# Time functions on Lorentzian length spaces

Annegret Burtscher∗ and Leonardo García-Heveling Department of Mathematics,
Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud
University Nijmegen, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Acknowledgements: LGH would like to thank Didier Solis for communication
related to their preprint [1]. The authors are also grateful to Ettore
Minguzzi for feedback on the introduction. AB’s research is supported by the
Dutch Research Council (NWO), Project number VI.Veni.192.208.

###### Abstract

In general relativity, time functions are crucial objects whose existence and
properties are intimately tied to the causal structure of a spacetime and also
to the initial value formulation of the Einstein equations. In this work we
establish all fundamental classical existence results on time functions in the
setting of Lorentzian (pre-)length spaces (including causally plain continuous
spacetimes, closed cone fields and even more singular spaces). More precisely,
we characterize the existence of time functions by $K$-causality, show that a
modified notion of Geroch’s volume functions are time functions if and only if
the space is causally continuous, and lastly, characterize global
hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our
results thus inevitably show that no manifold structure is needed in order to
obtain suitable time functions.

_2020 Mathematics Subject Classification:_ 53C23 (primary), 53C50, 53C75,
53C80, 83C05 (secondary).

_Keywords:_ general relativity, causality theory, time functions, volume
functions, spacetimes, Lorentzian length spaces.

##  1 Introduction

On a smooth spacetime $(M,g)$ a continuous function $t\colon M\to$ is called a
_time function_ if it satisfies

$p<q\implies t(p)<t(q)\ \text{ for all }\ p,q\in M,$

where $p<q$ means that there exists a causal curve from $p$ to $q$, and that
$p\neq q$. Time functions play a crucial role in Lorentzian causality theory
and Einstein’s general theory of relativity.

The study of time functions has a long history in general relativity. Their
origin can be traced back to the works of Geroch and Hawking in the late
1960s. Geroch introduced _volume time functions_ by normalizing the volume of
a spacetime to one, and by defining the time of a point $p$ as the volume of
its chronological past $I^{-}(p)$. In his seminal work [31] from 1970, Geroch
used these volume functions to characterize global hyperbolicity by the
existence of Cauchy surfaces and to obtain a topological splitting. Globlal
hyperbolicity is the strongest and most important causality condition in
general relativity. Cauchy surfaces represent the natural sets to pose initial
conditions for the Einstein equations (for a self-contained exposition see
[58]). Moreover, global hyperbolicity and its different characterizations play
a crucial role in the singularity theorems of Penrose and Hawking (see [36,
Section 8.2]), the Lorentzian splitting theorems (see [6, 25, 29] and follow-
up work) and the formulation of Quantum Field Theory on curved backgrounds [5,
Chapter 4].

Building upon Geroch’s idea, Hawking [35] showed that volume functions can be
“smeared out” to obtain time functions at a significantly lower step on the
causal ladder, namely stable causality (see also the work of Minguzzi [49, 50]
for the same result via the equivalent notion of $K$-causality and [53, Figure
20] for a depiction of the complete causal ladder). This result contributed to
Hawking’s program to find the minimal causality conditions that one should
impose on a spacetime in order to consider it as physically reasonable. As an
in-between result between stable causality and global hyperbolicity, Hawking
and Sachs [37] showed in 1974 that Geroch’s volume functions are continuous
themselves (that is, without the use of an averaging procedure) precisely when
the spacetime is causally continuous. Their proof, however, contained a
loophole that was later filled by Dieckmann [21, 23].

After these foundational works, the question remained whether time functions
and Cauchy surfaces can be chosen smooth, rather than just continuous. Despite
several attempts by Seifert [61], Sachs and Wu [59] and Dieckmann [21, 22],
this problem remained open for decades. Only in the early 2000s it was firmly
established by Bernal and Sánchez [7, 8, 9] that a spacetime that admits a
continuous time function also admits a smooth one, and that Geroch’s
topological splitting of globally hyperbolic spacetimes can be promoted to a
smooth, orthogonal splitting. While their work differs significantly from
previous approaches, it was recently established by Chruściel, Grant and
Minguzzi [20] that also a family of Geroch’s time functions are continuously
differentiable for globally hyperbolic metrics, and that Hawking’s time
functions can be smoothed out.

A radically different approach to show smoothness of time functions is that of
Fathi and Siconolfi [26, 27], which uses weak KAM theory. It has the added
advantage that it is formulated for smooth manifolds equipped with a
continuous field of closed convex tangent cones, of which Lorentzian manifolds
are just one class of examples. Note that also in the work of Chruściel, Grant
and Minguzzi [20], while staying in the more traditional framework of
Lorentzian manifolds, the metric tensor can be of regularity as low as
$C^{2,1}$. Similarly, global hyperbolicity was characterized by the existence
of Cauchy hypersurfaces and of Cauchy time functions on manifolds with
continuous Lorentzian metrics by Sämann [60] building on work by Chruściel and
Grant [19]. One should be aware that such low-regularity spacetimes already
exhibit considerable pathological behavior including “causal bubbles” (failure
of the push-up property) and not necessarily open chronological futures/pasts
[19, 30, 34, 45, 63]. Note that also the cone structure approach of Fathi and
Siconolfi has been developed further in very recent works of Bernard and Suhr
[10, 11] (using Conley theory) and Minguzzi [52] (using traditional
arguments). Besides the elimination of the need of a Lorentzian metric in the
theory of closed (causal) cones, also the smooth manifold structure should be
removable to some degree. Indeed, it was already pointed out by Minguzzi [51]
that sufficient conditions for the existence of time functions can be obtained
for more general topological spaces through Nachbin’s theory of closed ordered
spaces [55].

It remained open, however, if and to what extent general (and in particular,
all finer) results about the existence and properties of time functions known
for smooth spacetimes rely on the causal and topological structure alone. Our
present work answers this question fully in the abstract framework of
Lorentzian (pre-) length spaces by characterizing the existence of several
types of time functions with different steps on the causal ladder.

The framework of Lorentzian (pre-)length spaces is widely applicable as a
broad range of singular and regular “spacetimes” appearing in the literature
are encompassed, including the above-mentioned closed cone fields and causally
plain continuous spacetimes. Introduced by Kunzinger and Sämann [42] in 2018,
the notion of Lorentzian length spaces makes explicit what is already evident
in the early works of Weyl, Penrose and what has also been proposed by various
other authors (see, for instance, [12, 13, 14, 17, 24, 41, 66, 67] and [53,
Section 4.2.4]), namely that one should treat the causal structure as the most
fundamental geometric object in the general theory of relativity. At the same
time, an attempt has been made to translate metric geometric techniques à la
Gromov that have revolutionized Riemannian geometry to Lorentzian geometry. In
this spirit, Lorentzian (pre-)length spaces are defined as essentially metric
spaces equipped with a chronological and a causal order satisfying basic
order-theoretic properties such as open chronological futures/pasts and the
push-up property (see Section 2 for precise definitions). This general theory
of “spacetime-like” spaces encompasses basically all previously mentioned low-
regularity settings. Despite its youth the Lorentzian (pre-)length framework
has already celebrated important successes in the context of causality theory
[1, 42], inextendibility results [2, 33], synthetic curvature bounds [18, 48,
54] (related to energy conditions in general relativity) and stability [3, 43]
with respect to the null distance.

Nonetheless, the question of when a time function on Lorentzian (pre-)length
space exists has been neglected until now, the only exception being the recent
work of Kunzinger and Steinbauer [43] where it is shown that the existence of
certain time functions implies strong causality (the converse, however, is
false even in the manifold setting). In the present work, we fill this gap by
establishing several sharp existence results. The statements and a discussion
of our main results follows.

### Main results

In this paper we establish three major results relating the existence (and
properties) of time functions to three different steps on the causal ladder,
starting from the optimal condition for existence ($K$-causality) and building
it up to the top one (global hyperbolicity). All our results in the main body
of the paper are obtained for Lorentzian pre-length spaces obeying milder
axioms than those required of a Lorentzian length space (all definitions are
presented in Section 2 in a self-contained way). Establishing our results in
the context of Lorentzian pre-length spaces is important because they are more
widely applicable and often sufficient (see also [18, 43]). In this
introduction, however, we state a simplified version of our theorems in the
setting of Lorentzian length spaces (at the end we briefly comment on the pre-
length case). Generally it is useful to recall that on a smooth spacetime, the
local causal and topological structure is very rigid (all neighborhoods look
the same). Our proofs, on the other hand, rely almost exclusively on global
arguments. This way, we can reduce the local assumptions to the bare minimum,
doing away completely with the manifold structure.

Our first result characterizes the mere existence of time functions on
Lorentzian (pre-)length spaces by $K$-causality, thereby generalizing a result
of Hawking [35] and Minguzzi [50] for smooth spacetimes (note that
$K$-causality is equivalent to stable causality in the smooth setting [49]).

######  Theorem 1.1.

Suppose $X$ is a second countable, locally compact Lorentzian length space.
Then $X$ is $K$-causal if and only if $X$ admits a time function.

Here it is crucial that the $K$-relation is closed and transitive (by
Definition 3.1). Then, as already pointed out by Minguzzi [51], the general
theory of topological ordered spaces yields time functions on $K$-causal
spaces. To prove the converse statement, namely that existence of a time
function implies $K$-causality, we closely follow the approach of Minguzzi
[50], which requires the use of limit curve theorems.

Our second result is concerned with an explicit construction of time functions
on Lorentzian (pre-)length spaces that can be equipped with Borel probability
measures. Here, we are influenced by Geroch’s notion of volume functions [31]
as well as Hawking’s averaging procedure [35]. Since the boundaries of light
cones, however, are no longer hypersurfaces with measure zero as in the smooth
manifold setting, the definition of our _averaged volume functions_ as well as
the proof of their causal properties and continuity are significantly more
involved. The result we obtain is essentially a generalization of a theorem by
Hawking and Sachs [37] (and Dieckmann’s rigorous follow-up work [23]), and
thus the corresponding step on the causal ladder is that of causal continuity.

######  Theorem 1.2.

Let $X$ be a second countable, locally compact Lorentzian length space. Then
$X$ causally continuous if and only if the averaged volume functions on $X$
are time functions.

While the assumption on second countability of the underlying metric space is
crucial in order to have a suitable measure at hand, local compactness can be
removed by using a weaker (but in the smooth case equivalent) notion of causal
continuity.

Our third result characterizes globally hyperbolic Lorentzian (pre-)length
spaces by the existence of Cauchy time functions, whose level sets are Cauchy
sets that are intersected by every inextendible causal curve exactly once. The
smooth spacetime analogue is the seminal 1970 result of Geroch [31].

######  Theorem 1.3.

Let $X$ be a second countable Lorentzian length space with a proper metric
structure. Then the following are equivalent:

  1. (i)

$X$ is globally hyperbolic,

  2. (ii)

$X$ is non-totally imprisoning and the set of causal curves between any two
points is compact,

  3. (iii)

$X$ admits a Cauchy set,

  4. (iv)

$X$ admits a Cauchy time function.

To establish Theorem 1.3 we utilize our averaged volume functions introduced
already for Theorem 1.2, as well as the behavior of inextendible causal
curves. The use of _averaged_ volume functions poses an additional difficulty
in our proofs, compared to the smooth case. The other main challenge is the
fact that our Cauchy sets are not hypersurfaces, and in fact very little can
be deduced about their topology (hence the name Cauchy _set_ instead of
surface, see also our discussion below).

The above theorems follow immediately from their sharper versions, Theorems
3.2, 4.13 and 5.4, which are obtained for Lorentzian pre-length spaces. Some
essential conditions (which are part of the axioms of Lorentzian length
spaces), however, still need to be assumed. In Theorem 3.2 (generalizing
Theorem 1.1), for instance, we need to additionally impose the existence of
causal curves and their limit curves. The chronological relation, however, is
not needed in the proof at all. The proof of Theorem 4.13 (corresponding to
Theorem 1.2), on the other hand, does not require causal curves, but does make
use of both the causal and chronological relation. Furthermore, the two
relations need to satisfy a compatibility condition that we call
“approximating”, which simply means that the causal futures and pasts are
contained in the closure of the chronological ones. Finally, Theorem 5.4 (the
pre-length version of Theorem 1.3) builds upon Theorem 4.13, hence the
“approximating” condition is also needed here. Moreover, causal curves are
used already in the definition of Cauchy set and Cauchy time function, and the
limit curve theorems will be important again.

### Discussion and outlook

Since its inception, the framework of Lorentzian length spaces has seen a
rapid expansion [1, 2, 18, 33, 42, 43]. Notably, Cavalletti and Mondino [18]
recently introduced a notion of Ricci curvature bounds for Lorentzian pre-
length spaces, based on optimal transport theory, that mimics the strong
energy condition of general relativity, and implies a version of Hawking’s
singularity theorem. Our work completes another important milestone in
establishing the potential of this non-smooth causal theory by fully
characterizing the existence of time functions in terms of the causal ladder.
As an immediate application, we have now unambiguously established when one
can make use of time functions to define the null distance of Sormani and Vega
[64]. Very recently, this notion (as well as convergence) has been
investigated by Kunzinger and Steinbauer [43] in the context of Lorentzian
pre-length spaces (certain limits of examples in [3, Section 5] are also of
this type). With our new characterizations, in particular, of global
hyperbolicity via Cauchy time functions, more refined convergence/stability
results may be obtained. Moreover, it should be straightforward to carry over
the cosmological time function of Andersson, Galloway and Howard [4] to the
Lorentzian length space framework using the time separation function in place
of the Lorentzian distance.

While in broad terms we show that the classical results about time functions
admit direct generalizations for Lorentzian length spaces, we also find
interesting and not so subtle differences. In particular, although global
hyperbolicity is also characterized by the existence of Cauchy sets, these
Cauchy sets may not be homeomorphic to each other. This is in stark contrast
to the case of spacetimes, where Geroch’s celebrated splitting theorem [31]
(later refined by Bernal and Sánchez [7]) shows that all Cauchy surfaces on a
given spacetime must have the same topology. In fact, Geroch already showed in
[32] that transitions between compact spatial topologies not only contradict
global hyperbolicity, but in fact even violate the most basic of all
assumptions, chronology. While time travel is a no-go in any physically sound
theory, Sorkin [62] argues that topology change is a necessary feature of
_any_ convincing candidate theory of quantum gravity. Going beyond the setting
of smooth Lorentzian manifolds is thus a necessity in order to admit topology
change without violating chronology, a common approach being the use of
degenerate Lorentzian metrics [15, 38]. Current proposals for quantum gravity
also predict that physical spacetimes are represented by non-manifold-like
structures at small scales, such as causal sets [65], causal dynamical
triangulations [46], causal fermion systems [28], or spin foams [56]. An
interesting and important next step will be to see if and how the framework of
Lorentzian length spaces fits into these quantum gravitational theories.

