  1. (i)

$\tau(v,p)\leq\bar{\tau}(\bar{v},\bar{p})$, if $\tau(v,p)>0$.

  2. (ii)

$\tau(p,v)\leq\bar{\tau}(\bar{p},\bar{v})$, if $\tau(p,v)>0$;

where $\triangle\bar{x}\bar{y}\bar{z}$ is a comparison triangle of $(x,y,z)$
in $\mathbb{M}_{k}^{L}$ with corresponding sides $\bar{\alpha}$,
$\bar{\beta}$, $\bar{\gamma}$. Then $(X,d,\ll,\leq,\tau)$ has timelike
curvature bounded below by $k$.

###### Proof.

We will prove that $U$ is a comparison neighborhood with respect to
$\mathbb{M}_{k}^{L}$. Thus, take two points $p$, $q$ on the sides of triangle
$(x,y,z)$, so $\bar{p}$, $\bar{q}$ are the corresponding points on triangle
$\triangle\bar{x}\bar{y}\bar{z}$.

We first focus on vertex $y$. Suppose $p\in\beta$ and $q\in\alpha$. Notice
that $0<\tau(q,p)<\overline{\tau}(\overline{q},\overline{p})$ and observe that
$x\ll y\ll p$, then the sides of triangle $(x,y,p)$ satisfy timelike size
bounds for $k$. Thus, let $\triangle x_{1}y_{1}p_{1}$ be a comparison triangle
in $\mathbb{M}_{k}^{L}$ for triangle $(x,y,p)$.

Figure 4: Triangle $(x,y,z)$ and comparison triangles
$\triangle\bar{x}\bar{y}\bar{z}$ and $\triangle x_{1}y_{1}p_{1}$.

By hypothesis we have $\tau(x,p)\leq\bar{\tau}(\bar{x},\bar{p})$. Then

$\bar{\tau}(x_{1},p_{1})=\tau(x,p)\leq\bar{\tau}(\bar{x},\bar{p}),$

and therefore by Lema 2 we have
$\measuredangle\bar{p}\bar{y}\bar{x}\geq\measuredangle p_{1}y_{1}x_{1}$. On
the other hand, using Lemma 1 we deduce that

$\measuredangle\bar{p}\bar{y}\bar{q}=\frac{\bar{\tau}(\bar{q},\bar{y})}{\bar{\tau}(\bar{x},\bar{y})}\measuredangle\bar{p}\bar{y}\bar{x}\
\mbox{and}\ \measuredangle
p_{1}y_{1}q_{1}=\displaystyle\frac{\overline{\tau}(q_{1},y_{1})}{\overline{\tau}(x_{1},y_{1})}\measuredangle
p_{1}x_{1}y_{1}.$

Since $\bar{\tau}(\bar{x},\bar{y})=\tau(x,y)=\bar{\tau}(x_{1},y_{1})$ and
$\bar{\tau}(\bar{q},\bar{y})=\tau(q,y)=\bar{\tau}(q_{1},y_{1})$ we conclude
$\measuredangle\bar{p}\bar{y}\bar{q}\geq\measuredangle p_{1}y_{1}q_{1}$.
Again, by Lemma 2 we have
$\bar{\tau}(q_{1},p_{1})\leq\bar{\tau}(\bar{q},\bar{p})$ and therefore

$\tau(q,p)\leq\bar{\tau}(q_{1},p_{1})\leq\bar{\tau}(\bar{q},\bar{p}).$

Now we look at vertex $x$. Suppose $p\in\alpha$ and $q\in\gamma$. If
$\tau(p,q)=0$ then $\tau(p,q)\leq\bar{\tau}(\bar{p},\bar{q})$. So take
$\tau(p,q)>0$, which implies $x\ll p\ll q$ and the sides of triangle $(x,q,p)$
satisfy timelike size bounds for $k$. Let $(x_{1},p_{1},z_{1})$ be a
comparison triangle for $(x,p,z)$ and $q_{1}$ the corresponding point for $q$
in triangle $(x_{1},p_{1},z_{1})$.

Figure 5: Triangle $(x,y,z)$ and comparison triangles
$\triangle\bar{x}\bar{y}\bar{z}$ and $\triangle x_{1}p_{1}z_{1}$.

Then $\bar{\tau}(p_{1},z_{1})=\tau(p,z)\leq\bar{\tau}(\bar{p},\bar{z})$, which
implies $\measuredangle
p_{1}x_{1}z_{1}\leq\measuredangle\bar{p}\bar{x}\bar{z}$ because of Lemma 2.
Following a similar argument as in vertex $y$ we show that $\measuredangle
p_{1}x_{1}q_{1}\leq\measuredangle\bar{p}\bar{x}\bar{q}$ and therefore
$\bar{\tau}(p_{1},q_{1})\leq\bar{\tau}(\overline{p},\bar{q})$, again by Lemma
2. On the other hand $(x,p,z)$ we have $\tau(p,q)\leq\bar{\tau}(p_{1},q_{1})$,
thus

$\tau(p,q)\leq\overline{\tau}(p_{1},q_{1})\leq\bar{\tau}(\bar{p},\bar{q}).$

Again, the case $\tau(q,p)=0$ is trivial. If $\tau(q,p)>0$ we have $q\ll p\ll
y$ and therefore this case follows an analog path as the case $\tau(p,q)>0$
just analyzed, where the point $y$ plays the role of $x$. Thus
$\tau(q,p)\leq\bar{\tau}(\bar{q},\bar{p})$.

Finally, notice that the analysis of vertex $z$ is completely analogous to the
one preformed on vertex $x$, thus completing the proof. ∎

##  5 Alexandrov’s convexity property

Let us recall that if two future directed timelike curves
$\alpha,\beta:[0,a]\to M$ in a Lorentzian manifold
$(M,\langle\cdot,\cdot\rangle)$ meet at a point $p=\alpha(0)=\beta(0)$, then
the hyperbolic angle $\varphi$ spanned by $\alpha$ and $\beta$ at $p$ is given
by the relation

$-\cosh\varphi=\frac{\langle\alpha^{\prime}(0),\beta^{\prime}(0)\rangle}{|\alpha^{\prime}(0)|\
|\beta^{\prime}(0)|}.$

In this section we provide analog formulations adapted to the context of
Lorentzian pre-length spaces. The main idea is to construct a function that
plays the same role as $-\cosh\varphi$, being its monotonicity the most
relevant feature for comparison purposes.

######  Definition 7.

Let $\triangle\bar{p}\bar{q}\bar{r}$ be a comparison triangle in
$\mathbb{M}_{k}^{L}$ for a timelike geodesic triangle $(p,q,r)$ in a
Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$. We define the _comparison
angles at_ $p$, $q$, $r$ by

$\widetilde{\measuredangle}_{k}rpq=\measuredangle\bar{r}\bar{p}\bar{q}\quad\widetilde{\measuredangle}_{k}pqr=\measuredangle\bar{p}\bar{q}\bar{r},\quad\widetilde{\measuredangle}_{k}qrp=\measuredangle\bar{q}\bar{r}\bar{p},$

respectively.

######  Definition 8.

Given a compatible neighborhood $U$ in a Lorentzian pre-length space
$(X,d,\ll,\leq,\tau)$, take a timelike geodesic triangle $(x,y,z)$ in $U$,
realized by maximal causal curves ${\alpha}:[0,a]\to X$, ${\beta}:[0,b]\to X$,
${\gamma}:[0,c]\to X$ satisfying timelike size bounds for $k$. We define the
_angle comparison functions_
$\theta_{\alpha,\gamma}^{k}:(0,a]\times(0,c]\to\mathbb{R}$,
$\theta_{\beta,\alpha}^{k}:(0,b]\times(0,a]\to\mathbb{R}$ and
$\theta_{\gamma,\beta}^{k}:(0,c]\times(0,b]\to\mathbb{R}$ as follows:

  1. 1.

$\theta_{\alpha,\gamma}^{k}(s,t)=\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{st}$,
provided $\alpha(s)\ll\gamma(t)$ or $\gamma(t)\ll\alpha(s)$.

  2. 2.

$\theta_{\gamma,\beta}^{k}(s,t)=\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t)}{st}$,
provided ${\gamma}^{*}(s)\ll{\beta}^{*}(t)$ or
${\beta}^{*}(t)\ll{\gamma}^{*}(s)$.

  3. 3.

