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Event-horizon bifurcation data for the nontrapping theorem

Definition

For the Radu NT GR package, its SBS condition is that the future and past event boundaries $\partial_{M} I_{M}^{-} (M^{e n d})$ and $\partial_{M} I_{M}^{+} (M^{e n d})$ meet in the embedded sphere $S_{0} = Φ_{0} ({r = 1})$ . Near $S_{0}$ they are smooth transverse null hypersurfaces $H^{+}, H^{-}$ tangent to $T$ , with vanishing null expansion. The expansion is the induced positive-metric trace of $(X, Y) \mapsto g (\nabla_{X} L, Y)$ for a null generator $L$ and section tangent fields $X, Y$ .

Unlike the rotating SBS packages used in the two smallness papers, this source statement does not require $T ∣_{S_{0}} \equiv̸ 0$ . Its main theorem includes the case $T ∣_{S_{0}} \equiv 0$ , addressed by staticity in its §7. These are event-boundary hypotheses; a second globally defined Killing field is a conclusion of the rigidity theorem, not part of SBS here.

Facts & Assumptions

[F1]

Radu nontrapping global regularity package supplies the end and chosen sphere; Causal exterior, chronological boundaries and Cauchy data fixes the ambient boundary convention. Null expansions, screen quotients and focal data supplies the intrinsic null-expansion trace and normalization qualifications.

[F2]

Local null normal pairs and the positive screen quotient defines a smooth null hypersurface by its rank-two positive semidefinite induced metric, its radical generator line and spacelike sections; it also supplies the null-form rescaling law. Transverse embedded submanifolds defines transversality by $T_{p} H^{+} + T_{p} H^{-} = T_{p} M$ at intersection points. Only these mathematical interfaces are used; the choice convention required by the null-geometry supplier is inherited from Null expansions, screen quotients and focal data.

[F3]

The Radu SBS passage and section 1.2 opening supply the exact source-specific horizon predicates and the stated treatment of $T ∣_{S_{0}} \equiv 0$ . The two rotating smallness packages compared here are the Alexakis–Ionescu–Klainerman perturbative paper, section 1.1 SBS, and their small-horizon-field paper, section 1.1.2 SBS; each explicitly requires $T ∣_{S_{0}} \equiv̸ 0$ . Their complete source contracts are recorded in External input.

Justification

The ambient chronological boundaries and their specified local smooth embeddings determine the two horizon pieces. Their tangent spaces and induced section metrics make the nullness, transversality, tangency and trace predicates well-defined. Rescaling a null generator multiplies its expansion, so the zero-expansion condition is unchanged. The absence of a nonzero-horizon-field clause is kept explicitly because the source handles that case separately; no external staticity proof is reconstructed in this definition.

External input

Radu's section 1.1 SBS supplies the particular event-boundary assumptions, not just generic horizon terminology. Its section 1.2 opening attributes treatment of the identically-zero horizon field to staticity in section 7. This definition imports those formulation and attribution facts only; it does not apply or reconstruct the staticity or global rigidity proofs.

For the comparison with the rotating smallness papers, import only the non-identically-zero horizon-field clause from Alexakis–Ionescu–Klainerman, arXiv:0904.0982v1 section 1.1 SBS and arXiv:1304.0487v2 section 1.1.2 SBS. Their other hypotheses and conclusions are not added to the Radu SBS package.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, section 1.1 SBS and its two following explanatory paragraphs; Theorem 1.1; section 1.2 opening paragraph.

In the source GR setting, SBS specifies that the event boundaries $\partial_M I_M^-(M^{\rm end})$ and $\partial_M I_M^+(M^{\rm end})$ intersect at $S_0=\Phi_0(\{|x|=1\})$, and near this sphere are smooth transverse null hypersurfaces tangent to $T$ whose null second fundamental forms have zero induced-metric trace. SBS imposes no non-identically-zero condition on $T|_{S_0}$. Section 1.2 says the zero-field case is treated by staticity in section 7. The additional Killing symmetry is constructed, not assumed by SBS.

Hypotheses: This is a source-specific definition in the GR setting of a smooth oriented time-oriented stationary Ricci-flat four-manifold with the specified end, slice parametrization and exterior. Only the listed SBS predicates and the source treatment distinction are imported; no rigidity or staticity theorem is being proved or applied by this definition.

Reading limits: Retrieved and read the indicated version-1 HTML section 1.1 setup, GR, SBS, the two paragraphs after SBS, Theorem 1.1 and section 1.2 opening paragraph. Used for formulation and source attribution only. Section 7 staticity and global symmetry-construction proofs were not reviewed or reconstructed.

S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Version 1, section 1.1, SBS paragraph, especially its final sentence (after equation (1.3) and before PK).

The SBS assumptions include tangency of $T$ to both local null event-boundary hypersurfaces and $T|_{S_0}\not\equiv0$.

Hypotheses: A clause of the source stationary vacuum GR/SBS setup, with its specified bifurcation sphere and local event-boundary sheets. This clause is reported for comparison only; neither PK nor the Main Theorem is imported.

Reading limits: Retrieved the version-1 HTML and read section 1.1 setup, GR and the complete SBS paragraph including its final sentence and nonexpansion footnote. Checked only this assumption comparison; no smallness theorem or rigidity proof was reviewed.

S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · Version 2, section 1.1.2, SBS paragraph, final sentence immediately before section 1.1.3.

The SBS assumptions include tangency of $T$ to both local null event-boundary hypersurfaces and $T|_{S_0}\not\equiv0$.

Reading limits: Retrieved the version-2 HTML and read section 1.1 main objects and regularity assumptions through the complete SBS paragraph and its final sentence. Checked only the rotating SBS comparison; no elliptic estimate, staticity or rigidity theorem proof was reviewed.

Sources

Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, section 1.1 SBS and its two following explanatory paragraphs; Theorem 1.1; section 1.2 opening paragraph.
S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Version 1, section 1.1, SBS paragraph, especially its final sentence (after equation (1.3) and before PK).
S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · Version 2, section 1.1.2, SBS paragraph, final sentence immediately before section 1.1.3.