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Angle differential inequality near S

PDF p. 19: (3.13); [7, Theorem 3.1] and [36, Appendix B]

Statement

In the regular mass-one stationary vacuum/Harrison construction with $T ∣_{S} \equiv̸ 0$ , fix $δ > 0$ and the existing smooth spacelike maximal stationary sections $Σ_{q}$ , $q \in [δ, 1]$ , with the complete preceding boundary height and barrier hypotheses. On a sufficiently small common Gaussian collar of the bifurcation sphere $S$ , let $τ_{q}$ be future proper time along the normal geodesics of the reference section, let $U_{q} = - {grad}_{g_{q}} τ_{q}$ , and put $v_{q} = - g_{q} (n_{q}, U_{q})$ for the future candidate normal $n_{q}$ . Then $U_{q}$ is unit future timelike, $v_{q} \geq 1$ , and with $h_{q} = g_{q} ∣_{T Σ_{q}}$ one has $Δ_{h_{q}} \log v_{q} \geq \frac{1}{4} ∣ \nabla^{h_{q}} \log v_{q} ∣_{h_{q}}^{2} - C_{δ} v_{q}^{2}$ throughout the candidate collar, including $1 \leq v_{q} \leq 2$ . The constant and collar are independent of the candidates and of $q \in [δ, 1]$ . On $S$ , $U_{q} = {\overset{ˉ}{n}}_{q}$ and the previously established boundary boost bound gives $v_{q} \leq C_{δ}$ . This is a differential estimate, not yet an interior upper bound for $v_{q}$ .

Facts & Assumptions

[A1]

Adopt the supplied stationary vacuum model, regular data, mass-one normalization and existing maximal graph sections, and assume $T ∣_{S} \equiv̸ 0$ ; no eventual section-existence theorem is assumed. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Mass-one convention and Harrison denominator Stationary maximal graphs and normalized graph heights Work in ZFC. The Axiom of Choice

[F1]

Under [A1], the Harrison metrics are jointly smooth in spacetime and $q$ on one open neighborhood of $S$ . Proposition 2.1: Einstein–Maxwell equations and regularity The reference collar, induced metric and unit normal data are smooth and uniformly controlled on smaller compact collars. Lemma 3.3: common spacelike reference collar

[F2]

With the uniform Hawking-flow height bound from [F9], the completed boundary theorem gives the common one-to-one Gaussian normal graph domain, the cone bound and the boundary normal-angle estimate. Boundary angle bound and normal-graph cone Its Gaussian-flow helper constructs the normal coordinate family smoothly with one common compact parameter domain. Bounded boost times give a uniform Gaussian collar cone

[F3]

Spacelike normals, maximality and $k = - g_{q} (D n_{q}, \cdot)$ have their stated mathematical conventions. Spacelike sections, maximality and lapse–shift data

[F4]

The five local mathematical inputs give observer-frame bounds, exact Gauss–Codazzi angle differentiation, trace-free coercivity, uniform scalar estimates and the all-range logarithmic inequality. Boost-frame bounds for a smooth observer and arbitrary spacelike normals Pointwise Gauss–Codazzi identities for the normal angle Trace-free three-dimensional coercivity for the logarithmic gradient Uniform scalar angle inequalities from local background tensors Logarithmic angle inequality through the full range v at least one

[F5]

For the induced Riemannian metric $h_{q}$ , the intrinsic gradient is characterized by $h_{q} (\nabla^{h_{q}} f, X) = d f (X)$ for tangent $X$ . The Riemannian gradient is the metric dual of the differential

[F6]

The intrinsic Laplacian uses the positive-divergence convention $Δ_{h_{q}} f = {tr}_{h_{q}} {Hess}_{h_{q}} f = {div}_{h_{q}} (\nabla^{h_{q}} f)$ , with the one-sided extension for data smooth up to the boundary. Laplace–Beltrami operator as the trace of the Hessian

[F7]

The complete preceding boundary height/barrier hypotheses in the Statement are retained, including their actual original exterior-section maximality and selected-end access premises. The original maximality is $H_{g} = 0$ on $C = Σ_{0} \cap E$ , and the original reference section is $Σ_{1} = C \cup S$ , as specified by the common-collar contract. The independent spherical identification is not equated with the physical stationary end chart. These are inherited hypotheses for the candidate estimate, not consequences of regularity or candidate existence. Lemma 3.3: common spacelike reference collar Boundary angle bound and normal-graph cone

[F8]

The complete preceding height/barrier hypotheses in the Statement are those of the Hawking-height and barrier suppliers, for the same stationary product and selected single-primitive Harrison family. The supplied candidates are connected smooth spacelike maximal sections, smooth up to compact boundary $S$ , with one exterior end and interiors the stationary graphs over $Σ_{1}^{\circ}$ . Retain all their candidate-section, normalized-end and uniform-estimate hypotheses, including the radial height-decay and transformed-lapse hypotheses: normalized end heights and their first spatial derivatives tend to zero uniformly in direction for each fixed $q$ , the required estimates hold on a common end with constants independent of $q \in [δ, 1]$ , and $- g_{q} (T, n_{q}) > 0$ in the interior. In addition to the actual original-section maximality in [F7], the selected-end access premise means that for some $R_{a}$ and every $R \geq R_{a}$ , with $U_{R} = R \times {r > R}$ in the physical stationary end chart, $E \subset I_{(E, g)}^{-} (U_{R}) \cap I_{(E, g)}^{+} (U_{R})$ , with both witnessing timelike paths in the unchanged original exterior. These are the inherited complete height/barrier premises, not conclusions of regularity or candidate existence. Existence and boost-coordinate description of Hawking-flow heights K-flow barriers with boundary tangency comparison and uniform height bound

