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 Sect. 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 Sect. 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 Sect. 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 Sect. 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 \(\le \) (called causal relation) and a transitive relation \(\ll \) (called chronological or timelike relation) contained in \(\le \).

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

$$\begin I^+(p)&:= \left\ ,&J^+(p)&:= \left\ , \\ I^-(p)&:= \left\ ,&J^-(p)&:= \left\ , \\ I(p,q)&:= I^+(p) \cap I^-(q),&J(p,q)&:= J^+(p) \cap J^-(q), \end$$

and we write \(p < q\) if \(p \le q\) and \(p \ne q\).

Definition 2.2

[42, Definition 2.8]. A Lorentzian pre-length space is a causal set \((X,\ll ,\le )\) equipped with a metric d and a lower semicontinuous function \(\tau :X \times X \rightarrow [0,\infty ]\) satisfying, for all \(x,y,z \in X\)

(i)

\(\tau (x,y) > 0\) \(\Longleftrightarrow \) \(x \ll y\),

(ii)

\(\tau (x,y) = 0\) if \(x \not \le y\),

(iii)

\(\tau (x,z) \ge \tau (x,y) + \tau (y,z)\) if \(x \le y \le z\).

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 \((X,d,\ll ,\le ,\tau )\) be a Lorentzian pre-length space and \(x,y,z \in X\) with \(x \ll y \le z\) or \(x \le y \ll z\). 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 \((X,d,\ll ,\le ,\tau )\) be a Lorentzian pre-length space and I be any (open, half-open, or closed) interval in \(\mathbb \). A non-constant locally Lipschitz path \(\gamma :I \rightarrow X\) is called a

(i)

future-directed causal curve if \(\gamma (s_1) \le \gamma (s_2)\) for all \(s_1 < s_2 \in I\).

(ii)

past-directed causal curve if \(\gamma (s_2) \le \gamma (s_1)\) for all \(s_1 < s_2 \in I\).

Future- and past-directed timelike curves are defined analogously by replacing \(\le \) 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 \(\gamma :I \rightarrow X\) is parametrized by d-arclength iff

$$\begin L^d(\gamma \vert _) = b - a \text [a,b] \subseteq I. \end$$

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 \(\mathbb \).

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 \(\le _U\) by

$$\begin p \le _U q:\Longleftrightarrow \text to in .} \end$$

A neighborhood U is called weakly causally closed if \(\le _U\) 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 2.7 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 \(\le \) restricted to \(\overline \times \overline\) 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). Note also that weak causal closedness is most natural on causally path-connected spaces (as it would be a void condition on spaces with no causal curves at all), so one may even include causal path-connectedness in the definition, as done implicitly in [1].

Finally, we refine the concept of an inextendible curve.Footnote 1

Definition 2.8

Let X be a Lorentzian pre-length space and \(\gamma :(a,b) \rightarrow X\) be a future-directed causal curve. If there exists a causal curve \(\bar:(a,b] \rightarrow X\) such that \(\bar\vert _ = \gamma \), we say that \(\gamma \) is future-extendible. If there exists a causal curve \(\tilde:[a,b) \rightarrow X\) such that \(\tilde\vert _ = \gamma \), 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 \(\mathbb \), 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 Lorentzian pre-length space, let \(-\infty< a < b \le \infty \) and let \(\gamma :[a, b) \rightarrow X\) 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 \(\lim _ \gamma (t)\) does not exist.

Proof

First, assume that \(b = \infty \). If \(\gamma \) admitted an extension \(\bar\) as in Definition 2.8, then \(\bar\) would be locally Lipschitz and have two endpoints. Therefore, \(L^d(\bar) < \infty \), but also \(L^d(\bar)=L^d(\gamma ) = b-a\) (since we have only added one point), a contradiction. The rest of 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. \(\square \)

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 \(\gamma :[a,b] \rightarrow X\) is a future-directed causal curve, then its \(\tau \)-length \(L_(\gamma )\) is defined by (see [42, Definition 2.24])

$$\begin L_(\gamma ):= \inf \left\^ \tau (\gamma (t_i), \gamma ( t_)) \; \bigg | \; a = t_0< t_1< \cdots < t_N = b, N \in \mathbb \right\} . \end$$

In addition, the notion of localizability is needed.

Definition 2.10

[42, Definition 3.16] and [1, Definition 2.22].Footnote 2 We call a Lorentzian pre-length space \((X,d,\ll ,\le ,\tau )\) localizable if for every point \(p \in X\), there exists a neighborhood \(U_p\) of p such that

(i)

There exists a constant \(C > 0\) such that for all causal curves contained in \(U_p\) we have \(L^d(\gamma )\le C\).

(ii)

For every \(q \in U_p\) we have \(I^\pm (q) \cap U_p \ne \emptyset \).

(iii)

There exists a continuous function \(\omega _p :U_p \times U_p \rightarrow [0,\infty )\) such that \((U_p,d \vert _, \ll _, \le _, \omega _p)\) is a Lorentzian pre-length space. Moreover, for all \(x,y \in U_p\) with \(x < y\), it holds that

$$\begin \omega _p(x,y) = \max \ to } \}, \end$$

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

In the main sections of this paper, only assumptions (i) and (ii) of Definition 2.10 are needed; thus, we restate them separately in the upcoming sections (see Definitions 2.14 and 2.17, Lemma 2.19 and Proposition 2.20). 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 localizable Lorentzian pre-length space \((X,d,\ll ,\le ,\tau )\) is called a Lorentzian length space if for all \(p,q \in X\)

$$\begin \tau (p,q) = \sup \ to } \}. \end$$

2.2 Limit Curve Theorems

We revisit the limit curves theorems of Kunzinger and Sämann [42, Sect. 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 Sects. 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 \((\gamma _n)_\) be a sequence of future-directed causal curves \(\gamma _n :I \rightarrow X\) converging locally uniformly to a non-constant locally Lipschitz curve \(\gamma :I \rightarrow X\). Then \(\gamma \) is future-directed causal.

