LemmaStatement: AI-adaptedProof: AI-generated

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Quantitative lapse and shift bounds force annular timelikeness

Statement

For the AIK quantitative package and $\sup_{S_{0}} ∣ g (T, T) ∣ < ϵ^{2}$ , let $C_{1} = C_{1} (A)$ be the lapse constant from the explicit external estimate. For all sufficiently small $0 < ϵ < 1$ depending only on $A$ , the stationary field is strictly timelike on $1 + ϵ^{1 / 100} \leq r \leq R_{1}$ in $Σ_{0}$ . More explicitly, once the imported estimates hold, $ϵ^{7 / 150} < (8 A_{1} C_{1}^{2})^{- 1}$ suffices for this assertion.

Facts & Assumptions

[A1]

Work in ZFC, including the axiom of choice and its inherited countable and dependent choice uses. The Axiom of Choice

[F1]

The quantitative package includes $A_{1}^{- 1} I \leq (h_{i j}) \leq A_{1} I$ and the source section coordinates. Exact AIK global, horizon and quantitative hypotheses

[F2]

The norm identity and exact horizon-smallness translation are those of Horizon smallness is the induced norm of the tangential stationary field.

[F3]

Lapse and shift components use the hypersurface definitions in Spacelike sections, maximality and lapse–shift data.

[F4]

The explicit External input supplies $n \geq C_{1}^{- 1} (r - 1) / r$ and $\sum_{j} ∣ X_{j} ∣ \leq ϵ^{1 / 30}$ on the bounded annulus, for the covariant shift components $X_{j} = h_{j i} X^{i}$ .

[F5]

A self-adjoint endomorphism of a finite-dimensional real inner product space admits an orthonormal eigenbasis. Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

[F6]

In signature $(-, +, +, +)$ , a tangent vector is timelike exactly when its squared Lorentzian norm is negative. Timelike, null, spacelike and causal vectors of a Lorentz vector space

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the stated conditional conclusion.

1.1F1F3F4F5algebra

Put $δ = ϵ^{1 / 100} < 1$ . For $r \geq 1 + δ$ , monotonicity of $(r - 1) / r$ gives $n \geq C_{1}^{- 1} δ / (1 + δ) \geq δ / (2 C_{1})$ . Since $h^{- 1} \leq A_{1} I$ , the covariant-component estimate gives $h (X, X) = h^{i j} X_{i} X_{j} \leq A_{1} \sum_{j} X_{j}^{2} \leq A_{1} ϵ^{1 / 15}$ . The symmetric matrix $H = (h_{i j})$ defines a self-adjoint endomorphism of Euclidean $R^{3}$ , since $u^{T} H v = (H u)^{T} v$ . By [F5] it has an orthonormal eigenbasis. Applying [F1] to each unit eigenvector puts its eigenvalue in $[A_{1}^{- 1}, A_{1}]$ , so the eigenvalues of $H^{- 1}$ are the reciprocals of those of $H$ and are therefore at most $A_{1}$ , proving the inverse-metric bound.

2.1step 1.1F1F2F6algebra∎

The norm identity now yields $g (T, T) \leq - (4 C_{1}^{2})^{- 1} ϵ^{1 / 50} + A_{1} ϵ^{1 / 15}$ . Since $1 / 15 - 1 / 50 = 7 / 150$ , the threshold in the Statement bounds the second term by $(8 C_{1}^{2})^{- 1} ϵ^{1 / 50}$ , and hence $g (T, T) \leq - (8 C_{1}^{2})^{- 1} ϵ^{1 / 50} < 0$ . By [F6], this strict negativity makes $T$ timelike throughout the stated annulus. All constants in this condition depend only on the fixed source bounds.

External input

Import the source lapse lower bound (2.10) and shift estimate (2.15), with exactly the GR/SBS/maximal and quantitative assumptions stated in the package and the source horizon bound. Only these inequalities are used here; their elliptic, integral, interpolation and transport proofs remain external. The norm estimate uses covariant shift components and the inverse induced metric, avoiding an implicit identification with contravariant components.

Depends on

Used by · 1

The non-timelike region is confined to the local-rigidity collar

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

S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §2.2 Lemma 2.1, (2.9)–(2.10), PDF p. 7; Lemma 2.3, (2.15), PDF pp. 8–9

Under the source package and sup_{S0}|g(T,T)|<epsilon^2, with epsilon sufficiently small depending on A, 0<=n<1 and n>=C1(A)^-1(r-1)/r on Sigma1; moreover sup_{Sigma1 minus Sigma_R1}(sum_j|X_j|+sum_ij|nabla_i X_j|)<=epsilon^(1/30). The norms on X_j here are its covariant coordinate components as in the source transport formulas.

Hypotheses: The complete GR, SBS, maximality and quantitative bounds of §§1.1–1.2, with the horizon-smallness parameter sufficiently small depending on A. The covariant shift components are measured in the source section chart.

Reading limits: Read Lemmas 2.1, 2.2 and 2.3 and their printed arguments on PDF pp. 7–9. The maximum-principle/Hopf estimates, weighted integral estimate, interpolation and transport bounds are imported rather than reconstructed. Reading the printed arguments is not an independent proof review or verification of their cited PDE inputs.

Sources

S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §2.2 Lemma 2.1, (2.9)–(2.10), PDF p. 7; Lemma 2.3, (2.15), PDF pp. 8–9