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Remark 1.1: maximal slice and decay

PDF p. 4: Remark 1.1

Statement

The article states that a maximal slice in its regular non-degenerate stationary vacuum setting can be obtained from [15], and that the required decay can then be obtained by elliptic estimates. This records the existence/decay claim as stated, not an independent verification of [15].

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Facts & Assumptions

Given: The stationary-vacuum model and exactly the article regular-spacetime hypotheses.

Goal: Supply the maximal section and its required differentiated end decay while preserving the original attribution.

[F1]

The authoring convention is ZFC, including the Axiom of Choice. The Axiom of Choice

[F2]

The physical model assumes smooth four-dimensional general relativity with a stationary Ricci-flat geometry and the specified end normalization. Stationary vacuum model and asymptotic normalization

[F3]

The regular-spacetime hypotheses supply the exterior Cauchy section, complete stationary exterior flow and smooth transverse nonexpanding bifurcate boundary; maximality is not part of this definition. Regular non-degenerate stationary vacuum spacetime

[F4]

The authored completion chain matches all class-(b) and exclusion hypotheses of the precise CW external input and supplies a maximal bifurcation-boundary section with Cauchy interior in exactly the original exterior. Maximal bifurcation-boundary section under the original regular hypotheses Stationary maximal hypersurface existence with a horizon boundary

[F5]

The independent end-gauge chain supplies the required stationary chart adapted to that same maximal section, with differentiated metric decay through order six. The same maximal slice has the differentiated article end gauge and a limiting old time Differentiated asymptotic bounds and parameter uniformity

Proof

1.1F1F2F3F4

Apply the completion-chain conclusion to the regular vacuum geometry. Its positive-mass, full-jet reflection, vacuum development and abstract collar construction supply a strongly causal stationary ambient spacetime with complete stationary field, closed acausal section with boundary $S$ , and interior domain of dependence exactly $E$ . The precise CW input therefore gives a smooth maximal section whose boundary is $S$ and whose interior is Cauchy in $E$ . This uses neither a maximality hypothesis from MainTheorem1.1 nor global axisymmetry.

2.1F2F4F5step 1.1∎

Apply the end-gauge conclusion to this same maximal section and stationary vacuum end. It gives a stationary chart with $T = \partial_{t}$ and the section at $t = 0$ , satisfying $g_{00} = - 1 + 2 M / r + O_{6} (r^{- 2})$ , $g_{0 i} = O_{6} (r^{- 2})$ , and $g_{i j} = δ_{i j} + O_{6} (r^{- 1})$ , with the stated stationary mass normalization. These are the required differentiated bounds. Thus the maximal-slice existence and elliptic-decay claims have their separate matched inputs; the article citation only identifies the original remark.

Source notes

The maximality explicitly assumed in MainTheorem1.1 is not a premise of this proof. The existence theorem and the stronger end normalization are separate imported/local chains, with their actual source hypotheses and reading limits retained in those suppliers. No independent reconstruction of the cited nonlinear existence theorem, no separate proof review and no article-defect judgment is claimed.

Depends on

Dependency tree · 2 levels

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Sources

The uniqueness of smooth stationary black holes · PDF p4, Remark1.1; MainTheorem1.1 maximality assumption immediately above