LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A compact chronological target supplies a causal endpoint

Statement

Let $(E, g, N)$ be globally hyperbolic in the causal-exterior interface, and let $C \subset E$ be a compact spacelike two-surface in the null-screen interface. Let $τ$ be a temporal function and suppose $p \in \partial_{E} I_{E}^{-} (C)$ with $τ (p) < \min_{C} τ$ . Then some $c \in C$ satisfies $p \leq_{E} c$ and $p \neq c$ , whereas $p \notin I_{E}^{-} (C)$ . No acausality of $C$ is assumed.

Facts & Assumptions

[A1]

Work in ZFC; the axiom of choice includes the countable and dependent choice assumptions of the causal suppliers. The Axiom of Choice

[F1]

All chronological and causal relations, boundaries, temporal functions and global hyperbolicity are relative to $E$ . Causal exterior, chronological boundaries and Cauchy data

[F2]

A spacelike two-surface is a nonempty smooth embedded two-manifold without boundary with positive-definite induced metric. Null expansions, screen quotients and focal data

[F3]

In a globally hyperbolic spacetime, $J^{-} (C)$ is closed for compact $C$ . In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact

[F4]

The chronological past of each point is open. Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

[F5]

Arbitrary unions of open sets are open, by topology axiom (T2). Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

[F6]

A continuous real-valued function on a nonempty compact space attains its minimum, by the extreme-value clause (2). The restriction of a continuous function to a subspace is continuous. A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the asserted conclusion.

1.1A1F1F2F3F4F5given

The set $I_{E}^{-} (C) = ⋃_{c \in C} I_{E}^{-} (c)$ is open by [F4] and [F5], so its boundary is disjoint from it. Every point of that set lies in $J_{E}^{-} (C)$ ; its closure also lies there by [F3]. Thus $p \in J_{E}^{-} (C) ∖ I_{E}^{-} (C)$ .

2.1step 1.1F1F2F6given∎

The temporal function $τ$ is continuous by [F1], so $τ ∣_{C}$ is continuous by [F6]. The space $C$ is nonempty by [F2] and compact by hypothesis; clause (2) of [F6] therefore makes $\min_{C} τ$ well defined. By the definition of $J_{E}^{-} (C)$ there is a $c \in C$ with $p \leq_{E} c$ . If $p \in C$ , minimality would give $\min_{C} τ \leq τ (p)$ , contrary to the strict temporal inequality. Thus $p \notin C$ and $p \neq c$ . Together with step 1.1 this proves the assertion without any condition on causal relations among different points of $C$ .

Depends on

Used by · 5

Dependency tree · 2 levels

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