Truncated causal-boundary height for smooth approximation

PDF p. 25: Proof of Lemma 4.4, definitions of Ψ and u

Use the supplied near-horizon data with $C=C_t$ for a fixed $t$, the section $\mathrm{\Sigma }={\mathrm{\Sigma }}_{q,0}$, boundary $S$, distance function $\rho$, and section time $v$. All causal relations and boundaries below are relative to the exterior $\mathcal{E}$ with metric $g_q$. Thus $I^{-}(C)$ is the set of points joined to $C$ by a future-directed timelike curve in $\mathcal{E}$, and $\mathrm{\partial }{I}^{-}(C)$ is its boundary in $\mathcal{E}$. Let $W\subset \mathcal{E}$ be the specified portion inside the stationary cylinder $\mathcal{C}$.

Let $\eta >0$ and $\mathcal{R}=\{x\in \mathrm{\Sigma }:0\le \rho (x)\le \eta \}$. On the relevant translated section interiors use the future unit normal $n$, lapse $N>0$, and tangent shift $X$ of the stationary field $T$, so $T=Nn+X$. Put $V=Nn=T-X$. Assume a local flow ${\mathrm{\Psi }}_z$ of $V$ is supplied on $\mathcal{R}$ for every $\mathrm{\mid }z\mathrm{\mid }\le 2\delta$, for a chosen $\delta >0$, with ${\mathrm{\Psi }}_z(x)\in \mathcal{E}$ for $x\in \mathcal{R}\setminus S$. If $S$ is outside $\mathcal{E}$, this includes a separately supplied extension of the flow to $S$; membership in $I^{-}(C)$ is false for points outside $\mathcal{E}$. The parameter $\delta$ is the local smoothing parameter, distinct from any fixed lower bound on $q$. Define

$u(x)=\sup (\{-2\delta \}\cup \{z\in [-2\delta ,2\delta ]:{\mathrm{\Psi }}_z(x)\in I^{-}(C)\})\mathrm{.}$

For $\mathrm{\mid }z\mathrm{\mid }\le 2\delta$, let ${\mathrm{\Sigma }}_z^{\circ }=\{p\in \mathcal{E}:v(p)=z\}$ and define the cut $D_z={\mathrm{\Sigma }}_z^{\circ }\cap \mathrm{\partial }{I}^{-}(C)\cap W$. Assume $v$ is temporal: $dv$ is positive on every nonzero future-directed causal vector. Use the supplied family of past-directed affinely parametrized null geodesics ${\gamma }_c$ from $c\in C$, with the specified outward null-normal direction and with affine parameter increasing toward the past. Define $A_z=\{c\in C:{\gamma }_c\text{ meets }{D}_z\}$, and for $c\in A_z$ let $F_z(c)$ be that intersection point.

Compact remainder, causal cut and generator endpoint map

Stationary section time and translated maximal sections

Causal exterior, chronological boundaries and Cauchy data

Spacelike sections, maximality and lapse–shift data

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

Local and global flows generated by a vector field

Given: The section, cut data, temporal function, geodesic family, positive lapse and common flow domain specified in the Definition.

Goal: Check the flow normalization and the well-definedness of $u$, $D_z$, $A_z$ and $F_z$.

1.1 Since $X$ is tangent to a level of $v$, $X(v)=0$; hence $V(v)=T(v)-X(v)=1$. The positive multiple $Nn$ is future timelike on the relevant section interiors. The existence and common domain of its flow, including any extension to $S$, are hypotheses. [F2, F3, F6]

2.1 For each $x$, the set inside the supremum contains $-2\delta$ and is bounded above by $2\delta$. It therefore has a unique real supremum, lying in $[-2\delta ,2\delta ]$. The specified level set, relative boundary, and inside set determine $D_z$, and the intersection condition determines $A_z$. [F4, F5, step 1.1]

3.1 Along a past-directed null geodesic, the temporal function $v$ strictly decreases as its affine parameter increases. Thus each such geodesic meets the level $v=z$ at most once; every $c\in A_z$ consequently has exactly one intersection with $D_z$, which defines $F_z:A_z\to D_z$. No claim is made here that $F_z$ is onto, that the cuts are regular, or that $u$ is Lipschitz. [F1, F2, step 2.1] ∎

The article citation attributes the construction. The justification uses the stated supplier interfaces and supplied data, rather than the article's claim as a proof of itself.