Uniform localization of translated past-boundary cuts

By [F6], $L=B\cap \{\rho \ge b\}$ is closed in compact $B$, so [F5] makes it compact. It avoids $S$ and is contained in $P$ at time zero. Write $T_z(x)=(z,x)$. Since $P$ is open, the open section sets $O$ for which some $ϵ>0$ gives $T_z(O)\subset P$ whenever $\mathrm{\mid }z\mathrm{\mid }<ϵ$ cover $L$. Take finitely many such sets covering $L$ and, for each of these finitely many sets, one admissible $ϵ$. Their union $W$ and the positive minimum $\tau$ of these finitely many numbers satisfy $T_z(W)\subset P$ for $\mathrm{\mid }z\mathrm{\mid }<\tau$. This makes no simultaneous choice over all points of $L$. Every $x\in \mathrm{\Omega }$ with $\rho (x)\ge b$ belongs to $L$, so none of these points can give a boundary cut. Also $\mathrm{\Gamma }\cap \mathcal{R}=\mathrm{\varnothing }$ because $\mathcal{R}\subset {\mathrm{int}}_{\mathrm{\Sigma }}B$; hence $\mathrm{\Gamma }\subset L\subset W$. Thus this argument controls the entire specified inside and an open neighborhood of its outer boundary. No graph-section compact-core theorem is used.

Put $V_Q={\max }_{Q}v$, finite by [F5], and choose $0<z_0<{\min }_{Q}v\mathrm{/}2$. For $\mathrm{\mid }z\mathrm{\mid }\le z_0$, the stationary translate $T_{-z}Q$ is compact, has positive minimum time and maximum time $V_Q-z$, and has the same spatial projection as $Q$. Stationary translation preserves the metric, time orientation and causal curves in $E$, so [F2] applied to $T_{-z}Q$ excludes reachability from $(z,x)$ whenever $0<\rho (x)<a_0{e}^{-C(V_Q-z)}$. Set $d_{\ast }=a_0{e}^{-C(V_Q+z_0)}>0$ and choose $0<a<\min (b\mathrm{/}2,d_{\ast }\mathrm{/}4)$. The open spacetime region $\{\mathrm{\mid }v\mathrm{\mid }<z_0,0<\rho <d_{\ast }\}$ is disjoint from $P$, and therefore also from ${\mathrm{\partial }}_{E}P$: each of its points has an open neighborhood disjoint from $P$. In particular, all points with $\mathrm{\mid }v\mathrm{\mid }<z_0$ and $0<\rho \le 2a$ avoid both $P$ and its boundary. Points of $S$ are separately absent from $E$.

For $\mathrm{\mid }z\mathrm{\mid }<\min (\tau ,z_0)$, steps 1.1 and 1.2 put every cut point in $T_z\{2a<\rho <b\}$. The annulus $A=\{2a\le \rho \le b\}$ is a closed subset of compact $\mathcal{R}$, hence compact by [F5]. The full annulus lies in $\mathrm{\Omega }$ because $A\subset \mathcal{R}\subset {\mathrm{int}}_{\mathrm{\Sigma }}B$ and $A\cap S=\mathrm{\varnothing }$. Its stationary translate $T_z(A)$ is compact. In fact $D_z={\mathrm{\partial }}_{E}P\cap T_z(A)$: localization gives one inclusion, and the other follows from $A\subset \mathrm{\Omega }$. The boundary ${\mathrm{\partial }}_{E}P$ is closed in $E$, so $D_z$ is a closed subset of that compact translated annulus and is compact. This uses the full annulus supplied by the collar, not an annulus inferred from a graph-core result.

At time zero, $A$ is contained in the open collar annulus $\{a\mathrm{/}2<\rho <\eta \mathrm{/}2\}$, since $2a>a\mathrm{/}2$ and $b<\eta \mathrm{/}4$. Smooth inverse-flow dependence and compactness of $A$ keep ${\psi }_{-z}(A)$ in this annulus for all sufficiently small $\mathrm{\mid }z\mathrm{\mid }$. The spatial inverse-flow identity gives $T_z(A)\subset {\mathrm{\Psi }}_z(\{a\mathrm{/}2\le \rho \le \eta \mathrm{/}2\})$. This latter closed annulus lies in $\mathcal{R}\setminus \mathrm{\partial }\mathcal{R}$. Similarly, the open set ${\mathrm{int}}_{\mathrm{\Sigma }}B$ contains compact $\mathcal{R}$, so continuity and a finite subcover give a common time interval on which ${\psi }_z(\mathcal{R})\subset {\mathrm{int}}_{\mathrm{\Sigma }}B$. The spatial flow preserves $S$ and keeps interior points interior by [F1]. Thus the whole interior flow collar stays in ${\mathbb{R}}_v\times \mathrm{\Omega }$, and the two required localization inclusions follow.

By continuity of the spatial flow on compact $S$, preservation of $S$ and $\rho {\mathrm{\mid }}_S=0$, there are a fixed relative neighborhood $U_S$ of $S$ in $\mathcal{R}$ and a common small time interval with $\rho ({\psi }_{z}x)<2a$ for $x\in U_S$. The slice $S_{\eta }=\mathcal{R}\cap \{\rho =\eta \}$ is closed in compact $\mathcal{R}$ and hence compact. By continuity on $S_{\eta }$, where $\rho =\eta >b$, there are a fixed relative neighborhood $U_{\eta }$ of $S_{\eta }$ and a common small time interval with $\rho ({\psi }_{z}x)>b$ for $x\in U_{\eta }$. Step 3.1 keeps these spatial images in $B$. Step 1.2 therefore excludes the interior points of ${\mathrm{\Psi }}_z(U_S)$ from $P$, while step 1.1 puts ${\mathrm{\Psi }}_z(U_{\eta })$ in $P$, since its spatial images belong to $L$. The points of $S$ remain outside $P$ by the boundary convention. Choose ${\delta }_0>0$ so that $\mathrm{\mid }z\mathrm{\mid }\le 2{\delta }_0$ lies strictly within every common time interval just obtained, including $\mathrm{\mid }z\mathrm{\mid }<\min (\tau ,z_0)$. This gives both fixed endpoint neighborhoods, compact cuts and the asserted localization uniformly on the closed time range.