Alphabeta Math
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 vacuum KID development and complete stationary flow

Statement

Let smooth vacuum initial data on a complete boundaryless three-dimensional manifold have finitely many Euclidean ends, compact intervening region and h−δ=O4(r−α), K=O3(r−1−α), α>0. A smooth vacuum KID extends to a Killing field on the maximal globally hyperbolic vacuum development of these data, and that field has complete orbits. The lapse may have opposite signs on different ends. A globally hyperbolic vacuum development of an open part of the data embeds isometrically in the corresponding domain of dependence in this development. These are external Cauchy/KID/completeness inputs.

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: The asserted conclusion with its stated scope.

[F1]

Conditional reflection application: assume all hypotheses of the smooth vacuum data-double and stationary vacuum KID suppliers, including the specified canonical collar and signed double, data induced throughout the positive section by the same smooth Ricci-flat ambient metric, and an ambient Killing field commuting with the exact boost on that collar. Then the reflected canonical data and KID are smooth and satisfy the vacuum constraint and KID equations on the full double. These additional reflection hypotheses are not assumed for the general data of the Statement. Smooth vacuum constraint data obtained by reflection of a canonical end section Smooth reflection of the stationary vacuum Killing initial data

[F2]

Conditional geometric application: assume all hypotheses of the complete asymptotically flat double supplier, including the connected positive half with nonempty compact boundary, its specified canonical collar, compact smooth radial truncations exhausting that half, and h−δ=O6(r−1), K=O5(r−3) on its specified Euclidean end. Then the doubled initial geometry is complete, has two Euclidean ends and a compact intervening region, and retains these stronger differentiated end estimates. These additional hypotheses are required for this application. Complete two-ended geometry and decay of the reflected data

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 · 1

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

Robert Beig and Piotr T. Chruściel, Killing Initial Data · §2, vacuum Case2, pp4–5, equations(2.6) and(2.13)

Smooth vacuum constraint initial data admit a vacuum Cauchy development, and a vacuum KID extends to a Killing field in that development. No nonvanishing lapse hypothesis is used in vacuum Case2.

Hypotheses: Three-dimensional initial-data manifold; vacuum constraints rho=J=0, sufficiently regular data (here smooth), and vacuum KID stress tau_ij=0, meaning the stationary vacuum KID equations. The transversal lapse condition N nonzero belongs to separate Case1, not Case2.

Reading limits: Read actual §2 vacuum Case2 and the KID equations and conventions; no independent reconstruction of hyperbolic propagation or the cited Cauchy-existence proofs.

Piotr T. Chruściel, On completeness of orbits of Killing vector fields · MGHD existence/uniqueness discussion p2; Theorem1.1 p2; Corollary1.2(2) p3; Definition3.1 and sufficient falloff p12; Proposition2.4 pp9–10

Smooth three-dimensional vacuum initial data have a unique maximal globally hyperbolic vacuum development. On such a development a Killing field has complete orbits if its Cauchy data are complete asymptotically flat without boundary as in Definition3.1; sufficient bounds are h−delta O4(r^-alpha), K O3(r^-1-alpha), alpha>0. Any globally hyperbolic vacuum development of an open initial submanifold embeds into its MGHD, which in the full MGHD is D of that initial submanifold (Proposition2.4).

Hypotheses: Four-dimensional smooth vacuum maximal globally hyperbolic spacetime; complete boundaryless three-dimensional AF Cauchy hypersurface, compact core and finitely many Euclidean ends with the indicated falloff; smooth Killing field. There is no common future orientation requirement at all ends in Corollary1.2(2).

Reading limits: Read actual introduction, theorem/corollary, complete AF definition and Proposition2.4 statement/proof for its exact scope. The Cauchy development and completeness theorems are imported, not reconstructed or reviewed.

Sources