LemmaStatement: AI-generatedProof: AI-adapted

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Boost-frame bounds for a smooth observer and arbitrary spacelike normals

Statement

Let $g$ have signature $(-, +, +, +)$ , let $U$ be a future unit timelike vector field, and let $n$ be the future unit normal of a spacelike hypersurface. Put $v = - g (n, U)$ and $e = g + 2 U^{♭} \otimes U^{♭}$ , where $U^{♭} = g (U, \cdot)$ . Then $e$ is positive definite, $v \geq 1$ and $∣ n ∣_{e}^{2} = 2 v^{2} - 1$ . For any $g$ -orthonormal tangent basis $(e_{a})_{a = 1}^{3}$ set $Z_{a} = g (U, e_{a})$ . Then $\sum_{a} Z_{a}^{2} = v^{2} - 1$ and $∣ e_{a} ∣_{e} \leq \sqrt{2} v$ . These estimates do not require an a priori bound on the candidate normal. Tensor norms below are the operator norms obtained by using $e$ on every slot and its induced dual metric.

Facts & Assumptions

[F1]

Use the induced positive metric, future unit normal and Lorentzian orthogonal decomposition of the spacelike hypersurface interface. Spacelike sections, maximality and lapse–shift data

Proof

**Given:**The vectors and frame in the Statement. Goal: Establish the positive comparison metric and the displayed bounds.

1.1F1givenalgebra

In a Lorentz-orthonormal ambient frame with time axis $U$ , $g$ is $diag (- 1, 1, 1, 1)$ and $e$ is the identity matrix. Write $n = v U + z$ with $z ⊥ U$ . Future orientation makes $v > 0$ , and $- 1 = - v^{2} + ∣ z ∣^{2}$ gives $v \geq 1$ and $∣ n ∣_{e}^{2} = v^{2} + ∣ z ∣^{2} = 2 v^{2} - 1$ .

2.1F1step 1.1algebra∎

The tangent projection of $U$ onto $n^{⊥}$ is $U - v n$ , whose squared norm is $v^{2} - 1$ and whose components are $Z_{a}$ . Hence $\sum_{a} Z_{a}^{2} = v^{2} - 1$ . Also $∣ e_{a} ∣_{e}^{2} = g (e_{a}, e_{a}) + 2 g (U, e_{a})^{2} = 1 + 2 Z_{a}^{2} \leq 2 v^{2} - 1 \leq 2 v^{2}$ . This proves the claimed estimates at every $v \geq 1$ .

Depends on

Used by · 4

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, proof of Lemma B.1, printed/PDF pp.38–39; frame bound immediately after (B.8) and tangent-component identity immediately before (B.10)

In an ambient Lorentz-orthonormal frame with time axis V, the normal and tangent-frame components are bounded by a constant times v; the squared tangent components of V sum to v squared minus one.

Hypotheses: The pointwise setting uses a future unit timelike observer V, a future unit normal T0 to a spacelike hypersurface, a tangent orthonormal frame, and v=-g(T0,V).

Reading limits: Consulted the PDF text on pp.38–39, specifically the frame-bound sentence after (B.8) and the tangent-component identity before (B.10). Images were unavailable. This item's two proof steps are a local pointwise Lorentz-frame calculation, with no differentiation of v or logarithmic argument. Its exact comparison-metric formulas and validity for every v>=1 are established locally. The source's differential estimates, arcosh argument on v>2, and weighted maximum estimate are not imported.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, proof of Lemma B.1, printed/PDF pp.38–39; frame bound immediately after (B.8) and tangent-component identity immediately before (B.10)