Local causality and temporal functions on compact tubes

Put $B={\tilde{Q}}^{\circ }$. Every compact $C\subset B$ lies in $\tilde{Q}$, so the single finite bound in [F1] bounds every eligible curve in $C$, with the curve class and quantifiers required by [F2]. Apply [F2] to $g_{q_{\ast }}$ on $B$ and the compact set $Q$. This gives an open $O\subset B$ containing $Q$ and a smooth ${\tau }_{\ast }$ with past-timelike $g_{q_{\ast }}$-gradient.

Since $Q$ is compact and $g_{q_j}\to g_{q_{\ast }}$ uniformly there, the perturbation conclusion of [F2], with the consistent orientation in [F3], makes ${\mathrm{grad}}_{g_{q_j}}{\tau }_{\ast }$ past timelike on $Q$ for all sufficiently large $j$.

Use [A1] and [F5] to choose $h$. If $Q$ is empty all conclusions hold, so suppose it is nonempty. The $h$-unit future causal vectors based at $Q$ form a compact set: in a finite cover by relatively compact coordinate trivializations, the unit condition gives bounded fiber coordinates and the unit and future causal conditions are closed. By [F4], the continuous function $v\mapsto d{\tau }_{\ast }(v)$ is strictly positive there, so it has a minimum $c>0$. By positive homogeneity, $d{\tau }_{\ast }(v)\ge c\mathrm{\mid }v{\mathrm{\mid }}_h$ for every future causal $v$ over $Q$. For a future causal $\gamma :I\to Q$, put $f={\tau }_{\ast }\circ \gamma$. By [F4] and [F6], for any $a<b$ in $I$ one has $c{\int }_a^b\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_{h}dt\le f(b)-f(a)$. Since $f$ is increasing and bounded on $I$, it has finite endpoint limits, and the integral of $\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_h$ has arbitrarily small tails at either end. This argument uses the original parametrization, without invoking an arc-length theorem.

Here is the endpoint argument in local coordinates. Let $\beta =\sup I$ and let $p\in Q$ be a cluster point of $\gamma (t)$ as $t\to \beta$; such a point exists by compactness (the closures in $Q$ of the nested nonempty tails have nonempty intersection). Choose coordinates $x$ centered at $p$ whose image contains a closed ball of radius $2r>0$ lying in $O$, and take $D$ as in [F5] on this ball. Choose $a$ sufficiently late that $\mathrm{\mid }x(\gamma (a))\mathrm{\mid }<r\mathrm{/}2$ and $D{\int }_a^{\beta }\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_{h}dt<r\mathrm{/}2$. Until a first exit from the radius-$r$ ball, [F6] gives $\mathrm{\mid }x(\gamma (t))-x(\gamma (a))\mathrm{\mid }\le D{\int }_a^t\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_h<r\mathrm{/}2$, which forbids that exit. The entire tail therefore stays in that ball. The same inequality on any subinterval of the tail makes its coordinates Cauchy at $\beta$. Their limit is $x(p)$, since $p$ was a cluster point. Thus $\gamma (t)\to p\in Q$. Reversing the parameter gives the past endpoint as well. No completeness assumption on $h$ is used.

For completeness, these endpoints give extensions in the stated locally Lipschitz causal class, with its unparametrized meaning. The function $f$ is continuous and strictly increasing by [F4], and thus has an increasing continuous inverse on its image $J=f(I)$. Reparametrize the future tail by $u=f(t)$ and write $\eta (u)=\gamma (f^{-1}(u))$. For any two parameters on the tail, the chart estimate and step 2.2 give $\mathrm{\mid }x(\eta (v))-x(\eta (u))\mathrm{\mid }\le (D\mathrm{/}c)(v-u)$ when $u<v$. Thus $\eta$ extends Lipschitz continuously to the finite endpoint $u_{+}={\lim }_{t\to \beta }f(t)$ with value $p$. To verify its causal orientation, discard the null set of original parameters where $\gamma$ or $f$ is not differentiable, their chain rule fails, or $\dot{\gamma }$ is not nonzero future causal. The map $f$ is locally Lipschitz, since $\gamma$ is locally Lipschitz and ${\tau }_{\ast }$ is smooth; [F7] makes the image of this exceptional set null. At every remaining parameter, $f^{\mathrm{\prime }}=d{\tau }_{\ast }(\dot{\gamma })\ge c\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_h>0$. The elementary inverse difference quotient gives $(f^{-1}{)}^{\mathrm{\prime }}(f(t))=1\mathrm{/}{f}^{\mathrm{\prime }}(t)$, and hence ${\eta }^{\mathrm{\prime }}(f(t))=\dot{\gamma }(t)\mathrm{/}{f}^{\mathrm{\prime }}(t)$, a positive multiple of a nonzero future causal vector. A short smooth future timelike segment from $p$ exists: in a chart a straight line with an initially future timelike tangent stays future timelike for sufficiently small positive time by openness of the timelike cone. Adjoin it at $u_{+}$; both pieces are Lipschitz in the endpoint chart, and their common endpoint gives a Lipschitz concatenation. Off the single junction the velocities are nonzero future causal almost everywhere, so this extends the unparametrized future causal curve in $\mathcal{E}$. The past extension is identical with a past endpoint and a future-directed segment arriving there.

Consequently no causal curve contained in $Q$ is inextendible in $\mathcal{E}$. The conditional deformation assertion follows because its antecedent is false. Together with the temporal function of step 1.1 and its persistence from step 2.1 this proves all stated conclusions.