### Outline

The paper is structured as follows. In Section 2, we give a self-contained
account of the relevant aspects of the theory of Lorentzian (pre-)length
spaces, drawing from the existing literature but also introducing new
material. We then prove Theorems 1.1, 1.2 and 1.3 (more precisely, the
corresponding sharper Lorentzian pre-length space versions) in Sections 3, 4
and 5, respectively.

##  2 Lorentzian pre-length spaces

In this section we recall, and partly refine, the definition of Lorentzian
(pre-)length spaces, their causality conditions and the limit curve theorems.
We use the notation and results of [1, 42]. A reader familiar with the smooth
case will find that most classical concepts are defined in the same way for
Lorentzian pre-length spaces (with the difference that some important
properties do not follow automatically but have to be imposed separately, such
as causal curves themselves).

In Section 2.1 we recall the definition of Lorentzian pre-length space, and of
causal curve. This is standard material, except that we use a more precise
nomenclature for inextendible causal curves (Definition 2.8). In Section 2.2
we revisit the limit curve theorems of Kunzinger and Sämann [42] in the
slightly weaker framework of “local weak causal closedness” following a
suggestion of Aké et al. [1]. We also introduce the new notion of _Lorentzian
pre-length spaces with limit curves_ , which encompasses all necessary
assumptions needed for the application of the limit curve theorems. In Section
2.3 we introduce the notion of _approximating Lorentzian pre-length space_ ,
which can be seen as a much weaker version of Kunzinger and Sämann’s
“localizability”. Most importantly, we show that our newly introduced
properties are, in particular, satisfied by all Lorentzian length spaces.
Finally, in Section 2.4, we define time functions and introduce some elements
of causality theory for Lorentzian pre-length spaces. Notably, we give some
new characterizations of non-total imprisonment, both in terms of causal
curves and of time functions. Additional causality conditions, including
$K$-causality and global hyperbolicity, are introduced in later sections when
needed.

###  2.1 Basic definitions and properties

######  Definition 2.1 ([42, Definition 2.1]).

A _causal set_ is a set $X$ equipped with a preorder $\leq$ (called _causal_
relation) and a transitive relation $\ll$ (called _chronological_ or
_timelike_ relation) contained in $\leq$.

The following notation for the timelike/causal future or past of a point is
standard

$\displaystyle I^{+}(p)$ $\displaystyle:=\left\\{x\in X\mid p\ll x\right\\},$
$\displaystyle J^{+}(p)$ $\displaystyle:=\left\\{x\in X\mid p\leq x\right\\},$
$\displaystyle I^{-}(p)$ $\displaystyle:=\left\\{x\in X\mid x\ll p\right\\},$
$\displaystyle J^{-}(p)$ $\displaystyle:=\left\\{x\in X\mid x\leq p\right\\},$
$\displaystyle I(p,q)$ $\displaystyle:=I^{+}(p)\cap I^{-}(q),$ $\displaystyle
J(p,q)$ $\displaystyle:=J^{+}(p)\cap J^{-}(q),$

and we write $p<q$ if $p\leq q$ and $p\neq q$.

######  Definition 2.2 ([42, Definition 2.8]).

A _Lorentzian pre-length space_ is a causal set $(X,\ll,\leq)$ equipped with a
metric $d$ and a lower semicontinuous function $\tau\colon X\times
X\to[0,\infty]$ satisfying, for all $x,y,z\in X$

  1. (i)

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

  2. (ii)

$\tau(x,y)=0$ if $x\not\leq y$,

  3. (iii)

$\tau(x,z)\geq\tau(x,y)+\tau(y,z)$ if $x\leq y\leq z$.

Occasionally we denote a Lorentzian pre-length space simply by $X$. It follows
from Definition 2.2 that the sets $I^{\pm}(p)$ are open for all $p\in X$, a
fact that we will also refer to as the _openness of $\ll$_. The crucial push-
up property extends from the smooth situation.

######  Lemma 2.3 (Push-up [42, Lemma 2.10]).

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space and $x,y,z\in X$
with $x\ll y\leq z$ or $x\leq y\ll z$. Then $x\ll z$.

The function $\tau$ in Definition 2.2 is often called _time separation
function_ or _Lorentzian distance function_. We will not use this terminology.
In fact, we never need the function $\tau$ by itself but just the openness of
$\ll$ and the push-up property (we could also trivially set $\tau(p,q)=\infty$
if $p\ll q$ and $=0$ otherwise).

In smooth Lorentzian geometry the causal character of curves determines the
timelike and causal future and past, that is $I^{\pm}$ and $J^{\pm}$,
respectively. In Lorentzian pre-length spaces it is the other way round.

######  Definition 2.4 ([42, Definition 2.18]).

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space and $I$ be any
(open, half-open, or closed) interval in . A non-constant locally Lipschitz
path $\gamma\colon I\to X$ is called a

  1. (i)

_future-directed causal curve_ if $\gamma(s_{1})\leq\gamma(s_{2})$ for all
$s_{1}<s_{2}\in I$.

  2. (ii)

_past-directed causal curve_ if $\gamma(s_{2})\leq\gamma(s_{1})$ for all
$s_{1}<s_{2}\in I$.

Future- and past-directed _timelike_ curves are defined analogously by
replacing $\leq$ with $\ll$.

######  Remark 2.5.

By a result in metric geometry (see, for instance, [16, Proposition 2.5.9]),
we can parametrize any causal curve by $d$-arclength (the reference is for
closed intervals only, but the proof is in fact valid for any interval).
Recall that $\gamma\colon I\to X$ is parametrized by $d$-arclength iff

$L^{d}(\gamma|_{[a,b]})=b-a\text{ for all }[a,b]\subseteq I.$

Since a curve that is parametrized by $d$-arclength is automatically
$1$-Lipschitz continuous, causal curves remain causal when parametrizing them
by $d$-arclength. Hence, without loss of generality, we can assume that causal
curves are parametrized in a way so that they are not locally constant, i.e.,
not constant on any open subinterval of .

######  Definition 2.6 ([42, Definition 3.1]).

A Lorentzian pre-length space $X$ is called _causally path-connected_ if for
every $p<q$ there exists a future-directed causal curve connecting $p$ and
$q$, and for every $p\ll q$ a future-directed timelike curve connecting $p$
and $q$.

######  Definition 2.7 ([1, Definition 2.19]).

For a subset $U$ of a Lorentzian pre-length space $X$ we define the relation
$\leq_{U}$ by

$p\leq_{U}q:\Longleftrightarrow\text{there is a future-directed causal curve
from $p$ to $q$ in $U$.}$

A neighborhood $U$ is called _weakly causally closed_ if $\leq_{U}$ is closed,
and the Lorentzian pre-length space $X$ is called _locally weakly causally
closed_ if every point $p\in X$ is contained in a weakly causally closed
neighborhood $U$.

Definition 2.6 is satisfied on _any_ smooth Lorentzian manifold, and thus acts
as a replacement for regularity on a Lorentzian pre-length space. In contrast,
the “local causal closedness” condition of Kunzinger and Sämann [42,
Definition 3.4] is stronger than Definition 2.7 because it requires that
$\leq$ restricted to $U\times U$ is closed. For instance, in the smooth case
the latter notion would only be satisfied on strongly causal spacetimes (see
[1, p. 6] for a detailed discussion).

Finally, we refine the concept of an inextendible curve111Doubly-inextendible
causal curves are often simply called “inextendible”. On the other hand,
Kunzinger and Sämann [42] call a curve inextendible if it is either future- or
past-inextendible (or both), and have no need for the concept of double-
inextendibility. We will be more precise when needed..

######  Definition 2.8.

Let $X$ be a Lorentzian pre-length space and $\gamma\colon(a,b)\to X$ be a
future-directed causal curve. If there exists a causal curve
$\bar{\gamma}\colon(a,b]\to X$ such that $\bar{\gamma}|_{(a,b)}=\gamma$, we
say that $\gamma$ is _future-extendible_. If there exists a causal curve
$\tilde{\gamma}\colon[a,b)\to X$ such that $\tilde{\gamma}|_{(a,b)}=\gamma$,
we say that $\gamma$ is _past-extendible_. We say that $\gamma$ is
_future-(past-)inextendible_ if it is not future-(past-)extendible, and
_doubly-inextendible_ if it is neither future- nor past-extendible.

The analogous definition for past-directed causal curves is obtained by
interchanging future and past in Definition 2.8. The definition applies
accordingly to half-open intervals. If a path is defined on all of , we mean
extendibility to $\pm\infty$. Alternatively we can parametrize it by arclength
and apply the following lemma (which, of course, admits also a past version).

######  Lemma 2.9.

Let $X$ be a locally weakly causally closed, causally path-connected
Lorentzian pre-length space, let $-\infty<a<b\leq\infty$ and let
$\gamma\colon[a,b)\to X$ be a future-directed causal curve parametrized with
respect to $d$-arclength. If $(X,d)$ is a proper metric space or the curve
$\gamma$ is contained in a compact set, then $\gamma$ is future-inextendible
if and only if $b=\infty$. In this case $L^{d}(\gamma)=\infty$. Moreover,
$\gamma$ is future-inextendible if and only if $\lim_{t\nearrow b}\gamma(t)$
does not exist.

###### Proof.

The proof is the same as [42, Lemma 3.12]. Note that there, the assumption of
local “strong” causal closedness is only applied to points which lie on
$\gamma$. Hence that proof also works with our notion of weakly causally
closed neighborhood. ∎

In the remaining subsection we recall the definition of Lorentzian length
space (including necessary preliminary notions) as introduced in [42]. While
we will not directly work with Lorentzian length spaces in the main body of
this paper, our results about pre-length spaces immediately also lead to
useful Corollaries in this setting (see Introduction). A Lorentzian length
space is, in essence, just the Lorentzian analogue of length metric spaces
generalizing Riemannian manifolds where the time separation function $\tau$ is
used in place of a distance function to measure lengths and the admissible
class of curves respects causality. More precisely, if $\gamma\colon[a,b]\to
X$ is a future-directed causal curve, then its _$\tau$ -length_
$L_{\tau}(\gamma)$ is defined by (see [42, Definition 2.24])

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

In addition, the notion of localizability is needed. In the main sections of
this paper, only assumptions (i) and (ii) are needed, thus we restate them
separately below (see Definitions 2.14 and 2.17, Lemma 2.19 and Proposition
2.20).

######  Definition 2.10 ([42, Definition 3.16] and [1, Definition
2.22]222Similarly to the difference between weak and “strong” local causal
closedness, there is a difference between the notions of localizability in
[42, Definition 3.16] and [1, Definition 2.22] regarding the meaning of
$\ll_{U_{p}},\leq_{U_{p}}$.).

A Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is called _localizable_ if
for every point $p\in X$, there exists a neighborhood $U_{p}$ of $p$ such that

  1. (i)

There exists a constant $C>0$ such that for all causal curves contained in
$U_{p}$ we have $L^{d}(\gamma)\leq C$.

  2. (ii)

For every $q\in U_{p}$ we have $I^{\pm}(q)\cap U_{p}\neq\emptyset$.

  3. (iii)

There exists a continuous function $\omega_{p}\colon X\times X\to[0,\infty)$
such that $(U_{p},d|_{U_{p}\times U_{p}},\\\
\ll_{U_{p}},\leq_{U_{p}},\omega_{p})$ is a Lorentzian pre-length space.
Moreover, for all $x,y\in U_{p}$ with $x<y$, it holds that

$\omega_{p}(x,y)=\max\\{L_{\tau}(\gamma)\mid\gamma\colon[a,b]\to U_{p}\text{
fd.\ causal from $x$ to $y$ }\\},$

so in particular there exists a maximizing causal curve between $x$ and $y$.

By further assuming that also $\tau$ is given by length-maximization (but
without necessarily requiring the existence of global maximizers), one obtains
a Lorentzian length space.

######  Definition 2.11 ([42, Definition 3.22]).

A causally path-connected, locally (weakly) causally closed and localizing
Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is called a _Lorentzian
length space_ if for all $p,q\in X$

$\tau(p,q)=\sup\\{L_{\tau}(\gamma)\mid\gamma\text{ future-directed causal from
$p$ to $q$}\\}.$

###  2.2 Limit curve theorems

We revisit the limit curves theorems of Kunzinger and Sämann [42, Section 3.2]
and relax their assumption of local causal closedness to local weak causal
closedness (see Definition 2.7). That this extension is possible was already
pointed out by Aké et al. [1, p. 8]. The limit curve theorems are crucial for
Sections 3 and 5.

We start with [42, Lemma 3.6] where, instead of pointwise convergence, we need
to assume locally uniform convergence.

######  Lemma 2.12.

Let $X$ be a causally path-connected locally weakly causally closed Lorentzian
pre-length space and let $(\gamma_{n})_{n}$ be a sequence of future-directed
causal curves $\gamma_{n}\colon I\to X$ converging locally uniformly to a non-
constant curve $\gamma\colon I\to X$. Then $\gamma$ is future-directed causal.