[F9]

Under these matched hypotheses, the Hawking-height theorem supplies the unique Hawking-flow height $u_{q}^{K}$ , smooth individually up to $S$ . The barrier theorem supplies $\sup_{0 \leq ρ_{q} \leq ρ_{0}} ∣ u_{q}^{K} ∣ \leq B_{δ}$ on one common closed reference collar, including $S$ , with $B_{δ}$ independent of the candidates and of $q \in [δ, 1]$ . No uniform derivative bound for these heights is assumed. Existence and boost-coordinate description of Hawking-flow heights K-flow barriers with boundary tangency comparison and uniform height bound

Proof

Given: The conditional model and supplied smooth family, collar and candidate sections. Goal: Prove the all-range differential inequality with one uniform collar constant.

1.1A1F1F2F7F8F9

Match the height/barrier contracts before using the boundary theorem. The actual original reference section is the one in [F7]; [A1] supplies its standing nonvanishing scope, and [F8] supplies the same stationary product, selected single-primitive family, complete candidate/end hypotheses, uniform estimates and two-sided original-metric far-end access. The Hawking-height result therefore gives the unique heights and their individual smooth extensions through $S$ in [F9]. Apply the barrier result with precisely these hypotheses to obtain the bound $∣ u_{q}^{K} ∣ \leq B_{δ}$ on the common closed reference collar, including $S$ . This provides the uniform height premise of the boundary theorem in [F2]; that theorem now gives the common candidate graph domain, cone and boundary boost bound. No eventual candidate-existence theorem or normal-angle estimate is used to obtain this height bound.

2.1A1F1F2F3F4F7step 1.1

Retain the full preceding hypotheses and actual reference-section binding in [F7]. The nonvanishing hypothesis in [A1] meets the corresponding premise of both suppliers in [F1]. By [F1]–[F2], choose a smaller common closed Gaussian coordinate cylinder with its image in the smooth background neighborhood. Its parameter set, including $q \in [δ, 1]$ and the reference base points in a closed collar of compact $S$ , is compact. In these coordinates $g_{q} = - d τ_{q}^{2} + h_{q, τ_{q}}$ and $U_{q} = \partial_{τ_{q}} = - {grad}_{g_{q}} τ_{q}$ . Thus $U_{q}$ is smooth, future unit timelike. Smoothness of the background Gaussian maps and metrics gives common bounds for $D^{(q)} U_{q}$ , $(D^{(q)})^{2} U_{q}$ and $Ric (g_{q})$ in the positive metrics $e_{q} = g_{q} + 2 U_{q}^{♭} \otimes U_{q}^{♭}$ , using a finite cover. The inverse positive metric norms are bounded as well by compactness and positive definiteness. These are background bounds and use no derivative of a candidate graph.

3.1A1F2F3F4step 1.1step 2.1∎

Each candidate in this tube is smooth spacelike and maximal by [A1]–[F3]. Apply the all-range local theorem in [F4] with the background bound from step 2.1. It gives exactly the displayed inequality for every $v_{q} \geq 1$ , uniformly in the family. The local derivation divides only by $v_{q} \geq 1$ , so no missing passage through $v_{q} = 1$ or $1 \leq v_{q} \leq 2$ remains. It requires no compactness of a positive-height candidate portion and makes no use of a fixed temporal function across $q$ . On $S$ , [F2] identifies $U_{q}$ with the reference normal and supplies the uniform boundary boost bound. This proves the statement on the common collar and its individually smooth boundary.

Source notes

Radu Appendix B (B.8)–(B.10) motivates the local differentiation and trace-free argument. Its arcosh inequality is restricted to $v > 2$ ; the five local helpers instead retain the smooth angle v and derive the log inequality by division by v alone. No external all-v theorem is invented. The separate weighted-normal theorem, its compact nonnegative-height portions and its fixed-temporal-function hypotheses are not invoked. The Gaussian compact family supplies local coefficient bounds directly. No Einstein–Maxwell energy condition is needed for this kinematic differential calculation, since the ambient Ricci tensor is estimated as a bounded background tensor.

Depends on

Used by · 1

Cutoff maximum principle and uniform collar regularity

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

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1, printed/PDF pp.35–36 (hypotheses); proof pp.38–39, (B.8)–(B.11)

Method attribution: (B.8) differentiates the normal angle; Codazzi and trace-free algebra give the scalar and arcosh estimates (B.9)–(B.11), with the arcosh argument restricted to v>2.

Hypotheses: C2-convergent smooth Lorentz metrics near a compact set, a fixed smooth temporal function with uniformly timelike gradient, and smooth maximal hypersurfaces with compact nonnegative-height portions. These global hypotheses are not imported into the local logarithmic argument.

Reading limits: Retained independent source-input/background reading record: Consulted actual v1 PDF text of Lemma B.1 on pp.35–36 and its proof on pp.38–39 in this adjudication; no images were read. Method attribution only. The all-range logarithmic conclusion is supplied by the locally proved helpers, not by this source. Current maintenance checks the stated contract and its local application, without claiming new external-proof or PDF reading.

Sources

The uniqueness of smooth stationary black holes · PDF p.4, Theorem 1.1 and Remark 1.2 (standing nonvanishing scope); pp.18–19, (3.12)–(3.13) and Gaussian collar setup
Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1, printed/PDF pp.35–36 (hypotheses); proof pp.38–39, (B.8)–(B.11)