$\theta_{\beta,\alpha}^{k}(s,t)=\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y{\alpha}^{*}(t)}{st}$,
for every $(s,t)\in(0,b]\times(0,a]$.

where $\alpha^{*}:[0,a]\to X$, $\alpha(t)=\alpha(a-t)$ denotes the reverse
curve of $\alpha$.

It is immediate for the definition that $\theta_{\beta,\alpha}^{k}(s,t)>0$ and
$\theta_{\alpha,\gamma}^{k}(s,t)<0$, $\theta_{\gamma,\beta}^{k}(s,t)<0$.

######  Remark 4.

Notice that because the relation $\ll$ is open and $\tau$ is continuous in
$U$, we can always find small enough $s,t$ such that $\alpha(s)\ll\gamma(t)$
or $\gamma(t)\ll\alpha(s)$ so the conditions in part (1) of the above
definition are met. The same applies for parts (2) and (3).

An straightforward computation shows that when we apply Definition 8 to a
geodesic triangle in $\mathbb{R}^{2}_{1}$, viewed as a Lorentzian pre-length
space, we obtain that the value of $\theta_{\alpha,\gamma}^{0}(s,t)$ is
constant —independent of $(s,t)$— and
$\cosh^{-1}(-\theta_{\alpha,\gamma}^{0})$ is precisely the hyperbolic angle
$\varphi$ described above. A similar scenario holds for the functions
$\theta_{\alpha,\gamma}^{0}$ and $\theta_{\beta,\alpha}^{0}$. The same is true
for the model spaces $\mathbb{M}_{k}^{L}$ with $k\neq 0$. Thus, these
functions are natural candidates as comparison functions for Lorentzian length
spaces. To be able to show their monoticity, a lemma is in order.

######  Lemma 5.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space and suppose it has
timelike curvature bounded below by $k$. Let $U$ be a comparison neighborhood.
Suppose that $(x,y,z)$ is a timelike geodesic triangle in $U$, realized by
maximal causal curves $\alpha$, $\beta$, $\gamma$ whose lengths satisfy
timelike size bounds for $k$.

  1. 1.

For every $(s,t)\in(0,a]\times(0,c]$ such that $\alpha(s)\ll\gamma(t)$ or
$\gamma(t)\ll\alpha(s)$ the following inequalities hold:

    1. (a)

If $\alpha(s)\ll\gamma(t)$, then for all $s\geq s^{\prime}$ and
$t^{\prime}\geq t$ we have

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{s}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)}{s^{\prime}}\
\ \mbox{and}\ \
\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t^{\prime})}{t^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{t},$

    2. (b)

If $\gamma(t)\ll\alpha(s)$, then for all $s^{\prime}\geq s$ and $t\geq
t^{\prime}$ we have

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)}{s^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{s}\
\ \mbox{and}\ \
\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{t}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t^{\prime})}{t^{\prime}},$

  2. 2.

For every $(s,t)\in(0,c]\times(0,b]$ such that
${\gamma}^{*}(s)\ll{\beta}^{*}(t)$ or ${\beta}^{*}(t)\ll{\gamma}^{*}(s)$ the
following inequalities hold:

    1. (a)

If ${\gamma}^{*}(s)\ll{\beta}^{*}(t)$, then for all $s\geq s^{\prime}$ and
$t^{\prime}\geq t$ we have

$\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t)}{s}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s^{\prime})z{\beta}^{*}(t)}{s^{\prime}}\
\ \mbox{and}\ \
\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t^{\prime})}{t^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t)}{t},$

    2. (b)

If ${\beta}^{*}(t)\ll{\gamma}^{*}(s)$, then for all $s^{\prime}\geq s$ and
$t\geq t^{\prime}$ we have

$\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s^{\prime})z{\beta}^{*}(t)}{s^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t)}{s}\
\ \mbox{and}\ \
\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t)}{t}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}{\gamma}^{*}(s)z{\beta}^{*}(t^{\prime})}{t^{\prime}},$

  3. 3.

For every $(s,t)\in(0,b]\times t\in(0,a]$ we have

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y{\alpha}^{*}(t)}{s}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s^{\prime})y{\alpha}^{*}(t)}{s^{\prime}}\
\ \mbox{and}\ \
\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y{\alpha}^{*}(t^{\prime})}{t^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y{\alpha}^{*}(t)}{t},$

for all $s\geq s^{\prime}$ and $t^{\prime}\geq t$.

###### Proof.

For (1a), take $s\geq s^{\prime}$, then
$x\ll\alpha(s^{\prime})\ll\alpha(s)\ll\gamma(t)$ and therefore the sides of
triangle $(x,\alpha(s^{\prime}),\gamma(t))$ satisfy timelike size bounds for
$k$. Let $\triangle x_{1}p_{1}q_{1})$ and $\triangle x_{2}p_{2}q_{2}$ be
comparison triangles for $(x,\alpha(s),\gamma(t))$ and
$(x,\alpha(s^{\prime}),\gamma(t))$, respectively.

Figure 6: Triangle $(x,y,z)$ and comparison triangles $\triangle
x_{1}p_{1}q_{1}$ and $\triangle x_{2}p_{2}q_{2}$.

Let $p$ be the corresponding point in triangle $\triangle x_{1}p_{1}q_{1}$ for
$\alpha(s^{\prime})$ and denote $\theta_{1}=\measuredangle p_{1}x_{1}q_{1}$,
$\theta_{2}=\measuredangle p_{2}x_{2}q_{2}$. Since $U$ is a comparison
neighborhood with respect to $\mathbb{M}_{k}^{L}$ we have

$\bar{\tau}(p,q_{1})\geq\tau(\alpha(s^{\prime}),\gamma(t))=\bar{\tau}(p_{2},q_{2}),$

thus by Lemma 2 we get $\measuredangle px_{1}q_{1}\geq\measuredangle
p_{2}x_{2}q_{2}=\theta_{2}$. By Lemma 1 we have
$\displaystyle\frac{s^{\prime}}{s}\theta_{1}=\measuredangle_{k}px_{1}q_{1}\geq\theta_{2}$.
This implies

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{s}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)}{s^{\prime}}.$

For part of (1b), fix $t^{\prime}\geq t$. Then
$x\ll\alpha(s)\ll\gamma(t)\ll\gamma(t^{\prime})$ and the sides of
$(x,\alpha(s),\gamma(t^{\prime}))$ satisfy timelike size bounds for $k$. Let
$\triangle x_{1}p_{1}q_{1}$ and $\triangle x_{2}p_{2}q_{2}$ be comparison
triangles for triangles $(x,\alpha(s),\gamma(t^{\prime}))$ and
$(x,\alpha(s),\gamma(t))$, respectively.

Figure 7: Triangle $(x,y,z)$ and comparison triangles $\triangle
x_{1}p_{1}q_{1}$ and $\triangle x_{2}p_{2}q_{2}$.

Denote by $p$ the corresponding point for $\gamma(t)$ in triangle $\triangle
x_{1}p_{1}q_{1}$ and take $\theta_{1}=\measuredangle p_{1}x_{1}q_{1}$,
$\theta_{2}=\measuredangle p_{2}x_{2}q_{2}$. Just like in the previous case we
have

$\bar{\tau}(p_{1},p)\geq\tau(\alpha(s),\gamma(t))=\bar{\tau}(p_{2},q_{2})$

and applying Lemma 2 we deduce $\measuredangle p_{1}x_{1}p\geq\measuredangle
p_{2}x_{2}q_{2}=\theta_{2}$. Notice
$\displaystyle\frac{t}{t^{\prime}}\theta_{1}=\measuredangle
p_{1}x_{1}p\geq\theta_{2}$ by Lemma 1, which implies

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t^{\prime})}{t^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{t}.$

Alternatively, if $\gamma(t)\ll\alpha(s)$, then
$\gamma(t)\ll\alpha(s^{\prime})$ for all $s^{\prime}\geq s$. Let us take the
comparison triangles $\triangle x_{1}q_{1}p_{1}$ and $\triangle
x_{2}q_{2}p_{2}$ for triangles $(x,\gamma(t),\alpha(s^{\prime}))$ and
$(x,\gamma(t),\alpha(s))$, respectively. Also, the point $p$ is the
corresponding one for $\alpha(s)$ in triangle $\triangle x_{1}q_{1}p_{1}$.
Thus by curvature conditions we have

