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Independent regular nonrotating staticity and Schwarzschild theorem

Statement

Assume the following independent-source GR and SBS contract. Let $(M, g)$ be a connected oriented time-oriented smooth four-dimensional vacuum spacetime of signature $(-, +, +, +)$ with smooth stationary Killing field $T$ . Let $Σ^{'} \subset M$ be an embedded acausal edgeless spacelike partial Cauchy hypersurface, with a diffeomorphism $Φ_{0} : {x \in R^{3} : ∣ x ∣ > 1 / 2} \to Σ^{'}$ , and put $Σ_{1} = Φ_{0} ({∣ x ∣ > 1})$ . The designated end $M^{e n d} ≃ R \times {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ has $T = \partial_{t}$ future-directed, positive ADM rest mass $M$ , and $g_{00} = - 1 + 2 M / r + O_{6} (r^{- 2})$ , $g_{i j} = δ_{i j} + O_{6} (r^{- 1})$ , $g_{0 i} = O_{6} (r^{- 2})$ . Require that $Σ^{'}$ agrees with $t = 0$ on this end.

Define $E = I_{M}^{-} (M^{e n d}) \cap I_{M}^{+} (M^{e n d})$ and require that it is globally hyperbolic, with complete stationary orbits meeting $Σ_{1}$ , and that $Σ^{'} \cap I_{M}^{+} (M^{e n d}) = Σ^{'} \cap I_{M}^{-} (M^{e n d}) = Σ_{1}$ . The future and past event horizons $\partial_{M} I_{M}^{-} (M^{e n d})$ and $\partial_{M} I_{M}^{+} (M^{e n d})$ meet in the embedded sphere $S = Φ_{0} ({∣ x ∣ = 1})$ . Their portions near $S$ are the specified smooth transverse null sheets $H^{+}$ and $H^{-}$ , tangent to $T$ . Require that both sheets are nonexpanding: for each local smooth nonzero null generator $L$ and spacelike cross-section with induced metric $γ$ , the null second fundamental form $χ_{L} (U, V) = g (\nabla_{U} L, V)$ on tangent $U, V$ has ${tr}_{γ} χ_{L} = 0$ .

If $T ∣_{S} = 0$ , then $E$ is static and globally isometric to the Schwarzschild exterior of positive mass $M$ . Here staticity means that $T$ is timelike throughout $E$ and orthogonal to spacelike hypersurfaces, with $T^{♭} \land d T^{♭} = 0$ ; the Schwarzschild exterior is the mathematical model with areal radius $ρ > 2 M$ in geometrized units $G_{N} = c = 1$ . No nontrapping or analyticity hypothesis is required in this nonrotating branch.

The source proof also supplies an exterior-preserving auxiliary completion and a stationary-covariant maximal Cauchy foliation of $E$ . More precisely, the interiors $Σ_{s}$ of smooth spacelike hypersurfaces ${\overline{Σ}}_{s}$ with common compact bifurcation boundary $B$ , where $T = 0$ , are Cauchy surfaces in $E$ , foliate exactly $E$ , and satisfy $Σ_{s + t} = Φ_{t}^{T} (Σ_{s})$ for the stationary flow. Each ${\overline{Σ}}_{s}$ has a compact core, one asymptotically Euclidean end, induced metric asymptotically Euclidean with differentiated bounds, and is smooth up to $B$ and complete with $B$ included. The boundary $B$ belongs to the auxiliary completion; no identification of its ambient neighborhood with the original one is assumed.

On each ${\overline{Σ}}_{s}$ , let $n$ be the future unit normal, $h = g ∣_{T {\overline{Σ}}_{s}}$ , $k (U, V) = - g (\nabla_{U} n, V)$ , $N = - g (T, n)$ and $X = T - N n$ , so $T = N n + X$ with $X$ tangent; maximality means ${tr}_{h} k = 0$ . On the end the supplied estimates are $∣ k ∣_{h} = O (r^{- 2})$ , $∣ X ∣_{h} = O (r^{- 1})$ and $N \to 1$ . These are the auxiliary outputs available to the local application lemmas; they are conclusions of the independent source preparation, not extra assumptions on the original section.

