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Static vacuum Schwarzschild classification contract

Statement

In the zero-cosmological-constant vacuum Einstein model, let the spacetime exterior be the globally static product $g = - V^{2} d t^{2} + h$ on $R_{t} \times X$ . Assume the regular static exterior data: $\overline{X}$ is a connected oriented three-dimensional exterior complete up to nonempty compact smooth boundary $B$ , $V > 0$ inside and $V = 0$ on $B$ , the interior data are smooth and extend at least $C^{2}$ to $B$ , and $d V \neq 0$ there. The single asymptotically Euclidean end is the complement of a compact core, with $h - δ = O_{2} (r^{- 1})$ and $V = 1 - m / r + O_{2} (r^{- 2})$ , and there is no other end or deleted interior boundary. No connectedness of $B$ is assumed.

Then the exterior is globally isometric to the Schwarzschild exterior of mass $m > 0$ , and $B$ is a connected two-sphere. Staticity and this global static product must already have been established; this theorem does not turn an arbitrary non-rotating stationary spacetime into a static one. The classification is of the given exterior; it does not assert a maximal analytic extension beyond its horizon.

Statement references

Complete regular static exterior data and its normalized end

Vacuum curvature of a globally static product

The pointwise static end bounds imply the original weighted hypotheses

Regular static boundary geometry and positivity of the end coefficient

From a Schwarzschild spatial isometry to the normalized spacetime exterior

Adopted Einstein field equation in the direct-equation presentation

Facts & Assumptions

[A1]

The classical field law is adopted on the smooth spacetime interior with zero stress-energy and zero cosmological constant. Tracing $Ric (g) - \frac{1}{2} Scal (g) g = 0$ in dimension four gives $Ric (g) = 0$ . Adopted Einstein field equation in the direct-equation presentation

[F1]

The regular static data, their reduced spatial equations, original weighted-end conditions, boundary geometry, positive mass coefficient, and lifting from Schwarzschild spatial data are the following locally authored application suppliers: Complete regular static exterior data and its normalized end Vacuum curvature of a globally static product The pointwise static end bounds imply the original weighted hypotheses Regular static boundary geometry and positivity of the end coefficient From a Schwarzschild spatial isometry to the normalized spacetime exterior

[F2]

The complete static-data Schwarzschild classification is an established external input: Bunting–Masood-ul-Alam, p.148 hypotheses and Assumption 1, p.150 Theorem 2, and p.153 Riemannian conclusion, with Miao pp.1–2 giving its explicit spatial-data formulation.

External input

The classification theorem, including connectedness and the Schwarzschild spatial geometry, is imported from the two independent-source records. The five local prerequisites supply the application chain: the adopted vacuum law gives the spatial static equations, the differentiated pointwise decay implies Assumption 1 with $q = 6$ and $λ = 1 / 2$ , the boundary and energy identity give regular boundary data and $m > 0$ , and the global product lifts the spatial Schwarzschild classification to the stated spacetime exterior. The spatial classification is used on complete data, as explicitly restated by Miao; the “maximally extended” spacetime wording of the original Theorem 2 is not used to infer that an arbitrary ambient spacetime has such an extension. No local proof of the imported classification or of the positive-mass theorem is claimed.

Source notes

The original theorem’s weighted second derivatives were resolved by reading its source image, not by guessing from corrupted OCR. Miao’s later minimal-boundary theorem with a C3 metric is not substituted for the original C2 zero-lapse-boundary result.

Depends on

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External dependencies

G. L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, General Relativity and Gravitation 19 (1987), 147–154 · p.148, static-data assumptions, equations (2)–(5) and Assumption 1; p.150, Theorem 2; p.153, Riemannian doubling and conclusion of the main proof

The regular complete asymptotically Euclidean three-dimensional static vacuum data with positive lapse in the interior, zero lapse on compact boundary and normalized lapse at infinity have Schwarzschild spatial geometry; in particular the black-hole boundary is connected. Theorem 2 phrases the spacetime conclusion for a maximally extended static spacetime; the Riemannian data conclusion is the one imported here, with its standard explicit restatement in Miao, p.2.

Hypotheses: Connected oriented three-dimensional spatial exterior, compact C3 boundary; metric and lapse smooth in the interior and C2 on its closure; static vacuum equations; positive lapse inside and zero on boundary, nonzero boundary lapse gradient; complete spatial exterior with compact core and a single asymptotically Euclidean end. Assumption 1 requires metric perturbation O(r^-1), first derivatives O(r^-2), lapse 1-m/r+v with v=O(r^-2), first derivatives O(r^-3), second metric derivatives in L^q_{-lambda-2} and second v derivatives in L^q_{-lambda-3}, for some lambda>0 and 4<q<infinity; m>0. The local helpers establish the weighted conditions and positivity from the stated stronger pointwise bounds.

Reading limits: Read all eight original-paper HTML page transcriptions from the author upload; visually inspected the source images for p.148 weighted exponents and C2 regularity, p.150 Theorem 2 and its maximal-extension wording, and p.153 doubling regularity and Riemannian conclusion. The original source bytes/images are retained run-locally. This is a statement/hypothesis/application reading, not an audit or reconstruction of the positive-mass theorem, conformal doubling proof, or cited final classification argument.

P. Miao, A Remark on Boundary Effects in Static Vacuum Initial Data sets, arXiv:gr-qc/0504081v1 · pp.1–2, static equations (1) and paragraph beginning “A classic result of Bunting and Masood-ul-Alam”; p.3, boundary-regularity qualification for Miao’s different minimal-boundary theorem

For a one-ended asymptotically flat three-dimensional static vacuum manifold with nonempty compact boundary, if its normalized static potential vanishes on the boundary, the Bunting–Masood-ul-Alam theorem makes the spatial manifold isometric to (R3 minus the open ball of radius m/2, (1+m/(2|x|))^4 delta) with m>0; the boundary is a connected two-sphere.

Hypotheses: The Bunting–Masood-ul-Alam zero-lapse boundary case, with its source regularity and asymptotics verified from the original paper. Miao’s separate generalization assuming only minimal boundary is not imported; its p.3 C3 metric hypothesis is not substituted for the original theorem’s C2 boundary regularity.

Reading limits: Read the downloaded full PDF’s pp.1–3 by text extraction, especially the explicit classical Riemannian restatement and the separate C3 assumption on p.3. No proof of Miao’s generalized minimal-boundary theorem or any claimed weaker regularity result was read or reconstructed.

Sources

G. L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, General Relativity and Gravitation 19 (1987), 147–154 · p.148, static-data assumptions, equations (2)–(5) and Assumption 1; p.150, Theorem 2; p.153, Riemannian doubling and conclusion of the main proof
P. Miao, A Remark on Boundary Effects in Static Vacuum Initial Data sets, arXiv:gr-qc/0504081v1 · pp.1–2, static equations (1) and paragraph beginning “A classic result of Bunting and Masood-ul-Alam”; p.3, boundary-regularity qualification for Miao’s different minimal-boundary theorem