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Vacuum metrics embed as stationary electrovacuum data with zero field

Statement

Work in the ZFC convention of the stationary Lorentzian data. Let $(M, g, N, o, T)$ be smooth oriented time-oriented stationary Lorentzian data with $Ric (g) = 0$ . Set $F = 0$ and $A = 0$ , the global zero one-form. Then $d A = F$ , $d F = d (⋆_{g} F) = 0$ and ${Ric}_{a b} = 2 (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ .

The Maxwell field is invariant under every isometry and satisfies $L_{T} F = 0$ . Its electric and magnetic fluxes vanish on every compact oriented smooth embedded spacelike two-surface without boundary, with the flux conventions of the geometric electrovacuum data.

If an exterior chart $(t, x)$ with coordinate range $I \times E$ , where $I$ is an open interval and $E = {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ with $R_{0} > 0$ , is additionally specified, write $A = A_{0} d t + \sum_{i = 1}^{3} A_{i} d x^{i}$ . For every nonnegative integer $m$ and every real $b$ , each component is $O_{m} (r^{- b})$ in the Cartesian spatial derivatives, uniformly for $t \in I$ : for all $∣ β ∣ \leq m$ , $∣ \partial_{x}^{β} A_{μ} (t, x) ∣ \leq 0 r^{- b - ∣ β ∣}$ . For every $0 < α < 1$ , its weighted $C^{m, α}$ bound of decay order $b$ also has bound zero, uniformly in $t$ . These are the differentiated and weighted estimates of the asymptotic-decay convention; no existence of an end or such a chart is asserted.

Thus the vacuum datum supplies the vacuum specialization of the electrovacuum framework without adding a physical matter field, and the zero potential satisfies every decay order in the specified differentiated and weighted classes whenever the additional chart is supplied.

Facts & Assumptions

[F1]

The geometric electrovacuum equations, Hodge map and electric and magnetic flux normalizations are explicitly defined. Geometric electrovacuum equations and charge convention

[F2]

The metric, Ricci convention, chosen time orientation and stationary Killing condition have the fixed smooth Lorentzian meanings. Stationary Lorentzian data and tensor conventions

[F3]

The differentiated $O_{m}$ and weighted $C^{m, α}$ bounds use a specified Cartesian exterior chart, spatial multi-indices, annular Hölder seminorms and common constants for a parameter family. Differentiated asymptotic bounds and parameter uniformity

[F4]

Exterior differentiation is defined by the invariant vector-field formula: its terms are derivatives of evaluations of the form and evaluations of the form on Lie brackets. The exterior derivative by the invariant vector-field formula

[F5]

For a smooth map $h$ and a $k$ -form $ω$ , $(h^{*} ω)_{p} (v_{1}, \dots, v_{k}) = ω_{h (p)} (d h_{p} v_{1}, \dots, d h_{p} v_{k})$ . The pullback of a differential form

Proof

Given: The data and assumptions in the Statement.

Goal: Establish the zero-field embedding, invariance, vanishing fluxes and conditional decay estimates.

1.1givenF1F4algebra

For a zero form, every evaluation in the invariant exterior-derivative formula is the zero scalar function, its derivative by any vector field is zero, and every bracket-evaluation term is zero. Hence $d A = 0 = F$ and $d F = 0$ . The Hodge map in [F1] is fiberwise linear, so $⋆_{g} F = 0$ and the same derivative calculation gives $d (⋆_{g} F) = 0$ .

2.1givenF1F2step 1.1algebra

All components of $F$ and all their raised components vanish. Both quadratic terms in the electrovacuum Ricci equation are therefore zero, and the hypothesis $Ric (g) = 0$ gives that equation with the metric, orientation and Ricci convention inherited unchanged. Together with step 1.1 this proves that $(M, g, N, o, 0)$ is a geometric electrovacuum datum with the global potential $A = 0$ .

2.2F1F2F5step 1.1algebra

The pullback formula in [F5] evaluates a zero form to zero for any smooth map. In particular, every isometry preserves $F = 0$ , and every local flow map $Φ_{s}$ of $T$ has $Φ_{s}^{*} F = 0$ . Defining $L_{T} F = \frac{d}{d s} ∣_{s = 0} Φ_{s}^{*} F$ gives $L_{T} F = 0$ , with no completeness requirement on the flow. For the inclusion of any surface in the Statement, the pullbacks of $F$ and $⋆_{g} F$ are zero. Their flux integrals in [F1] are integrals of the zero top form and hence vanish.

3.1givenF3step 2.1step 2.2algebra∎

In any additional chart specified in the Statement all four coefficient functions $A_{μ}$ and every spatial derivative of them are identically zero. The bounds of [F3] therefore hold for every $m$ and $b$ with $C_{β} = 0$ and any common $R_{1} > R_{0}$ , independent of $t$ . Every derivative Hölder seminorm is zero, since its numerator is a difference of zero values, so the weighted annular sums are also zero for every $0 < α < 1$ . This proves all the asserted conditional decay orders. Along with steps 2.1 and 2.2, it establishes the zero-field specialization; it makes no existence or classification assertion.

Depends on

Used by · 1

Axisymmetric vacuum uniqueness with complete global hypotheses

Dependency tree · 2 levels

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