###### Proof.

For every $s\in I$ there exists a weakly causally closed neighborhood
$U_{\gamma(s)}$. By continuity, we can pick $s_{1}<s<s_{2}$ such that
$\gamma([s_{1},s_{2}])\subseteq U_{\gamma(s)}$ (if $s$ is a boundary point of
$I$, then $s_{1}=s$ or $s_{2}=s$ is chosen). Assume additionally that we
choose $s_{1},s_{2}$ close enough such that $(\gamma_{n})_{n}$ converges
uniformly on $[s_{1},s_{2}]$. This implies that there exists
$n_{0}\in\mathbb{N}$ such that for all $n>n_{0}$,
$\gamma_{n}([s_{1},s_{2}])\subseteq U_{\gamma(s)}$. Now it follows from the
definition of weakly causally closed neighborhood that $\gamma$ restricted to
$[s_{1},s_{2}]$ is future-directed causal. Since $s$ was arbitrary, we can
decompose $\gamma$ as a concatenation of future-directed causal curves, hence
by transitivity of $\leq$, $\gamma$ is future-directed causal on $I$. ∎

######  Theorem 2.13 (Limit curve theorem).

Let X be a causally path-connected locally weakly causally closed Lorentzian
pre-length space. Let $(\gamma_{n})_{n}$ be a sequence of future-directed
causal curves $\gamma_{n}\colon[a,b]\to X$ that are uniformly Lipschitz
continuous, i.e., there is an $L>0$ such that
$\operatorname{Lip}(\gamma_{n})\leq L$ for all $n\in\mathbb{N}$. Suppose that
there exists a compact set that contains every $\gamma_{n}$ or that $d$ is
proper and that the curves $(\gamma_{n})_{n}$ accumulate at some point, i.e.,
there is a $t_{0}\in[a,b]$ such that $\gamma_{n}(t_{0})\to x_{0}\in X$. Then
there exists a subsequence $(\gamma_{n_{k}})_{k}$ of $(\gamma_{n})_{n}$ and a
Lipschitz continuous curve $\gamma\colon[a,b]\to X$ such that
$\gamma_{n_{k}}\to\gamma$ uniformly. If $\gamma$ is non-constant, then
$\gamma$ is a future-directed causal curve.

###### Proof.

The proof of [42, Theorem 3.7] goes through. The assumption of local “strong”
causal closedness is only used to invoke [42, Lemma 3.6], but since the
convergence is uniform, we can replace it by our Lemma 2.12. ∎

This first limit curve theorem is already very useful. In order to formulate
our second limit curve theorem, we need a better control over the $d$-length
of causal curves.

######  Definition 2.14 ([42, Definition 3.13]).

A Lorentzian pre-length space $(X,d,\allowbreak\ll,\leq,\tau)$ is called _$d$
-compatible_ if for every $p\in X$ there exists a neighborhood $U$ of $p$ and
a constant $C>0$ such that $L^{d}(\gamma)\leq C$ for all causal curves
$\gamma$ contained in $U$.

To ease the nomenclature, we group some of our previous assumptions into the
following definition.

######  Definition 2.15.

A _Lorentzian pre-length space with limit curves_ is a causally path-
connected, locally weakly causally closed, and $d$-compatible Lorentzian pre-
length space.

It is an easy consequence that Lorentzian length spaces are particular cases
of Lorentzian pre-length spaces with limit curves (see also Proposition 2.20
below).

######  Theorem 2.16 (Limit curve theorem for inextendible curves).

Let $X$ be a Lorentzian pre-length space with limit curves. Let
$(\gamma_{n})_{n}$ be a sequence of future-directed causal curves
$\gamma_{n}\colon[0,L_{n}]\to X$ which are parametrized with respect to
$d$-arclength and satisfy $L_{n}:=L^{d}(\gamma_{n})\to\infty$. If there exists
a compact set that contains every curve $\gamma_{n}([0,L_{n}])$ or if $d$ is
proper and $\gamma_{n}(0)\to x$ for some $x\in X$, then there exists a
subsequence $(\gamma_{n_{k}})_{k}$ of $(\gamma_{n})_{n}$ and a future-directed
causal curve $\gamma\colon[0,\infty)\to X$ such that $\gamma_{n_{k}}\to\gamma$
locally uniformly. Moreover, $\gamma$ is future-inextendible.

###### Proof.

The proof of [42, Theorem 3.14] goes through. The assumption of local “strong”
causal closedness is only used to invoke [42, Lemmas 3.6 and 3.12, Theorem
3.7], so we can replace them by our Lemma 2.12, Lemma 2.9 and Theorem 2.13
respectively. ∎

While the limit curve theorems are stated for future-directed curves, they of
course also hold for past-directed ones.

###  2.3 Approximating Lorentzian pre-length spaces

In this subsection we introduce our new “approximating” condition relating the
causal structure and the topology on $X$. It is satisfied on all spacetimes
regardless of their place in the causal ladder, and will be crucial in Section
4. In Proposition 2.20 we show that all Lorentzian length spaces automatically
fulfill the “approximating” condition, and also our earlier Definition 2.15.

######  Definition 2.17.

A Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is called _approximating_
if for all points $p\in X$ it holds that
$J^{\pm}(p)\subseteq\overline{I^{\pm}(p)}$.

It is called _future-(past-)approximating_ if the approximating property holds
for $+$($-$).

The approximating property can equivalently be characterized via sequences as
follows.

######  Lemma 2.18.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space. Then $X$ is
(future-/past-)approximating if and only if for every point $p\in X$ there
exists a sequence $(p^{\pm}_{n})_{n}$ in $I^{\pm}(p)$ such that
$p^{\pm}_{n}\to p$ as $n\to\infty$.

We say that the sequence $(p^{+}_{n})_{n}$ approximates $p$ from the future,
and that the sequence $(p^{-}_{n})_{n}$ approximates $p$ from the past.

###### Proof.

That such sequences exist on approximating spaces is obvious, because $p\in
J^{\pm}(p)\subseteq\overline{I^{\pm}(p)}$. To show the converse, suppose $q\in
J^{+}(p)$ and $(q_{n}^{+})$ is a sequence in $I^{+}(q)$ approximating $q$ from
the future. By the push-up Lemma 2.3, $q_{n}^{+}\in I^{+}(p)$ for all
$n\in\mathbb{N}$, hence $q\in\overline{I^{+}(p)}$. ∎

Assuming that $X$ is causally path-connected, we get even more
characterizations, which in the smooth case are in fact the most widely used
ones.

######  Lemma 2.19.

Let $(X,d,\ll,\leq,\tau)$ be a causally path-connected Lorentzian pre-length
space. Then the following are equivalent:

  1. (i)

$X$ is future-(past-)approximating,

  2. (ii)

$I^{+}(p)\neq\emptyset$ ($I^{-}(p)\neq\emptyset$) for all $p\in X$,

  3. (iii)

for every point $p\in X$ is there exists a future-(past-)directed timelike
curve $\gamma\colon[a,b)\to X$ with $\gamma(a)=p$.

Note that if $X$ is approximating, we can always join the future- and past-
directed curves from point (iii) to find a timelike curve
$\gamma\colon(a,b)\to X$ through $p$.

###### Proof.

(i) $\implies$ (ii) Let $p\in X$ be any point. Then $p\in J^{+}(p)$, so if $X$
is future-approximating, we get that $\emptyset\neq
J^{+}(p)\subseteq\overline{I^{+}(p)}$. This implies that
$I^{+}(p)\neq\emptyset$.

(ii) $\implies$ (iii) By assumption, there exists points $q\in I^{+}(p)$. By
causal path-connectedness, there exists a future-directed timelike curve
$\gamma$ from $p$ to $q$, which by Definition 2.4 must be non-constant, even
if $p=q$. We can then remove the appropriate endpoint of $\gamma$ to get the
desired curve.

(iii) $\implies$ (i) Let $p\in X$ and $\gamma\colon[a,b)\to X$ be a future-
directed timelike curve with $\gamma(a)=p$. By continuity of $\gamma$, we have
$p=\lim_{s\to a}\gamma(s)$, so $p\in\overline{I^{+}(p)}$.

The past statements are proved analogously. ∎

We can now easily see how Lorentzian length spaces are particular cases of the
more general pre-length spaces that we will be working with in the rest of the
paper.

######  Proposition 2.20.

If $X$ is a localizing, causally path-connected Lorentzian pre-length space,
then $X$ is approximating and $d$-compatible. If $X$ is a Lorentzian length
space, then $X$ is an approximating Lorentzian pre-length space with limit
curves.

###### Proof.

If $X$ is localizing, then by property (i) in Definition 2.10, $X$ is
$d$-compatible. Furthermore, by property (ii), every point $q\in X$ has
$I^{\pm}(q)\neq\emptyset$, and then by Lemma 2.19, $X$ is approximating. The
second statement follows trivially from the definitions. ∎

######  Remark 2.21.

In connection with the null distance on Lorentzian pre-length spaces,
Kunzinger and Steinbauer [43, Definition 3.4] introduced the notion of
_sufficiently causally connectedness (scc)_. A Lorentzian pre-length space is
scc if it is path-connected (in the sense of metric spaces), causally path-
connected (Definition 2.6) and every point $p\in X$ lies on some timelike
curve $\gamma$. While the last condition is reminiscent of property (iii) in
our Lemma 2.19, it is in fact weaker, since scc puts no restriction on whether
$p$ should be a future, past, or no endpoint of $\gamma$. On the other hand,
we do not need to assume path-connectedness.

Having established the existence of causal (even timelike) curves through
every point in Lemma 2.19, the question remains whether one can find a
(doubly-)inextendible causal curve through every point (see Definition 2.8).
The following proposition and corollary answer this question in the
affirmative, which will be crucial in Section 5 when studying Cauchy sets. We
need to assume that $(X,d)$ is proper in order to invoke the limit curve
theorem. The use of the latter is also the reason why we only prove the
existence of intextendible causal (and not timelike) curves.

######  Proposition 2.22 (Existence of maximal extensions of causal curves).

Let $X$ be an approximating Lorentzian pre-length space with limit curves.
Suppose, in addition, that $(X,d)$ is proper. Then, for every
future-(past-)directed causal curve $\gamma\colon[a,b)\to X$ with $b<\infty$,
there exists a $c\geq b$ and a future-(past-)inextendible causal curve
$\lambda:[a,c)\to X$ such that $\lambda|_{[a,b)}=\gamma$.

###### Proof.

Consider, without loss of generality, the case that $\gamma$ is future-
directed. If $\gamma$ is already inextendible, there is nothing to prove.
Hence we consider the case of $\gamma$ being extendible. Then $\gamma$ has an
endpoint, which we will, by abuse of notation, denote as $\gamma(b)$. Since
$X$ is approximating, by Lemma 2.19 there is a future-directed timelike curve
starting at $\gamma(b)$. Concatenating it with $\gamma$, we get a proper
extension $\tilde{\gamma}\colon[a,c)\to X$ of $\gamma$, where $c>b$. If we can
choose $\tilde{\gamma}$ to be inextendible, we are done. Hence, suppose for
the sake of contradiction that all extensions of $\gamma$ are themselves
extendible. There are two possible cases:

  1. 1.

There exists a constant $C>0$ such that the $d$-arclength of all extensions of
$\gamma$ is bounded by $C$. Suppose that we have chosen $C$ as small as
possible. Then there exists a sequence $(\gamma_{n})_{n}$ of extensions such
that $L^{d}(\gamma_{n})\to C$. Since all the $\gamma_{n}$ are extendible
(hence we can add their future-endpoints) and agree at the point $\gamma(b)$,
by Theorem 2.13 a subsequence converges to a limit curve
$\gamma_{\infty}\colon[a,c]\to X$ of arclength $L^{d}(\gamma_{\infty})=C$. But
then, by the above, $\gamma_{\infty}$ admits a future extension, which is then
also an extension of $\gamma$ and has arclength greater than $C$, a
contradiction.

  2. 2.

There exists a sequence $(\gamma_{n})_{n}$ of extensions of $\gamma$ such that
$L^{d}(\gamma_{n})\to\infty$. In this case we can apply Theorem 2.16 to find
an inextendible limit curve $\gamma_{\infty}$ of a subsequence. This
$\gamma_{\infty}$ is then the desired inextendible extension of $\gamma$. ∎

Combining Lemma 2.19 and Proposition 2.22 gives us an important conclusion.

######  Corollary 2.23.

Let $X$ satisfy the assumptions of Proposition 2.22. Then, for every point in
$p\in X$, there exists a doubly-inextendible causal curve passing through $p$.
∎

###  2.4 Causality conditions and time functions

The conditions in the previous subsections relating the topology and the
causal structure (such as approximating) are satisfied automatically when the
topology is that of a manifold, and the causal structure is induced by a
Lorentzian metric. They can thus be thought of as making our Lorentzian pre-
length spaces more “manifold-like”, while still being much more general. In
this subsection, on the other hand, we are going to discuss causality
conditions, i.e., steps on the causal ladder, which are not satisfied by all
smooth spacetimes and hence also not by all Lorentzian pre-length spaces. They
should be thought of as criteria for physical reasonability.

In this section, we consider the notions of causality and non-total
imprisonment, and the definition of time functions (for an in-depth treatment
of the causal ladder for Lorentzian length spaces, see [1]). Most of the
material is standard, but Theorem 2.27 and Proposition 2.28 are new. The goal
of this paper is to characterize the existence of (certain kinds of) time
functions by suitable causality conditions, which will be introduced in the
main sections. The causality conditions in this section are weaker, but also
play an important role.

A smooth spacetime is called causal if it contains no closed causal curves.
The following equivalent definition is better suited for Lorentzian pre-length
spaces.

######  Definition 2.24 ([42, Definition 2.35]).

A Lorentzian pre-length space is called _causal_ if for any two points $p,q\in
X$, $p<q$ implies $q\not<p$.

Time functions too, can be defined either via causal curves (time functions
are then required to be strictly increasing on future-directed causal curves),
or in the following, more order-theoretic manner.

######  Definition 2.25.

A function $f\colon X\to$ on a Lorentzian pre-length space
$(X,d,\ll,\leq,\tau)$ is called a _generalized time function_ if for all
$p,q\in X$,

$p<q\implies f(p)<f(q).$

It is called a _time function_ if it is also continuous.

Clearly, the existence of a (generalized) time function requires that the
underlying space is at least causal.

In the smooth case, non-total imprisonment is equivalent to the following
definition (see, for instance, [53, Theorem 4.39]).

######  Definition 2.26 ([42, Definition 2.35]).

A Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is called _non-totally
imprisoning_ if for every compact set $K\subseteq X$ there exists a constant
$C>0$ such that for every causal curve $\gamma$ with image in $K$,
$L^{d}(\gamma)\leq C$.

As a corollary to the limit curve theorems, we obtain the following
alternative characterizations.

######  Theorem 2.27.

Let $X$ be a Lorentzian pre-length space with limit curves. Then the following
are equivalent.

  1. (i)

$X$ is non-totally imprisoning.

  2. (ii)

No compact set in $X$ contains a future-inextendible causal curve.

  3. (iii)

No compact set in $X$ contains a past-inextendible causal curve.

  4. (iv)

No compact set in $X$ contains a doubly-inextendible causal curve.

###### Proof.

The equivalence between (i), (ii) and (iii) is shown in [42, Corollary 3.15].
As a consequence of Lemma 2.9, any doubly-inextendible curve has infinite
arclength. Thus (i) implies (iv).

