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The mass-one Killing form has electric seed flux minus four pi

Statement

In a stationary end with $T = \partial_{t}$ , suppose $g_{00} = - 1 + 2 / r + O_{1} (r^{- 2})$ , $g_{0 i} = O_{1} (r^{- 2})$ and $g_{i j} = δ_{i j} + O_{1} (r^{- 1})$ . Put $F = \frac{1}{2} d T^{♭}$ and use signature $(-, +, +, +)$ and the orientation for which $d t \land d x^{1} \land d x^{2} \land d x^{3}$ is positive. On the outward oriented coordinate sphere $S_{R}$ in $t = 0$ , $\int_{S_{R}} F = O (R^{- 1})$ and $\int_{S_{R}} ⋆_{g} F = - 4 π + O (R^{- 1})$ . In fact the first integral is exactly zero, since $F$ is an exact spacetime form. No specified leading angular-momentum coefficient of $g_{0 i}$ is required.

Facts & Assumptions

Given: the hypotheses in the Statement; work in ZFC.

Goal: establish the Statement.

[F1]

The regular end, spatial derivative notation and Killing-form convention are fixed. Regular non-degenerate stationary vacuum spacetime Killing two-form and normalized Ernst notation Differentiated asymptotic bounds and parameter uniformity

[F2]

The geometric charge convention uses the oriented Lorentzian Hodge star and the factor four pi. Work in ZFC; its choice axiom supplies the countable choice required by general Stokes. Geometric electrovacuum equations and charge convention The Axiom of Choice

[F3]

On the standard smooth unit sphere with outward orientation, the area form $d ω$ satisfies $\int_{S^{2}} d ω = 4 π$ . Unit-sphere area form and outward radial flux

[F4]

Exterior differentiation commutes with pullback by a smooth map. General Stokes applies to an oriented smooth manifold of dimension at least one and a compactly supported smooth form of one lower degree; an empty boundary contributes zero. The exterior derivative commutes with pullback The general Stokes theorem

Proof

Given: the hypotheses in the Statement; work in ZFC.

Goal: establish the Statement.

[F1] The regular end, spatial derivative notation and Killing-form convention are fixed. Regular non-degenerate stationary vacuum spacetime Killing two-form and normalized Ernst notation Differentiated asymptotic bounds and parameter uniformity

[F2] The geometric charge convention uses the oriented Lorentzian Hodge star and the factor four pi. Work in ZFC; its choice axiom supplies the countable choice required by general Stokes. Geometric electrovacuum equations and charge convention The Axiom of Choice

[F3] On the standard smooth unit sphere with outward orientation, the area form $d ω$ satisfies $\int_{S^{2}} d ω = 4 π$ . Unit-sphere area form and outward radial flux

[F4] Exterior differentiation commutes with pullback by a smooth map. General Stokes applies to an oriented smooth manifold of dimension at least one and a compactly supported smooth form of one lower degree; an empty boundary contributes zero. The exterior derivative commutes with pullback The general Stokes theorem

Proof technique: direct.

1.1F1F2

Stationarity gives $F_{0 i} = - \frac{1}{2} \partial_{i} g_{00} = x^{i} / r^{3} + O (r^{- 3})$ and $F_{i j} = \frac{1}{2} (\partial_{i} g_{0 j} - \partial_{j} g_{0 i}) = O (r^{- 3})$ . Write $η = - d t^{2} + \sum_{i} (d x^{i})^{2}$ . The inverse-matrix identity and determinant continuity give $g^{- 1} - η^{- 1} = O (r^{- 1})$ and $\sqrt{∣ \det g ∣} = 1 + O (r^{- 1})$ , so the Hodge component formula yields $(⋆_{g} F - ⋆_{η} F)_{i j} = O (r^{- 3})$ . The time-space leading part of $F$ is $r^{- 2} d t \land d r$ , while its other coefficients and the time-space remainders are $O (r^{- 3})$ .

2.1step 1.1F2F3

On an oriented Minkowski orthonormal coframe, $⋆_{η} (d t \land d r) = - R^{2} d ω$ on $S_{R}$ , since the two-form has norm squared minus one and its wedge with its star must be minus the oriented volume. Here $d ω$ is the outward unit-sphere area form, and the sphere is identified with $S^{2}$ by the orientation-preserving map $ω \mapsto (0, R ω)$ . Thus the leading sphere pullback is $- d ω$ . A two-form with Cartesian coefficients $O (R^{- 3})$ pulls back under this map with coefficients $O (R^{- 1})$ , since each of its two tangent slots contributes a factor $R$ . Its integral is $O (R^{- 1})$ on the fixed unit sphere. The area identity $\int_{S^{2}} d ω = 4 π$ therefore proves both estimates.

3.1F1F2F4step 2.1∎

Let $i_{R} : S_{R} ↪ M^{e n d}$ be the smooth inclusion and let $α_{R} = i_{R}^{*} T^{♭}$ . Naturality of the exterior derivative gives $i_{R}^{*} F = \frac{1}{2} d α_{R}$ . The coordinate sphere $S_{R}$ is a compact oriented smooth two-manifold without boundary. The smooth one-form $α_{R}$ has compact support because its support is closed in compact $S_{R}$ . Applying general Stokes in dimension two, with countable choice supplied by ZFC, gives $\int_{S_{R}} F = \frac{1}{2} \int_{S_{R}} d α_{R} = \frac{1}{2} \int_{\partial S_{R}} α_{R} = 0$ . The form $T^{♭}$ need only be defined on the end; no filling ball inside the coordinate domain is used. This proves the exact vanishing and completes the Statement without specifying a leading angular-momentum coefficient.

Depends on

Used by · 3

Dependency tree · 2 levels

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