If \(\gamma \) is a nearly timelike curve from p to q such that \(L(\gamma ) = \tau (p,q)\) where \(\tau \) is the time separation function introduced in the previous section (defined with respect to nearly timelike curves), then we call \(\gamma \) a nearly timelike maximizer from p to q. Note that \(\tau (p,q) < \infty \) whenever a nearly timelike maximizer from p to q exists.

In this section we establish sufficient conditions ensuring the existence of a nearly timelike maximizer between two points p and q in a \(C^0\) spacetime (M, g), see Theorem 4.3. We adopt the notation

$$\begin \mathcal (p,q) :=\, \mathcal ^+(p) \cap \mathcal ^-(q). \end$$

Also, we need the following two definitions:

(M, g) is strongly causal if for every \(p \in M\) and every neighborhood U of p, there is a neighborhood \(V \subset U\) of p such that

$$\begin \gamma (a), \gamma (b) \in V \quad \Longrightarrow \quad \gamma \subset U \end$$

whenever \(\gamma :[a,b] \rightarrow M\) is a causal curve.

A subset \(E \subset M\) is called causally plain if

$$\begin q \in J^+(p) \quad \Longrightarrow \quad q \in \overline \,\, \text \,\, p \in \overline \end$$

for all \(p,q \in E\).

Remark 4.1

An assumption used throughout this section is that \(\mathcal (p,q)\) is causally plain. We emphasize that this does not imply that no bubbling from p or q exists, but it does imply the lack of bubbling from within \(\mathcal (p,q)\). Proposition 4.5 outlines the class of bubbling spacetimes to which the results in this section are relevant; a specific example based on the spacetime in [14] is provided after that proposition. Also, we do not need the full strength of strong causality for the results in this section. It would be sufficient to consider strong causality on \(\mathcal (p,q)\), and, in fact, it would be sufficient to consider only nearly timelike curves instead of causal curves in the definition.

The following lemma establishes a limit curve argument for nearly timelike curves. Recall from the example after Corollary 3.6 that a limit curve argument for nearly timelike curves does not hold in general, so additional assumptions need to be imposed.

Lemma 4.2

Suppose (M, g) is a strongly causal \(C^0\) spacetime. Assume \(\mathcal (p,q)\) is compact and causally plain. Let \(\gamma _n :[0,b_n] \rightarrow M\) be a sequence of nearly timelike curves from \(p_n\) to \(q_n\) parameterized by h-arclength. Assume

$$\begin p_n \rightarrow p, \quad q_n \rightarrow q, \quad p_n \in \mathcal ^+(p), \quad q_n \in \mathcal ^-(q),\quad \text \quad p \ne q. \end$$

Then there is a \(b \in (0, \infty )\) and a nearly timelike curve \(\gamma :[0,b] \rightarrow M\) from p to q such that for each \(t \in (0,b)\), there is a subsequence \(\gamma _\) which converges to \(\gamma \) uniformly on [0, t].

Proof

We first show \(\sup _\ < \infty \). By assumption, \(p_n \in \mathcal ^+(p)\). Therefore \(\gamma _n(t) \in \mathcal ^+(p)\) by Theorem 3.3(2) for each \(t \in [0,b_n]\). Likewise \(\gamma _n(t) \in \mathcal ^-(q)\). Therefore each \(\gamma _n\) is contained in the compact set \(\mathcal (p,q)\). By [28, Prop. 2.17], for each \(x \in \mathcal (p,q)\), there is a neighborhood \(U_x\) such that \(L_h(\lambda ) \le 1\) for any causal (and hence any nearly timelike) curve \( \lambda \subset U_x\). By strong causality, there are neighborhoods \(V_x \subset U_x\) such that \(\lambda \subset U_x\) whenever \(\lambda :[a,b] \rightarrow M\) is a causal curve with endpoints in \(V_x\). Since \(\mathcal (p,q)\) is covered by \(\_(p,q)}\), there is a finite subcover \(V_1, \dotsc , V_N\). It follows that any nearly timelike curve with image contained in \(\mathcal (p,q)\) has h-length bounded by N. Thus \(\sup _\ \le N\).

Since every sequence in \(}\) contains a monotone subsequence, we can assume \(b_n\) is monotone by restricting to a subsequence. Then either (1) \(b_n \rightarrow \infty \) or (2) \(b_n \rightarrow b < \infty \). The first scenario is ruled out by the paragraph above. Therefore the second scenario must hold. Moreover, \(b > 0\). Indeed, we have \(d_h(p_n,q_n) \le b_n\), and taking \(n \rightarrow \infty \) gives \(d_h(p,q) \le b\). Thus the assumption \(p \ne q\) implies \(b > 0\).

