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The exposed boundary point controls the complement: J^-(q) cap U^c cap J^+(Sigma) is confined to a prescribed neighbourhood

Statement

Assume the Axioms of Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $(M, g, N, j, ψ)$ and $(M^{'}, g^{'}, N^{'}, j^{'}, ψ^{'})$ be developments of the same data, let $(U, φ)$ be a common subdevelopment in the sense of Common subdevelopments, maximal common developments and corresponding boundary points clause (a), write $Σ : = j (Σ)$ , and let $p \in \partial U$ satisfy $J^{-} (p) \cap \partial U \cap J^{+} (Σ) = {p}$ as in An exposed boundary point: the last boundary point of a boundary null geodesic has no other boundary point in its past. Then for every open neighbourhood $W$ of $p$ in $M$ there exists $q \in I^{+} (p)$ with $J^{-} (q) \cap U^{c} \cap J^{+} (Σ) \subseteq W .$

Facts & Assumptions

Given: developments $M, M^{'}$ of the same data, a common subdevelopment $(U, φ)$ , and a point $p \in \partial U$ with the displayed exposed-point property, and an open neighbourhood $W$ of $p$ .

[L1]

Global hyperbolicity: $M$ is globally hyperbolic with closed causal relation, $M = I^{-} (Σ) ⊔ Σ ⊔ I^{+} (Σ)$ and $J^{+} (Σ) \cap J^{-} (Σ) = Σ$ (The existence of a Cauchy set implies strong causality and compact causal diamonds, In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact clause (a), Structure of corresponding boundary points: achronality of the future boundary, equality of pasts, and uniqueness [L1] as used there).

[L2]

Compact truncated pasts: $J^{-} (r) \cap J^{+} (Σ)$ is compact for every $r$ , and it is empty for $r \in I^{-} (Σ)$ by [L1] (The causal past of a point truncated by the causal future of the initial Cauchy surface is compact).

[L3]

Subdevelopment convexity and boundary structure: the achronality of $\partial U \cap I^{+} (Σ)$ and the convexity of A development sharing the initial Cauchy surface with a larger development is causally convex in it are available (Structure of corresponding boundary points: achronality of the future boundary, equality of pasts, and uniqueness clause (a), A development sharing the initial Cauchy surface with a larger development is causally convex in it).

[L4]

Past convexity of the subdevelopment: every causal curve in $M$ from $Σ$ to a point of $U \cap I^{+} (Σ)$ is contained in $U$ (A development sharing the initial Cauchy surface with a larger development is causally convex in it clause (a)); and a causal curve in the globally hyperbolic $M$ between two chronologically related points of $U$ is contained in $U$ (A development sharing the initial Cauchy surface with a larger development is causally convex in it clause (c)).

No choice principle beyond the two declared axioms is used in this item.

Proof

technique · direct

1.1givenL1choose

(Setup.) Let $γ : [0, ε] \to M$ be a future timelike curve with $γ (0) = p$ (it exists since every point of a time-oriented Lorentzian manifold has a future timelike curve). If the conclusion failed for every $q = γ (t)$ , there would be a sequence $t_{j} \in (0, ε]$ with $t_{j} \to 0$ and points $q_{j} \in J^{-} (γ (t_{j})) \cap U^{c} \cap J^{+} (Σ) \cap W^{c}$ .

1.2L1L2L3L4choosecontradictionalgebra

(The correctly directed complement exclusion.) Suppose $r \in int (U^{c}) \cap J^{+} (Σ)$ and $r \leq p$ . Since $Σ \subset U$ , $r \notin Σ$ and hence $r \in I^{+} (Σ)$ . Choose a short past timelike perturbation $r_{-} ≪ r$ inside the open set $int (U^{c}) \cap I^{+} (Σ)$ . Push-up gives $r_{-} ≪ r \leq p$ , so $I^{+} (r_{-}) \cap I^{+} (Σ)$ is an open neighborhood of the boundary point $p$ . It meets $U$ at $s$ . A timelike curve from $Σ$ to $r_{-}$ followed by one from $r_{-}$ to $s \in U \cap I^{+} (Σ)$ contradicts past convexity [L4], because it passes through $r_{-} \notin U$ . Thus there is no such $r$ . This uses a past perturbation; no future perturbation below a null-related endpoint is inferred.

2.1step 1.1L1L2algebra

(Compactness and a limit.) Fix any $t_{*} \in (0, ε]$ with $t_{*} \geq t_{j}$ for all $j$ (equivalently pass to the subsequence with $t_{j} \leq ε$ ); then $q_{j} \in J^{-} (γ (ε)) \cap U^{c} \cap J^{+} (Σ) \cap W^{c}$ for all $j$ . Since $U^{c}$ and $W^{c}$ are closed and $J^{+} (Σ)$ is closed [L1], the set $J^{-} (γ (ε)) \cap U^{c} \cap J^{+} (Σ) \cap W^{c}$ is closed in the compact set $J^{-} (γ (ε)) \cap J^{+} (Σ)$ of [L2], hence compact; passing to a subsequence, $q_{j} \to q$ with $q \in J^{-} (γ (ε)) \cap U^{c} \cap J^{+} (Σ) \cap W^{c}$ .

3.1step 2.1L1algebra

( $q \leq p$ .) Since $q_{j} \leq γ (t_{j})$ for every $j$ , $q_{j} \to q$ and $γ (t_{j}) \to p$ , closedness of the causal relation [L1] gives $q \leq p$ .

4.1step 2.1step 3.1step 1.2contradiction∎

( $q = p$ , hence $q \in W$ , contradiction.) If $q \in \partial U$ , then $q \in J^{-} (p) \cap \partial U \cap J^{+} (Σ) = {p}$ by the exposed-point property, so $q = p \in W$ , contradicting $q \in W^{c}$ of step 2.1. Hence $q \in U^{c} ∖ \partial U$ ; but $q \leq p$ by step 3.1 and $q \in J^{+} (Σ)$ by step 2.1, contradicting step 1.2. Therefore the assumption of step 1.1 is false: for every sufficiently small $t$ one has $J^{-} (γ (t)) \cap U^{c} \cap J^{+} (Σ) \subseteq W$ , which is the assertion with $q : = γ (t)$ .

Remarks

This is Sbierski's Lemma 3.11. Its content is that the exposed boundary point $p$ is "the last point of the complement in its own past": past the exposed point, the complement recedes into every neighbourhood of $p$ , so the time separation from a nearby $q$ is maximized exactly on the boundary; this is used in The spacelike restart surface at an exposed boundary point: maximizers on the boundary, the level hypersurface and its radial tube. Step 2.1 reproduces the source's exclusion of $U^{c} ∖ \partial U$ points causally below $p$ in the same two-case form (timelike case and null case) as the corresponding argument in An exposed boundary point: the last boundary point of a boundary null geodesic has no other boundary point in its past; the source's phrase "contradicts the global hyperbolicity of $U$ " is here expanded into the use of past convexity of the subdevelopment, A development sharing the initial Cauchy surface with a larger development is causally convex in it clause (a).

The exposed boundary point controls the complement: J^-(q) cap U^c cap J^+(Sigma) is confined to a prescribed neighbourhood

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources