Basic limit curve lemma: arclength-parametrized causal curves have causal limit curves on compact parameter intervals and future-inextendible limits on half-lines

(Bounded ranges and equicontinuity.) By [L1] each ${\alpha }_k$ is $1$-Lipschitz for $d_h$, and since ${\alpha }_k(0)\to p$ there is $k_0$ with $d_h({\alpha }_k(0),p)\le 1$ for all $k\ge k_0$. Then for $k\ge k_0$ the image ${\alpha }_k([0,T])$ is contained in the closed ball $\bar{B}(p,T+1)$, which is compact by [L2]; discarding finitely many terms, all curves take values in one compact ball.

(Arclength parametrization.) Let $\beta :[r_0,r_1]\to M$ be any future causal curve with $r_0<r_1$. By [L9], its speed $w(t):=\sqrt{h_{\beta (t)}({\beta }^{\mathrm{\prime }}(t),{\beta }^{\mathrm{\prime }}(t))}$ is measurable, integrable and positive almost everywhere. Define $s(t):={\int }_{r_0}^{t}w(u)du$ and $L:=s(r_1)>0$. Then $s$ is absolutely continuous and strictly increasing, hence has a continuous inverse $\theta :[0,L]\to [r_0,r_1]$. To verify that $\theta$ is absolutely continuous, fix $\epsilon >0$ and choose $\delta >0$ so the measurable set $E:=\{t:w(t)<\delta \}$ has measure less than $\epsilon \mathrm{/}2$; this is possible because $w>0$ almost everywhere. For any finite disjoint family of intervals $[u_i,v_i]\subseteq [0,L]$ with total length less than $\delta \epsilon \mathrm{/}2$, their inverse intervals $[\theta (u_i),\theta (v_i)]$ have disjoint interiors and ${\int }_{\theta (u_i)}^{\theta (v_i)}w=v_i-u_i$ by the definition of $s$. The total length of their intersections with $\{w\ge \delta \}$ is therefore less than $\epsilon \mathrm{/}2$, and the total length of their intersections with $E$ is at most $\mathrm{\mid }E\mathrm{\mid }<\epsilon \mathrm{/}2$. Thus $\theta$ satisfies the absolute-continuity criterion. Since $\theta$ is increasing and absolutely continuous, composition of the locally absolutely continuous chart representatives of $\beta$ with $\theta$ is absolutely continuous by the epsilon-delta definition on disjoint intervals. The exceptional null sets where ${\beta }^{\mathrm{\prime }}$ or $s^{\mathrm{\prime }}$ fails to exist or $w=0$ have null images under the absolutely continuous map $s$; outside those images the inverse derivative formula gives ${\theta }^{\mathrm{\prime }}(s)=1\mathrm{/}w(\theta (s))>0$. The chain rule then gives ${\tilde{\beta }}^{\mathrm{\prime }}(s)={\beta }^{\mathrm{\prime }}(\theta (s)){\theta }^{\mathrm{\prime }}(s)$ almost everywhere: it is nonzero and future causal, and its $h$-speed equals $1$. Integrating this speed on each subinterval proves ${\mathrm{\ell }}_h(\tilde{\beta }{\mathrm{\mid }}_{[u,v]})=v-u$.

(Diagonal extraction on a countable dense set.) Let $D:=[0,T]\cap \mathbb{Q}=\{q_1,q_2,\dots \}$. Applying [L2] to the compact set $\bar{B}(p,T+1)$ and extracting recursively: from any subsequence one can select a further subsequence whose values at a prescribed $q_i$ converge. Performing this for $i=1,2,\dots$ and taking the diagonal subsequence yields a subsequence, still written ${\alpha }_k$, for which ${\alpha }_k(q)$ converges for every $q\in D$; the nested-subsequence recursion uses Dependent Choice in [A1]. Put $\alpha (q):={\lim }_k{\alpha }_k(q)$ for $q\in D$.

(The limit on the dense set is Lipschitz, hence extends.) For $q,q^{\mathrm{\prime }}\in D$, continuity of $d_h$ gives $d_h(\alpha (q),\alpha (q^{\mathrm{\prime }}))={\lim }_k{d}_h({\alpha }_k(q),{\alpha }_k(q^{\mathrm{\prime }}))\le \mathrm{\mid }q-q^{\mathrm{\prime }}\mathrm{\mid }$, so $\alpha :D\to M$ is $1$-Lipschitz. For $t\in [0,T]$ and any sequence $q_n\in D$ with $q_n\to t$, the values $\alpha (q_n)$ form a Cauchy sequence in the compact ball $\bar{B}(\alpha (t_0),1)$ for suitable $t_0\in D$ (or in $\bar{B}(p,T+1)$), hence converge by [L2]; the limit does not depend on the sequence, again by the Lipschitz bound. This defines $\alpha :[0,T]\to M$ extending the dense-set values, and the bound $d_h(\alpha (s),\alpha (t))\le \mathrm{\mid }t-s\mathrm{\mid }$ passes to the limit; in particular $\alpha$ is continuous and (a1) holds.

