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.

Mass-one Ernst and radial primitive asymptotics

PDF p. 9: §2.2, before and at (2.10)

Statement

Assume the regular stationary vacuum model and its stationary end coordinates $(t, x^{1}, x^{2}, x^{3})$ with mass parameter one. Put $r = ∣ x ∣$ , $f = - g (T, T)$ , $F = \frac{1}{2} d T^{♭}$ , and let $σ = f + i ψ$ be the Ernst potential normalized to tend to one at infinity. Let $⋆_{g}$ be the oriented Lorentzian Hodge star and $Θ_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ . Then, in these coordinates, $f = 1 - 2 / r + O_{6} (r^{- 2}), F_{i j} = \frac{1}{2} (\partial_{i} g_{0 j} - \partial_{j} g_{0 i}) = O_{5} (r^{- 3}), (⋆_{g} F)_{i j} = O_{5} (r^{- 2}) .$ Moreover $\partial_{i} ψ = O_{5} (r^{- 3})$ , $ψ = O_{6} (r^{- 2})$ , $σ = 1 - 2 / r + O_{6} (r^{- 2})$ , and $(Θ_{0})_{i j} = O_{5} (r^{- 4})$ . The spatial restriction of $Θ_{0}$ is closed, and its radial primitive has components $(W_{0})_{i} (x) = - \int_{1}^{\infty} s x^{j} (Θ_{0})_{j i} (s x) d s = O_{5} (r^{- 3}) .$ Here $O_{m} (r^{- b})$ means the stated bound and corresponding bounds after each Cartesian spatial derivative of order at most $m$ . The radial formula specifies spatial components on the end; it does not by itself assert a global or horizon-regular choice of $W_{0}$ .

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Stationary Lorentzian data and tensor conventions

Differentiated asymptotic bounds and parameter uniformity

Complex self-dual forms and metric-dependent contraction

Killing two-form and normalized Ernst notation

Closed Ernst form and existence of normalized potential

Facts & Assumptions

[A1]

Adopt the stationary vacuum model and the regular-spacetime hypotheses, including the fixed stationary Cartesian end and mass parameter $M_{0} = 1$ . Thus $T = \partial_{t}$ , the metric is independent of $t$ , $g_{00} = - 1 + 2 / r + O_{6} (r^{- 2})$ , $g_{0 i} = O_{6} (r^{- 2})$ , and $g_{i j} = δ_{i j} + O_{6} (r^{- 1})$ . Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Differentiated asymptotic bounds and parameter uniformity

[F1]

The stationary and Ernst conventions define $T^{♭} = g (T, \cdot)$ , $f = - g (T, T)$ , $F = \frac{1}{2} d T^{♭}$ , $F = F + i ⋆_{g} F$ , and the normalized potential by $d σ = 2 ι_{T} F$ and $σ \to 1$ . The potential exists under [A1]. Stationary Lorentzian data and tensor conventions Complex self-dual forms and metric-dependent contraction Killing two-form and normalized Ernst notation Closed Ernst form and existence of normalized potential

[F2]

In vacuum, $d F = d ⋆_{g} F = 0$ . On the end, define $Θ_{0}$ directly by $Θ_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ , using the fields in [F1]. Vacuum Killing and Ernst differential identities

[F3]

A closed spatial $C^{5}$ two-form with coefficients $O_{5} (r^{- 4})$ has the displayed radial primitive, with components $O_{5} (r^{- 3})$ . Radial primitive and five differentiated decay bounds

[F4]

On an oriented, time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , the Hodge star is characterized by $P \land ⋆_{g} Q = ⟨ P, Q ⟩_{g} {v o l}_{g}$ , where for real two-forms $⟨ P, Q ⟩_{g} = \frac{1}{2} P_{a b} Q^{a b}$ , and $⋆_{g}^{2} = - id$ on two-forms. Its coordinate component formula and the coordinate formula for $d$ apply, and $d (α \land β) = d α \land β + (- 1)^{k} α \land d β$ for a $k$ -form $α$ . Lorentzian exterior calculus and Hodge star

[F5]

For a smooth vector field $X$ and a $k$ -form $α$ , contraction satisfies $ι_{X} (α \land β) = (ι_{X} α) \land β + (- 1)^{k} α \land ι_{X} β$ . Interior product on forms is a graded antiderivation

Proof

**Given:**The model, end coordinates, and normalized potential in the Statement.

Goal: Establish the differentiated estimates and the radial primitive.

