TheoremStatement: AI-adaptedProof: Literature-sourced

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.

External local boost collar and canonical temporal splice

Statement

Assume the source GR and SBS hypotheses, including positive ADM mass and the acausal edgeless partial section with both chronological end intersections equal to its exterior part. Then near the bifurcation surface $S$ there are smooth double-normal coordinates $(U, V, p)$ , a constant $κ > 0$ and a Killing field $K$ with $K = κ (V \partial_{V} - U \partial_{U})$ and $K ∣_{S} = 0$ . In rotation $K = T + Ω Z$ with a commuting local circle field $Z$ preserving $U, V$ ; in the nonrotating case $K = T$ . The outward section $Σ_{l o c} = {U = V = r, 0 < r < δ}$ is spacelike and acausal, and its complete $T$ -saturation is an embedded product collar in $E$ . As $r ↓ 0$ this collar leaves every compact subset of the stationary quotient. The source cutoff construction gives a temporal function on $D (Σ_{0})$ equal to $(V - U) / 2$ near $S$ and to a Cauchy temporal function with zero level $Σ_{0}$ outside a compact neighborhood. No global horizon-generating field on $E$ is claimed.

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: The asserted conclusion with its stated scope.

[F1]

The adopted working collar supplies the required partial-Cauchy section and the original exterior. A working partial-Cauchy collar satisfies the source GR section equalities Regular non-degenerate stationary vacuum spacetime

[F2]

The general regular vacuum geometry has the positive stationary ADM mass needed in GR. Positive stationary mass from the bifurcate vacuum boundary

External input

Import the cited external results only under the exact hypotheses in the Statement and source records. Their proofs are not reconstructed or independently reviewed here.

Depends on

Used by · 5

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · GR/SBS pp3–4; Proposition2.1 p7; Lemma2.2 statement pp9–10, exact boost coordinates (2.4) pp10–11 and cutoff construction pp11–12, quotient-escape conclusion p13; nonrotating specialization pp13–14

Under GR and SBS, a local Hawking field K=T+Omega Z has exact boost form K=kappa(V partial_V−U partial_U), kappa>0, with K=0 on S; in rotation Z is a commuting periodic local field. There is an acausal canonical outward section and a proper stationary collar R×Sigma_loc in E; its r→0 end leaves every compact quotient set. The proof constructs a temporal function equal to (V−U)/2 near S and the original Cauchy temporal function away from a compact neighborhood.

Hypotheses: The source GR package requires a smooth oriented time-oriented Ricci-flat four-manifold, positive ADM mass, the stated O6 stationary end with T future, globally hyperbolic DOC E, complete T-orbits in E meeting Sigma1, and an acausal edgeless partial section with both chronological end intersections Sigma1. SBS requires the smooth transverse nonexpanding bifurcate horizons and T tangent. The specified local conclusions use GR and SBS only, not NT or global axisymmetry. In the nonrotating case set K=T, Omega=0 and omit Z.

Reading limits: Read full actual GR/SBS, Lemma2.2 statement and the displayed coordinate/cutoff and properness passages in its proof, plus the nonrotating specialization. Imported local results and construction; no proof review, nontrapping conclusion or global rotational field is asserted.

Sources

Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · GR/SBS pp3–4; Proposition2.1 p7; Lemma2.2 statement pp9–10, exact boost coordinates (2.4) pp10–11 and cutoff construction pp11–12, quotient-escape conclusion p13; nonrotating specialization pp13–14