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.

Stable-horizon flux-energy and area-charge inequalities

PDF p. 27: Proof of Theorem 4.2; proof of Corollary 4.1

Statement

Assume the regular non-degenerate stationary vacuum model, with original exterior section maximal, ADM mass normalized to one and stationary Killing field $T$ nontrivial on its bifurcation sphere $S$ . Fix $q \in (0, 1]$ and an individually admissible smooth spacelike maximal section $Σ_{q}$ as in Nonnegative principal eigenvalue of the horizon stability operator, with a smooth spacelike extension and outward normal-graph tube as in Horizon expansion variation and Einstein–Maxwell curvature. Work in ZFC. Let $g_{q}, F_{q}$ be the Harrison metric and electromagnetic two-form, and let $γ_{q} = g_{q} ∣_{T S}$ and $d μ_{q}$ be the induced metric and oriented area form. Define electric and magnetic flux densities by $(⋆_{g_{q}} F_{q}) ∣_{T S} = e_{q} d μ_{q}$ and $F_{q} ∣_{T S} = b_{q} d μ_{q}$ . Then $\int_{S} (e_{q}^{2} + b_{q}^{2}) d μ_{q} \leq 4 π, ∣ S ∣_{g_{q}} \geq \frac{1}{4 π} {(\int_{S} e_{q} d μ_{q})}^{2} .$ Here $∣ S ∣_{g_{q}} = \int_{S} d μ_{q}$ . In particular, both inequalities hold for sufficiently small positive $q$ under the same hypotheses. The second inequality follows directly from the same stability equation and Gauss–Bonnet.

Facts & Assumptions

Given: The model, parameter, admissible section, extension and graph data in the Statement.

Goal: Prove the two displayed inequalities.

[A1]

The regularity and normalization conventions and the meaning of a spacelike maximal section are those of Regular non-degenerate stationary vacuum spacetime, Mass-one convention and Harrison denominator and Spacelike sections, maximality and lapse–shift data, in the conditional model Stationary vacuum model and asymptotic normalization. Work in ZFC. The Axiom of Choice

[F1]

The horizon and surface-data definitions supply the smooth positive surface measure, null gauge, flux densities and intrinsic gradient, divergence, Laplacian and Gauss curvature. Horizon null frame, flux densities and stability operator notation Surface measure, flux densities and null normal gauge

[F2]

For the stated section, $S$ is a connected closed two-sphere and the outward graph stability operator has nonnegative principal eigenvalue $λ$ . The principal-eigenvalue theorem supplies a smooth eigenfunction $ϕ > 0$ with $L_{q} ϕ = λ ϕ$ . Nonnegative principal eigenvalue of the horizon stability operator Principal eigenfunctions and MOTS spectral stability

[F3]

In this gauge the operator is $L_{q} ϕ = - Δ ϕ + 2 ζ \cdot \nabla ϕ + (K + div ζ - ∣ ζ ∣^{2} - e_{q}^{2} - b_{q}^{2}) ϕ$ , with all surface operations taken for $γ_{q}$ . Horizon expansion variation and Einstein–Maxwell curvature

[F4]

On the closed oriented surface, the integral of the divergence of a smooth vector field is zero. Gauss–Bonnet gives $\int_{S} K d μ_{q} = 2 π χ (S) = 4 π$ ; a tetrahedral triangulation of the two-sphere has Euler characteristic $4 - 6 + 4 = 2$ . Riemannian divergence theorem Global Gauss-Bonnet for closed oriented surfaces

[F5]

Cauchy–Schwarz applies to square-integrable functions for the finite surface measure. Cauchy-Schwarz inequality for L²

[F6]

The construction in Proposition 2.1: Einstein–Maxwell equations and regularity applies under [A1] and supplies the normalized smooth Ernst potential and a stationary one-form family satisfying the defining curl and contraction identities, with $d_{q} = D_{q} (p) > 0$ at a zero $p \in S$ of $T$ , on the exterior and an open neighborhood of $S$ where $D_{q} > 0$ . The symbols $g_{q}, F_{q}$ denote the resulting smooth Lorentzian metric and real two-form of Stationary Harrison one-form and metric and Harrison electromagnetic form, using the same fields, spacetime orientation and time orientation in [F1]–[F3].

Proof

technique · direct

1.1givenF1F2F3F4F6algebra

Set $u = \log ϕ$ and let $ζ^{♯}$ be the surface metric-dual vector field. Dividing the eigenvalue equation in [F3] by $ϕ$ and using $Δ ϕ / ϕ = Δ u + ∣ \nabla u ∣^{2}$ gives $λ = K - e_{q}^{2} - b_{q}^{2} + div (ζ^{♯} - \nabla u) - ∣ ζ^{♯} - \nabla u ∣^{2}$ . The positive eigenfunction makes $u$ smooth. Integrating, applying [F4], and using $λ \geq 0$ yields $\int_{S} (e_{q}^{2} + b_{q}^{2}) d μ_{q} = 4 π - λ ∣ S ∣_{g_{q}} - \int_{S} ∣ ζ^{♯} - \nabla u ∣^{2} d μ_{q} \leq 4 π$ .

2.1F1F5F6step 1.1algebra∎

Smoothness and compactness put $e_{q}$ and the constant function $1$ in $L^{2} (d μ_{q})$ . By [F5], $∣ \int_{S} e_{q} d μ_{q} ∣^{2} \leq ∣ S ∣_{g_{q}} \int_{S} e_{q}^{2} d μ_{q}$ . Since $b_{q}^{2} \geq 0$ , step 1.1 bounds the last integral by $4 π$ . Dividing by $4 π$ gives the area–charge inequality. This also proves its derivation from the same stability equation and Gauss–Bonnet. The fixed $q$ was arbitrary in $(0, 1]$ , so the asserted sufficiently-small-positive-parameter case follows.

Source notes

The article reference records attribution. The proof uses the listed suppliers; it does not use the article's assertion as a premise.

Depends on

Used by · 2

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 p. 27, proof of Theorem 4.2 and proof of Corollary 4.1