$\overline{\tau}(q_{2},p_{2})=\tau(\gamma(t),\alpha(s))\leq\overline{\tau}(\gamma(t),\alpha(s^{\prime})).$

Therefore

$\frac{s}{s^{\prime}}\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)=\frac{s}{s^{\prime}}\measuredangle
p_{1}x_{1}q_{1}=\measuredangle px_{1}q_{1}\geq\measuredangle
p_{2}x_{2}q_{2}=\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t),$

and hence

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)}{s^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{s}.$

On the other hand, if $t\geq t^{\prime}$ then
$\gamma(t^{\prime})\ll\alpha(s)$. Choose comparison triangles $\triangle
x_{1}q_{1}p_{1}$ and $\triangle x_{2}q_{2}p_{2}$ for triangles
$(x,\gamma(t),\alpha(s))$ and $(x,\gamma(t^{\prime}),\alpha(s))$,
respectively. Now, the point $p$ on triangle $\triangle x_{1}q_{1}p_{1}$ the
corresponding point for $\gamma(t^{\prime})$. Since

$\overline{\tau}(q_{2},p_{2})=\tau(\gamma(t^{\prime}),\alpha(s))\leq\overline{\tau}(p,p_{1}),$

we obtain

$\frac{t^{\prime}}{t}\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)=\frac{t^{\prime}}{t}\measuredangle
p_{1}x_{1}q_{1}=\measuredangle p_{1}x_{1}p\geq\measuredangle
p_{2}x_{2}q_{2}=\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t^{\prime}).$

In conclusion

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{t}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t^{\prime})}{t^{\prime}}.$

Now, let us focus in cases (3a) and (3b). If $s\geq s^{\prime}$ then
${\alpha}^{*}(t)\ll\beta(s^{\prime})$. Let us take the comparison triangles
$\triangle p_{1}y_{1}q_{1}$ and $\triangle p_{2}y_{2}q_{2}$ for triangles
$({\alpha}^{*}(t),y,\beta(2))$ and $({\alpha}*(t),y,\beta(s^{\prime}))$,
respectively (here observe that $\tau({\alpha}^{*}(t),y)=t$ by definition).
Set $p$ the corresponding point for $\beta(s^{\prime})$ in triangle $\triangle
p_{1}y_{1}q_{1}$. Then

$\overline{\tau}(p_{2},q_{2})=\tau({\alpha}^{*}(t),\beta(s^{\prime}))\leq\overline{\tau}(p_{1},p),$

it means

$\frac{s^{\prime}}{s}\widetilde{\measuredangle}\beta(s)y{\alpha}^{*}(t)=\frac{s^{\prime}}{s}\measuredangle
p_{1}y_{1}q_{1}=\measuredangle p_{1}y_{1}p\geq\measuredangle
p_{2}y_{2}q_{2}=\widetilde{\measuredangle}\beta(s^{\prime})y{\alpha}^{*}(t).$

Hence

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y{\alpha}^{*}(t)}{s}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s^{\prime})y{\alpha}^{*}(t)}{s^{\prime}}.$

In case that $t^{\prime}\geq t$ we take the comparison triangles $\triangle
p_{1}y_{1}q_{1}$ and $\triangle p_{2}y_{2}q_{2}$ for triangles
$({\alpha}^{*}(t^{\prime}),y,\beta(s))$ and $(\alpha(t),y,\beta(s))$,
respectively (here $\tau({\alpha}^{*}(t^{\prime}),y)=t^{\prime}$ and
$\tau({\alpha}^{*}(t),y)=t$). If $p$ is the corresponding point for
$\alpha(t)$ in triangle $\triangle p_{1}y_{1}q_{1}$, then

$\overline{\tau}(p_{2},q_{2})=\tau({\alpha}^{*}(t),\beta(s))\leq\overline{\tau}(p,q_{1}).$

This last inequality implies

$\frac{t}{t^{\prime}}\widetilde{\measuredangle}{\alpha}^{*}(t^{\prime})y\beta(s)=\frac{t}{t^{\prime}}\measuredangle
p_{1}y_{1}q_{1}=\measuredangle py_{1}q_{1}\geq\measuredangle
p_{2}y_{2}q_{2}=\widetilde{\measuredangle}{\alpha}^{*}(t)y\beta(s),$

thus

$\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y\tilde{\alpha}(t^{\prime})}{t^{\prime}}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\beta(s)y\tilde{\alpha}(t)}{t}.$

Finally, cases (2a) and (2b) are analogous to (1a) and (1b) and the proof is
complete.

∎

In Alexandrov geometry the monotonicity of the angle comparison functions is
equivalent to the definition of curvature bounds (see for example Definition
4.3.1 of [7] or Section 2.2 of [16]). Hence, this monotonicity property is
termed as _the local version of the Alexandrov convexity_. We proceed to
establish a similar result in the Lorentzian context.

######  Theorem 6 (Angle monotonicity).

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space and suppose it has
timelike curvature bounded below by $k$. Let $U$ be a comparison neighborhood.
Suppose that $(x,y,z)$ is a timelike geodesic triangle in $U$, realized by
maximal causal curves $\alpha$, $\beta$, $\gamma$ whose lengths satisfy
timelike size bounds for $k$. For every $s,s^{\prime}\in(0,\tau(x,y)]$ and
$t,t^{\prime}\in(0,\tau(x,z)]$ such that $s^{\prime}\leq s$, $t^{\prime}\leq
t$ and

  1. 1.

$\alpha(s)\ll\gamma(t)$ or $\gamma(t)\ll\alpha(s)$,

  2. 2.

$\alpha(s^{\prime})\ll\gamma(t^{\prime})$ or
$\gamma(t^{\prime})\ll\alpha(s^{\prime})$,

we have the following monotonicity condition

$\theta^{k}_{\alpha,\gamma}(s^{\prime},t^{\prime})\leq\theta^{k}_{\alpha,\gamma}(s,t).$

Similar inequalities apply for functions $\theta^{k}_{\gamma,\beta}$ and
$\theta^{k}_{\beta,\alpha}$.

###### Proof.

Here we will deal with the case when $\alpha(s)\ll\gamma(t)$ and
$\alpha(s^{\prime})\ll\gamma(t^{\prime})$ and similar ideas apply for the
other cases. Thus, using Proposition 5 we get

$t\theta^{k}_{\alpha,\gamma}(s,t)=\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{s}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)}{s^{\prime}}=t\theta^{k}_{\alpha,\gamma}(s^{\prime},t),$

which implies
$\theta^{k}_{\alpha,\gamma}(s,t)\geq\theta^{k}_{\alpha,\gamma}(s^{\prime},t)$.
On the other hand

$s^{\prime}\theta^{k}_{\alpha,\gamma}(s^{\prime},t)=\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t)}{t}\geq\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s^{\prime})x\gamma(t^{\prime})}{t^{\prime}}=s^{\prime}\theta^{k}_{\alpha,\gamma}(s^{\prime},t^{\prime}),$

and therefore
$\theta^{k}_{\alpha,\gamma}(s^{\prime},t)\geq\theta^{k}_{\alpha,\gamma}(s^{\prime},t^{\prime})$.
Finally

$\theta^{k}_{\alpha,\gamma}(s,t)\geq\theta^{k}_{\alpha,\gamma}(s^{\prime},t)\geq\theta^{k}_{\alpha,\gamma}(s^{\prime},t^{\prime}).$

∎

As in the Alexandrov case, by assuming the monotonicity of
$\theta^{k}_{\alpha,\gamma}$, $\theta^{k}_{\gamma,\beta}$
$\theta^{k}_{\beta,\alpha}$ we get the converse of Theorem 6.

######  Theorem 7.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space and $(x,y,z)$ be a
timelike geodesic triangle in a compatible neighborhood $U$, realized by
maximal timelike curves $\alpha$, $\beta$, $\gamma$ whose lengths satisfy
timelike size bounds for $k$. If $\theta^{k}_{\alpha,\gamma}$,
$\theta^{k}_{\gamma,\beta}$ and $\theta^{k}_{\beta,\alpha}$ are increasing
functions, then $(X,d,\ll,\leq,\tau)$ has timelike curvature bounded below by
$k$.

