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A maximal stationary exterior has future access to its far tail

Statement

Assume the Axiom of Choice. Let $E = R \times C$ be a smooth stationary time-oriented Lorentz product as in the inaccessible-set lemma, with complete time-translation Killing field $T = \partial_{t}$ and spacelike slices. Write $n$ for the unit future normal, $h = g ∣_{T C}$ , $k (U, V) = - g (\nabla_{U} n, V)$ , $N = - g (T, n)$ and $X = T - N n$ . Suppose $C$ is connected and is the interior of a smooth spherical exterior $\overline{C}$ : by this we mean a smooth manifold with compact boundary $S$ supplied with a diffeomorphism $F : \overline{C} \to {y \in R^{3} : ∣ y ∣ \geq 1}$ , with $C = \overline{C} ∖ S$ . Its stationary slice is maximal, ${tr}_{h} k = 0$ . Assume its induced positive metric is smooth up to compact boundary $S$ and asymptotically Euclidean. Near $S$ , assume the ambient time-oriented Lorentz metric $g$ , the field $T$ and the slice extend smoothly across $S$ , with the extended slice spacelike and $T$ tangent to $S$ . These are extensions of the original geometric data, not an extension of the stationary product coordinates through $S$ . Assume $N > 0$ throughout $C$ . Assume $T$ is future directed sufficiently far out. Specify the asymptotic stationary end by an open set $A \subset C$ and a diffeomorphism $x : A \to {z \in R^{3} : ∣ z ∣ > R_{0}}$ for some $R_{0} > 0$ , such that $K_{R} = \overline{C} ∖ x^{- 1} ({∣ z ∣ > R})$ is compact for every $R > R_{0}$ . Put $r (p) = ∣ x (p) ∣$ on $A$ ; the notation ${r > R}$ means $x^{- 1} ({∣ z ∣ > R})$ . In the stationary coordinates $(t, x)$ on this end let $g_{00} = - 1 + O (r^{- 1})$ , $g_{0 i} = O (r^{- 2})$ , and $g_{i j} = δ_{i j} + O (r^{- 1})$ . For any sufficiently large $R_{f}$ , put $U_{f} = R \times {r > R_{f}}$ . Then every point of $E$ belongs to $I_{E}^{-} (U_{f})$ , where chronological reachability is by locally absolutely continuous future timelike curves contained in $E$ . No timelikeness assumption on the entire designated end, or invariance of a separately defined domain of outer communications under end truncation, is used.

Statement references

Spacelike sections, maximality and lapse–shift data

Causal curves, chronological and causal reachability, and inextendibility

Inaccessible set is the closure of its relative interior

The Axiom of Choice

Complete volume preserving shift flow on a spherical exterior

Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right

Facts & Assumptions

[A1]

The Axiom of Choice is assumed, meeting the foundational hypotheses of the flow, measure and inaccessible-set suppliers. The Axiom of Choice

[F1]

Maximality, Killing's equation and smooth extension of the section and fields across $S$ , with $T$ tangent to $S$ , give zero shift divergence, boundary tangency and asymptotic decay. Maximal stationary shift has zero divergence and tangent boundary

[F2]

On a smooth spherical exterior supplied with $F$ and an end chart $x$ whose complements $K_{R}$ are compact, as explicitly specified in the Statement, a smooth shift up to $S$ , tangent there, divergence free and $O (r^{- 2})$ on the asymptotically Euclidean end has a complete volume-preserving flow. Borel subsets of bounded exterior annuli ${p \in C : 1 < ∣ F (p) ∣ \leq L}$ , $L < \infty$ , have finite volume even when their closures meet $S$ . Complete volume preserving shift flow on a spherical exterior

[F3]

For a stationary product with spacelike slices, globally positive lapse and complete $- X$ flow, the inaccessible set is closed and is the closure of its relative interior, and its forward images lie in that interior. Its proof also establishes time-translation invariance of the chronological past of a translation-invariant target. Inaccessible set is the closure of its relative interior

[F4]

