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.

The complete stationary flow preserves the selected causal exterior

Statement

Use the global stationary comparison data, and set $M_{e x t} = ⋃_{t \in R} ϕ_{t} (S_{e x t})$ and $D = I_{M}^{+} (M_{e x t}) \cap I_{M}^{-} (M_{e x t})$ . For every $s \in R$ , $ϕ_{s} (M_{e x t}) = M_{e x t}$ and $ϕ_{s} (D) = D$ . Thus the stationary symmetry acts on the entire specified exterior; a restriction of its flow to a small neighborhood would not supply this conclusion for arbitrary times.

Facts & Assumptions

[F1]

The comparison data supply a smooth time-oriented spacetime $(M, g, N)$ and a complete stationary Killing flow $ϕ_{t}$ generated by $T$ , defined on all of $M$ for every real time. The Killing condition is $L_{T} g = 0$ . Smooth global data for the axisymmetric comparison Causal exterior, chronological boundaries and Cauchy data

[F2]

Chronological futures and pasts are defined by future timelike curves in the specified ambient manifold. Causal exterior, chronological boundaries and Cauchy data

[F3]

A global flow is smooth on $R \times M$ , satisfies $ϕ_{0} = {i d}_{M}$ and $ϕ_{s} \circ ϕ_{t} = ϕ_{s + t}$ , and hence each $ϕ_{s}$ has smooth inverse $ϕ_{- s}$ . Local and global flows generated by a vector field

[F4]

For a smooth tensor field $A$ and a smooth vector field $X$ with flow $Φ_{t}$ , the condition $L_{X} A = 0$ implies $Φ_{t}^{*} A = A$ at all defined times. Apply this to $A = g$ and $X = T$ using the Killing condition in F1. A tensor field is flow-invariant exactly when its Lie derivative vanishes

Proof

Given: The data and assumptions in the Statement.

Goal: Establish the claimed conditional conclusion.

1.1F1F3F4givenalgebra

The global flow law gives $ϕ_{s} (⋃_{t} ϕ_{t} (S_{e x t})) = ⋃_{t} ϕ_{s + t} (S_{e x t}) = M_{e x t}$ . Applying F4 to $g$ and $T$ gives $ϕ_{s}^{*} g = g$ for every real $s$ , so each $ϕ_{s}$ is an isometry. Each $ϕ_{s}$ preserves time orientation: its derivative maps timelike vectors continuously as $s$ varies from zero, where $ϕ_{0} = {i d}_{M}$ , so a future timelike vector cannot cross to the other timelike component.

2.1F2F3step 1.1algebra∎

Composing a future timelike curve with the time-orientation-preserving isometry $ϕ_{s}$ gives such a curve with the corresponding image endpoints. Applying also its inverse $ϕ_{- s}$ yields $ϕ_{s} (I_{M}^{\pm} (M_{e x t})) = I_{M}^{\pm} (M_{e x t})$ . Intersect the two equalities. Completeness is used in the first step and in the inverse for all real $s$ ; no local-to-global extension is proved.

Depends on

Used by · 1

Axisymmetric vacuum uniqueness with complete global hypotheses

Dependency tree · 2 levels

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