###### Proof.

We rely on Proposition 4 in order to prove that $(X,d,\ll,\leq,\tau)$ has
timelike curvature bounded below by $k$. Now suppose $p\in\alpha$, then $p\ll
y\ll z$ and $\bar{p}\in\bar{\alpha}$ so that
$\tau(x,p)=\bar{\tau}(\bar{x},\bar{p})=s$. Let $\triangle x_{1}p_{1}z_{1}$ be
a comparison triangle for $(x,p,z)$ in $\mathbb{M}_{k}^{L}$.

Figure 8: Triangle $(x,y,z)$ and comparison triangles
$\triangle\bar{x}\bar{y}\bar{z}$ and $\triangle x_{1}p_{1}z_{1}$.

Note that

$\displaystyle\frac{\measuredangle
p_{1}x_{1}z_{1}}{sc}=\displaystyle\frac{\widetilde{\measuredangle}_{k}pxz}{sc}=\theta^{k}_{\alpha,\gamma}(s,c),$

and due to the monotonicity of $\theta^{k}_{\alpha,\gamma}$ we conclude

$\displaystyle\frac{\measuredangle
p_{1}x_{1}z_{1}}{sc}=\theta^{k}_{\alpha,\gamma}(s,c)\leq\theta^{k}_{\alpha,\gamma}(a,c)=\displaystyle\frac{\widetilde{\measuredangle}_{k}yxz}{ac}=\displaystyle\frac{\measuredangle\bar{y}\bar{x}\bar{z}}{ac},$

Therefore $\displaystyle\frac{\measuredangle
p_{1}x_{1}z_{1}}{sc}\leq\displaystyle\frac{\measuredangle\bar{y}\bar{x}\bar{z}}{ac}$.
On the other hand, by Lemma 1 it follows that

$\measuredangle\bar{y}\bar{x}\bar{z}=\frac{a}{s}\measuredangle\bar{p}\bar{x}\bar{z},$

thus

$\displaystyle\frac{\measuredangle
p_{1}x_{1}z_{1}}{sc}\leq\displaystyle\frac{\measuredangle\bar{y}\bar{x}\bar{z}}{ac}=\displaystyle\frac{\measuredangle\bar{p}\bar{x}\bar{z}}{sc},$

then $\measuredangle p_{1}x_{1}z_{1}\leq\measuredangle\bar{p}\bar{x}\bar{z}$.
Finally, applying Lemma 2 in $\mathbb{M}_{k}^{L}$ we get the desired
inequality $\tau(p,z)\leq\bar{\tau}(\bar{p},\bar{z})$. All the remaining cases
can be worked out similarly. ∎

##  6 Normalized angles in Lorentzian pre-length spaces

So far, we have been able to find suitable angle comparison functions in our
non-smooth context. Now we are ready to define the main notion of our work,
namely, normalized angles for Lorentzian pre-length spaces.

######  Definition 9.

Following the same assumptions of Definition 8 we set

$\measuredangle yxz=\displaystyle\lim_{s,t\to
0}\theta^{k}_{\alpha,\gamma}(s,t),\ \measuredangle
xyz=\displaystyle\lim_{s,t\to 0}\theta^{k}_{\beta,\alpha}(s,t)$ $\mbox{ and }\
\measuredangle xzy=\displaystyle\lim_{s,t\to
0}\theta^{k}_{\gamma,\beta}(s,t).$

######  Remark 5.

Notice that the limits for $\measuredangle yxz$ and $\measuredangle xzy$
described above need not exist, even in the presence of lower curvature
bounds, since in this case the limits may diverge to $-\infty$. On the other
hand, since $\theta^{k}_{\alpha,\beta}(s,t)$ is positive, monotonicity
guarantees that $\measuredangle xyz$ always exists.

Furthermore, as we will prove, all normalized angles exist when the vertices
of a geodesic triangle are not conjugate points along their sides. In other
words if in a timelike geodesic triangle the sides can be extended past their
vertices as maximizing timelike curves then the normalized angles exist (refer
to Proposition 13 below). In the next results we will not be assuming this
mild extra hypothesis, but rather require the existence of the normalized
angles.

As an immediate consequence of the definition of normalized angles we have a
property called the _local Lorentzian Toponogov property_.

######  Theorem 8.

For any comparison neighborhood $U$ in a Lorentzian pre-length space
$(X,d,\ll,\leq,\tau)$ with curvature bounded below by $k$ and any triangle
$(x,y,z)$ in $U$ we have

$\measuredangle
yxz\leq\displaystyle\frac{\widetilde{\measuredangle}_{k}yxz}{ac},\
\measuredangle
xyz\leq\displaystyle\frac{\widetilde{\measuredangle}_{k}xyz}{ab},\
\measuredangle
xzy\leq\displaystyle\frac{\widetilde{\measuredangle}_{k}xzy}{bc}$

Once we have a notion of normalized angle in a Lorentzian length space, we
show an adapted version of the hinge theorem.

######  Theorem 9.

Let $U$ be a comparison neighborhood of a Lorentzian pre-length space
$(X,d,\ll,\leq,\tau)$ with curvature bounded below by $k$. Let us consider a
timelike geodesic triangle $(x,y,z)$ in $U$, realized by maximal causal curves
$\alpha$, $\beta$, $\gamma$ whose lengths satisfy timelike size bounds for $k$
and such that $\measuredangle yxz$ is finite and
$\measuredangle\bar{y}\bar{x}\bar{z}=ac\measuredangle yxz$. Then, for any
$s,t\in(0,a]\times(0,c]$ satisfaying $\alpha(s)\ll\gamma(t)$ or
$\gamma(t)\ll\alpha(s)$, we have

$\tau(\alpha(s),\gamma(t))\geq\bar{\tau}(\bar{\alpha}(s),\bar{\gamma}(t)).$

###### Proof.

Let us take $(s,t)\in(0,a]\times(0,c]$ such that $\alpha(s)\ll\gamma(t)$. If
$\bar{\tau}(\bar{\alpha}(s),\bar{\gamma}(t))=0$ then we are done. Otherwise,
suppose $\bar{\tau}(\bar{\alpha}(s),\bar{\gamma}(t))>0$. Let
$trianglex_{1}y_{1}z_{1}$ be a comparison triangle for triangle
$(x,\alpha(s),\gamma(t))$. Applying Lemma 1 we get

$\measuredangle\bar{\alpha}(s)\bar{x}\bar{\gamma}(t)=\frac{st}{ac}\measuredangle\bar{y}\bar{x}\bar{z}=\frac{st}{ac}\left(ac\measuredangle
yxz\right)=st\measuredangle yxz$

Because of the definition of the normalized angle we have

$\measuredangle\bar{\alpha}(s)\bar{x}\bar{\gamma}(t)=st\measuredangle yxz\leq
st\theta^{k}_{\alpha,\gamma}(s,t)=st\left(\displaystyle\frac{\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)}{st}\right)=\widetilde{\measuredangle}_{k}\alpha(s)x\gamma(t)=\measuredangle
y_{1}x_{1}z_{1},$

therefore
$\measuredangle\bar{\alpha}(s)\bar{x}\bar{\gamma}(t)\leq\measuredangle
y_{1}x_{1}z_{1}$. By Lemma 2 we conclude
$\bar{\tau}(y_{1},z_{1})\geq\bar{\tau}(\bar{\alpha}(s),\bar{\gamma}(t))$. But
$\tau(\alpha(s),\gamma(t))=\bar{\tau}(y_{1},z_{1})$, thus

$\tau(\alpha(s),\gamma(t))\geq\bar{\tau}(\bar{\alpha}(s),\bar{\gamma}(t)).$

∎

For metric length spaces of bounded curvature the following fact about the sum
of adjacent angles is well known: for any geodesic segment $\gamma$ from $x$
to $y$, a point $p\in\gamma$ and a point $q\not\in\gamma$ we have

$\measuredangle xpq+\measuredangle qpy=\pi$ (1)

In the case of Lorentzian model spaces of constant curvature we have the next
lemma. A detailed proof can be found in [12].

######  Lemma 10.

Let us consider a triangle $\triangle pqr$ satisfying size bounds for $k$ in
$\mathbb{M}_{k}^{L}$. Let $m$ be a point on the side $\beta$ joining $p$ with
$r$ and suppose this geodesic $\beta$ is parametrized by $[0,1]$, so there
exists $\lambda\in[0,1]$ such that $m=\beta(\lambda)$. Then

$(1-\lambda)\measuredangle qmp+\lambda\measuredangle rmq=0.$

One of the most important issues that need to be addressed in comparison
geometry consists on relating the measures of the angles of the comparison
triangles coming from the subdivision of a given triangle. The following
results deals with this situation.