It remains to be shown that (iv) implies (i). Suppose $X$ is totally
imprisoning. Then there exists a compact set $K$ and a sequence of future-
directed causal curves $\gamma_{n}\colon[0,L_{n}]\to X$, parametrized by
arclength and contained in $K$, such that $L_{n}=L^{d}(\gamma_{n})\to\infty$.
Consider the sequence of future-directed causal curves
$\bar{\gamma}_{n}\colon[0,L_{n}/2]\to X$ given by

$\bar{\gamma}_{n}(s):=\gamma_{n}\left(\frac{L_{n}}{2}+s\right).$

Then also $L^{d}(\bar{\gamma}_{n})=L_{n}/2\to\infty$ and we can apply Theorem
2.16 to find a converging subsequence $(\bar{\gamma}_{n_{k}})_{k}$ and a
future-inextendible causal limit curve $\bar{\gamma}\colon[0,\infty)\to X$.
Next consider the sequence of past-directed causal curves
$\tilde{\gamma}_{k}\colon[-L_{n}/2,0]\to X$ given by

$\tilde{\gamma}_{k}(s):=\gamma_{n_{k}}\left(\frac{L_{n}}{2}+s\right).$

Again we can apply Theorem 2.16 to find a converging subsequence
$(\tilde{\gamma}_{k_{m}})_{m}$ and a past-inextendible limit curve
$\tilde{\gamma}\colon(-\infty,0]\to X$. Note that

$\tilde{\gamma}(0)=\lim_{m\to\infty}\gamma_{n_{k_{m}}}(L_{n}/2)=\bar{\gamma}(0),$

where the limit in the middle exists by compactness of $K$, a fact that we had
already used implicitly when applying the Limit Curve Theorem 2.16. Thus the
curve $\tilde{\gamma}$ joined with $\bar{\gamma}$ is a doubly-inextendible
causal curve $(-\infty,\infty)\to X$ contained in $K$. This contradicts our
assumption (iv). ∎

The following result was shown by Kunzinger and Sämann for Lorentzian length
spaces, but only the assumption of causal path-connectedness is used in the
proof.

######  Proposition 2.28 ([42, Theorem 3.26]).

Suppose $(X,d,\ll,\leq,\tau)$ is a causally path-connected Lorentzian pre-
length space. If $X$ is non-totally imprisoning, then $X$ is causal.

For spacetimes, it is well-known that the existence of a time function implies
strong causality, and that strong causality implies non-total imprisonment.
This result has been shown by Kunzinger and Steinbauer for Lorentzian length
spaces [43, Theorem 3.13], under the additional assumption that the time
function must be topologically locally anti-Lipschitz. We instead give a
direct proof of the fact that for Lorentzian pre-length spaces with limit
curves, the existence of any kind of time function implies non-total
imprisonment.

######  Lemma 2.29.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space with limit curves.
If $X$ admits a time function, then $X$ is non-totally imprisoning.

###### Proof.

Suppose $X$ is totally imprisoning. Then by Theorem 2.27 there exists a
compact set $K\subseteq X$ and a future-inextendible future-directed causal
curve $\gamma\colon[0,\infty)\to K$. Note the following two facts:

  1. (i)

By Lemma 2.9, because $\gamma$ is inextendible, $\lim_{s\to\infty}\gamma(s)$
does not exist.

  2. (ii)

By compactness of $K$, for every sequence $(s_{i})_{i}$ in $[0,\infty)$, there
exists a subsequence of $\left(\gamma(s_{i})\right)_{i}$ that converges in
$K$.

Thus we can find two sequences $(r_{i})_{i}$ and $(s_{i})_{i}$ in $[0,\infty)$
such that $p:=\lim_{i\to\infty}\gamma(r_{i})\\\
\neq\lim_{i\to\infty}\gamma(s_{i})=:q$ (in particular, both limits exist).

For these $p,q$, pick $\delta>0$ small enough so that $B_{\delta}(p)$ is
contained in a weakly causally closed neighborhood and $q\not\in
B_{\delta}(p)$. Assume w.l.o.g. that $r_{1}<s_{1}<r_{2}<s_{2}\ldots$ and that
for all $i\in\mathbb{N}$ we have $\gamma(r_{i})\in B_{\delta/2}(p)$ and
$\gamma(s_{i})\not\in B_{\delta/2}(p)$. Now define a third sequence
$(a_{i})_{i}$ with $r_{i}<a_{i}<s_{i}$ and such that $a_{i}$ is the value at
which $\gamma|_{[r_{i},s_{i}]}$ first intersects $\partial B_{\delta/2}(p)$.
We then have $r_{i}<a_{i}<s_{i}<r_{i+1}$. By compactness of $\partial
B_{\delta/2}(p)$, there exists a subsequence of $(\gamma(a_{i}))_{i}$ that
converges to a point $q^{\prime}\neq p$. Since $r_{i}<a_{i}$, we have
$\gamma(r_{i})\leq\gamma(a_{i})$. Because $B_{\delta}(p)$ is contained in a
weakly causally closed neighborhood, we have $p<q^{\prime}$. By assumption,
$X$ admits a time function $t\colon X\to$, for which it holds that
$t(p)<t(q^{\prime})$. On the other hand, since $a_{i}<r_{i+1}$, we have
$\gamma(a_{i})\leq\gamma(r_{i+1})$. Thus $t(\gamma(a_{i}))\leq
t(\gamma(r_{i+1}))$ for all $i\in\mathbb{N}$ and by continuity of $t$ and
$\gamma$ also $t(q^{\prime})\leq t(p)$. Combining this with the previous
inequality, we obtain

$t(p)<t(q^{\prime})\leq t(p),$

which is a contradiction. ∎

##  3 Time functions and $K$-causality

The notion of $K$-causality was first introduced by Sorkin and Woolgar [63] to
study spacetimes with continuous Lorentzian metrics. Among the multiple
applications of this concept, we emphasize the work of Minguzzi [50], who
showed that for smooth spacetimes, $K$-causality is equivalent to stable
causality. Since Hawking [35] had shown earlier that stable causality is
equivalent to the existence of a time function, so is $K$-causality. Minguzzi
in [50] also gave a direct proof of the equivalence between $K$-causality and
the existence of time functions, which is more mathematically rigorous and
less dependent on the Lorentzian manifold structure. Since $K$-causality is a
purely order-theoretical notion, it can be used verbatim333In [63] the
$K^{+}$-relation is only required to contain $I^{+}$. Our definition with
$J^{+}$ can be traced back to [1, 50]. On spacetimes and approximating
Lorentzian pre-length spaces, we have $J^{+}\subseteq\overline{I^{+}}$, hence
there it makes no difference. for Lorentzian pre-length spaces.

######  Definition 3.1 ([63, Definitions 8 and 9]).

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space. The _$K^{+}$
-relation_ on $X$ is defined as the (unique) smallest transitive relation that
contains $\leq$ and is (topologically) closed.

A Lorentzian pre-length space $X$ is called _$K$ -causal_444In [1, 42] a
Lorentzian pre-length space with this property is called _stably causal_
because in the smooth case $K$-causality and stable causality are equivalent
[49]. The definition of stable causality [35, p. 433], however, requires
knowledge about causal properties of “nearby” Lorentzian metrics. Since
(causal) stability of Lorentzian pre-length spaces has not yet been
investigated in this sense, we prefer to use the standard term $K$-causal. if
the $K^{+}$-relation is antisymmetric.

In this section we establish the equivalence of the existence of time
functions and $K$-causality on certain Lorentzian pre-length spaces (see
Theorem 3.2 below and Theorem 1.1 formulated for Lorentzian length spaces).
This result generalizes the analogous theorem known for smooth spacetimes by
Minguzzi [50, Theorem 7] and the proof is obtained along the same lines.

######  Theorem 3.2.

Suppose $\tau$ is a second countable, locally compact Lorentzian pre-length
space with limit curves. Then $X$ is $K$-causal if and only if $X$ admits a
time function.

Before proving the theorem, we show that time functions can exist more
generally also on Lorentzian pre-length spaces that are not $K$-causal if they
do not satisfy the limit curve property (Definition 2.15).

######  Example 3.3 (There exist Lorentzian pre-length spaces that admit a
time function but are neither strongly causal nor $K$-causal).

Let $(X,d)$ be the Euclidean plane with coordinates $(t,x)$. For any pair of
points $p_{i}=(t_{i},x_{i})$, $i=1,2$, let

$\displaystyle p_{1}\ll p_{2}$ $\displaystyle:\iff t_{1}<t_{2},$
$\displaystyle p_{1}\leq p_{2}$ $\displaystyle:\iff t_{1}<t_{2}\text{ or
}p_{1}=p_{2},$

as depicted in Figure 1, and

$\tau(p_{1},p_{2})=t_{2}-t_{1}.$

This equips $(X,d)$ with the structure of a Lorentzian pre-length space.
Clearly, the $t$-coordinate is a time function.

$I^{+}(p)$$J^{+}(p)$$p$$x$$t$ Figure 1: The sets $I^{+}(p)$ (blue, without
boundary) and $J^{+}(p)=I^{+}(p)\cup\\{p\\}$ for a point $p$ in Example 3.3.

However, this space is not $K$-causal, as any closed relation containing
$\leq$ (or $\ll$) must contain the relation $\leq_{R}$ given by

$p_{1}\leq_{R}p_{2}\iff t_{1}\leq t_{2}.$

But $\leq_{R}$ is not antisymmetric, so the $K$-relation will not be
antisymmetric either, and our space is therefore not $K$-causal.

Note that while $K$-causality implies strong causality on smooth spacetimes
and locally compact Lorentzian length spaces [1, Proposition 3.16] this need
not be the case for Lorentzian pre-length spaces. To see that $X$ is also not
strongly causal, recall that a Lorentzian pre-length space $X$ is called
_strongly causal_ if the Alexandrov topology, generated by

$I(p,q)=\\{r\in X\mid p\ll r\ll q\\},\qquad p,q\in X,$

agrees with the metric topology [42, Definitions 2.4 and 2.35].

The Alexandrov topology of the above example is generated by the open
horizontal stripes, hence is strictly coarser than the Euclidean topology and
$X$ therefore not strongly causal.

###  3.1 Proof of Theorem 3.2

We follow Minguzzi’s proof for smooth spacetimes [50]. A key element is the
following theorem from utility theory, a branch of mathematical economics with
resemblances to causality theory.

######  Theorem 3.4 (Levin’s Theorem [44]).

Let $X$ be a second countable, locally compact Hausdorff topological space and
$R$ be a closed preorder on $X$. Then there exists a continuous function
$f:X\to$ such that

$(x,y)\in R\implies f(x)\leq f(y),$

with equality if and only if $x=y$.

That $K$-causality implies the existence of a time function is a direct
consequence of Levin’s Theorem, and is in fact true in an even more general
setting than ours, as already pointed out by Minguzzi [51].

It remains to show the converse. The next lemma gives us a more explicit
characterization of the $K$-relation. Its proof is not significantly different
to its smooth counterpart [50, Lemma 3] but included for the sake of clarity.

######  Lemma 3.5.

Let $(X,d,\ll,\leq,\tau)$ be a non-totally imprisoning, locally compact
Lorentzian pre-length space with limit curves. If $(p,q)\in K^{+}\subseteq
X\times X$, then either $p\leq q$ or for every relatively compact open set $B$
containing $p$, there exists $r\in\partial B$ such that $p<r$ and $(r,q)\in
K^{+}$.

###### Proof.

For the purposes of this proof, it will be more convenient to denote relations
as subsets of $X\times X$; in particular, $J^{+}:=\\{(p,q)\in X\times X\mid
p\leq q\\}$. Consider the relation

$\displaystyle R^{+}:=\\{(p,q)\in K^{+}\mid$ $\displaystyle\,(p,q)\in
J^{+}\text{ or for every relatively compact }$ $\displaystyle\text{ open set
}B\text{ containing }p\text{ there is an }r\in\partial B$ $\displaystyle\text{
such that }p<r\text{ and }(r,q)\in K^{+}\\}.$

Clearly, $J^{+}\subseteq R^{+}\subseteq K^{+}$. We will show that $R^{+}$ is
closed and transitive, which then implies $R^{+}=K^{+}$, and in turn proves
the Lemma. Transitivity can be proven exactly in the same way as in the smooth
case, so we refer to [50, Lemma 3].

To show closedness or $R^{+}$, consider $(p_{n},q_{n})\to(p,q)$ with
$(p_{n},q_{n})\in R^{+}$ for all $n\in\mathbb{N}$. If $p=q$ then $(p,q)\in
J^{+}\subseteq R^{+}$. We can therefore assume that $p\neq q$ and it remains
to be shown that $(p,q)\in R^{+}$ as well. Let $B$ be an open relatively
compact neighborhood of $p$. For sufficiently large $n$, $p_{n}\neq q_{n}$ and
$p_{n}\in B$. By passing to a subsequence if necessary, we can assume that
either $(p_{n},q_{n})\in J^{+}$ (case 1) or $(p_{n},q_{n})\in R^{+}\setminus
J^{+}$ (case 2) for all $n\in\mathbb{N}$:

  1. 1.

Suppose $(p_{n},q_{n})\in J^{+}$ for all $n\in\mathbb{N}$. By assumption $X$
is causally path-connected (and $p_{n}\neq q_{n}$), thus there exist future-
directed causal curves $\gamma_{n}\colon[0,1]\to X$ from $p_{n}$ to $q_{n}$.
By passing to a subsequence if necessary, we can assume that $\gamma_{n}$
either lies entirely in $\overline{B}$ for all $n$, or leaves $\overline{B}$
for all $n$.

If all the $\gamma_{n}$ lie inside $\overline{B}$, then by non-total
imprisonment their lengths (and by linear reparametrization also their
Lipschitz constants) are bounded above by a positive constant $C$ independent
of $n$. Thus we are in conditions to apply Theorem 2.13, which shows the
existence of a limit causal curve connecting $p$ and $q$, and hence $(p,q)\in
J^{+}\subseteq R^{+}$.