1.1A1F1F4algebra

Since $T = \partial_{t}$ , $f = - g_{00}$ and $T^{♭} = g_{0 μ} d x^{μ}$ . Stationarity and [A1] give the claimed expansion for $f$ , $F_{i j} = \frac{1}{2} (\partial_{i} g_{0 j} - \partial_{j} g_{0 i}) = O_{5} (r^{- 3})$ , and $F_{0 i} = - \frac{1}{2} \partial_{i} g_{00} = O_{5} (r^{- 2})$ . Inverting the metric near its Minkowski limit gives $g^{00} = - 1 + O_{6} (r^{- 1})$ , $g^{0 i} = O_{6} (r^{- 2})$ , and $g^{i j} = δ^{i j} + O_{6} (r^{- 1})$ ; the same derivative bounds follow by differentiating $g^{- 1} g = I$ . The metric volume coefficients and their spatial derivatives through order five are bounded. In the Hodge component formula, the spatial components involve $F^{0 k} = O_{5} (r^{- 2})$ , whereas $(⋆_{g} F)_{0 i}$ involves $F^{j k} = O_{5} (r^{- 3})$ . Consequently $(⋆_{g} F)_{i j} = O_{5} (r^{- 2})$ and $(⋆_{g} F)_{0 i} = O_{5} (r^{- 3})$ .

1.2F4algebra

For real two-forms, the component pairing in [F4] is symmetric because $g^{- 1}$ is symmetric. Apply its Hodge characterization first to $(P, Q) = (⋆_{g} F, F)$ and then to $(P, Q) = (F, ⋆_{g} F)$ to obtain $⋆_{g} F \land ⋆_{g} F = ⟨ ⋆_{g} F, F ⟩_{g} {v o l}_{g} = ⟨ F, ⋆_{g} F ⟩_{g} {v o l}_{g} = F \land ⋆_{g} (⋆_{g} F) = - F \land F$ . Thus $F \land F + ⋆_{g} F \land ⋆_{g} F = 0$ .

2.1F1step 1.1algebra

Taking the imaginary part of $d σ = 2 ι_{T} (F + i ⋆_{g} F)$ yields $\partial_{i} ψ = 2 (⋆_{g} F)_{0 i} = O_{5} (r^{- 3})$ . Since $ψ \to 0$ , integration along the ray from $r$ to infinity gives $ψ = O (r^{- 2})$ uniformly in angle. Every nonzero spatial derivative of order at most six is a derivative of some $\partial_{i} ψ$ , giving $ψ = O_{6} (r^{- 2})$ . Combining this with the expansion of $f$ gives the stated expansion of $σ$ . The product rule in $Θ_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ now gives $(Θ_{0})_{i j} = O_{5} (r^{- 4})$ .

3.1F1F2F3F4F5step 2.1step 1.2algebra∎

By [F1], $d f = 2 ι_{T} F$ and $d ψ = 2 ι_{T} ⋆_{g} F$ . The derivative product rule in [F4] and the vacuum identities in [F2] give $d Θ_{0} = 8 ((ι_{T} F) \land F + (ι_{T} ⋆_{g} F) \land ⋆_{g} F)$ . Moving a one-form past a two-form makes two sign changes, so the contraction rule [F5] gives $ι_{T} (F \land F + ⋆_{g} F \land ⋆_{g} F) = 2 ((ι_{T} F) \land F + (ι_{T} ⋆_{g} F) \land ⋆_{g} F)$ . The left side vanishes by step 1.2, hence $d Θ_{0} = 0$ . In the coordinate formula for $d$ , its purely spatial coefficients are exactly the closure equations for the spatial restriction of $Θ_{0}$ . That restriction has smooth coefficients and the $O_{5} (r^{- 4})$ bounds from step 2.1, so [F3] applies on a sufficiently far-out exterior and proves both the displayed integral formula and its five differentiated decay bounds.

The article citation records attribution of the original claim. The argument above uses the declared model, geometric definitions, vacuum identities, normalized-potential result, and radial primitive lemma.

Source notes

Depends on

Used by · 5

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 page 9, §2.2, before and at equation (2.10)