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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An exterior-contained null segment to a distant target cannot start inward

Statement

Work in ZFC. Use the compact spacelike surface $S$ , future/outward frame $(n, ν)$ , oriented exterior wedge $E$ and smooth spacelike section extension $(Σ, h)$ of the outward-graph initial-jets data. For a fixed smooth function $ϕ > 0$ on $S$ , set $F_{ϵ} (y) = \exp_{y}^{h} (ϵ ϕ (y) ν_{y})$ and $S_{ϵ} = F_{ϵ} (S)$ . Let $n_{ϵ}$ be the future unit normal of $Σ$ restricted to $S_{ϵ}$ , and let $ν_{ϵ}$ be the unit normal to $S_{ϵ}$ tangent to $Σ$ with the outward orientation continued from $ν$ .

Choose $ϵ > 0$ sufficiently small for the inward local crossing lemma, and let $K$ and $τ_{0} > 0$ be its fixed compact tube and common short affine time. If a future null geodesic segment starts on $S_{ϵ}$ , remains entirely in the open exterior $E$ , and ends outside $K$ , its initial tangent cannot be a positive multiple of the inward null normal $V_{ϵ} = n_{ϵ} - ν_{ϵ}$ . The conclusion is specific to this sufficiently small graph family near the bifurcation surface, rather than an assertion for arbitrary outer trapped surfaces.

Facts & Assumptions

[F1]

The surface $a = 0$ , $b > 0$ is outside the open exterior and is its local future boundary. A bifurcate future horizon has an oriented defining pair near its section

[F2]

For the sufficiently small graphs in the Statement, the ambient affine geodesics starting at $F_{ϵ} (y)$ with vector $V_{ϵ} (y)$ stay in the fixed compact tube $K$ for $0 \leq λ \leq τ_{0}$ and cross $a = 0, b > 0$ at a parameter $λ_{*} \leq C ϵ < τ_{0}$ . The inward null family reaches the future local horizon in short affine time

[F3]

Affine geodesics with the same initial data are unique on their common interval, and positive affine rescaling obeys $γ_{p, c V} (t) = γ_{p, V} (c t)$ for $c > 0$ . The geodesic initial-value problem and the exponential map of the metric-selected connection

[F4]

The oriented defining-pair data and smooth spacelike section extension supply the graph family $F_{ϵ} (y) = \exp_{y}^{h} (ϵ ϕ (y) ν_{y})$ for fixed smooth $ϕ > 0$ , its future unit normal $n_{ϵ}$ , and its smooth outward unit graph normal $ν_{ϵ}$ . The vector $V_{ϵ} = n_{ϵ} - ν_{ϵ}$ is its specified unhalved future inward null normal, and the sufficiently small positive graphs lie in $E$ . Small outward normal graphs have controlled inward null initial jets

[A1]

Work in ZFC, including the countable and dependent choice required by the invoked geodesic, compactness and causal interfaces. The Axiom of Choice

Proof

Given: The assumptions in the Statement.

Goal: Exclude the inward initial ray for the stated exterior-contained segment.

1.1F2F3F4givenalgebra

Suppose the initial tangent were $c V_{ϵ} (y)$ with $c > 0$ . By [F3], rescale the affine parameter positively so that the initial tangent is exactly the vector $V_{ϵ} (y)$ specified in [F4], and let $ℓ$ be the rescaled segment length. Uniqueness identifies this segment with the ambient geodesic in [F2] on $[0, \min {ℓ, τ_{0}}]$ . If $ℓ \leq τ_{0}$ , its endpoint lies in $K$ by [F2], contrary to the given target condition. Thus $ℓ > τ_{0}$ .

2.1F1F2step 1.1contradiction∎

The segment therefore contains the crossing supplied by [F2], at $λ_{*} < τ_{0}$ . That point has $a = 0$ and $b > 0$ and hence does not belong to the open exterior by [F1]. This contradicts the assumed exterior containment of the segment and excludes the inward initial ray. No global claim about causal escape after crossing the horizon is needed.

Depends on

Used by · 2

Dependency tree · 2 levels

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