If, on the other hand, none of the $\gamma_{n}$ lie entirely inside
$\overline{B}$, then there is a first parameter value $s_{n}$ at which
$\gamma_{n}$ leaves $B$ (and $\gamma_{n}(s_{n})\in\partial B$ by
connectedness), and define a new sequence of curves
$\tilde{\gamma}_{n}:=\gamma_{n}|_{[0,s_{n}]}$. Linear reparametrization so
that $\tilde{\gamma}_{n}\colon[0,1]\to X$ together with Theorem 2.13 shows the
existence of a limit causal curve that connects $p$ with a point
$r=\lim_{n\to\infty}\gamma_{n}(s_{n})\in\partial B$ (again w.l.o.g. by passing
to a subsequence). Because $\gamma_{n}(s_{n})\leq q$ for all $n$ it follows
that $(r,q)\in\overline{J^{+}}\subseteq K^{+}$ by closedness of the
$K$-relation. Since also $p<r$, we again conclude that $(p,q)\in R^{+}$.

  2. 2.

Suppose $(p_{n},q_{n})\in R^{+}\setminus J^{+}$ for all $n$. Then there exist
points $r_{n}\in\partial B$ such that $p_{n}<r_{n}$ and $(r_{n},q_{n})\in
K^{+}$. By passing to a subsequence if necessary, we can assume that $r_{n}\to
r\in\partial B$. Arguing as in case 1 (for the sequence $(p_{n},r_{n})$),
either $p<r$, or there exists $r^{\prime}\in\partial B$ such that
$p<r^{\prime}$ and $(r^{\prime},r)\in\overline{J^{+}}\subseteq K^{+}$.
Combining this with the fact that $(r,q)\in K^{+}$ by closedness of the
$K$-relation, it follows that $(p,q)\in R^{+}$.

In both cases we have thus shown that $(p,q)\in R^{+}$, which concludes the
proof. ∎

The next and final lemma is key, and tells us that time functions are
$K$-utilities, in the language of economics. In other words, a time function
with respect to the causal relation $\leq$ is automatically also a time
function with respect to the $K^{+}$-relation.

######  Lemma 3.6.

Let $(X,d,\ll,\leq,\tau)$ be a locally compact Lorentzian pre-length space
with limit curves and $t\colon X\to$ be a time function. If $(p,q)\in K^{+}$,
then either $p=q$ or $t(p)<t(q)$.

###### Proof.

The proof is the same as the one of [50, Lemma 4], replacing [50, Lemma 2] and
[50, Lemma 3] by our Lemmas 2.29 and 3.5, respectively. ∎

###### Proof of Theorem 3.2.

That $K$-causal spaces admit time functions is a direct consequence of Levin’s
Theorem 3.4. Conversely, if $X$ admits a a time function, then the
$K^{+}$-relation must be antisymmetric, as otherwise it would contradict Lemma
3.6. ∎

######  Remark 3.7.

It is worth pointing out that, throughout this section, we have not made use
of the chronological relation $\ll$, nor of timelike curves. Hence, Theorem
3.2 is still valid if in Definition 2.6 we only require the existence of
causal (and not of timelike) curves, or even if $\ll$ is empty.

##  4 Volume time functions

In this section we introduce and explicitly construct a special type of
functions, called _averaged volume functions_ , on Lorentzian pre-length
spaces that are equipped with probability measures. While the existence of a
suitable measure solely depends on the topology (in fact, the metric
structure) it is the causal structure that determines whether these functions
are time functions. More precisely, we will see that the averaged volume
functions are time functions if and only if the underlying Lorentzian pre-
length space is causally continuous in Theorem 4.13 (see Theorem 1.2 for the
Lorentzian length space version thereof).

Our results generalize a classical theorem of Dieckmann [23, 21] (also stated
earlier by Hawking and Sachs [37], but with an incomplete proof). Volume
functions had already been introduced earlier by Geroch [31, Sec. 5] to study
global hyperbolicity; we will replicate those results in Section 5. In this
section, we follow the approach of Dieckmann, but also make use of an
averaging procedure similar to that used by Hawking [35] to study stable
causality and time functions. Besides that, our methods in this section are
based on order- and measure-theoretical arguments, and we do not need to
assume the existence of causal curves.

###  4.1 Averaged volume functions

To construct averaged volume functions on a Lorentzian pre-length space $X$,
we equip $X$ with a Borel measure $\mu$ satisfying

  1. (i)

$\mu(X)=1$, i.e., $\mu$ is a probability measure, and

  2. (ii)

$\operatorname{supp}(\mu)=X$, i.e., $\mu$ has full support.

When $X$ is finite the construction of $\mu$ is trivial. Otherwise such a
measure exists precisely when $(X,d)$ is a separable metric space (which is
automatically satisfied for all compact spaces).

######  Proposition 4.1.

A metric space $(X,d)$ admits a Borel probability measure $\mu$ with
$\operatorname{supp}(\mu)=X$ if and only if it is separable (equivalently,
second-countable).

###### Proof.

Suppose $(X,d)$ is separable. Then it contains a countable dense subset
$D=\\{p_{n}\mid n\in\mathbb{N}\\}$. Denote by $\delta_{n}$ the Dirac delta
measure centered at $p_{n}$, and define

$\mu:=\sum_{n\in\mathbb{N}}2^{-n}\delta_{n}.$

The measure $\mu$ has the desired properties since (i)
$\mu(X)=\sum_{n}2^{-n}=1$ and (ii) for all open sets $A$, $A\cap
D\neq\emptyset$ by denseness and hence $\mu(A)>0$. The proof of the converse
can be found in [47, p. 134].

Finally, secound countability implies separability, and on metric spaces the
two notions in fact are equivalent. ∎

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space equipped with a
Borel probability measure $\mu$ of full support. For $r\in(0,1)$ and $p\in X$,
let

$\displaystyle I^{\pm}_{r}(p)$ $\displaystyle:=\left\\{x\in X\mid
d\left(x,I^{\pm}(p)\right)<r\right\\},$ $\displaystyle V^{\pm}_{r}(p)$
$\displaystyle:=\mu\left(I^{\pm}_{r}(p)\right).$

We call $I^{\pm}_{r}(p)$ the $r$-thickening of $I^{\pm}(p)$, as depicted in
Figure 2, and $V^{\pm}_{r}(p)$ its volume.

$I^{-}(p)$$r$$I_{r}^{-}(p)$$p$ Figure 2: The sets $I^{-}(p)$ (dark blue) and
$I^{-}_{r}(p)$ (light and dark blue) for some point $p$ in Minkowski
spacetime, with $d$ the Euclidean distance.

######  Definition 4.2.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space equipped with a
Borel probability measure $\mu$ of full support. The _future_ ($+$) and _past_
($-$) _averaged volume functions_ of $\mu$ are defined by

$t^{\pm}(p):=\mp\int_{0}^{1}V^{\pm}_{r}(p)\,dr,\qquad p\in X.$

Note that the integral exists for all points $p\in X$ because the function
$r\mapsto V^{\pm}_{r}(p)$ is increasing and bounded.

######  Remark 4.3 (Comparison to previous definitions of volume functions).

The classical definition of a volume function by Geroch [31, Section 5] is
simpler and reads $t_{\mathrm{cl}}^{\pm}(p)=\mp\mu\left(I^{\pm}(p)\right)$.
However, it was discovered by Dieckmann [23, Def. 1.2] that in order to show
continuity, one has to require that $\mu$ also satisfies the property (iii)
$\mu(\partial I^{\pm}(p))=0$. On a smooth spacetime $(M,g)$ one can always
construct such an admissible measure $\mu$ from the volume form using a
partition of unity and utilize that $\partial I^{\pm}(p)$ is a hypersurface
having zero Lebesgue measure in charts [23, Prop. 1.1]. Since we do not have a
manifold structure and the Lebesgue measure at our disposal, we instead
integrate over $r$ to “average out” discontinuities, hence the addition of
“averaged” in the naming of volume functions in Definition 4.2. This averaging
procedure is inspired by the work of Hawking [35] on stable causality and time
functions.

######  Remark 4.4 ($\mu$-dependence).

It is clear that the above constructions of $I^{\pm}_{r}$, $V^{\pm}_{r}$ and
$t^{\pm}$ depend crucially on the choice of $d$ and $\mu$. We will, however,
see that the _existence_ of (generalized) time functions is at this point
independent of the particular choice of $\mu$ and also of $d$ (as long as the
metric is second countable). More precisely, whether the averaged volume
functions $t^{\pm}$ are indeed (continuous) time functions depends only on the
causal structure of $X$.

We end this subsection by proving that $t^{\pm}$ are _isotone_ or _causal_
functions (see [53, Def. 1.17]), a property that is weaker than being a time
function.

######  Lemma 4.5.

Let $X$ be a Lorentzian pre-length space as in Definition 4.2 and let $p,q\in
X$. If $p\leq q$, then $V^{-}_{r}(p)\leq V^{-}_{r}(q)$ and $V^{+}_{r}(p)\geq
V^{+}_{r}(q)$. In particular,

$p\leq q\Longrightarrow t^{\pm}(p)\leq t^{\pm}(q).$

###### Proof.

By the push-up Lemma 2.3, $p\leq q$ implies $I^{-}(p)\subseteq I^{-}(q)$, and
hence $I^{-}_{r}(p)\subseteq I^{-}_{r}(q)$ for all $r\in(0,1)$. The first
conclusion thus follows from the monotonicity of $\mu$, and the second one
from the monoticity of the integral. ∎

###  4.2 Averaged volume functions as generalized time functions

Finally, in order to show that $t^{\pm}$ are generalized time functions we
need to apply the standard distinguishing causality condition, or at least our
own weaker version therefore.

######  Definition 4.6 ([41, p. 486]).

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space. We say that

  1. (i)

$X$ is _past-distinguishing_ if

$I^{-}(p)=I^{-}(q)\Longrightarrow p=q,\qquad p,q\in X,$

  2. (ii)

$X$ is _future-distinguishing_ if

$I^{+}(p)=I^{+}(q)\Longrightarrow p=q,\qquad p,q\in X.$

We call $X$ _distinguishing_ if it is both past- and future-distinguishing.

######  Definition 4.7.

We say that $X$ is _causally_ (_past_ \- or _future_ -) _distinguishing_ if
the conditions of Definition 4.6 are only required to hold for all $p,q\in X$
with $p\leq q$.

Furthermore, we assume that the Lorentzian pre-length spaces are approximating
(see Section 2.3). This avoids the pathological situation where the future or
past of a point could be “far away” from the point itself, or empty.

Having equipped our spaces with sufficient causal and topological conditions,
we are in a position to establish the well-known classical result about
generalized time functions [23, Prop. 2.2] also for separable Lorentzian pre-
length spaces (for which averaged volume functions $t^{\pm}$ from Definition
4.2 are well-defined by Proposition 4.1).

######  Proposition 4.8.

Let $(X,d,\ll,\leq,\tau)$ be a past-(future-)approximating Lorentzian pre-
length space equipped with a Borel probability measure of full support. Then
$X$ is causally past-(future-)distinguishing if and only if $t^{-}$ ($t^{+}$)
is a generalized time function i.e., for all $p,q\in X$

$p<q\Longrightarrow t^{\mp}(p)<t^{\mp}(q).$

###### Proof.

We prove the past version. If $X$ is causally past-distinguishing, then for
all $p,q\in X$ with $p<q$,

$I^{-}(p)\subsetneq I^{-}(q).$

Clearly, $q\not\in I^{-}(p)$, and we show that also $q\not\in\partial
I^{-}(p)$: For any $x\in I^{-}(q)$ the future $I^{+}(x)$ is open and thus
contains an open set $V$ around $q$. If $q\in\partial I^{-}(p)$ then there is
a point $y\in V\cap I^{-}(p)$, thus $x\ll y\ll p$ and by transitivity $x\in
I^{-}(p)$. Therefore $I^{-}(q)\subseteq I^{-}(p)$, a contradiction.

Since any metric space is normal, the disjoint closed sets $\\{q\\}$ and
$\overline{I^{-}(p)}$ can be separated by disjoint open sets. In particular,
there exists an open neighborhood $U$ around $q$ such that
$r_{0}:=d(U,I^{-}(p))>0$. The intersection $U\cap I^{-}(q)$ is open, and by
the past-approximating property of $X$ at $q$ (see Lemma 2.18) also non-empty.
Thus

$\displaystyle t^{-}(q)-t^{-}(p)$
$\displaystyle=\int_{0}^{1}\mu\left(I_{r}^{-}(q)\setminus
I_{r}^{-}(p)\right)dr$ $\displaystyle\geq\int_{0}^{r_{0}}\mu\left(U\cap
I^{-}(q)\right)dr>0.$

To prove the converse, assume that for all $p<q$ we have
$t^{-}(q)-t^{-}(p)>0$. In order to show that $X$ is causally past-
distinguishing furthermore assume that $I^{-}(p)=I^{-}(q)$. But then, clearly
$I^{-}_{r}(p)=I^{-}_{r}(q)$ for all $r\in[0,1]$, and therefore
$t^{-}(p)=t^{-}(q)$, a contradiction. Hence $p=q$. ∎

Example 3.3 is a Lorentzian pre-length space that is causally distinguishing
but not distinguishing (any two points on the same level set of $t$ have the
same past, but are not causally related to one another). One can how that
under certain conditions the two notions agree, in particular, for smooth
spacetimes.

######  Proposition 4.9.

Let $(X,d,\ll,\leq,\tau)$ be a locally compact, causally path-connected,
locally weakly causally closed, (past-/future-)approximating Lorentzian pre-
length space. If $X$ is causally (past-/future-)distinguishing, then it is
also (past-/future-)distinguishing.

###### Proof.

Suppose $X$ is causally past-distinguishing but not past-distinguishing (the
future case is analogous). Then there exist two distinct points $p,q\in X$
such that $I^{-}(p)=I^{-}(q)$. By Lemma 2.18 there is a sequence
$(p_{n}^{-})_{n}$ that approximates $p$ from the past. Because $p_{n}^{-}\in
I^{-}(p)=I^{-}(q)$ and $X$ is causally path-connected, there exists a sequence
of causal curves $(\gamma_{n})_{n}$ connecting $p_{n}^{-}$ and $q$. Let
$\delta$ be small enough so that $B_{\delta}(p)$ is compact and does not
contain $q$. Consider the sequence of points $(r_{n})_{n}$ where $\gamma_{n}$
first intersects $\partial B_{\delta/2}(p)$. By compactness, we may assume it
converges to a point $r\in\partial B_{\delta/2}(p)$. Without loss of
generality suppose that $B_{\delta}(p)$ is contained in a weakly causally
closed neighborhood, and so furthermore $p\leq r$ and $I^{-}(p)\subseteq
I^{-}(r)$ by the push-up Lemma 2.3. Moreover, by openness of $\ll$ we have
$I^{-}(r)\subseteq\bigcup_{n}I^{-}(r_{n})$, and since $r_{n}\ll q$, by
transitivity we have $I^{-}(r_{n})\subseteq I^{-}(q)$. Hence
$I^{-}(p)\subseteq I^{-}(r)\subseteq I^{-}(q)$, which together with our
initial assumption $I^{-}(p)=I^{-}(q)$ implies that $I^{-}(p)=I^{-}(r)$. Since
$X$ is causally past-distinguishing we thus know that $p=r$, which contradicts
the fact that $d(p,r)=\frac{\delta}{2}>0$. Thus $X$ must indeed also be past-
distinguishing. ∎

###  4.3 Continuity of averaged volume functions

We conclude this section by generalizing the equivalence between causal
continuity and the continuity of volume functions (shown in the smooth case by
Dieckmann [23, Proposition 2.5]) to establish when $t^{\pm}$ of Definition 4.2
are time functions.