######  Lemma 11 (Straightening lemma for shoulder angles).

Let $\triangle qpr$, $\triangle q_{1}p_{1}m_{1}$ and $\triangle
q_{2}m_{2}r_{2}$ be three timelike geodesic triangles satisfying curvature
bounds for $k$ in $\mathbb{M}_{k}^{L}$, and $m$ a point on the side joining
$p$ to $r$. Suppose $\overline{\tau}(q,p)=\overline{\tau}(q_{1},p_{1})$,
$\overline{\tau}(q,r)=\overline{\tau}(q_{2},r_{2})$,
$\overline{\tau}(p,m)=\overline{\tau}(p_{1},m_{1})$,
$\overline{\tau}(m,r)=\overline{\tau}(m_{2},r_{2})$ and
$\overline{\tau}(q_{1},m_{1})=\overline{\tau}(q_{2},m_{2})$. If

$\left(1-\frac{\overline{\tau}(p,m)}{\overline{\tau}(p,r)}\right)\widetilde{\measuredangle}q_{1}m_{1}p_{1}+\frac{\overline{\tau}(p,m)}{\overline{\tau}(p,r)}\widetilde{\measuredangle}q_{2}m_{2}r_{2}\geq
0,$

then

$\widetilde{\measuredangle}qpm\geq\widetilde{\measuredangle}q_{1}p_{1}m_{1}\
and\
\widetilde{\measuredangle}qrm\geq\widetilde{\measuredangle}q_{2}r_{2}m_{2}.$

######  Remark 6.

  1. 1.

The statement obtained by reversing all inequalities in the above Lemma holds
true as well.

  2. 2.

We have analogous results when the point $m$ is located in any of the
remaining sides of the triangle $m$.

We end our analysis on non-normalized angles by proving a converse of the
straightening lemma that will prove useful in establishing the existence of
angles in

######  Lemma 12.

Suppose $\triangle{q}{p}{r}$ is a triangle satisfying size bounds for $K$ in a
model space of curvature $K$. Let ${m}$ be a point on the side joining ${p}$
to ${r}$, and set $\lambda=\lambda_{{m}}\in[0,1]$ the affine parameter
corresponding to point $m$ when the side is parametrized by $[0,1]$. Let
$\triangle q_{1}p_{1}m_{1}$ and $\triangle q_{2}m_{2}r_{2}$ be triangles in
respective model spaces of curvature $K$, where
$\bar{\tau}(q_{1},m_{1})=\tau|(q_{2},m_{2})$,
$\overline{\tau}(q_{1},p_{1})=\overline{\tau}\tilde{q},\tilde{p})$,
$\overline{\tau}(q_{2},r_{2})=\overline{\tau}({q},{r})$,
$\overline{\tau}(p_{1},m_{1})=\overline{\tau}({p},{m})$ and
$\overline{\tau}(m_{2},r_{2})=\overline{\tau}({m},{r})$. If

$\measuredangle{q}{p}{m}\geq\measuredangle q_{1}p_{1}m_{1}\ \mbox{and}\
\measuredangle{q}{r}{m}\geq\measuredangle q_{2}r_{2}m_{2}$

then

$(1-\lambda)\measuredangle p_{1}m_{1}q_{1}+\lambda\measuredangle
r_{2}m_{2}q_{2}\geq 0.$

###### Proof.

Because $\measuredangle{q}{p}{m}\geq\measuredangle q_{1}p_{1}m_{1}$, then
applying Lemma 2 we have $\bar{\tau}(q,m)\leq\bar{\tau}(q_{1},m_{1})$. But by
Lemma 3, the shoulder angles in triangles $\triangle{q}{p}{m}$ and $\triangle
q_{1}p_{1}m_{1}$ satisfy $\measuredangle{p}{m}{q}\leq\measuredangle
p_{1}m_{1}q_{1}$. Similarly, $\measuredangle{q}{m}{r}\leq\measuredangle
q_{2}m_{2}r_{2}$. In conclusion

$(1-\lambda)\measuredangle p_{1}m_{1}q_{1}+\lambda\measuredangle
r_{2}m_{2}q_{2}\geq(1-\lambda)\measuredangle\tilde{p}\tilde{m}\tilde{q}+\lambda\measuredangle\tilde{q}\tilde{m}\tilde{r}=0.$

∎

We now establish the same inequality as above for Lorentzian pre-length spaces
with lower curvature bounds.

######  Proposition 13.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space with timelike
curvature bounded below by $k$. For a timelike geodesic triangle $(x,y,z)$
realized by maximal causal curves $\alpha$, $\beta$, $\gamma$ whose side
lengths satisfy timelike size bounds for $k$, fix a point $m$ in $\beta$ and a
maximal timelike curve $\sigma$ connecting $m$ with $x$. Then

$\measuredangle ymx+\measuredangle xmz\geq 0.$

Similar inequalities hold if $m$ is on $\alpha$ or $\gamma$.

###### Proof.

Let $U$ be a comparison neighborhood around $m$. Let the points $q$, $p$,
$r\in U$ lie on sides $\beta_{my}$, $\beta_{mz}$ and $\sigma$, where the
latter is a maximal geodesic connecting $m$ and $x$, such that $p\ll q$. Thus,
$(p,q,r)$ is a timelike geodesic triangle satisfying curvature bounds for $k$.
Furthermore, set $\tau(q,m)=s$, $\tau(m,r)=t$ and $\tau(p,m)=h$.

Figure 9: Timelike geodesic triangle $(x,y,z)$ in $X$

Let $\triangle p_{1}q_{1}r_{1}$, $\triangle p_{2}q_{2}m_{2}$ and $\triangle
p_{3}m_{3}r_{3}$ comparison triangles in $\mathbb{M}^{L}_{k}$ for triangles
$(q,p,r)$, $(p,q,m)$ and $(p,m,r)$, respectively. Let us denote
$\bar{\tau}(p_{1},m_{1})=h_{1}$.

Figure 10: Comparison triangles in $\mathbb{M}_{k}^{L}$

By the curvature condition we have $h_{1}\geq h$, thus by Lemma 2 we have
$\measuredangle p_{1}q_{1}m_{1}\geq\measuredangle p_{2}q_{2}m_{2}$ and
$\measuredangle p_{1}r_{1}m_{1}\geq\measuredangle p_{3}r_{3}m_{3}$. Now
applying Lemma 12 we deduce

$\begin{array}[]{rcl}\left(1-\frac{s}{s+t}\right)\measuredangle
q_{2}m_{2}p_{2}+\left(\frac{s}{s+t}\right)\measuredangle
r_{3}m_{3}p_{3}&\geq&0\\\
\left(\frac{t}{s+t}\right)(sh\theta^{k}_{\sigma,\beta_{my}}(h,s))+\left(\frac{s}{s+t}\right)(th\theta^{k}_{\beta_{mz},\sigma}(t,h))&\geq&0\\\
\theta^{k}_{\sigma,\beta_{my}}(h,s)+\theta^{k}_{\beta_{mz},\sigma}(t,h)&\geq&0\end{array}$

Since $\angle xmz=\lim\limits_{{t,h}\to 0}\theta^{k}_{\beta_{mz},\sigma}(t,h)$
is non-negative, the above inequality provides a lower bound for
$\theta^{k}_{\sigma,\beta_{my}}(h,s)$. Thus $\angle ymx$ exists also. Taking
the limit when $s,t,h$ go to $0$ we conclude the desired inequality

$\measuredangle ymx+\measuredangle xmz\geq 0.$

∎

######  Remark 7.

We emphasize that Proposition 13 is in fact a global result (as long as
timelike size bounds for $k$ are satisfied). We also would like to remark that
in virtue of this result, we can show that in any Lorentzian pre-length space
with lower timelike curvature bounds all three normalized angles in a geodesic
triangle satisfying timelike size bounds exist and are finite, provided that
its sides can be extended past their vertices as maximal timelike curves.

