Compact remainder, causal cut and generator endpoint map

PDF p. 24: Lemma 4.3 notation; PDF p. 25: paragraph before Lemma 4.4

Fix a sufficiently small $q>0$ and use the supplied stationary section data, writing $\mathrm{\Sigma }={\mathrm{\Sigma }}_{q,0}$, ${\mathrm{\Sigma }}^{\circ }=\mathrm{\Sigma }\setminus S$, and $v=v_q$. Thus ${\mathrm{\Sigma }}^{\circ }$ is the level set $v=0$ and has induced Riemannian metric $h_q$. Assume that $\mathrm{\Sigma }$ is a connected smooth Riemannian manifold with boundary $S$, with metric extending $h_q$ to the boundary. Let $\rho :\mathrm{\Sigma }\to [0,+\mathrm{\infty }]$ be given by $\rho (x)={\inf }_{y\in S}{d}_{h_q}(x,y)$, with the infimum taken in the extended real line and $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. Here $d_{h_q}$ is intrinsic distance on $\mathrm{\Sigma }$. Define $S_{\eta }=\{x\in \mathrm{\Sigma }:\rho (x)=\eta \}$ for finite $\eta >0$. If $S\mathrm{\ne }\mathrm{\varnothing }$, then $\rho$ is the usual finite nonnegative boundary distance; if $S=\mathrm{\varnothing }$, then $\rho \equiv +\mathrm{\infty }$ and $S_{\eta }=\mathrm{\varnothing }$.

Use the comparison sphere $C=C_q$, its stationary translate $C_t$, and the stationary cylinder $\mathcal{C}$. Supply the closed region $B\subset \mathrm{\Sigma }$ described as the portion inside $\mathcal{C}$, including $S$; the choice of inside is part of the data. All chronological relations are relative to the exterior $\mathcal{E}$ with metric $g_q$: $I_{\mathcal{E}}^{-}(C_t)$ consists of points joined to a point of $C_t$ by a future-directed timelike curve in $\mathcal{E}$. Define $K=B\setminus I_{\mathcal{E}}^{-}(C_t)$, regarding $S$ as disjoint from this subset of $\mathcal{E}$, and define $D={\mathrm{\partial }}_{{\mathrm{\Sigma }}^{\circ }}(K\cap {\mathrm{\Sigma }}^{\circ })$, the boundary in the topology of ${\mathrm{\Sigma }}^{\circ }$.

For the endpoint construction, assume $v$ is temporal on $\mathcal{E}$, meaning its differential is positive on every nonzero future-directed causal vector. Assume $C_t$ is a spacelike two-surface with a specified outward future null normal direction at each point, and supply for each $c\in C_t$ a past-directed affinely parametrized null geodesic ${\gamma }_c$ starting at $c$ in the opposite direction, on an interval containing zero and staying in $\mathcal{E}$ on the portion considered. Here a null geodesic has nonzero null tangent and zero covariant acceleration; its affine parameter increases toward the past. Put $A=\{c\in C_t:{\gamma }_c\text{ meets }D\}$, and for $c\in A$ define $F(c)$ to be its point of intersection with $D$. Once $t$ is fixed, $C_t$ may be denoted by $C$. No compactness, coverage of $D$, smoothness, or area property of these sets or of $F$ is asserted by these definitions.

Stationary section time and translated maximal sections

Comparison spheres and their stationary cylinders

Causal exterior, chronological boundaries and Cauchy data

Null expansions, screen quotients and focal data

Spacelike sections, maximality and lapse–shift data

Riemannian distance on a connected manifold

Every subset of ℝ̄ has a least upper bound and a greatest lower bound in ℝ̄, agreeing with the real supremum and infimum on nonempty sets bounded in ℝ

Given: The section, boundary metric, inside region, temporal function and chosen geodesic family in the Definition.

Goal: Verify that the displayed sets and endpoint map are unambiguously defined.

1.1 For each $x\in \mathrm{\Sigma }$, the set $\{d_{h_q}(x,y):y\in S\}$ consists of finite nonnegative reals. If $S\mathrm{\ne }\mathrm{\varnothing }$, this set is nonempty and bounded below by zero, so its extended-real infimum agrees with its real infimum and is finite, bounded above by $d_{h_q}(x,y_0)$ for any fixed $y_0\in S$. If $S=\mathrm{\varnothing }$, its extended-real infimum is $+\mathrm{\infty }$, so every finite level set $S_{\eta }$ is empty. This defines $\rho :\mathrm{\Sigma }\to [0,+\mathrm{\infty }]$ and all the displayed level sets without a nonempty-boundary assumption. The specified inside region and chronological past determine $K$; taking closure minus interior in ${\mathrm{\Sigma }}^{\circ }$ determines $D$, including if either set is empty. [given, F1, F2, F3, F4, F6]

2.1 Along each past-directed null geodesic, $v$ strictly decreases as its affine parameter increases, because its tangent is the negative of a future-directed causal vector and $dv$ is positive on such vectors. Consequently the geodesic meets the level set ${\mathrm{\Sigma }}^{\circ }=\{v=0\}$ at most once. Since $D\subset {\mathrm{\Sigma }}^{\circ }$, every $c\in A$ has exactly one intersection point with $D$, so $F:A\to D$ is a function. This argument assumes the displayed family and does not prove its existence or that every point of $D$ is reached. [F1, F5, step 1.1] ∎