Statement references

Stationary vacuum model and asymptotic normalization

Causal exterior, chronological boundaries and Cauchy data

Bifurcate Killing horizons, rotation and normalization

Differentiated asymptotic bounds and parameter uniformity

Spacelike sections, maximality and lapse–shift data

Null expansions, screen quotients and focal data

The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

Facts & Assumptions

[A1]

Adopt the vacuum model and the causal, horizon and differentiated asymptotic conventions, with precisely the additional GR/SBS relationships stated above. Stationary vacuum model and asymptotic normalization Causal exterior, chronological boundaries and Cauchy data Bifurcate Killing horizons, rotation and normalization Differentiated asymptotic bounds and parameter uniformity

[F1]

The independent nonrotating branch supplies the completion, foliation, bounds and staticity/Schwarzschild conclusion under those GR/SBS hypotheses, as specified in source reference 0. Its cited maximal-slice theorem supplies the diffeomorphism type and stationary covariance under the auxiliary-spacetime hypotheses, as specified in source reference 1.

[F2]

The spacelike-hypersurface interface fixes the future unit normal, induced metric, negative-sign second fundamental form, maximality, lapse, shift and completeness including a compact boundary. Spacelike sections, maximality and lapse–shift data

[F3]

Null second fundamental form and expansion on spacelike cross-sections have the stated trace convention. Null expansions, screen quotients and focal data

[F4]

The Schwarzschild exterior model is a mathematical datum; take its positive mass parameter to be $M$ and its constants $G_{N} = c = 1$ . The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation

External input

Import the nonrotating branch of Radu’s proof of Theorem 1.1, PDF pp. 69–70, under GR and SBS on pp. 3–4, including the end-section alignment and the chronological definitions of the exterior and event horizons. Lemma 2.13, p. 25, and Appendix A, pp. 71–74, supply the exterior-preserving horizon completion. The nonrotating branch invokes the further auxiliary preparation for maximal existence, records the lapse/shift bounds, proves staticity and applies static uniqueness. Its use here is an explicit external input; the rotating branch’s NT assumption is not imported.

Chruściel–Wald Theorem 4.2, PDF p. 25 with (4.1)–(4.3) on pp. 24–25 and the class-(b) definition on p. 11, is the cited maximal-slice input: same boundary, diffeomorphism type, stationary covariance, Cauchy interiors and, under timelike convergence, foliation. Radu’s pp. 69–70 supply applicability through the auxiliary completion and vacuum equation. The differentiated asymptotic language uses Chruściel–Wald Definition 2.1; Remark 3 on p. 25 permits logarithmic height. No stronger constant-height conclusion or local reconstruction of maximal existence is claimed.

Depends on

Used by · 1

Non-rotating staticity and Schwarzschild uniqueness

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

A. F. Radu, On rigidity of stationary black holes under a nontrapping assumption, arXiv:2609.23872v2 · Section 1.1, GR and SBS, PDF/printed pp. 3–4; Lemma 2.13, p. 25; nonrotating branch of the proof of Theorem 1.1, pp. 69–70; Appendix A, pp. 71–74.

GR and SBS imply the exterior-preserving horizon completion in Lemma 2.13. The nonrotating branch T|S0=0 on pp. 69–70 imports the auxiliary bifurcate completion and maximal foliation, proves staticity, and identifies the exterior with Schwarzschild. It records compact boundary where T=0, one asymptotically flat end, interior domain of dependence E, stationary translates foliating E, lapse tending to one, and |k|=O(r^-2), |X|=O(r^-1). SBS defines nonexpansion as zero trace of g(D_U L,V) on spacelike horizon cross-sections. NT is used in the rotating branch, not this branch.