######  Remark 8.

When the normalized angles exist, their value is independent of the comparison
model space $\mathbb{M}_{k}^{L}$ in which the angle comparison functions
$\theta^{k}_{\alpha,\gamma}$, $\theta^{k}_{\gamma,\beta}$,
$\theta^{k}_{\beta,\alpha}$ are defined. Intuitively, this is expected in the
Riemannian setting since all Riemannian model spaces are conformally
equivalent and angles between curves are preserved under conformal
transformations. In practice, the proof relies heavily on the Law of cosines
for the model spaces. The same is true for Lorentzian spaceforms, thus
following closely the argument in Proposition 2.9 in [6] and using the
Lorentzian Laws of cosines (see for example [5], [9]) we can show the
aforementioned independence in the context of Lorentzian pre-length spaces as
well.

As it turns out, the inequality of Proposition 13 enables us to have a partial
converse to Theorem 8.

######  Theorem 14.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space such that for any
geodesic triangle $(x,y,z)$, realized by maximal timelike curves $\alpha$,
$\beta$, $\gamma$ in a compatible neighborhood $U$, all normalized angles
exist and satisfy

$\measuredangle
yxz\leq\displaystyle\frac{\widetilde{\measuredangle}_{k}yxz}{ac},\
\measuredangle
xyz\leq\displaystyle\frac{\widetilde{\measuredangle}_{k}xyz}{ab},\
\measuredangle
xzy\leq\displaystyle\frac{\widetilde{\measuredangle}_{k}xzy}{bc}$

Furthermore,

  1. 1.

If $p\in\beta$ then $\measuredangle ypx+\measuredangle xpz\geq 0$.

  2. 2.

If $p\in\alpha$ then $\measuredangle xpz+\measuredangle zpy\geq 0$.

  3. 3.

If $p\in\gamma$ with $y\ll p$ or $p\ll y$ then $\measuredangle
xpy+\measuredangle ypz\geq 0$.

Then $(X,d,\ll,\leq,\tau)$ has timelike curvature bounded below by $k$.

###### Proof.

We show that $U$ satisfies the conditions of Proposition 4. Here we handle the
case $p\in\beta$ and since the remaining ones are completely analogous. Just
to simplify notation set $b_{1}=\tau(y,p)$, $b_{2}=\tau(p,z)$ (thus
$b=b_{1}+b_{2}$), $h=\tau(x,p)$ and $\bar{h}=\bar{\tau}(\bar{x},\bar{p})$. We
want to prove that $\bar{h}\geq h$. Let $\triangle x_{1}y_{1}p_{1}$ and
$\triangle x_{2}p_{2}z_{2}$ comparison triangles in $\mathbb{M}_{k}^{L}$ for
triangles $(x,y,p)$ and $(x,p,z)$, respectively. Thus

$\begin{array}[]{rcl}\left(1-\frac{b_{1}}{b}\right)\measuredangle
y_{1}p_{1}x_{1}+\left(\frac{b_{1}}{b}\right)\measuredangle
x_{2}p_{2}y_{2}&=&\left(\frac{b_{2}}{b}\right)\measuredangle
y_{1}p_{1}x_{1}+\left(\frac{b_{1}}{b}\right)\measuredangle x_{2}p_{2}y_{2}\\\
&\geq&\left(\frac{b_{2}}{b}\right)\left(b_{1}h\measuredangle
ypx\right)+\left(\frac{b_{1}}{b}\right)\left(b_{2}h\measuredangle
xpz\right)\\\ &=&\left(\frac{b_{1}b_{2}h}{b}\right)\left(\measuredangle
ypx+\measuredangle xpz\right)\geq 0,\\\ \end{array}$

therefore $\left(1-\frac{b_{1}}{b}\right)\measuredangle
y_{1}p_{1}x_{1}+\left(\frac{b_{1}}{b}\right)\measuredangle x_{2}p_{2}y_{2}\geq
0$. Then by Lemma 11 we get
$\measuredangle\bar{x}\bar{y}\bar{p}\geq\measuredangle x_{1}y_{1}p_{1}$. Using
Lemma 2 we conclude $\bar{\tau}(\bar{x},\bar{p})\geq\bar{\tau}(x_{1},p_{1})$,
which implies $\bar{h}\geq h$. ∎

##  7 First variation for nonnegatively curved Lorentzian length spaces

This section is devoted to proving a first variation formula for globally
hyperbolic length spaces with timelike curvature bounded below by zero. We
first show a local version for pre-length spaces and then use it for the
global result. Throughout this section, we use repeatedily the Law of Cosines
in Minkowski space $\mathbb{R}^{2}_{1}$, thus we state it for ease of
reference.

######  Lemma 15 (Law of Cosines for $\mathbb{R}^{2}_{1}$).

Given a timelike geodesic triangle $\triangle xyz$ satisfying curvature bounds
in $\mathbb{R}^{2}_{1}$ with $x\ll y\ll z$ and $\tau_{\mathbb{R}^{2}_{1}}$ the
time separation function in $\mathbb{R}^{2}_{1}$ we have

$\begin{array}[]{rcl}\tau_{\mathbb{R}^{2}_{1}}(y,z)^{2}&=&\tau_{\mathbb{R}^{2}_{1}}(x,y)^{2}+\tau_{\mathbb{R}^{2}_{1}}(x,z)^{2}+2\measuredangle
yxz,\\\
\tau_{\mathbb{R}^{2}_{1}}(x,y)^{2}&=&\tau_{\mathbb{R}^{2}_{1}}(x,z)^{2}+\tau_{\mathbb{R}^{2}_{1}}(y,z)^{2}+2\measuredangle
xzy,\\\
\tau_{\mathbb{R}^{2}_{1}}(x,z)^{2}&=&\tau_{\mathbb{R}^{2}_{1}}(x,y)^{2}+\tau_{\mathbb{R}^{2}_{1}}(y,z)^{2}+2\measuredangle
zyx,\\\ \end{array}$

As an immediate consequence we have the following result

######  Lemma 16.

Let $\\{\triangle x_{i}y_{i}z_{i}\\}_{i=1}^{\infty}$ a sequence of timelike
geodesic triangles in $\mathbb{R}^{2}_{1}=\mathbb{M}^{L}_{0}$ such that

  1. 1.

$\tau_{\mathbb{R}^{2}_{1}}(x_{i},y_{i})=a$ and
$\tau_{\mathbb{R}^{2}_{1}}(x_{i},z_{i})=c$.

  2. 2.

$\displaystyle\lim_{i\to\infty}x_{i}=x,\displaystyle\lim_{i\to\infty}y_{i}=y,\displaystyle\lim_{i\to\infty}z_{i}=z$,

  3. 3.

$\lim\limits_{i\to\infty}\tau_{\mathbb{R}^{2}_{1}}(y_{i},z_{i})=\tau_{\mathbb{R}^{2}_{1}}(y,z)$.

Then $\displaystyle\lim_{i\to\infty}\measuredangle
y_{i}x_{i}z_{i}=\measuredangle yxz$ .

######  Proposition 17 (Semi-continuity of angles).

Let $(X,d,\ll,\leq,\tau,)$ a Lorentzian pre-length space with timelike
curvature bounded by $k=0$ and $U$ a comparison neighborhood with respect to
$\mathbb{R}^{2}_{1}$. Let $\\{(x_{i},y_{i},z_{i})\\}_{i=1}^{\infty}$ a
sequence of timelike geodesic triangles realized by maximal curves
$\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$ whose side lengths satisfy timelike
size bounds for $0$. Suppose

$\displaystyle\lim_{i\to\infty}x_{i}=x,\displaystyle\lim_{i\to\infty}y_{i}=y,\displaystyle\lim_{i\to\infty}z_{i}=z,\displaystyle\lim_{i\to\infty}\alpha_{i}=\alpha,\displaystyle\lim_{i\to\infty}\beta_{i}=\beta,\displaystyle\lim_{i\to\infty}\gamma_{i}=\gamma,$

and $\alpha_{i}\to\alpha$, $\beta_{i}\to\beta$, $\gamma_{i}\to\gamma$
uniformly where $\alpha$, $\beta$, $\gamma$ are maximal timelike curves that
satisfy timelike size bounds for $k=0$. Finally, suppose all normalized angles
are finite. Then

$\displaystyle\limsup_{i\to\infty}\measuredangle
y_{i}x_{i}z_{i}\leq\measuredangle yxz.$

###### Proof.

Fix $\varepsilon>0$. Since the definition of $\measuredangle yxz$, there
exists $\delta>0$ with the next property: for every $t,s<\delta$ such that
$\alpha(s)\ll\gamma(t)$ or $\gamma(t)\ll\alpha(s)$ we have