######  Definition 4.10.

Let $\tau$ be a Lorentzian pre-length space. If for all $p,q\in X$,

  1. (i)

$I^{+}(p)\subseteq I^{+}(q)$ implies $I^{-}(q)\subseteq I^{-}(p)$, we say that
$X$ is _past reflecting_.

  2. (ii)

$I^{-}(p)\subseteq I^{-}(q)$ implies $I^{+}(q)\subseteq I^{+}(p)$, we say that
$X$ is _future reflecting_.

If $X$ is both past and future reflecting, we say that it is _reflecting_.

We define causal continuity as in the smooth case. Aké et al. showed that this
notion of causal continuity implies strong causality [1, Proposition 3.15], as
it does in the smooth case. However, it turns out that the optimal condition
for our Theorem 4.13 is a weaker version thereof. This weaker version is not
(trivially) sufficient for the results of Aké et al. to still hold. In any
case, both definitions are equivalent when the conditions of Proposition 4.9
are met (in particular, in the smooth case).

######  Definition 4.11 ([1, Definition 3.9]).

A Lorentzian pre-length space is called _causally continuous_ if it is
reflecting and distinguishing.

######  Definition 4.12.

A Lorentzian pre-length space is called _weakly causally continuous_ if it is
reflecting and causally distinguishing.

Using this terminology we can state the main result of this section (see
Theorem 1.2 for the corresponding Lorentzian length space version).

######  Theorem 4.13.

Let $X$ be an approximating Lorentzian pre-length space equipped with a Borel
probability measure of full support. Then $X$ is weakly causally continuous if
and only if the averaged volume functions $t^{\pm}$ are time functions.

Note that also in this section we combine a topological condition (second
countable) and a causal condition ((weakly) causally continuous) to
characterize the existence of volume time functions.

###### Proof.

By Proposition 4.8 the averaged volume functions are generalized time
functions if and only if $X$ is causally distinguishing. In Lemma 4.16 below
we show that their continuity is characterized by $X$ being reflecting. ∎

First we show that property (iii) discussed in Remark 4.3 holds for almost all
thickenings of $I^{\pm}(p)$.

######  Lemma 4.14.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space equipped with a
Borel probability measure $\mu$ of full support. For $p\in X$ consider the
functions $r\mapsto V^{\pm}_{r}(p)=\mu(I^{\pm}_{r}(p))$ as defined in Section
4.1 and

$r\mapsto\overline{V^{\pm}_{r}}(p):=\mu\left(\overline{I^{\pm}_{r}(p)}\right).$

Then the sets

$S^{\pm}(p):=\left\\{r\in(0,1)\mid\overline{V^{\pm}_{r}}(p)\neq
V^{\pm}_{r}(p)\right\\}$

are countable in $(0,1)$ for all $p\in X$.

###### Proof.

Let $p\in X$. By definition and additivity of $\mu$ we have

$\displaystyle\overline{V^{\pm}_{r}}(p)-V^{\pm}_{r}(p)$
$\displaystyle=\mu(\overline{I^{\pm}_{r}(p)})-\mu(I^{\pm}_{r}(p))$
$\displaystyle=\mu\left(\overline{I^{\pm}_{r}(p)}\setminus
I^{\pm}_{r}(p)\right)=\mu(\partial I^{\pm}_{r}(p)).$

Hence we can rewrite $S^{\pm}(p)$ as

$S^{\pm}(p)=\left\\{r\in(0,1)\ \middle|\ \mu\left(\partial
I^{\pm}_{r}(p)\right)>0\right\\}.$

To see that $S^{\pm}(p)$ is countable, consider the sets

$S_{n}^{\pm}(p):=\left\\{r\in(0,1)\ \middle|\ \mu\left(\partial
I^{\pm}_{r}(p)\right)>\frac{1}{n}\right\\},$

for $n\in\mathbb{N}$. We show that $|S_{n}^{\pm}(p)|\leq n$. Otherwise there
would exist at least $n+1$ distinct $r_{i}\in S_{n}^{\pm}(p)$. Since all
$\partial I^{\pm}_{r_{i}}(p)$ are disjoint, this would imply that

$\frac{n+1}{n}<\sum_{i=1}^{n+1}\mu(\partial I^{\pm}_{r_{i}}(p))\leq\mu(X)=1,$

a contradiction. Finally, because
$S^{\pm}(p)=\bigcup_{n\in\mathbb{N}}S_{n}^{\pm}(p)$, we deduce that the sets
$S^{\pm}(p)$ are countable for any $p\in X$. ∎

In addition to Lemma 4.14, the continuity of the averaged volume functions
rests on the following general result which is based on Lebesgue’s dominated
convergence theorem (in [39] formulated for $X=^{d}$ but true for all
sequential spaces).

######  Theorem 4.15 ([39, Theorem 16.10]).

Let $(X,d)$ be a metric space, $U\subseteq X$ and $y_{0}\in U$. Consider a
function $f\colon^{n}\times U\to\cup\\{\pm\infty\\}$. Assume that

  1. (i)

for every fixed $y\in U$ the function $x\mapsto f(x,y)$ is integrable,

  2. (ii)

for almost all $x\in^{n}$ the function $y\mapsto f(x,y)$ is continuous at
$y_{0}$,

  3. (iii)

there exists an integrable function $F\colon^{n}\to\cup\\{+\infty\\}$ with the
property that for every $y\in U$,

$|f(x,y)|\leq F(x)$

holds almost everywhere on n.

Then the function

$g(y):=\int_{{}^{n}}f(x,y)\,dx$

is continuous at the point $y_{0}$.

With this, we can prove the last lemma of this section (compare with [23,
Proposition 1.6]). Together with Proposition 4.8, it constitutes the proof of
Theorem 4.13.

######  Lemma 4.16.

Let $(X,d,\ll,\leq,\tau)$ be an approximating Lorentzian pre-length space
equipped with a Borel probability measure $\mu$ of full support. For any point
$q\in X$, the following are equivalent:

  1. (i)

$t^{-}$ ($t^{+}$) is continuous at $q$.

  2. (ii)

$I^{+}(q)\subseteq I^{+}(p)\implies I(\tilde{p},p)\cap I^{-}(q)\neq\emptyset$
for all $\tilde{p}\ll p$
($I^{-}(q)\subseteq I^{-}(p)\implies I(p,\tilde{p})\cap I^{-}(q)\neq\emptyset$
for all $p\ll\tilde{p}$).

  3. (iii)

$X$ is past (future) reflecting at $q$.

  4. (iv)

$\bigcap_{q\ll x}I^{-}(x)\subseteq\overline{I^{-}(q)}$
($\bigcap_{x\ll q}I^{+}(x)\subseteq\overline{I^{+}(q)}$).

###### Proof.

We show the past versions. Fix $q\in X$.

(i) $\Longrightarrow$ (ii) Assume (i) holds but not (ii). Then there exist
points $p$ and $\tilde{p}$ such that $\tilde{p}\ll p$ and $I^{+}(q)\subseteq
I^{+}(p)$ but $I(\tilde{p},p)\cap I^{-}(q)=\emptyset$. Note that
$I(\tilde{p},p)$ is open and non-empty, because we can approximate $p$ from
the past, and by openness of $\ll$, any past-approximating sequence must enter
$I^{+}(\tilde{p})$. Let $r_{0}>0$ be small enough, then also the set

$B:=\left\\{x\in I(\tilde{p},p)\mid
d(x,I^{-}(q))>r_{0}\right\\}=I(\tilde{p},p)\setminus\overline{I^{-}_{r_{0}}(q)}.$

is open and non-empty: If not, then
$I(\tilde{p},p)\subseteq\overline{I^{-}_{r_{0}}(q)}$. But since
$I(\tilde{p},p)$ is open, this would mean that $I(\tilde{p},p)\cap
I^{-}(q)\neq\emptyset$, contradicting our earlier assumption. Clearly, $B\cap
I_{r}^{-}(q)=\emptyset$ for all $r<r_{0}$. On the other hand, since
$q_{n}^{+}\in I^{+}(q)\subseteq I^{+}(p)$, transitivity of $\ll$ implies that
$I(\tilde{p},p)\subseteq I^{-}(p)\subseteq I^{-}(q_{n}^{+})$. For a sequence
$(q_{n}^{+})$ approximating $q$ from the future we have

$t^{-}(q_{n}^{+})-t^{-}(q)=\int_{0}^{1}\mu\left(I^{-}_{r}(q_{n}^{+})\setminus
I^{-}_{r}(q)\right)dr.$

Since $B\subseteq I^{-}_{r}(q_{n}^{+})\setminus I^{-}_{r}(q)$ for $r<r_{0}$,
it follows that

$t^{-}(q_{n}^{+})-t^{-}(q)\geq\int_{0}^{r_{0}}\mu(B)\,dr>0,$

showing discontinuity of $t^{-}$ at $q$, a contradiction to (i).

(ii) $\Longrightarrow$ (iii) Assume (ii) holds but not (iii). If $X$ is not
past reflecting at $q$, then there exists a $p\in X$ with $I^{+}(q)\subseteq
I^{+}(p)$ but $I^{-}(p)\not\subseteq I^{-}(q)$. Thus there is a $\tilde{p}\in
I^{-}(p)\setminus I^{-}(q)$ and by (ii) there exists a point $\hat{p}\in
I(\tilde{p},p)\cap I^{-}(q)$. But then $\tilde{p}\ll\hat{p}\ll q$, and by
transitivity $\tilde{p}\ll q$, a contradiction.

(iii) $\Longrightarrow$ (iv) If $\bigcap_{q\ll x}I^{-}(x)=\emptyset$ the
conclusion is trivial. Suppose $p\in\bigcap_{q\ll x}I^{-}(x)$. Then $x\in
I^{+}(p)$ for all $x\gg q$, i.e., $I^{+}(q)\subseteq I^{+}(p)$. By (iii) $X$
is past reflecting at $q$, hence $I^{-}(p)\subseteq I^{-}(q)$. Since $X$ is
past-approximating, $p\in
J^{-}(p)\subseteq\overline{I^{-}(p)}\subseteq\overline{I^{-}(q)}$. Since $p$
was an arbitrary point in the intersection, (iv) follows.

(iv) $\Longrightarrow$ (i) Fix $r\in(0,1)$ and let $\epsilon>0$. Because $X$
is past-approximating, we can find a sequence $(q^{-}_{i})_{i}$ that
approximates $q$ from the past. By openness of $\ll$, it then follows that

$\bigcup_{i=1}^{\infty}I^{-}_{r}(q^{-}_{i})=I^{-}_{r}(q),$

and by standard measure theory [40, Theorem 1.2.5] there exists
$i_{0}\in\mathbb{N}$, such that

$\mu\left(I^{-}_{r}(q)\right)-\mu\left(I^{-}_{r}(q^{-}_{i_{0}})\right)<\epsilon.$

By Lemma 4.5, we deduce

$V^{-}_{r}(q)-V^{-}_{r}(p)<\epsilon\quad\text{ for all }\quad p\in
I^{+}(q^{-}_{i_{0}}).$ (1)

Next, consider $(q^{+}_{j})_{j}$ a sequence approximating $q$ from the future.
Assumption (iv) implies that

$\bigcap_{j=1}^{\infty}I_{r}^{-}(q^{+}_{j})\subseteq\overline{I_{r}^{-}(q)},$

and hence

$\mu\left(\bigcap_{j=1}^{\infty}I_{r}^{-}(q^{+}_{j})\right)\leq\mu\left(\overline{I_{r}^{-}(q)}\right).$

By [40, Theorem 1.2.5], for $r$ given, there exists $j_{0}\in\mathbb{N}$ such
that

$\mu\left(I_{r}^{-}(q^{+}_{j_{0}})\right)-\mu\left(\overline{I_{r}^{-}(q)}\right)<\epsilon.$

Then, using Lemma 4.5 we deduce

$V^{-}_{r}(p)-\overline{V^{-}_{r}}(q)<\epsilon\quad\text{ for all }\quad p\in
I^{-}(q^{+}_{j_{0}}).$ (2)

By Lemma 4.14, $\overline{V^{-}_{r}}(q)={V^{-}_{r}}(q)$ for all but countably
many $r$. Hence for almost all $r\in(0,1)$, we can combine (1) and (2) to
write

$|V_{r}^{-}(q)-V_{r}^{-}(p)|<\epsilon\quad\text{ for all }\quad p\in
I(q^{-}_{i_{0}},q^{+}_{j_{0}}).$

We conclude that almost all functions $V^{-}_{r}\colon X\to$, $r\in(0,1)$, are
continuous at $q$. Thus it follows from Theorem 4.15 that

$t^{-}(p)=\int_{0}^{1}V^{-}_{r}(p)\,dr$

is continuous at $q$. ∎

##  5 Global hyperbolicity and Cauchy time functions

The highest step on the causal ladder, namely global hyperbolicity, is
fundamental for a number of deep and important results in general relativity,
such as the study of the Cauchy problem of the Einstein equations, the
singularity theorems, and Lorentzian splitting theorems. This is due to the
fact global hyperbolicity is equivalent to the existence of a Cauchy time
function, whose level sets in turn are Cauchy surfaces, i.e., domains suitable
for specifying initial data for hyperbolic PDEs, and for imposing focusing
conditions for geodesics. This characterization of global hyperbolicity was
first obtained by Geroch [31] in 1970 and makes use of volume time functions.
In the same vein, in this section we characterize global hyperbolicity for
Lorentzian pre-length spaces in four different ways by also utilizing our
constructions from Section 4.