(Uniform convergence.) Let $\epsilon >0$. Since all ${\alpha }_k$ and $\alpha$ are $1$-Lipschitz, choose finitely many points $q_1,\dots ,q_N\in D$ such that every $t\in [0,T]$ has some $q_j$ with $\mathrm{\mid }t-q_j\mathrm{\mid }<\epsilon \mathrm{/}4$. For each $j$, ${\alpha }_k(q_j)\to \alpha (q_j)$, so there is $k_1$ with $d_h({\alpha }_k(q_j),\alpha (q_j))<\epsilon \mathrm{/}2$ for all $k\ge k_1$ and all $j\le N$. Then for $k\ge k_1$ and arbitrary $t$, with $\mathrm{\mid }t-q_j\mathrm{\mid }<\epsilon \mathrm{/}4$, $d_h({\alpha }_k(t),\alpha (t))\le d_h({\alpha }_k(t),{\alpha }_k(q_j))+d_h({\alpha }_k(q_j),\alpha (q_j))+d_h(\alpha (q_j),\alpha (t))<\epsilon ,$ using the Lipschitz bounds for ${\alpha }_k$ and $\alpha$. Hence ${\alpha }_k\to \alpha$ uniformly on $[0,T]$.

(Geodesic connectibility.) Let $C$ be a strictly convex normal ball and $s<t$ with $\alpha ([s,t])\subseteq C$. For large $k$ also ${\alpha }_k([s,t])\subseteq C$ by uniform convergence. By [L4] the position vector $w_k:={\exp }_{{\alpha }_k(s)}^{-1}({\alpha }_k(t))$ lies in the closed future cone ${\mathcal{C}}_{{\alpha }_k(s)}^{+}$, and smoothness of the two-point inverse velocity (The joint exponential map: smoothness, invertible differential at the diagonal, and the small-velocity two-point inverse clause (c)) gives $w_k\to w:={\exp }_{\alpha (s)}^{-1}(\alpha (t))$ because ${\alpha }_k(s)\to \alpha (s)$ and ${\alpha }_k(t)\to \alpha (t)$. The future cones vary continuously in the base point and are closed, so $w$ lies in the closed future cone at $\alpha (s)$; hence the radial geodesic $u\mapsto {\exp }_{\alpha (s)}(uw)$, $u\in [0,1]$, is a future causal geodesic contained in $C$ joining $\alpha (s)$ to $\alpha (t)$, since the radial geodesic of a conic vector stays in $C$ by [L4]. This proves (a2).

(The chart bound for the limit.) Let $U$ be the chart of [L6] with constant $K$, and let $s<t$ with $\alpha ([s,t])\subseteq U$; by uniform convergence on $[s,t]$ we may assume ${\alpha }_k([s,t])\subseteq U$ for all large $k$. As long as a future causal curve stays in $U$, its tangent satisfies $\mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}}\le \sqrt{1+3K^2}{v}^0$ by [L6], and the operator norm of the matrix $(h_{ab})$ is bounded on $U$ by a constant $\mathrm{\Lambda }$, so $\sqrt{h({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}\le c(x^0\circ \gamma {)}^{\mathrm{\prime }}$ almost everywhere with $c:=\sqrt{\mathrm{\Lambda }(1+3K^2)}$; integrating over $[s,t]$ gives $t-s={\mathrm{\ell }}_h({\alpha }_k{\mathrm{\mid }}_{[s,t]})\le c(x^0({\alpha }_k(t))-x^0({\alpha }_k(s)))$. Passing to the limit $k\to \mathrm{\infty }$ yields $t-s\le c(x^0(\alpha (t))-x^0(\alpha (s)))$, which is the bound of (a3) with $C:=c\ge 1$ (enlarging $c$ to at least one if necessary).

