Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions

(Finiteness.) Let $r\in J^{-}(q)$ and let $\alpha$ be any future causal curve from $r$ to $q$. Every point of $\alpha$ lies in $J^{+}(r)\cap J^{-}(q)$, which is compact, so by [L1] the $h$-arclength of $\alpha$ and hence $L_g(\alpha )$ is bounded by a constant depending only on $r,q$; therefore ${\tau }_q(r)<\mathrm{\infty }$. If $r\mathrm{\notin }{J}^{-}(q)$ no such curve exists and ${\tau }_q(r)=0$; if $r=q$ causality excludes a nonconstant return curve and the stipulated zero-separation convention gives ${\tau }_q(q)=0$, represented by the constant degenerate geodesic as specified in (a); and ${\tau }_q(r)>0$ iff there is a future timelike curve from $r$ to $q$, i.e. $r\in I^{-}(q)$.

(Attainment and geodesy of maximizers.) Let $r\in J^{-}(q)\setminus \{q\}$ and, using Countable Choice, let ${\alpha }_n$ be causal curves from $r$ to $q$ with $L_g({\alpha }_n)\to {\tau }_q(r)$. All images lie in the compact set $J^{+}(r)\cap J^{-}(q)$; parametrize them by $h$-arclength as ${\alpha }_n:[0,b_n]\to M$. After a subsequence $b_n\to b<\mathrm{\infty }$ by [L1], and $b\ge d_h(r,q)>0$. The bounded two-endpoint theorem [L2] gives a joining limit $\alpha :[0,b]\to M$ and uniform convergence of the actual normalized causal representatives ${\tilde{\alpha }}_n(t):={\alpha }_n((b_n\mathrm{/}b)t)$ on $[0,b]$. Set ${\beta }_n(s):={\tilde{\alpha }}_n(bs)={\alpha }_n(b_{n}s)$ and $\beta (s):=\alpha (bs)$ on $[0,1]$; their uniform convergence is exactly that of the supplied representatives after the fixed parameter change. The ${\beta }_n$ and $\beta$ are causal curves on a single fixed domain; increasing affine reparametrization preserves Lorentzian length. Thus [L2]'s length semicontinuity applies to these curves, and, with $\gamma :=\beta$, gives $L_g(\gamma )\ge \mathrm{lim\hspace{0.17em}sup}{L}_g({\alpha }_n)={\tau }_q(r)$, so $L_g(\gamma )={\tau }_q(r)$. In every sufficiently small convex normal neighborhood, the local Lorentzian normal-maximum theorem [L3] bounds the length of a subarc by that of its radial causal geodesic, with equality only along that geodesic up to increasing reparametrization. If any subarc failed this equality condition, finite replacement would strictly increase the total length, contradicting maximality. Thus all local subarcs have geodesic images; on overlaps they have the same tangent ray and geodesic uniqueness joins them into a single causal geodesic up to monotone reparametrization, and its image lies in $J^{+}(r)\cap J^{-}(q)$.

(The radial formula.) Let $V,W,q$ be as in clause (b) and let $r\in W\cap I^{-}(q)$. Let $\gamma$ be a maximizer from $r$ to $q$ (step 2.1). Since $W$ is causally convex, $\gamma \subseteq W\subseteq V$; by [L3] applied in $V$ with base point $q$, every causal curve in $V$ from $r$ to $q$ has length at most $\sqrt{-g_q({\exp }_q^{-1}(r),{\exp }_q^{-1}(r))}$, with equality exactly for the radial geodesic, which lies in $V$; since all competitors from $r$ to $q$ lie in $W\subseteq V$ and the radial segment itself is a competitor with that length, ${\tau }_q(r)=\sqrt{-g_q({\exp }_q^{-1}(r),{\exp }_q^{-1}(r))}$. The right-hand side is smooth on $I^{-}(q)\cap W$ by [L3] and the smoothness of the inverse exponential map; its continuous extension by $0$ at points of $W\setminus J^{-}(q)$ and at $q$ is exactly ${\tau }_q$ there by step 1.1.

(Upper semicontinuity on compact sets.) Let $r_n\to r$ in a compact $K$. If $\mathrm{lim\hspace{0.17em}sup}{\tau }_q(r_n)=0$, upper semicontinuity is immediate from nonnegativity. Otherwise take a subsequence approaching that positive limsup; discard zero-separation terms, so every remaining $r_n\mathrm{\ne }q$ is causally related to $q$. Choose, by Countable Choice, ${\alpha }_n$ be causal curves from $r_n$ to $q$ with $L_g({\alpha }_n)\ge {\tau }_q(r_n)-1\mathrm{/}n$. Passing to the compact diamond $J^{+}(K^{\mathrm{\prime }})\cap J^{-}(q)$ for a compact $K^{\mathrm{\prime }}$ containing the $r_n$ ([L1]), parametrize ${\alpha }_n:[0,b_n]\to M$ by $h$-arclength and pass to $b_n\to b<\mathrm{\infty }$ using [L1]. The case $b=0$ cannot occur on this positive-limsup subsequence: the compact-set metric comparison gives $L_g({\alpha }_n)\le C{b}_n\to 0$, contradicting $L_g({\alpha }_n)\ge {\tau }_q(r_n)-1\mathrm{/}n$. For $b>0$ the bounded two-endpoint limit gives $\alpha :[0,b]\to M$. As in step 2.1, the actual causal curves ${\beta }_n(s)={\alpha }_n(b_{n}s)$ converge uniformly on $[0,1]$ to $\gamma (s)=\alpha (bs)$, and retain exactly the original Lorentzian lengths. Hence the common-domain length theorem applies, giving ${\mathrm{lim\hspace{0.17em}sup}}_n{\tau }_q(r_n)\le {\mathrm{lim\hspace{0.17em}sup}}_n{L}_g({\alpha }_n)\le L_g(\gamma )\le {\tau }_q(r)$ by upper semicontinuity [L2] and step 2.1. If $r\in I^{-}(q)$ then a maximizer ${\gamma }_r$ from $r$ to $q$ exists (step 2.1); since $I^{-}(q)$ is open, for $n$ large the inner segment of ${\gamma }_r$ can be joined to $r_n$ by a short timelike segment inside $I^{-}(q)$, giving ${\mathrm{lim\hspace{0.17em}inf}}_n{\tau }_q(r_n)\ge {\tau }_q(r)$ by letting the inner cut shrink; combined with the upper bound this gives continuity at $r$.