###  5.1 Definitions and main result

The causality conditions we use for Lorentzian pre-length space are defined
analogously to the smooth case as follows (cf. [53]).

######  Definition 5.1 ([42, Definition 2.35]).

A Lorentzian pre-length space $(X,d,\ll,\leq,\tau)$ is called _globally
hyperbolic_ if it is non-totally imprisoning and the causal diamonds $J(p,q)$
are compact for all $p,q\in X$.

######  Definition 5.2.

A time function $t\colon X\to$ on a Lorentzian pre-length space
$(X,d,\ll,\leq,\tau)$ is called a _Cauchy time function_ if for every doubly-
inextendible causal curve $\gamma$ we have $\operatorname{Im}(t\circ\gamma)=$.

For smooth and continuous Lorentzian metrics, global hyperbolicity is also
characterized by the existence of a Cauchy surface, which is then a
topological (even smooth) hypersurface. We extend the definition verbatim, but
adopt the name _Cauchy set_ to emphasize that we are not in the manifold
setting.

######  Definition 5.3.

Let $(X,d,\ll,\leq,\tau)$ be a Lorentzian pre-length space. A _Cauchy set_ is
a subset $\mathcal{S}\subseteq X$ such that every doubly-inextendible causal
curve intersects $\mathcal{S}$ exactly once.

Geroch [31, Section 4] makes use of Leray’s notion of global hyperbolicity
which is formulated in terms of the topology on the collection of certain
curves. We will show that Definition 5.1 is equivalent to this notion also for
Lorentzian pre-length spaces. To this end, for any two points $p,q\in X$, we
consider the set $\mathcal{C}(p,q)$ the equivalence class of future-directed
causal curves from $p$ to $q$ with continuous, strictly monotonically
increasing reparametrizations, equipped with the Hausdorff distance between
the images of the curves as subsets in $X$, i.e.,

$d_{H}(\gamma_{1},\gamma_{2})=\max\\{\sup_{x\in\gamma_{1}}d(x,\gamma_{2}),\sup_{y\in\gamma_{2}}d(\gamma_{1},y)\\}$

(we write $\gamma_{i}$ also for the image $\operatorname{Im}(\gamma_{i})$,
since the parametrization does not matter).

In this section we establish the third, and last, main result of this paper
(which via 2.20 immediately implies the Lorentzian length version stated in
Theorem 1.3 of the Introduction).

######  Theorem 5.4.

Let $(X,d,\ll,\leq,\tau)$ be an approximating Lorentzian pre-length space with
limit curves. Suppose, in addition, that $(X,d)$ is second countable and
proper. Then the following are equivalent:

  1. (i)

$X$ is globally hyperbolic,

  2. (ii)

$X$ is non-totally imprisoning and $\mathcal{C}(p,q)$ is compact, for any
$p,q\in X$,

  3. (iii)

$X$ admits a Cauchy time function,

  4. (iv)

$X$ admits a Cauchy set.

######  Remark 5.5 (Smooth spacetimes).

Manifolds are second-countable by definition. Moreover, any differentiable
manifold admits a complete Riemannian metric [57], and hence, by the
Hopf–Rinow Theorem, a proper distance. For spacetimes with continuous metrics,
however, Theorem 5.4 cannot be applied unrestrictedly, since only causally
plain $C^{0}$-spacetimes satisfy the axioms of Lorentzian pre-length spaces
[42, Example 5.2] (but a characterization of global hyperbolicity on _all_
$C^{0}$-spacetimes has been obtained in [60]). On the other hand, Theorem 5.4
is of course valid well beyond the manifold setting. For instance, by the
Hopf–Rinow–Cohn-Vossen Theorem it is sufficient that $(X,d)$ is a complete,
locally compact length metric space for it to be proper.

######  Remark 5.6 (Topology change).

On a spacetime $(M,g)$, if $\mathcal{S}$ is a smooth Cauchy surface, then any
other Cauchy surface is diffeomorphic to $\mathcal{S}$ [7] (similarly, if one
works with continuous Cauchy surfaces, then they are homeomorphic). The
diffeomorphism can be constructed by following the flow of the time-
orientation vector field. Further, $M$ is foliated by Cauchy surfaces.

For Lorentzian pre-length spaces $X$, it is still true that in the setting of
Theorem 5.4 the level sets of Cauchy time functions yield a decomposition of
$X$ as a disjoint union of Cauchy sets. Different Cauchy sets, however, need
not be homeomorphic, nor even homotopy equivalent as the next examples show.

######  Example 5.7 (Degenerate subsets of Minkowski space).

Let $(M,\eta)$ be $(n+1)$-dimensional Minkowski spacetime, with coordinates
$(t,x)\in\times^{n}$. Define

$X:=\\{(t,x)\mid t\leq 0,x=0\text{ or }t>0,|x|=t/2\\}$

(here $|x|$ denotes the Euclidean norm of $x\in^{n}$), equipped with the
causal and chronological relations induced by $\eta$ and the Euclidean
distance. Then $X$ is a globally hyperbolic Lorentzian pre-length space and
satisfies the assumptions of Theorem 5.4. The function $f(t,x)=t$ is a Cauchy
time function, with some of its level sets being points (if $t\leq 0$) and
some being $(n-1)$-spheres (if $t>0$).

######  Example 5.8 (Degenerate generalized cones).

If $(X_{1},d_{1})$, $(X_{2},d_{2})$ be separable, proper geodesic length
spaces one can construct generalized cones $Y_{1}$ and $Y_{2}$ in the sense of
[2] over them and glue them together at the tip. The resulting space can be
equipped with the structure of a Lorentzian length space in the usual way,
which is then globally hyperbolic by [2, Prop. 4.10], and has Cauchy sets
homeomorphic to $X_{1}$, $X_{2}$ and $\\{\text{tip}\\}$.

###  5.2 Properties of Cauchy sets

In what follows we prove several results involving Cauchy sets that are
crucial in the proof of Theorem 5.4 for the implication (iv) $\Longrightarrow$
(ii), but are also of independent interest. We make use of several results
about inextendible causal curves, in particular, about the existence of
maximal extension of causal curves obtained in Section 2.1.

######  Proposition 5.9 (Basic properties of Cauchy sets).

Let $(X,d,\ll,\leq,\tau)$ be an approximating Lorentzian pre-length space with
limit curves such that $(X,d)$ is proper. If $\mathcal{S}\subseteq X$ is a
Cauchy set, then the following properties hold:

  1. (i)

$\mathcal{S}$ is _acausal_ , i.e., distinct points on $\mathcal{S}$ are not
causally related,

  2. (ii)

$X=J^{-}(\mathcal{S})\cup J^{+}(\mathcal{S})$,

  3. (iii)

$X=I^{-}(\mathcal{S})\sqcup\mathcal{S}\sqcup I^{+}(\mathcal{S})$, where
$\sqcup$ denotes the disjoint union,

  4. (iv)

$J^{\pm}(\mathcal{S})=\overline{I^{\pm}(\mathcal{S})}=I^{\pm}(\mathcal{S})\sqcup\mathcal{S}$,
and $\mathcal{S}$ and $J^{\pm}(\mathcal{S})$ are closed.

###### Proof.

(i) Suppose $p<q$ and $p,q\in\mathcal{S}$. Then by causal path-connectedness,
there exists a causal curve $\gamma\colon[a,b]\to X$ from $p$ to $q$. By
Proposition 2.22 there exists a maximal doubly-inextendible extension
$\lambda$ of $\gamma$, which therefore intersects $\mathcal{S}$ more than
once, a contradiction to $\mathcal{S}$ being a Cauchy set.

(ii) Let $p\in X$. By Corollary 2.23, there exists a doubly-inextendible
causal curve $\gamma$ through $p$. Since $\mathcal{S}$ is a Cauchy set, it
intersects $\gamma$ exactly once, say at $\gamma(0)$. Hence $p\leq\gamma(0)$
or $\gamma(0)\leq p$.

(iii) Let $p\in X$. By Lemma 2.19, we can assume that the curve $\gamma$ in
(ii) is timelike in a neighborhood of $p$. Then, if $p\leq\gamma(0)$, we have
either $p=\gamma(0)\in\mathcal{S}$ or $p\ll\gamma(-\epsilon)<\gamma(0)$ for
some $\epsilon>0$, which then by the push-up Lemma 2.3 implies
$p\ll\gamma(0)$. Applying the same argument in the case $\gamma(0)\leq p$, we
conclude that $p\in I^{-}(\mathcal{S})\cup\mathcal{S}\cup I^{+}(\mathcal{S})$.

It remains to be shown that the union is disjoint. Since $\mathcal{S}$ is
acausal by (i) and the push-up property it is clear that $\mathcal{S}$ and
$I^{\pm}(\mathcal{S})$ are disjoint. Moreover, $I^{+}(\mathcal{S})$ and
$I^{-}(\mathcal{S})$ are disjoint since otherwise, there exists a causal curve
that intersects $\mathcal{S}$ at two different points or is closed timelike
(if the points coincide), and hence contradict $\mathcal{S}$ being Cauchy.

(iv) The sets $I^{\pm}(\mathcal{S})=\bigcup_{p\in\mathcal{S}}I^{\pm}(p)$ are
unions of open sets and hence open. By (iii) is
$\mathcal{S}=X\setminus(I^{-}(\mathcal{S})\sqcup I^{+}(\mathcal{S}))$ the
complement of open sets and thus closed. By the same argument is $X\setminus
I^{\mp}(\mathcal{S})=I^{\pm}(\mathcal{S})\sqcup\mathcal{S}$ closed, and hence

$\overline{I^{\pm}(\mathcal{S})}\subseteq\overline{I^{\pm}(\mathcal{S})\sqcup\mathcal{S}}=I^{\pm}(\mathcal{S})\sqcup\mathcal{S}\subseteq
J^{\pm}(\mathcal{S})\subseteq\overline{I^{\pm}(\mathcal{S})},$

where the last inclusion is due to the assumption that $X$ is approximating.
Thus $J^{\pm}(\mathcal{S})=\overline{I^{\pm}(\mathcal{S})}$, and hence closed.
∎

Closedness of Cauchy sets, in particular, will be key in establishing (iv)
$\Longrightarrow$ (ii) of Theorem 5.4. In the remaining lemmas of this
subsection we prove this implication in steps.

######  Lemma 5.10.

Let $(X,d,\ll,\leq,\tau)$ be an approximating Lorentzian pre-length space with
limit curves. Suppose, in addition, that $(X,d)$ is proper. If $X$ contains a
Cauchy set $\mathcal{S}$, then $X$ is non-totally imprisoning.

###### Proof.

Suppose $X$ is totally imprisoning. Then by Theorem 2.27, there is a compact
set $K\subseteq X$ and a doubly-inextendible future-directed causal curve
$\gamma\colon\to K$. By Definition 5.3 we know that, without loss of
generality, $\gamma\cap\mathcal{S}=\\{\gamma(0)\\}$.

First we show that $\tilde{\gamma}:=\gamma|_{[1,\infty)}$ must come
arbitrarily close to $\mathcal{S}$, that is, there exist parameter values
$s_{n}$ such that $d(\tilde{\gamma}(s_{n}),\mathcal{S})<1/n$, for all
$n\in\mathbb{N}$. Suppose this is not the case: then there exists a $\delta$
such that $d(\tilde{\gamma}(s),\mathcal{S})>\delta$ for all $s\in[1,\infty)$,
and hence $\tilde{\gamma}$ is contained in the compact set
$K\setminus\mathcal{S}_{\delta}$, where $\mathcal{S}_{\delta}:=\\{x\in X\mid
d(x,\mathcal{S})<\delta\\}$. But then, since $\tilde{\gamma}$ is future-
inextendible, by the proof of Theorem 2.27, there also exists a doubly-
inextendible causal curve contained in $K\setminus\mathcal{S}_{\delta}$, in
contradiction to $\mathcal{S}$ being a Cauchy set. Thus we conclude that there
must exist a sequence $(s_{n})_{n}$ such that
$d(\tilde{\gamma}(s_{n}),\mathcal{S})\to 0$. Moreover, since $\mathcal{S}$ is
closed by Proposition 5.9 (iv), $K\cap\mathcal{S}$ is compact, and hence
$\left(\tilde{\gamma}(s_{n})\right)_{n}$ converges, up to a subsequence, to
some point $p\in\mathcal{S}$. This implies, since
$\gamma\cap\mathcal{S}=\\{\gamma(0)\\}$, that $s_{n}\to\infty$.

Having established the existence of a sequence $s_{n}\to\infty$ in such that
$\gamma(s_{n})\to p\in\mathcal{S}$, we may consider the sequence of past-
directed causal curves $\lambda_{n}:[0,s_{n}]\to X$ given by

$\lambda_{n}(s)=\gamma(s_{n}-s).$

By Lemma 2.9, $\gamma$ has infinite arclength, and thus
$L^{d}(\lambda_{n})\to\infty$. Since $\lambda_{n}(0)\to p$, we can apply
Theorem 2.16 to find a past-directed limit curve $\lambda:[0,\infty)\to X$
with $\lambda(0)=p$, hence $\lambda([0,\infty))\subseteq J^{-}(p)\subseteq
J^{-}(\mathcal{S})$. On the other hand, we have that
$\gamma([0,\infty))\subseteq J^{+}(\gamma(0))\subseteq J^{+}(\mathcal{S})$,
therefore $\lambda_{n}([0,s_{n}])\subseteq J^{+}(\mathcal{S})$. By Proposition
5.9(iv), $J^{+}(\mathcal{S})$ is closed, so as a limit curve, $\lambda$ is
contained in it. But then $\lambda([0,\infty))\subseteq J^{-}(\mathcal{S})\cap
J^{+}(\mathcal{S})$. This is a contradiction since, by Proposition 5.9 (iv),
$J^{-}(\mathcal{S})\cap J^{+}(\mathcal{S})=\mathcal{S}$, and by Proposition
5.9 (i), Cauchy surfaces are acausal. ∎

Finally, the last two lemmas prove compactness of $\mathcal{C}(p,q)$. They are
adapted from Geroch’s original proof for smooth spacetimes [31]. Similarly to
$\mathcal{C}(p,q)$, we denote by $\mathcal{C}(p,\mathcal{S})$ the space of
causal curves from a point $p\in X$ to a Cauchy set $\mathcal{S}$, equipped
with the Hausdorff distance.