Hypotheses: A connected oriented time-oriented smooth four-dimensional spacetime of signature (-,+,+,+), Ric(g)=0, and smooth stationary Killing field T. An embedded acausal edgeless spacelike partial Cauchy section Sigma-prime is parametrized by Phi0:{|x|>1/2}->Sigma-prime; Sigma1=Phi0({|x|>1}). The designated end is R times a Cartesian exterior, T=partial_t future-directed, with positive ADM rest mass M and g00=-1+2M/r+O6(r^-2), gij=delta+O6(r^-1), g0i=O6(r^-2). Sigma-prime agrees with t=0 on this end. E=I-minus(Mend) intersect I-plus(Mend) is globally hyperbolic, its stationary orbits are complete and meet Sigma1, and Sigma-prime intersect I-plus(Mend)=Sigma-prime intersect I-minus(Mend)=Sigma1. The event horizons partial I-minus(Mend) and partial I-plus(Mend) intersect in S=Phi0({|x|=1}); near S they are smooth transverse nonexpanding null hypersurfaces tangent to T. Restrict to T|S=0.

Reading limits: In this adjudication, read the locally retained full-PDF text of pp. 3–4, 25, 69–70 and 71–74, and opened the independent arXiv PDF. The cited auxiliary completion and maximal existence are accepted as external inputs under the displayed GR/SBS hypotheses; no reconstruction, independent proof audit, or review of the rotating/nontrapping argument or cited static-uniqueness proofs is claimed. No visual-inspection credit from prior work is claimed. The studied article supplies no premise here.

P. T. Chruściel and R. M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes, arXiv:gr-qc/9304009 · Section 2, Definition 2.1 and its Remarks 1–2, PDF pp. 8–9; Definition 2.3, p. 11; Section 4, equations (4.1)–(4.3), Theorem 4.2 and Remark 3, pp. 24–25.

For a class-(b) spacetime with no black or white hole in the interior domain of dependence of its section, Theorem 4.2 supplies maximal asymptotically flat hypersurfaces with the same boundary, diffeomorphic to that section and covariant under stationary translation as in (4.2); their interiors are Cauchy surfaces. Under timelike convergence they foliate the interior domain of dependence. Definition 2.3 specifies a closed acausal section with compact core and asymptotic ends; its induced metric is complete including the boundary. Asymptotic regularity and differentiated falloff use Definition 2.1. Remark 3 allows logarithmic height, rather than an assumed constant asymptotic height.

Hypotheses: A strongly causal smooth time-oriented class-(b) spacetime with a connected closed acausal spacelike section having compact boundary, compact core and finitely many (k,alpha)-asymptotically stationary ends, k>=2 and 0<alpha<=1; a complete stationary field tangent to the boundary; and (B union W) intersect int D(Sigma)=empty. The foliation conclusion additionally requires the stationary field to be timelike on int D(Sigma) or Ric(Z,Z)>=0 for every timelike Z. Here the exterior-preserving preparation and applicability are imported from Radu pp. 69–70; vacuum provides timelike convergence and there is one end.

Reading limits: Read the local PDF text of pp. 24–25 and the extracted PDF text of Definitions 2.1 and 2.3 on pp. 8–11. The theorem statement, definitions and Remark 3 are used, not a reconstruction of the maximal-existence proof or its cited elliptic estimates. Applicability of the auxiliary-spacetime preparation to GR/SBS is accepted at the precise independent Radu locator, not derived solely from the four formulation suppliers. No constant-height or improved orthogonal-end expansion is inferred here.

Sources

A. F. Radu, On rigidity of stationary black holes under a nontrapping assumption, arXiv:2609.23872v2 · Section 1.1, GR and SBS, PDF/printed pp. 3–4; Lemma 2.13, p. 25; nonrotating branch of the proof of Theorem 1.1, pp. 69–70; Appendix A, pp. 71–74.
P. T. Chruściel and R. M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes, arXiv:gr-qc/9304009 · Section 2, Definition 2.1 and its Remarks 1–2, PDF pp. 8–9; Definition 2.3, p. 11; Section 4, equations (4.1)–(4.3), Theorem 4.2 and Remark 3, pp. 24–25.