Proof

For every \(s \in I\) there exists a weakly causally closed neighborhood \(U_\). By continuity, we can pick \(s_1< s < s_2\) such that \(\gamma ([s_1,s_2]) \subseteq U_\) (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 \((\gamma _n)_\) converges uniformly on \([s_1,s_2]\). This implies that there exists \(n_0 \in \mathbb \) such that for all \(n > n_0\), \(\gamma _n([s_1,s_2]) \subseteq U_\). 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 \(\le \), \(\gamma \) is future-directed causal on I. \(\square \)

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 \(\gamma _n :[a, b] \rightarrow X\) that are uniformly Lipschitz continuous, i.e., there is an \(L > 0\) such that \(}(\gamma _n ) \le L\) for all \(n \in \mathbb \). 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 \(\gamma _n (t_0 ) \rightarrow x_0 \in X\). Then there exists a subsequence \((\gamma _)_k\) of \((\gamma _n)_n\) and a Lipschitz continuous curve \(\gamma :[a, b] \rightarrow X\) such that \(\gamma _ \rightarrow \gamma \) 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. \(\square \)

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 \((X,d,\ll ,\le ,\tau )\) is called d-compatible if for every \(p \in X\) there exists a neighborhood U of p and a constant \(C > 0\) such that \(L^d(\gamma ) \le C\) 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 \(\gamma _n :[0, L_n ] \rightarrow X\) which are parametrized with respect to d-arclength and satisfy \(L_n:= L^d (\gamma _n ) \rightarrow \infty \). If there exists a compact set that contains every curve \(\gamma _n ([0, L_n ])\) or if d is proper and \(\gamma _n (0) \rightarrow x\) for some \(x \in X\), then there exists a subsequence \((\gamma _)_k\) of \((\gamma _n )_n\) and a future-directed causal curve \(\gamma :[0, \infty ) \rightarrow X\) such that \(\gamma _ \rightarrow \gamma \) 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 Lemmas 2.12, 2.9 and Theorem 2.13 respectively. \(\square \)

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 Sect. 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 \((X,d,\ll ,\le ,\tau )\) is called approximating if for all points \(p \in X\) it holds that \(J^\pm (p) \subseteq \overline\).

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 \((X,d,\ll ,\le ,\tau )\) 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 \(p^\pm _n \rightarrow p\) as \(n\rightarrow \infty \).

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 \(p \in J^\pm (p) \subseteq \overline\). 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 \(n \in \mathbb \), hence \(q \in \overline\). \(\square \)

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

(i)

X is future-(past-)approximating,

(ii)

\(I^+(p) \ne \emptyset \) (\(I^-(p) \ne \emptyset \)) for all \(p \in X\),

(iii)

for every point \(p \in X\) there exists a future-(past-)directed timelike curve \(\gamma :[a,b) \rightarrow X\) 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 \(\gamma :(a,b) \rightarrow X\) 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 \(\emptyset \ne J^+(p) \subseteq \overline\). This implies that \(I^+(p) \ne \emptyset \).

(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 \(\gamma :[a,b) \rightarrow X\) be a future-directed timelike curve with \(\gamma (a) = p\). By continuity of \(\gamma \), we have \(p = \lim _ \gamma (s)\), so \(p \in \overline\).

The past statements are proved analogously. \(\square \)

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 localizable, 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 localizable, then by property (i) in Definition 2.10, X is d-compatible. Furthermore, by property (ii), every point \(q \in X\) has \(I^\pm (q) \ne \emptyset \), and then by Lemma 2.19, X is approximating. The second statement follows trivially from the definitions. \(\square \)

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 (or past) 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 Sect. 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 \(\gamma :[a,b) \rightarrow X\) with \(b < \infty \), there exists \(c \in [b,\infty ]\) and a future-(past-) inextendible causal curve \(\lambda : [a,c) \rightarrow X\) such that \(\lambda \vert _ = \gamma \).

Proof

Consider, without loss of generality, the case that \(\gamma \) is future-directed. If \(\gamma \) is already inextendible, there is nothing to prove since we can just choose \(c=b\). 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 \(\tilde:[a,c) \rightarrow X\) of \(\gamma \), where \(c > b\). If we can choose \(\tilde\) 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.

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 \(L^d(\gamma _n) \rightarrow C\). 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 \(\gamma _\infty :[a,c] \rightarrow X\) 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.

There exists a sequence \((\gamma _n)_n\) of extensions of \(\gamma \) such that \(L^d(\gamma _n) \rightarrow \infty \). 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 \(\gamma \). \(\square \)

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 \(p \in X\), there exists a doubly inextendible causal curve passing through p. \(\square \)

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 \(f :X \rightarrow \mathbb \) on a Lorentzian pre-length space \((X,d,\ll ,\le ,\tau )\) is called a generalized time function if for all \(p,q \in X\),

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

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, [52, Theorem 4.39]).

Definition 2.26

[42, Definition 2.35]. A Lorentzian pre-length space \((X,d,\ll ,\le ,\tau )\) 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, \(L^d(\gamma ) \le C\).

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.

(i)

X is non-totally imprisoning.

(ii)

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

(iii)

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

(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 not non-totally imprisoning. Then there exists a compact set K and a sequence of future-directed causal curves \(\gamma _n :[0,L_n] \rightarrow X\), parametrized by arclength and contained in K, such that \(L_n = L^d(\gamma _n) \rig