$\measuredangle yxz>\theta^{k}_{\alpha,\gamma}(s,t)-\varepsilon.$

Take $s,t<\delta$ with such property and denote $p=\alpha(s)$, $q=\gamma(t)$,
$p_{i}=\alpha_{i}(s)$ and $q_{i}=\gamma_{i}(t)$. If there exists a subsequence
$\\{p_{i_{j}},q_{i_{j}}\\}$ such that $\tau(p_{i_{j}},q_{i_{j}})=0$, then by
the continuity of $\tau$ in $U$ we would have $\tau(p,q)=0$, which is a
contradiction. Thus, we can find $N\in\mathbb{N}$ such that
$\tau(p_{i},q_{i})>0$ for all $i\geq N$. Let
$\triangle\bar{x}_{i}\bar{p}_{i}\bar{q}_{i}$ and
$\triangle\bar{x}\bar{p}\bar{q}$ comparison triangles for
$(x_{i},p_{i},q_{i})$ and $(x,p,q)$, respectively. Further
$\displaystyle\lim_{i\to\infty}\tau(p_{i},q_{i})=\tau(p,q)$. On the other hand
observe that
$\displaystyle\lim_{i\to\infty}\theta^{k}_{\alpha_{i},\gamma_{i}}(s,t)=\theta^{k}_{\alpha,\gamma}(s,t)$
since $\measuredangle p_{i}x_{i}q_{i}\to\measuredangle pxq$ as $i\to\infty$
because of Lemma 16. So there exists $M\in\mathbb{N}$ such that

$\theta^{k}_{\alpha,\gamma}(s,t)>\theta^{k}_{\alpha_{i},\gamma_{i}}(s,t)-\varepsilon,$

for all $i\geq M$. In conclusion, for all $i\geq\max\\{N,M\\}$ we have

$\measuredangle
yxz\geq\theta^{k}_{\alpha,\gamma}(s,t)-\varepsilon>\theta^{k}_{\alpha_{i},\gamma_{i}}(s,t)-2\varepsilon\geq\measuredangle
y_{i}x_{i}z_{i}-2\varepsilon.$

Hence

$\displaystyle\limsup_{i\to\infty}\measuredangle
y_{i}x_{i}z_{i}\leq\measuredangle yxz.$

∎

######  Proposition 18.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space with timelike
curvature bounded by $0$ and $U$ a comparison neighborhood with respect to
$\mathbb{R}^{2}_{1}$. Set a timelike geodesic triangle $(x,y,z)$ in $U$
realized by maximal curves $\alpha$, $\beta$, $\gamma$ whose side lengths
satisfy timelike size bounds for $0$. For every $0\leq t\leq\tau(x,y)$ we
define $\ell(t)=\tau(\alpha(t),z)$. Then

$\displaystyle\liminf_{t\to
0^{+}}\displaystyle\frac{\ell(t)-\ell(0)}{t}\geq\measuredangle yxz.$

###### Proof.

Let $c=\tau(x,z)$. We only consider the case when $s\in(0,c]$ satisfies
$\alpha(t)\ll\gamma(s)$, since the case $\gamma(s)\ll\alpha(t)$ is similar.
Let $\triangle x_{1}p_{1}q_{1}$ and $\triangle x_{2}p_{2}q_{2}$ be comparison
triangles for $(x,\alpha(t),z)$ and $(x,\alpha(t),\gamma(s))$.

Figure 11: Variation Formula.

Now observe that

$\theta_{\alpha,\gamma}(t,s)=\displaystyle\frac{\measuredangle
p_{2}x_{2}q_{2}}{ts}.,$

and using Lemma 1 we also have $\measuredangle
p_{1}x_{1}q_{1}=\frac{s}{c}\measuredangle p_{1}x_{1}z_{1}$. Thus, applying
Lemma 15 in triangle $\triangle p_{1}x_{1}z_{1}$ we get

$\begin{array}[]{rcl}\displaystyle\frac{\measuredangle
p_{1}x_{1}q_{1}}{ts}&=&\displaystyle\frac{\frac{s}{c}\measuredangle
p_{1}x_{1}z_{1}}{ts}\\\ &=&\displaystyle\frac{\measuredangle
p_{1}x_{1}z_{1}}{ct}\\\ &=&\displaystyle\frac{\ell(t)^{2}-c^{2}-t^{2}}{2ct}\\\
&=&\left(\displaystyle\frac{\ell(t)-c}{t}\right)\left(\displaystyle\frac{\ell(t)+c}{2c}\right)-\displaystyle\frac{t}{2c},\end{array}$

therefore $\displaystyle\frac{\measuredangle
p_{1}x_{1}q_{1}}{ts}=\left(\displaystyle\frac{\ell(t)-c}{t}\right)\left(\displaystyle\frac{\ell(t)+c}{2c}\right)-\displaystyle\frac{t}{2c}$.
On the other hand, by the curvature conditions we have

$\overline{\tau}(p_{2},q_{2})=\tau(\alpha(t),\gamma(s))\leq\overline{\tau}(p_{1},q_{1}),$

which means that $\measuredangle p_{2}x_{2}q_{2}\leq\measuredangle
p_{1}x_{1}q_{1}$ because of Lemma 2. In conclusion

$\begin{array}[]{rcl}\left(\displaystyle\frac{\ell(t)-c}{t}\right)\left(\displaystyle\frac{\ell(t)+c}{2c}\right)-\displaystyle\frac{t}{2c}&=&\displaystyle\frac{\measuredangle
p_{1}x_{1}q_{1}}{ts}\\\ &\geq&\displaystyle\frac{\measuredangle
p_{2}x_{2}q_{2}}{ts}\\\ &=&\theta_{\alpha,\gamma}(s,t)\\\
&\geq&\theta_{\alpha,\gamma}(s,t)-\displaystyle\frac{t}{2c}\\\
&\geq&\measuredangle yxz-\displaystyle\frac{t}{2c}.\end{array}$

Taking the limit when $t\to 0^{+}$ we have that $\displaystyle\frac{t}{2c}$
goes to $0$ and the value of $\ell(t)$ tends to $c$, thus
$\displaystyle\frac{\ell(t)+c}{2c}$ approximates $1$, thus establishing the
desired inequality. ∎

######  Remark 9.

Here we have to note two important things. First, in Proposition 18 we can
obtain similar inequalities for angles $\measuredangle xyz$ and $yzx$ using
the corresponding distance functions $\ell$. Second, observe that angle
$\measuredangle yxz$ is a function depending on the geodesics $\alpha$ and
$\gamma$, but the function $\ell$ does not depend on $\gamma$. So, if
$\max_{\gamma}(\measuredangle yxz)$ denotes the supremum of angles
$\measuredangle yxz$ over all the timelike maximal geodesics $\gamma$
connecting $x$ with $z$ we conclude

$\displaystyle\liminf_{t\to
0^{+}}\displaystyle\frac{\ell(t)-\ell(0)}{t}\geq\max_{\gamma}(\measuredangle
yxz).$

######  Theorem 19 (First variation formula).

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space with timelike
curvature bounded by $0$ and $U$ a comparison neighborhood. Set a timelike
geodesic triangle $(x,y,z)$ in $U$ realized by maximal curves $\alpha$,
$\beta$, $\gamma$ whose side lengths satisfy timelike size bounds for $0$. For
every $0\leq t\leq\tau(x,y)=a$ we define $\ell(t)=\tau(\alpha(t),z)$. Assume
that a sequence of future directed causal curves
$\\{\gamma_{i}\\}_{i=1}^{\infty}$ converges uniformly to $\gamma$, where
$\gamma_{i}(0)=\alpha(t_{i})$ for some sequence $\\{t_{i}\\}_{i=1}^{\infty}$,
$t_{i}\to 0$ as $i\to\infty$. Then

$\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell(t_{i})-\ell(0)}{t_{i}}=\measuredangle
yxz.$

###### Proof.

Set $p_{i}=\alpha(t_{i})$ and take a comparison triangle
$\triangle\tilde{x}_{i}\tilde{p}_{i}\tilde{z}_{i}$ for $(x,p_{i},z)$. Thus, by
Lemma 15 applied to triangle
$\triangle\tilde{x}_{i}\tilde{p}_{i}\tilde{z}_{i}$ we get