######  Lemma 5.11.

Let $(X,d,\ll,\leq,\tau)$ be an approximating Lorentzian pre-length space with
limit curves. Suppose, in addition, that $(X,d)$ is proper. Let $\mathcal{S}$
be a Cauchy surface in $X$. Then for all $p\in X$, the space
$\mathcal{C}(p,\mathcal{S})$ is compact.

###### Proof.

Suppose $\mathcal{C}(p,\mathcal{S})\neq\emptyset$, otherwise the statement is
trivial. In particular, this means that $p\in J^{-}(\mathcal{S})$. The set
$\mathcal{C}(p,\mathcal{S})$ together with the Hausdorff distance $d_{H}$ is a
metric space, and hence it remains to prove sequential compactness. Let
$(\gamma_{n})_{n}$ be a sequence in $\mathcal{C}(p,\mathcal{S})$. We
distinguish between two cases:

  1. 1.

There exists a constant $C>0$ such that $L^{d}(\gamma_{n})<C$ for all
$n\in\mathbb{N}$. Then by Theorem 2.13, a subsequence of $(\gamma_{n})_{n}$
converges uniformly (and thus also with respect to $d_{H}$) to a future-
directed causal limit curve $\gamma\colon[0,1]\to X$. Since $\gamma_{n}(0)=p$,
also $\gamma(0)=p$. Moreover, $\gamma_{n}(1)\in\mathcal{S}$ for all
$n\in\mathbb{N}$, and $\mathcal{S}$ is closed by Proposition 5.9 (iv), thus
also $\gamma(1)\in\mathcal{S}$. Hence $\gamma\in\mathcal{C}(p,\mathcal{S})$.

  2. 2.

On the other hand, assume that $L^{d}(\gamma_{n})\to\infty$. By Theorem 2.16 a
subsequence of $(\gamma_{n})_{n}$ converges locally uniformly to a future-
inextendible causal limit curve $\gamma\colon[0,\infty)\to X$. Since
$\gamma_{n}\subseteq J^{-}(\mathcal{S})$, and by Proposition 5.9(iv),
$J^{-}(\mathcal{S})$ is closed, we have $\gamma\subseteq J^{-}(\mathcal{S})$.
By Proposition 2.22, there exists a doubly-inextendible enxtension
$\tilde{\gamma}\colon\to X$ of $\gamma$, which by transitivity of $\leq$ is
also contained in $J^{-}(\mathcal{S})$. Then $\tilde{\gamma}$ must intersect
$\mathcal{S}$, say at $\tilde{\gamma}(s_{0})$. This implies that for all
$s>s_{0}$, $\gamma(s)\in J^{-}(\mathcal{S})\cap
J^{+}(\mathcal{S})=\mathcal{S}$. Moreover, $\gamma|_{(s_{0},\infty)}$ cannot
be constant, because then by Lemma 2.9, $\gamma$ would be future-extendible.
Since by Proposition 5.9 (i) $\mathcal{S}$ is acausal, we arrive at a
contradiction. ∎

######  Lemma 5.12.

Let $(X,d,\ll,\leq,\tau)$ be a approximating Lorentzian pre-length space with
limit curves. Suppose, in addition, that $(X,d)$ is proper. If $X$ contains a
Cauchy set $\mathcal{S}$, then $\mathcal{C}(p,q)$ is compact for all $p,q\in
X$.

###### Proof.

By Lemma 5.10, if $X$ contains a Cauchy set $\mathcal{S}$, then $X$ is non-
totally imprisoning and hence causal. Thus we may assume without loss of
generality that $p<q$ so that $\mathcal{C}(p,q)$ is nontrivial. We prove
sequential compactness, with $(\gamma_{n})_{n}$ always denoting a sequence in
$\mathcal{C}(p,q)$. In view of Proposition 5.9 (i) and (iii), we can
distinguish three cases:

  * 1.

$p\in\mathcal{S}$ and $q\in I^{+}(\mathcal{S})$ (or analogously,
$q\in\mathcal{S}$ and $p\in I^{-}(\mathcal{S})$): Then $(\gamma_{n})_{n}$ can
be seen as a sequence in $\mathcal{C}(\mathcal{S},q)$. By Lemma 5.11, there
exists a limit curve $\gamma$ in $\mathcal{C}(\mathcal{S},q)$. But since
$\gamma_{n}$ starts at $p$ for all $n$, also $\gamma$ must start at $p$, hence
it is an element of $\mathcal{C}(p,q)$.

  * 2.

$p,q\in I^{+}(\mathcal{S})$ (or analogously, $p,q\in I^{-}(\mathcal{S})$):
There exists a future-directed timelike curve $\lambda$ from $\mathcal{S}$ to
$p$. Construct a new sequence $(\tilde{\gamma}_{n})_{n}$ by concatenating
$\lambda$ with $\gamma_{n}$. Then $(\tilde{\gamma}_{n})_{n}$ is a sequence in
$\mathcal{C}(\mathcal{S},q)$ and thus has a causal limit curve
$\tilde{\gamma}$ by Lemma 5.11. Because of how the sequence was constructed,
$(\tilde{\gamma}_{n})_{n}$ must be the concatenation of $\lambda$ with a
causal curve $\gamma\in\mathcal{C}(p,q)$ which is the Hausdorff limit of
$(\gamma_{n})_{n}$.

  * 3.

$p\in I^{-}(\mathcal{S})$ and $q\in I^{+}(\mathcal{S})$: By Proposition 2.22
we can extend each $\gamma_{n}$, and the maximal extension must intersect
$\mathcal{S}$ exactly once, say at $\gamma(0)$. Because of Proposition 5.9
(iv), $\gamma(0)$ cannot lie to the past of $p$ nor to the future of $q$,
hence it must in fact lie on $\gamma_{n}$. Consider the sequence
$(\bar{\gamma}_{n})_{n}$ where $\bar{\gamma}_{n}$ is the restriction of
$\gamma_{n}$ from $p$ to $\gamma_{n}(0)$. By Lemma 5.11 a subsequence
$(\bar{\gamma}_{n_{k}})_{k}$ converges to a limit curve $\bar{\gamma}$ in
$\mathcal{C}(p,\mathcal{S})$. Similarly, consider the sequence
$(\tilde{\gamma}_{n_{k}})_{k}$ of the restrictions of the original curves
$\gamma_{n_{k}}$ from $\gamma_{n_{k}}(0)$ to $q$. By Lemma 5.11, we may assume
it converges to a limit curve $\tilde{\gamma}$ in
$\mathcal{C}(\mathcal{S},q)$. Since by construction the endpoints of
$\bar{\gamma}$ and $\tilde{\gamma}$ agree on $\mathcal{S}$, we can join them
to obtain a limit causal curve of (a subsequence of) the original sequence
$(\gamma_{n})_{n}$ from $p$ to $q$. ∎

###  5.3 Proof of Theorem 5.4

We prove Theorem 5.4 in several steps. The implications (ii) $\Longrightarrow$
(i) and (iii) $\Longrightarrow$ (iv) are straightforward. The most involved
step (iv) $\Longrightarrow$ (ii) follows from our results in Section 5.2 about
properties of Cauchy sets. Finally, for the implication (i) $\Longrightarrow$
(iii) we show that thee averaged volume functions of Section 4 have additional
properties on globally hyperbolic Lorentzian pre-length spaces.

###### Proof of Theorem 5.4.

(ii) $\Longrightarrow$ (i) By (ii), $X$ is already non-totally imprisoning,
and thus it remains to be shown that the causal diamonds $J(p,q)$ are compact
for all $p\leq q$. If $p=q$, then $J(p,q)=\\{p\\}$ because non-total
imprisonment implies causality. Suppose that $p<q$. Let $(x_{n})_{n}$ be any
sequence in $J(p,q)$. By causal path-connectedness of $X$, every $x_{n}$ lies
on a causal curve $\gamma_{n}\colon[0,1]\to X$ from $p$ to $q$. By (ii), the
space of curves $\mathcal{C}(p,q)$ is compact, and hence a subsequence
$(\gamma_{n_{k}})_{k}$ of $(\gamma_{n})_{n}$ converges to a causal curve
$\gamma\in\mathcal{C}(p,q)$ in the Hausdorff sense. In particular, for the
corresponding points, $d(x_{n_{k}},\gamma)\to 0$ as $k\to\infty$. Since
$\gamma$ itself is compact, a subsequence of $(x_{n_{k}})_{k}$ must converge
to a point on $\gamma$. Hence $J(p,q)$ is compact.

(iii) $\Longrightarrow$ (iv) Suppose $t$ is a Cauchy time function. Then the
level sets of $t$ are Cauchy sets: For $s\in$ consider the preimage
$\mathcal{S}=t^{-1}(\\{s\\})$. Since $t$ is a time function, any future-
directed causal curve $\gamma$ intersects $\mathcal{S}$ at most once. If
$\gamma$ is furthermore doubly-inextendible, and since $t$ is Cauchy,
$\operatorname{Im}(t\circ\gamma)=$ and thus
$\mathcal{S}\cap\gamma\neq\emptyset$. Hence doubly-inexdendible causal curves
intersect $\mathcal{S}$ at exactly one point.

(iv) $\Longrightarrow$ (ii) Suppose $X$ admits a Cauchy set. By Lemma 5.10,
$X$ is non-totally imprisoning. By Lemma 5.12, the set of future-directed
causal curves $\mathcal{C}(p,q)$ between $p$ and $q$ is compact for any
$p,q\in X$.

(i) $\Longrightarrow$ (iii) Let

$t:=\ln\left(-\frac{t^{-}}{t^{+}}\right).$

By (i), $X$ is globally is hyperbolic, and thus it follows immediately from
Lemma 5.14 (see below) that $t$ is a Cauchy time function. ∎

We finish the remaining parts of the proof of (i) $\Longrightarrow$ (iii). By
definition, $I^{\pm}(p)\subseteq J^{\pm}(p)$ for all points $p$ in a
Lorentzian pre-lenghth space $X$. If $X$ is approximating then furthermore
$J^{\pm}(p)\subseteq\overline{I^{\pm}(p)}$, and thus
$\overline{I^{\pm}(p)}=\overline{J^{\pm}(p)}$. If $X$ is globally hyperbolic,
we can say even more, which will allow us to apply our results of Section 4.

######  Proposition 5.13.

Let $(X,d,\ll,\leq,\tau)$ be a causally path-connected, approximating
Lorentzian pre-length space. If $X$ is globally hyperbolic, then $X$ is
_causally simple_ , meaning that $J^{\pm}(p)$ is closed and thus
$J^{\pm}(p)=\overline{I^{\pm}(p)}$ for all $p\in X$. Moreover, if $X$ is
causally simple, then $X$ is causally continuous.

###### Proof.

These statements were shown by Aké et al. for Lorentzian length spaces [1,
Propositions 3.13 & 3.14]. The same proof goes through for our assumptions,
because the assumption of localizability is only needed to invoke [1, Sequence
Lemma 2.25] which in our case is replaced by Lemma 2.18. ∎

We also need the following result.

######  Lemma 5.14.

Let $(X,d,\ll,\leq,\tau)$ be an approximating Lorentzian pre-length space with
limit curves. Suppose, in addition, that $(X,d)$ is a proper metric space
equipped with a Borel probability measure of full support and corresponding
averaged volume functions $t^{+}\colon X\to[-\infty,0]$ and $t^{-}\colon
X\to[0,\infty]$ (see Definition 4.2). If $X$ is globally hyperbolic, then
$t^{\pm}$ are time functions. Moreover, for every doubly-inextendible future-
directed causal curve $\gamma\colon(a,b)\to X$ it holds that

$\lim_{s\to b}t^{+}(\gamma(s))=\lim_{s\to a}t^{-}(\gamma(s))=0.$

###### Proof.

We follow [21, Satz II.20]. Suppose $X$ is globally hyperbolic, approximating
Lorentzian pre-length space with limit curves. Then by Proposition 5.13 $X$ is
causally continuous, and thus by Theorem 4.13 $t^{\pm}$ are time functions.

To show the second part of the statement, assume for contradiction that
$\gamma\colon(a,b]\to X$ is a future-directed past-inextendible causal curve
with

$\lim_{s\to a}t^{-}(\gamma(s))>0.$ (3)

On the other hand, by standard measure theory [40, Thm. 1.2.5] and Lebesgue’s
dominated convergence theorem,

$\displaystyle\lim_{s\to a}t^{-}(\gamma(s))$ $\displaystyle=\lim_{s\to
a}\int_{0}^{1}\mu\left(I^{-}_{r}(\gamma(s))\right)dr$
$\displaystyle=\int_{0}^{1}\lim_{s\to
a}\mu\left(I^{-}_{r}(\gamma(s))\right)dr$
$\displaystyle=\int_{0}^{1}\mu\left(\bigcap_{s>a}I^{-}_{r}(\gamma(s))\right)dr.$

Assumption (3) implies that there exists an $r\in(0,1)$ and a point
$p\in\bigcap_{s>a}I^{-}_{r}(\gamma(s))$. This means that for every $s>a$,
there exists a $q\in I^{-}(\gamma(s))$ such that $d(q,p)<r$. In particular, we
can find a sequence $q_{n}\in I^{-}\left(\gamma(a+1/n)\right)$ such that
$d(q_{n},p)<r$ for all $n\in\mathbb{N}$. Because $d$ is proper, the closed
ball of radius $r$ around $p$ is compact, and hence $q_{n}$ converges to a
limit point $q$ (up to a subsequence). Now for a given $s>a$, we choose
$n_{0}$ such that $s>a+1/n_{0}$. Then, by the push-up Lemma 2.3, $q_{n}\in
I^{-}(\gamma(s))$ for all $n\geq n_{0}$. Because of this,
$q\in\overline{I^{-}(\gamma(s))}=J^{-}(\gamma(s))$, where the last equality
follows by global hyperbolicity and Proposition 5.13. Since $s>a$ was
arbitrary, we have that $\gamma\subseteq J^{+}(q)$. In particular,
$q\leq\gamma(b)$, and by global hyperbolicity, $J(q,\gamma(b))$ is compact.
But then the curve $\gamma$ is imprisoned in $J(q,\gamma(b))$, in
contradiction to global hyperbolicity. ∎

This concludes our characterization of global hyperbolicity with the existence
of Cauchy time functions (and Cauchy sets) in the setting of Lorentzian pre-
length spaces.

## References

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