Extend each \(\gamma _n\) to inextendible causal curves \(\tilde_n :}\rightarrow M\) by, for example, concatenating each \(\gamma _n\) with the maximal integral curve of a timelike vector field and then reparamterizing by h-arclength. By the usual limit curve theorem [28, Thm. 2.21], there exists a subsequence (still denoted by \(\tilde_n\)) and a causal curve \(\tilde:}\rightarrow M\) with \(\tilde(0) = p\) such that \(\tilde_n\) converges to \(\tilde\) uniformly on compact subsets of \(}\). The triangle inequality gives

$$\begin d_h\big (q, \tilde_(b)\big ) \,&\le \, d_h\big (q, \gamma _(b_)\big ) \,+\, d_h\big (\gamma _(b_), \tilde_(b)\big ) \\&\le \, d_h\big (q, \gamma _(b_)\big ) \,+\, |b_n-b|. \end$$

Since \(\gamma _(b_) \rightarrow q\) and \(b_ \rightarrow b\), the right-hand side limits to 0. Thus \(\tilde_(b) \rightarrow q\). Therefore \(\tilde|_\) is a causal curve from p to q. Set \(\gamma = \tilde|_\). Fix \(t \in (0,b)\). There is a subsequence (still denoted by \(b_n\)) such that \(b_n \ge t\) for all n. Therefore, for this subsequence, we have \(\gamma _n = \tilde_n\) on [0, t]; hence \(\gamma _n\) converges uniformly to \(\gamma \) on [0, t].

It remains to show that \(\gamma \) is a nearly timelike curve. Fix \(s < t\) in [0, b]. By compactness, we have \(\gamma \subset \mathcal (p,q)\). Therefore \(\gamma (t) \in \overline\) since \(\mathcal (p,q)\) is causally plain. Likewise \(\gamma (s) \in \overline\). \(\square \)

Theorem 4.3

Suppose (M, g) is a strong causal \(C^0\) spacetime. Assume \(\mathcal (p,q)\) is compact and causally plain. If \(q \in \mathcal ^+(p)\) with \(q \ne p\), then there is a nearly timelike maximizer \(\gamma \) from p to q, i.e., \(L(\gamma ) = \tau (p,q)\).

Proof

By definition of \(\tau \), there is a sequence of nearly timelike curves \(\gamma _n :[0,b_n] \rightarrow M\) from p to q satisfying \(\tau (p,q) \le L(\gamma _n) + 1/n\). Assume each \(\gamma _n\) is parameterized by h-arclength. Let \(\gamma :[0, b] \rightarrow M\) be the nearly timelike curve from p to q appearing in the conclusion of Lemma 4.2. As in the proof of that lemma, let \(\tilde_n :}\rightarrow M\) be the inextendible causal curve extensions of \(\gamma _n\) and let \(\tilde:}\rightarrow M\) be the resulting limit curve so that \(\gamma = \tilde|_\).

It suffices to show \(L(\gamma ) \ge \tau (p,q)\). There is a subsequence \(\tilde_|_\) which converges uniformly to \(\gamma \); moreover, \(b_\) limits to b monotonically as \(k \rightarrow \infty \). Fix \(\varepsilon > 0\). By upper semicontinuity of the Lorentzian length functional [28, Prop. 3.7], there is an N such that \(k \ge N\) implies

$$\begin L(\gamma ) + \varepsilon \,&\ge \, L\big (\tilde_|_\big ) \\&=\, L(\gamma _) + \int _}^b \sqrt_',\tilde_'\big )} \\&\ge \, \big (\tau (p,q) - 1/n_k\big ) + \int _}^b\sqrt_',\tilde_'\big )}. \end$$

As \(k \rightarrow \infty \), the above integral limits to zero. This follows since (1) \(b_ \rightarrow b\) and (2) there is a neighborhood U of q such that \(-g(\lambda ',\lambda ')\) is bounded on U for any h-arclength parmeterized curve \(\lambda \) contained in U. (1) is clear. To prove (2), let U be a coordinate neighborhood of q with coordinates \(x^\mu \), and assume U is h-convex and has compact closure. Using similar triangle inequality arguments as in the proof of [28, Prop. 2.2], it follows that the component functions \(\lambda ^\mu = x^\mu \circ \lambda \) of any h-arclength parameterized curve \(\lambda \) are Lipschitz with the same Lipschitz constant; this proves (2). Thus, taking \(k \rightarrow \infty \), we have \(L(\gamma ) + \varepsilon \ge \tau (p,q)\). Since \(\varepsilon > 0\) was arbitrary, we have \(L(\gamma ) \ge \tau (p,q)\). \(\square \)

The next theorem proves a sequential continuity result for \(\tau \) but only from directions within \(\mathcal (p,q)\) and under the assumption that \(\mathcal (p,q)\) is compact and causally plain. It could have applications to synthetic approaches of Lorentzian geometry which require a continuous \(\tau \) as in [35].