Finite measure and those exact topological properties force a nonempty-complement inaccessible set to be empty. Finite volume and interior density force exact flow invariance

[F5]

The hypersurface convention gives the smooth future unit normal, induced metric, second fundamental form, maximality and lapse–shift splitting. Apply its normal construction on the stipulated smooth spacelike extension near $S$ . Spacelike sections, maximality and lapse–shift data

[F6]

Chronological past consists of points joined to the target by a locally absolutely continuous future timelike curve in the specified Lorentz manifold. Causal curves, chronological and causal reachability, and inextendibility

[F7]

A compact subset is compact in its subspace topology. Equivalently, every family of ambient open sets covering it has a finite covering subfamily. This applies to the compact sets $K_{R}$ and the ambient manifold $\overline{C}$ . Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it

[F8]

A closed subset of $C$ has open complement. The Borel sigma-algebra of $C$ contains every open subset and, being a sigma-algebra, is closed under complements relative to $C$ . Hence every closed subset of $C$ is Borel. Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison The Borel sigma-algebra of a topological space Sigma-algebras

Proof

Given: The hypotheses in the Statement.

Goal: Establish future chronological access to every sufficiently far stationary tail.

1.1givenF6

Choose $R_{f}$ so large that $g (T, T) = g_{00} < 0$ on $U_{f}$ and $T$ is future directed there. Each point of $U_{f}$ reaches another point of $U_{f}$ by a positive time translation, so $U_{f} \subset I_{E}^{-} (U_{f})$ . Define $B = {x \in C : (0, x) \notin I_{E}^{-} (U_{f})}$ . It is disjoint from ${r > R_{f}}$ , so its complement is nonempty.

1.2givenF1F5algebra

By [F5] applied to the smooth spacelike extension near $S$ , $n$ extends smoothly across $S$ . The formulas $N = - g (T, n)$ and $X = T - N n$ therefore give smooth extensions of the lapse and shift along that slice. Since $T$ is tangent to $S$ , orthogonality gives $N ∣_{S} = 0$ and $X ∣_{S} = T ∣_{S} \in T S$ . Together with maximality, [F1] gives ${div}_{h} X = 0$ and $X^{i} = O (r^{- 2})$ . Thus the boundary regularity needed by [F1] and [F2] follows from the explicit ambient and slice extensions, rather than from smoothness of $h$ alone.

2.1givenA1F2F3F7F8step 1.1step 1.2

Apply [F2] to obtain a complete flow $ψ_{s}$ of $- X$ preserving $μ_{h}$ , which is positive on nonempty open sets. Apply [F3] with this very $U_{f}$ and the globally positive lapse: $B$ is closed, $B = {\overline{{int}_{C} B}}^{C}$ , and $ψ_{s} (B) \subset {int}_{C} B$ for $s > 0$ . In particular $B$ is Borel by [F8]. By step 1.1, $B \subset K_{R_{f}} \cap C$ . The sets ${p \in \overline{C} : ∣ F (p) ∣ < m}$ for integers $m \geq 2$ form an increasing open cover of the compact $K_{R_{f}}$ , so [F7] gives a finite covering subfamily and hence $∣ F (p) ∣ < L$ there for some finite $L > 1$ . Hence $B$ is contained in the bounded exterior annulus ${p \in C : 1 < ∣ F (p) ∣ \leq L}$ of [F2], which gives $μ_{h} (B) < \infty$ even if $B$ accumulates at $S$ . All flow hypotheses are imposed on the original slice up to its boundary; no orbit is extended through a horizon. The choice assumptions of both suppliers are met by [A1].

3.1F3F4F6step 1.1step 2.1∎

Apply [F4] on the connected $C$ to obtain $B = \emptyset$ . Time-translation invariance of $I_{E}^{-} (U_{f})$ from [F3] extends access from $t = 0$ to every point of $E$ : translate the timelike curve supplied at $(0, x)$ by $t$ to obtain one from $(t, x)$ to $U_{f}$ . This proves the stated relative-exterior chronological-access assertion without replacing the designated end in the definition of $E$ .

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