LemmaStatement: AI-generatedProof: AI-generated

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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.

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Electrovac contraction in the normalized normal plane

Statement

Under the geometric electrovac equations of Geometric electrovacuum equations and charge convention and the oriented surface gauge of Surface measure, flux densities and null normal gauge, one has ${Scal}_{g} = 0$ , $G = Ric$ and $Ric (L, \underline{L}) = e^{2} + b^{2}$ . Here $F ∣_{T S} = b d μ_{q}$ and $(⋆ F) ∣_{T S} = e d μ_{q}$ are the actual oriented flux densities. This identity needs no vanishing assumption on the tangential electromagnetic components.

Facts & Assumptions

[F1]

The geometric electrovac equations use ${Ric}_{a b} = 2 (F_{a c} F_{b}^{c} - g_{a b} F_{c d} F^{c d} / 4)$ , and the Hodge and surface orientations are fixed. Geometric electrovacuum equations and charge convention Surface measure, flux densities and null normal gauge

[F2]

Scalar curvature and the Einstein tensor mean ${Scal}_{g} = g^{a b} {Ric}_{a b}$ and $G_{a b} = {Ric}_{a b} - \frac{1}{2} {Scal}_{g} g_{a b}$ , with the same curvature and Ricci contraction convention as in [F1]. The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Proof

Given: The data and explicit hypotheses of the Statement.

Goal: Establish its identities with the indicated normalizations.

1.1givenF1F2algebra

At a point choose an orthonormal frame $(n, ν, E_{1}, E_{2})$ with $(E_{1}, E_{2})$ positively oriented on $S$ . Put $a = F (n, ν)$ , $x_{A} = F (n, E_{A})$ , $y_{A} = F (ν, E_{A})$ and $c = F (E_{1}, E_{2})$ . Then $F_{a b} F^{a b} = 2 (- a^{2} - ∣ x ∣^{2} + ∣ y ∣^{2} + c^{2})$ . Direct substitution gives $Ric (n, n) = a^{2} + ∣ x ∣^{2} + ∣ y ∣^{2} + c^{2}$ and $Ric (ν, ν) = - a^{2} + ∣ x ∣^{2} + ∣ y ∣^{2} - c^{2}$ . The trace of the defining electrovac equation is ${Scal}_{g} = 2 (F_{a c} F^{a c} - 4 F_{c d} F^{c d} / 4) = 0$ .

2.1step 1.1F1F2algebra∎

Symmetry of Ric and $L = n + ν$ , $\underline{L} = (n - ν) / 2$ yield $Ric (L, \underline{L}) = (Ric (n, n) - Ric (ν, ν)) / 2 = a^{2} + c^{2}$ . By the flux definitions $c = b$ , while the Hodge definition in this orthonormal frame gives $e = \pm a$ , with the sign fixed by the independently chosen spacetime orientation. Therefore $a^{2} + c^{2} = e^{2} + b^{2}$ ; the definition of $G$ and the zero scalar trace also give $G = Ric$ .

Depends on

Used by · 2

Dependency tree · 2 levels

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