Theorem 4.4

Suppose (M, g) is a strongly causal \(C^0\) spacetime. Assume \(\mathcal (p,q)\) is compact and causally plain. If

$$\begin p_n \rightarrow p, \quad q_n \rightarrow q, \quad p_n \in \mathcal ^+(p), \quad q_n \in \mathcal ^-(q),\quad \text \quad p \ne q, \end$$

then

$$\begin \lim _ \tau (p_n, q_n) \,=\, \tau (p,q). \end$$

Proof

By lower semicontinuity of \(\tau \), we have \(\tau (p,q) \le \liminf \tau (p_n, q_n)\). It suffices to show \(\tau (p,q) \ge \limsup \tau (p_n, q_n)\). Set \(t:= \limsup \tau (p_n, q_n)\). If \(t = 0\), then \(\tau (p,q) \ge t\) is immediate. Therefore we can assume \(t > 0\). Seeking a contradiction, suppose \(\tau (p,q) < t\). Then there are subsequences (still denoted by \(p_n\) and \(q_n\)) and an \(\varepsilon > 0\) such that \(\tau (p,q) < \tau (p_, q_) -2\varepsilon \) for all n and \(\tau (p_, q_) \rightarrow t\) as \(n \rightarrow \infty \). Since \(t > 0\), we can assume \(\tau (p_, q_) > 0\) for all n by restricting to a further subsequence. Let \(\gamma _n :[0,b_n] \rightarrow M\) be a sequence of nearly timelike curves from \(p_\) to \(q_\) such that \(L(\gamma _) > \tau (p_, q_) - 1/n\). Let \(\gamma :[0, b] \rightarrow M\) be the nearly timelike curve from p to q appearing in the conclusion of Lemma 4.2. As in the proof of that lemma, let \(\tilde_n :}\rightarrow M\) be the inextendible causal curve extensions of \(\gamma _n\) and let \(\tilde:}\rightarrow M\) be the resulting limit curve so that \(\gamma = \tilde|_\). There is a subsequence \(\tilde_|_\) which converges uniformly to \(\gamma \); moreover, \(b_\) limits to b monotonically as \(k \rightarrow \infty \). By upper semicontinuity of the Lorentzian length functional [28, Prop. 3.7], there is an N such that \(k \ge N\) implies

$$\begin L(\gamma ) + \varepsilon \,&\ge \, L\big (\tilde_|_\big ) \\&=\, L(\gamma _) + \int _}^b \sqrt_',\tilde_'\big )} \\&>\, \big (\tau (p_,q_) - 1/n_k\big ) + \int _}^b\sqrt_',\tilde_'\big )} \\&>\, \big (\tau (p,q) + 2\varepsilon - 1/n_k\big ) + \int _}^b\sqrt_',\tilde_'\big )}. \end$$

As in the proof of Theorem 4.3, the integral term vanishes as \(k \rightarrow \infty \). Therefore, we obtain \(L(\gamma ) \ge \tau (p,q) + \varepsilon \), which is a contradiction. \(\square \)

The previous results rely on \(\mathcal (p,q)\) being compact and causally plain. The following proposition gives sufficient conditions ensuring this and summarizes the results in this section. A specific example for which the proposition applies follows afterward. Recall that a \(C^0\) spacetime (M, g) is globally hyperbolic if it is strongly causal and \(J^+(p) \cap J^-(q)\) is compact for all \(p,q \in M\).

Proposition 4.5

Let (M, g) be a globally hyperbolic \(C^0\) spacetime. For an open subset \(M' \subset M\), assume g is smooth on \(M'\) (locally Lipschitz is sufficient) and that \(J^+(M') \subset M'\). If \(\overline \setminus \ \subset M'\) for some \(p \in \overline\), then for all \(q \in \mathcal ^+(p)\) with \(q \ne p\), the following hold:

(1):

\(\mathcal (p,q)\) is compact and causally plain.

(2):

There is a nearly timelike maximizer from p to q.

(3):

\(\displaystyle \lim \nolimits _ \tau (p_n, q_n) = \tau (p,q)\) if \(p_n \rightarrow p\) and \(q_n \rightarrow q\) with \(p_n \in \mathcal ^+(p)\) and \(q_n \in \mathcal ^-(q)\).