(Future causality of the limit.) By (a1) and [L3], $\alpha$ is locally Lipschitz in charts, hence locally absolutely continuous and differentiable almost everywhere with derivative ${\alpha }^{\mathrm{\prime }}$ (Causal curves, chronological and causal reachability, and inextendibility clause (a), [L7]). Let $t$ be a point at which $\alpha$ is differentiable; choose the chart $U$ around $\alpha (t)$ and $h>0$ so small that $\alpha ([t,t+h^{\mathrm{\prime }}])\subseteq U$ for all $h^{\mathrm{\prime }}\le h$. By step 5.2, $x^0(\alpha (t+h^{\mathrm{\prime }}))-x^0(\alpha (t))\ge h^{\mathrm{\prime }}\mathrm{/}c>0$ for every $0<h^{\mathrm{\prime }}\le h$, so the difference quotients of $x^0\circ \alpha$ at $t$ from the right are at least $1\mathrm{/}c>0$, and dividing by $h^{\mathrm{\prime }}$ and letting $h^{\mathrm{\prime }}\downarrow 0$ gives $(x^0\circ \alpha {)}^{\mathrm{\prime }}(t)\ge 1\mathrm{/}c>0$. At the same time, by step 5.1, for small $h^{\mathrm{\prime }}>0$ the position vector $w_{h^{\mathrm{\prime }}}:={\exp }_{\alpha (t)}^{-1}(\alpha (t+h^{\mathrm{\prime }}))$ lies in the closed future cone, and by [L4] $w_{h^{\mathrm{\prime }}}\mathrm{/}{h}^{\mathrm{\prime }}\to {\alpha }^{\mathrm{\prime }}(t)$ as $h^{\mathrm{\prime }}\downarrow 0$; the cone is closed under positive scaling, so ${\alpha }^{\mathrm{\prime }}(t)$ lies in the closed future cone and has positive $x^0$-component, hence is a nonzero future causal vector by [L5]. Therefore at almost every parameter value the derivative of $\alpha$ exists, is nonzero and is future causal, that is, $\alpha$ is a future causal curve in the sense of Causal curves, chronological and causal reachability, and inextendibility.

(Nonconstancy.) Let $s<t$ and suppose $\alpha$ were constant on $[s,t]$, with image $\{q\}$. Applying step 5.2 with a chart $U$ around $q$ (whose existence is the content of Causal cone domination in adapted charts and Lipschitz-graph bounds for causal curves clause (a)) gives $t-s\le c(x^0(\alpha (t))-x^0(\alpha (s)))=0$, a contradiction. Hence $\alpha$ is not constant on any nondegenerate interval, which is (a4). Clauses (a1)-(a4) are proven, so (a) holds.

(Eventual half-line domains.) Let ${\gamma }_k:I_k\to M$ satisfy (b). Start with the infinite index set $S_0=\mathbb{N}$. Recursively, after $S_{n-1}$ is chosen, discard its finitely many indices for which $[0,n]\subseteq ̸I_k$; the eventual-domain hypothesis leaves an infinite tail. Apply (a) to those restrictions on $[0,n]$ and choose an infinite subsequence $S_n\subseteq S_{n-1}$ on which they converge uniformly to ${\gamma }^{(n)}:[0,n]\to M$. The choices depend on the preceding subsequence and are supplied by Dependent Choice in [A1]. Since $S_{n+1}\subseteq S_n$, uniqueness of the uniform limit gives ${\gamma }^{(n+1)}{\mathrm{\mid }}_{[0,n]}={\gamma }^{(n)}$. Choose increasing indices $k_n\in S_n$ recursively, for instance the least element greater than $k_{n-1}$. For each fixed $m$ and all $n\ge m$, $k_n\in S_m$ and $[0,m]\subseteq I_{k_n}$, so the actual restrictions ${\gamma }_{k_n}{\mathrm{\mid }}_{[0,m]}$ converge uniformly to ${\gamma }^{(m)}$. The consistent limits define $\gamma :[0,\mathrm{\infty })\to M$, satisfying (a1)-(a4) on every compact interval. Reversing the parameter and orientation applies this same eventual-tail construction to the past-domain assertion. This proves the convergence part of (b) for its full stated varying-domain scope.

(Future-inextendibility.) Suppose $\gamma (t)\to x\in M$ as $t\to \mathrm{\infty }$. Choose the chart $U$ of [L6] around $x$ and $t_0$ with $\gamma ([t_0,\mathrm{\infty }))\subseteq U$, and let $c$ be the constant of step 5.2 for this chart. Then for all $t\ge t_0$ the bound $t-t_0\le c(x^0(\gamma (t))-x^0(\gamma (t_0)))$ holds, so $x^0(\gamma (t))\ge x^0(\gamma (t_0))+(t-t_0)\mathrm{/}c\to +\mathrm{\infty }$ as $t\to \mathrm{\infty }$, contradicting $x^0(\gamma (t))\to x^0(x)\in \mathbb{R}$. Hence $\gamma$ does not converge at $+\mathrm{\infty }$ and is future-inextendible by Causal curves, chronological and causal reachability, and inextendibility clause (d). This proves (b).

(Conclusion.) Part (a) is steps 1.1-6.2 and part (b) is steps 7.1 and 8.1; Step 1.2 supplies the stated arclength parametrization.