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Lorenz potential wave contains Ricci coupling

Statement

Let $(M, g, N, o)$ be the Lorentzian geometry and use the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. Let $A_{a}$ be a smooth one-form, $A^{a} : = g^{a b} A_{b}$ , and $F : = d A$ , so $F_{a b} = \nabla_{a} A_{b} - \nabla_{b} A_{a}$ by torsion-freeness. Define $□_{g} A_{b} : = \nabla^{a} \nabla_{a} A_{b}$ and $div A : = \nabla_{a} A^{a}$ . Then

$\nabla^{a} F_{a b} = □_{g} A_{b} - \nabla_{b} (div A) - {Ric}_{b c} A^{c} .$

Consequently, if $\nabla_{a} F^{a b} = - μ_{0} J^{b}$ , the Lorenz condition $div A = 0$ makes the potential equation

$□_{g} A_{b} - {Ric}_{b c} A^{c} = - μ_{0} J_{b} .$

The curvature term is part of the identity; this is not an uncoupled collection of scalar metric wave equations. The statement is local in the sense that it assumes a potential $A$ is already defined on the open set under consideration; it does not assert a global potential for every closed $F$ .

Facts & Assumptions

Given: The smooth oriented, time-oriented Lorentzian geometry, a smooth one-form $A$ , its exterior derivative $F = d A$ , and, in the last display, the indicated Maxwell source equation.

[F1]

Oriented Lorentzian fields, observers and coordinate conventions: $\nabla$ is the torsion-free metric-compatible connection with $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ and ${Ric}_{b d} = R^{a}_{b a d}$ .

[F2]

Lorentzian exterior calculus and Hodge star: $d A$ has coordinate components $2 \partial_{[a} A_{b]}$ .

[F3]

The commutator of covariant derivatives on a vector field and its contracted component identity: for a smooth vector field $Z$ , $\nabla_{a} \nabla_{b} Z^{a} - \nabla_{b} \nabla_{a} Z^{a} = {Ric}_{b d} Z^{d}$ in the same curvature convention.

[A1]

The source convention in Electrovacuum tensor Cauchy system is $\nabla_{a} F^{a b} = - μ_{0} J^{b}$ ; the source-free case has $J = 0$ .

Proof

1.1F1F2algebra

By [F2], $F_{a b} = 2 \partial_{[a} A_{b]}$ . The Levi-Civita Christoffel symbols are symmetric in their lower indices by [F1], so their contribution cancels under antisymmetrization; hence $F_{a b} = \nabla_{a} A_{b} - \nabla_{b} A_{a}$ . Metric compatibility then gives $F^{a b} = \nabla^{a} A^{b} - \nabla^{b} A^{a}$ .

2.1step 1.1F1F3algebra

Taking the divergence of the raised identity in 1.1 yields $\nabla_{a} F^{a b} = \nabla_{a} \nabla^{a} A^{b} - \nabla_{a} \nabla^{b} A^{a}$ . Apply [F3] with the free index raised by $g^{b c}$ : $\nabla_{a} \nabla^{b} A^{a} = \nabla^{b} \nabla_{a} A^{a} + {Ric}^{b}_{c} A^{c}$ . Substitution gives $\nabla_{a} F^{a b} = □_{g} A^{b} - \nabla^{b} (div A) - {Ric}^{b}_{c} A^{c}$ .

3.1step 2.1A1algebra∎

Lower the free index in 2.1 using $\nabla g = 0$ . This proves the first display. If the source Maxwell equation [A1] holds, then imposing $div A = 0$ removes exactly its gradient term and gives the stated Lorenz-gauge equation with the Ricci coupling retained.

Lorenz potential wave contains Ricci coupling

Statement

Facts & Assumptions

Proof

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