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Extending smooth equation residuals from a dense nonzero-field locus

Statement

Work in ZFC, including the Axiom of Choice. Let $U$ be an open region of a smooth Hausdorff second-countable four-manifold without boundary, equipped with a smooth Lorentzian metric $h$ of signature $(-, +, +, +)$ , an independently specified orientation $o$ , and a smooth $h$ -timelike vector field $N$ selecting a time orientation. Let $T$ be a smooth vector field on $U$ (a section of $T U$ ), and let $H$ be a smooth real two-form there. Suppose $D : = {x \in U : T_{x} \neq 0}$ is dense in $U$ and the source-free geometric electrovacuum equations hold on $D$ . Then they hold on all of $U$ , including the zeros of $T$ . The Hodge star uses $h$ and $o$ , and all raised indices use $h^{- 1}$ . Density is a required hypothesis; smoothness alone does not imply it for an arbitrary vector field. No Killing-field or stationarity condition on $T$ is required.

Facts & Assumptions

[A1]

Use The Axiom of Choice, as in the geometric electrovacuum interface. This includes the countable choice assumed in the smooth Lorentzian and Hodge suppliers; the dense-set argument below makes no sequence choices.

[F1]

Use the signature, index and curvature conventions of Stationary Lorentzian data and tensor conventions only. The lemma does not assume that $(h, T)$ is stationary; the regularity suppliers below apply to the specified Lorentzian metric without a Killing field.

[F2]

The equations and metric-specific Hodge star have the meanings in Geometric electrovacuum equations and charge convention, applied to $(U, h, N, o, H)$ . Write their residuals as

$R_{a b} : = Ric (h)_{a b} - 2 (H_{a c} H_{b}^{c} - \frac{1}{4} h_{a b} H_{c d} H^{c d}), P : = d H, Q : = d (⋆_{h} H) .$

The equations are exactly $R = 0$ , $P = 0$ and $Q = 0$ .

[F3]

On an oriented, time-oriented smooth Lorentzian four-manifold, Lorentzian exterior calculus and Hodge star supplies the smooth Hodge bundle map and the exterior derivative of smooth forms, including its local coordinate formula.

[F4]

The inverse metric, musical isomorphisms and index raising supplies the smooth inverse metric and index raising. The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors defines the smooth curvature and Ricci fields of the metric-selected connection, with the conventions in [F1]. These are mathematical interfaces for arbitrary smooth Lorentzian metrics.

[F5]

Two continuous maps into a Hausdorff space that agree on a dense subset are equal states that continuous maps from a topological space to a Hausdorff space which agree on a dense subset agree everywhere.

Proof

Given: The geometric data and dense-locus equation hypotheses in the Statement.

Goal: Show that all three electrovacuum residuals vanish throughout $U$ .

1.1givenA1F1F2F3F4

The data $(U, h, N, o, H)$ satisfy the dimensional, smoothness, orientation and time-orientation hypotheses of [F2] and [F3]. By [F4], $h^{- 1}$ and $Ric (h)$ are smooth; finite contractions and products with $H$ make $R$ smooth. By [F3], $⋆_{h} H$ , $P = d H$ and $Q = d (⋆_{h} H)$ are smooth forms. Thus all three residuals have continuous real coordinate components in every chart. The choice convention in [A1] covers that inherited by these geometric suppliers.

2.1givenF2F5step 1.1∎

Let $V \subset U$ be any coordinate-chart domain. The set $D \cap V$ is dense in $V$ : every nonempty relatively open subset of $V$ is open in $U$ , because $V$ is open, and therefore meets $D$ . For each coordinate component $r : V \to R$ of $R$ , $P$ or $Q$ , step 1.1 gives continuity, and the equation hypothesis gives $r = 0$ on $D \cap V$ . The constant zero map is continuous and $R$ is Hausdorff, so [F5] implies $r = 0$ on $V$ . Every point lies in such a chart, hence all three residuals vanish on $U$ , proving the asserted equations also at zeros of $T$ . This argument uses the stipulated density, not merely nonvanishing at one point.

Depends on

Used by · 1

Proposition 2.1: Einstein–Maxwell equations and regularity

Dependency tree · 2 levels

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