Proof

(2) and (3) follow from (1) via Theorems 4.3 and 4.4, respectively, so it suffices to show (1). We first show that \(\mathcal (p,q)\) is causally plain. Fix \(x,y \in \mathcal (p,q)\) with \(y \in J^+(x)\). We want to show \(y \in \overline\) and \(x \in \overline\). Either \(x = p\) or \(x \ne p\). If \(x = p\), then \(y \in \mathcal ^+(x)\) and so the result follows. Now assume \(x \ne p\). Then \(x \in \overline \ \subset M'\). Therefore any causal curve from x to y will be contained in \(M'\) since \(J^+(M') \subset M'\). The metric is smooth on \(M'\) and so the push-up property holds on \(M'\) from which the result follows. (In fact this argument proves that \(\mathcal ^+(p)\) is causally plain.)

Now we show \(\mathcal (p,q)\) is compact. First note that \(J(p,q):= J^+(p) \cap J^-(q)\) is compact by global hyperbolicity. By the Hopf-Rinow theorem, J(p, q) is closed and bounded (with respect to the Riemannian distance function \(d_h\)). Since \(\mathcal (p,q) \subset J(p,q)\), it follows that \(\mathcal (p,q)\) is also bounded. Therefore it suffices to show that \(\mathcal (p,q)\) is closed. Let r be a limit point of \(\mathcal (p,q)\). We can assume \(r \ne p,q\). Let \(r_n \in \mathcal (p,q)\) be a sequence with \(r_n \rightarrow r\). Let \(\gamma _n :[0,b_n] \rightarrow M\) be a sequence of h-arclength parameterized nearly timelike curves from p to \(r_n\). Since J(p, q) is compact, [28, Prop. 3.4] and its proof imply that there is a \(b \in (0,\infty )\) with \(b_n \rightarrow \infty \) and a causal curve \(\gamma :[0, b] \rightarrow M\) from p to r such that for any \(t \in (0,b)\), there is a subsequence of \(\gamma _n\) which converges to \(\gamma \) uniformly on [0, t]. \(\square \)

Claim: \(\gamma \) is a nearly timelike curve from p to r.

To prove the claim, the following will be useful. Fact: \(\gamma (t) \in M'\) for all \(t \in (0,b]\).

We first prove the fact. It suffices to show \(\gamma (t) \in \overline \\). \(\gamma (t) \ne p\) since there are no closed causal curves in M. If \(t < b\), then there is a subsequence \(\gamma _n(t) \rightarrow \gamma (t)\). If \(t = b\), then \(r_n \rightarrow \gamma (b)\).

Now we prove the claim. Fix \(s < t\) in [0, b]. We want to show \(\gamma (t) \in \overline\) and \(\gamma (s) \in \overline\). Either \(s = 0\) or \(s \ne 0\). Consider first \(s \ne 0\). Then \(\gamma (s), \gamma (t) \in M'\), by the fact, and so the result follows since the push-up property applies on \(M'\). Now assume \(s = 0\). The proof of the fact shows that \(\gamma (t) \in \overline\). Now we show \(p \in \overline\). If U is any neighborhood of p, choose \(0< \varepsilon < t\) small enough so that \(\gamma (\varepsilon ) \in U\). The fact implies \(\gamma (\varepsilon ) \in \overline\), hence U intersects \(I^-\big (\gamma (t)\big )\). This proves the claim.

Thus \(r \in \mathcal ^+(p)\). An easier argument shows that \(r \in \mathcal ^-(q)\). \(\square \)

Example 4.6

Let (M, g) be the García-Heveling-Soultanis \(C^0\) spacetime [14] which is globally hyperbolic. Let P denote the null cone \(t = |x|\). The metric is smooth on \(M':= I^+(P)\), and since the lightcones of g are narrower than those for the Minkowski metric, it follows that \(J^+(M') \subset M'\). Moreover, for each \(p \in P\), direct calculations as in [14] show that \(\overline \ \subset M'\). Therefore the previous proposition applies. In particular, for any \(p \in P\) and \(q \in \mathcal ^+(p)\) with \(q \ne p\), there is a nearly timelike maximizer from p to q. This spacetime also demonstrates that global hyperbolicitiy does not necessarily imply compactness of \(\mathcal (p,q)\) for all p, q. In (t, x) coordinates, let \(p = (-1,1)\) and \(q = (1,1)\). Then \(\mathcal (p,q)\) is not compact since it does not contain the origin \(0 = (0,0)\) which is a limit point of \(\mathcal (p,q)\). See Fig. 4.

Fig. 4

The García-Heveling-Soultanis \(C^0\) spacetime (M, g) is globally hyperbolic, and so the diamond J(p, q) is compact. But this does not imply compactness of \(\mathcal (p,q)\)