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.

MOTS boosts preserve the spacetime stability sign

Statement

For a MOTS with null pair $(ℓ, k)$ normalized by $g (ℓ, k) = - 1$ , let $h : S \to (0, \infty)$ be smooth and set $ℓ^{'} = h ℓ$ , $k^{'} = h^{- 1} k$ . For the same normal velocity $X$ , the variation satisfies $δ_{X} θ_{(ℓ^{'})} = h δ_{X} θ_{(ℓ)}$ . Consequently if $X = γ ℓ - ψ k$ with $γ \geq 0$ and $ψ > 0$ , then $X = γ^{'} ℓ^{'} - ψ^{'} k^{'}$ with $γ^{'} = γ / h \geq 0$ , $ψ^{'} = h ψ > 0$ , and the inequality $δ_{X} θ \geq 0$ is unchanged by the boost.

Statement references

General spacelike two-surfaces, null gauges and fluxes

Outgoing expansion variation at a general MOTS

Facts & Assumptions

[F1]

The null pairing and tangential expansion use the general surface convention. General spacelike two-surfaces, null gauges and fluxes

[F2]

The variation at a MOTS is well defined and independent of extensions of its base null gauge. Outgoing expansion variation at a general MOTS

Proof

**Given:**The objects and hypotheses in the Statement.

Goal: Establish the assertions and well-definedness stated there.

1.1F1F2

Choose any actual deformation realizing $X$ and extend $h$ smoothly along its parameter with its prescribed initial value. Orthogonality of $ℓ_{t}$ to the tangent surface gives $θ_{(h_{t} ℓ_{t})} = h_{t} θ_{(ℓ_{t})}$ . Differentiating at zero gives $δ_{X} θ_{(ℓ^{'})} = h δ_{X} θ_{(ℓ)} + (\partial_{t} h_{t}) ∣_{0} θ_{(ℓ)} = h δ_{X} θ_{(ℓ)}$ . Its sign is unchanged since $h > 0$ .

2.1F1step 1.1∎

Substitution of $ℓ = ℓ^{'} / h$ and $k = h k^{'}$ into $X$ gives the stated new coefficients and signs. Moreover $g (X, X) = 2 γ ψ \geq 0$ , so these are outgoing non-timelike normal directions, including the past null direction $- k$ when $γ = 0$ . This is a pointwise identity and does not assume that $X$ is tangent to a spacelike slice.

Depends on

Used by · 2

Dependency tree · 2 levels

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