Physics theoremStatement: AI-adaptedProof: AI-generated

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Proposition 5.1: recovery of Ionescu–Klainerman conditions

PDF p. 28: Proposition 5.1; (5.1); PDF p. 29: Proposition 5.1; proof

Statement

Adopt the stationary vacuum model and regular non-degenerate stationary construction, with original exterior section spacelike and maximal, ADM mass one, and stationary Killing field $T$ not identically zero on the bifurcation sphere $S$ . Use the supplied normalized Ernst potential $σ = f + i ψ$ , the single regular Harrison family, and its chosen pole $p \in S$ , as in the two recovery suppliers. Here $p$ is the common pole of the local Hawking and axial fields $K, Z$ supplied under these hypotheses: $K = T + Ω_{H} Z$ , $K ∣_{S} = 0$ , $T (p) = Z (p) = 0$ , $Z$ has least positive common period $2 π$ , and $K$ has positive normal boost rate $κ$ in the future/outward normal basis. The Harrison family uses this same pole. Here $f = - g (T, T)$ and $ψ$ is real. Define $F_{a b} = \nabla_{a} T_{b}$ and $F = F + i ⋆_{g} F$ , where $\nabla$ and $⋆_{g}$ are the original metric's Levi–Civita connection and oriented Hodge star. Contract complex bilinearly, without conjugation or a factor of one half, and raise both indices using the original metric $g$ .

Then $- 4 F_{a b} F^{a b} = (1 - σ)^{4}$ everywhere on $S$ , and $Re (1 / (1 - σ (p))) > 1 / 2$ .

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Spacelike sections, maximality and lapse–shift data

Mass-one convention and Harrison denominator

Killing two-form and normalized Ernst notation

Complex self-dual forms and metric-dependent contraction

Local Hawking and axial fields, boost and pole

Inverse Harrison relation and Killing-scalar identity

Axial averaging and positive boost at the pole

Facts & Assumptions

[A1]

The stationary vacuum model, regular construction, maximal original exterior section, mass-one normalization, nontriviality of $T$ on $S$ , supplied potential, Harrison family and pole are those in the Statement. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Spacelike sections, maximality and lapse–shift data Mass-one convention and Harrison denominator

[F1]

The normalized Ernst and complex-form conventions give the displayed definitions of $σ$ , $F$ and $F$ , with the original metric used to raise indices and a complex bilinear contraction. Killing two-form and normalized Ernst notation Complex self-dual forms and metric-dependent contraction

[F2]

Under the stated construction hypotheses, the inverse Harrison contraction gives $F_{a b} F_{g}^{a b} = - \frac{1}{4} (1 - σ)^{4}$ on $S$ . Inverse Harrison relation and Killing-scalar identity

[F3]

With the local fields and common pole in [F4], the same normalized potential and regular Harrison family, and $d_{0} = ∣ 1 - σ (p) ∣^{2} > 0$ , axial averaging gives $2 Re (1 / (1 - σ (p))) - 1 > 0$ ; in particular the displayed reciprocal is defined. Axial averaging and positive boost at the pole

[F4]

The regular stationary vacuum data, maximal original exterior section and $T ∣_{S} \equiv̸ 0$ supply local commuting Killing fields $K, Z$ with $K = T + Ω_{H} Z$ , $K ∣_{S} = 0$ , a circle action of least positive common period $2 π$ , a common pole $p \in S$ with $T (p) = Z (p) = 0$ , and a constant positive normal boost rate $κ$ for $K$ in the future/outward normal basis. These are the fields and pole used for the stated single Harrison family. Local Hawking and axial fields, boost and pole

Proof

Given: The model, construction and conventions in the Statement.

Goal: Establish the two displayed conditions on $S$ and at $p$ .

1.1A1F4

The regular stationary vacuum data, maximal original exterior section and nontriviality of $T$ in [A1] match [F4]. Use its local fields $K, Z$ and common pole $p$ , with the normalized circle action and positive future/outward boost. By the stated family choice this is the same pole used throughout the two recovery suppliers, rather than a separately chosen zero of $T$ .

2.1A1F1F2step 1.1algebra

The construction hypotheses and original-metric contraction convention match [F2] for this family. Multiplying its equality by $- 4$ gives $- 4 F_{a b} F^{a b} = (1 - σ)^{4}$ at every point of $S$ .

3.1A1F1F3F4step 1.1step 2.1algebra∎

At the common pole, $T (p) = 0$ gives $f (p) = - g (T, T) (p) = 0$ , so $σ (p) = i ψ (p)$ and $d_{0} = ∣ 1 - σ (p) ∣^{2} = 1 + ψ (p)^{2} > 0$ . Together with the local fields, circle normalization and positive boost from step 1.1 and the same potential and family, this verifies the premises of [F3]. Adding one to its strict inequality and dividing by the positive number two gives $Re (1 / (1 - σ (p))) > 1 / 2$ . Together with step 2.1 this proves both conditions.

Source notes

The article is cited for attribution of Proposition 5.1. The deduction above uses the declared local-field and recovery suppliers. The article's assertion is not a proof premise.

Depends on

Used by · 1

Conditional horizon recovery and Kerr implications

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.

Sources

The uniqueness of smooth stationary black holes · PDF pages 28–29, Proposition 5.1 and its proof