$\measuredangle
zp_{i}x\leq\displaystyle\frac{\measuredangle\tilde{z}_{i}\tilde{p}_{i}\tilde{x}_{i}}{\ell(t_{i})\cdot
t_{i}}=\displaystyle\frac{c^{2}-\ell(t_{i})^{2}-t_{i}^{2}}{2\ell(t_{i})\cdot
t_{i}},$

therefore

$\left(\displaystyle\frac{\ell(t_{i})-c}{t_{i}}\right)\left(\displaystyle\frac{\ell(t_{i})+c}{2\ell(t_{i})}\right)+\displaystyle\frac{t_{i}}{2\ell(t_{i})}\leq-\measuredangle
zp_{i}x.$

Because of Proposition 13 we conclude

$\left(\displaystyle\frac{\ell(t_{i})-c}{t_{i}}\right)\left(\displaystyle\frac{\ell(t_{i})+c}{2\ell(t_{i})}\right)+\displaystyle\frac{t_{i}}{2\ell(t_{i})}\leq-\measuredangle
zp_{i}x\leq\measuredangle yp_{i}z,$

hence

$\begin{array}[]{rcl}\displaystyle\limsup_{i\to\infty}\displaystyle\frac{\ell(t)-\ell(0)}{t_{i}}&=&\displaystyle\limsup_{i\to\infty}\left[\left(\displaystyle\frac{\ell(t_{i})-c}{t_{i}}\right)\left(\displaystyle\frac{\ell(t_{i})+c}{2\ell(t_{i})}\right)+\displaystyle\frac{t_{i}}{2\ell(t_{i})}\right]\\\
&\leq&\displaystyle\limsup_{i\to\infty}\measuredangle
yp_{i}z\leq\measuredangle yxz,\end{array}$

this last inequality holds because of Proposition 17. Finally, the result
follows from Proposition 18. ∎

######  Remark 10.

As in Remark 9 we are able to conclude that in fact

$\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell(t_{i})-\ell(0)}{t_{i}}=\max_{\gamma}{\measuredangle
yxz},$

because

$\displaystyle\limsup_{i\to\infty}\displaystyle\frac{\ell(t)-\ell(0)}{t_{i}}\leq\measuredangle
yxz\leq\max_{\gamma}(\measuredangle yxz).$

We emphasize the first variation formula holds for timelike geodesic triangles
of arbitrary size. Recall that in globally hyperbolic length spaces, the time
separation $\tau$ is continuous and any two causally related points can be
joined by a maximal causal curve.

######  Theorem 20.

Let $(X,d,\ll,\leq,\tau)$ be a globally hyperbolic Lorentzian length space
with timelike curvature bounded below by $0$. Set a timelike geodesic triangle
$(x,y,z)$ in $X$ realized by maximal curves $\alpha$, $\beta$, $\gamma$ whose
side lengths satisfy timelike size bounds for $0$. For every $0\leq
t\leq\tau(x,y)=a$ we define $\ell(t)=\tau(\alpha(t),z)$. Assume that a
sequence of future directed causal curves $\\{\gamma_{i}\\}_{i=1}^{\infty}$
converges uniformly to $\gamma$, where $\gamma_{i}(0)=\alpha(t_{i})$ for some
sequence $\\{t_{i}\\}_{i=1}^{\infty}$, $t_{i}\to 0$ as $i\to\infty$. Then

$\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell(t_{i})-\ell(0)}{t_{i}}=\measuredangle
yxz.$

###### Proof.

The proof is divided in two parts. First, let us take $U$ a comparison
neighborhood with respect to $\mathbb{R}^{2}_{1}$ around $x$. Fix to points
$y_{0}\in\alpha$ and $z_{0}\in\gamma$ in $U$ such that $y_{0}\ll z_{0}$ and
define $\ell_{0}(t)=\tau(\alpha(t),z_{0})$ for $0\leq t\leq\tau(x,y_{0})$.
Then by Theorem 19 we have

$\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell_{0}(t_{i})-\ell_{0}(0)}{t_{i}}=\measuredangle
y_{0}xz_{0}=\measuredangle yxz.$

Now observe that $\ell(t_{i})\geq\ell_{0}(t_{i})+\tau(z_{0},z)$, then

$\displaystyle\frac{\ell(t_{i})-\tau(x,z)}{t_{i}}\geq\displaystyle\frac{\ell_{0}(t_{i})-(\tau(x,z)-\tau(z_{0},z))}{t_{i}},$

and therefore

$\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell(t_{i})-\ell(0)}{t_{i}}\geq\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell_{0}(t_{i})-\ell_{0}(0)}{t_{i}}=\measuredangle
yxz.$

For the second part and using de continuity of $\tau$ with respect to $d$ take
a large enough $i$ such that $\alpha(t_{i})\in U$ and $z_{i}\in\gamma(i)$ with
$\tau(\alpha(t_{i}),z_{i})=r$ for a fix $r>0$. Let
$(\bar{x}_{i},\bar{p}_{i},\bar{z}_{i})$ a comparison triangle in
$\mathbb{R}^{2}_{1}$ for triangle $(x,\alpha(t_{i}),z_{i})$. Set
$c_{i}=\tau(x,z_{i})$, then by curvature conditions in $U$ and applying The
Law of Cosines in $\mathbb{R}^{2}_{1}$ we have

$\measuredangle
z_{i}\alpha(t_{i})x\leq\displaystyle\frac{\measuredangle\bar{x}_{i}\bar{p}_{i}\bar{z}_{i}}{rt_{i}}=\displaystyle\frac{c_{i}^{2}-r^{2}-t_{i}^{2}}{2rt_{i}},$

which is equivalent to

$\left(\displaystyle\frac{r-c_{i}}{t_{i}}\right)\left(\displaystyle\frac{r+c_{i}}{2r}\right)+\displaystyle\frac{t_{i}}{2r}\leq-\measuredangle
z_{i}\alpha(t_{i})x\leq\measuredangle y\alpha(t_{i})z_{i}=\measuredangle
y\alpha(t_{i})z,$

this last inequalities on behalf of Proposition 13. On the other hand, since
$x\ll\alpha(t_{i})\ll z_{i}\ll z$ we obtain $(\ell(t_{i})-r)+c_{i}\leq c$ and
$r\leq c_{i}$, thus

$\left(\displaystyle\frac{\ell(t_{i})-c}{t_{i}}\right)\left(\displaystyle\frac{r+r}{2r}\right)+\displaystyle\frac{t_{i}}{2r}\leq\left(\displaystyle\frac{r-c_{i}}{t_{i}}\right)\left(\displaystyle\frac{r+c_{i}}{2r}\right)+\displaystyle\frac{t_{i}}{2r}\leq\measuredangle
y\alpha(t_{i})z.$

When $t_{i}\to 0$ and using Proposition 17 we have

$\displaystyle\lim\sup_{i\to\infty}\displaystyle\frac{\ell(t_{i})-c}{t_{i}}\leq\displaystyle\lim\sup_{i\to\infty}\measuredangle
y\alpha(t_{i})z\leq\measuredangle yxz,$

and we are done. ∎

######  Remark 11.

Here we have the same situations as in Remarks 9, namely

$\displaystyle\lim_{i\to\infty}\displaystyle\frac{\ell(t_{i})-\ell(0)}{t_{i}}=\max_{\gamma}(\measuredangle
yxz).$

## Acknowledgements

W. Barrera was partially supported by Conacyt under grants SNI 45382 and
Ciencia de Frontera 21100. D. Solis was partially supported by Conacyt under
grant SNI 38368. The authors are very thankful to T. Beran for insightful
comments on an earlier version of this work. The authors are very thankful to
the organizers of _SCRI21, a tribute to Roger Penrose_ for this outstanding
event.

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Waldemar Barrera. Facultad de Matemáticas, Universidad Autónoma de Yucatán,
Periférico Norte Tablaje 13615, C.P. 97110, Mérida, México.
<EMAIL_ADDRESS>
Didier A. Solis. Facultad de Matemáticas, Universidad Autónoma de Yucatán,
Periférico Norte Tablaje 13615, C.P. 97110, Mérida, México.
<EMAIL_ADDRESS>
Luis M. Montes de Oca. Facultad de Matemáticas, Universidad Autónoma de
Yucatán, Periférico Norte Tablaje 13615, C.P. 97110, Mérida, México.
<EMAIL_ADDRESS>