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Curved electromagnetic stress divergence

Statement

Let $U$ be an oriented open subset of a smooth time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , with metric-selected connection $\nabla$ . Let $F \in Ω^{2} (U)$ and $j \in Γ (T U)$ be smooth, and let $μ_{0} > 0$ . Assume

$d F = 0, \nabla_{a} F^{a b} = - μ_{0} j^{b},$

with the sign convention of Sourced Maxwell sign and current conservation. Define the Maxwell Hilbert tensor by the compact metric variation of Compact potential and metric variation of the Maxwell action,

$T_{a b}^{E M} : = \frac{1}{μ_{0}} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}) .$

Then

$\nabla_{a} T_{E M}^{a b} = - F^{b c} j_{c} .$

All fields and derivatives are smooth on $U$ . In length-coordinate SI bookkeeping, assign $[F_{a b}] = W b m^{- 2}$ , $[μ_{0}] = H m^{- 1}$ , $[j^{a}] = A m^{- 2}$ , and $[T_{E M}] = J m^{- 3}$ ; these are formal units and not observed values. No quadratic stress is assigned to singular point self-fields or distributional sources.

Facts & Assumptions

Given: The Lorentzian metric, its metric-compatible torsion-free connection, a smooth real two-form $F$ , a smooth vector current $j$ , $μ_{0} > 0$ , and the two Maxwell equations in the Statement.

[F1]

In the selected sourced convention, $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ and $d F = 0$ ; equivalently the first equation is $d ⋆ F = μ_{0} ι_{j} {v o l}_{g}$ . (Sourced Maxwell sign and current conservation)

[F2]

The Maxwell stress tensor obtained by compact variation at fixed lower $F$ is $T_{a b}^{E M} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ . (Compact potential and metric variation of the Maxwell action)

[F3]

The induced connection on tensor fields satisfies the product rule and commutes with contractions; it is torsion-free and metric-compatible, so $\nabla_{a} g_{b c} = 0$ . (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields)

[A1]

The geometric interface carries the ${A C}_{ω}$ convention through its canonical smooth tangent-bundle supplier. This proof uses no choice after the smooth manifold, metric and fields are given. (Oriented Lorentzian fields, observers and coordinate conventions, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

technique · differentiate the variational Maxwell tensor, contract the homogeneous Maxwell identity with $F$, and cancel the derivative terms

1.1A1F1F3givenalgebra

Write $F^{2} : = F_{c d} F^{c d}$ . Because $\nabla$ is torsion-free, $d F = 0$ is equivalent to $\nabla_{a} F_{d c} + \nabla_{d} F_{c a} + \nabla_{c} F_{a d} = 0$ . Metric compatibility and the product rule in [F3] give the raised-index derivatives used below.

2.1step 1.1F3algebra

Multiply the cyclic identity in step 1.1 by $F^{a c}$ . The first term is $X_{d} : = F^{a c} \nabla_{a} F_{d c}$ . In the third term, swapping the dummy indices $a, c$ and using $F^{c a} = - F^{a c}$ and $F_{a d} = - F_{d a}$ gives another $X_{d}$ . The middle term is $F^{a c} \nabla_{d} F_{c a} = - \frac{1}{2} \nabla_{d} (F_{a c} F^{a c}) = - \frac{1}{2} \nabla_{d} F^{2}$ . Thus $2 X_{d} - \frac{1}{2} \nabla_{d} F^{2} = 0$ , so $F^{a c} \nabla_{a} F_{d c} = \frac{1}{4} \nabla_{d} F^{2}$ .

3.1F1F2F3step 2.1algebra

Raise $d$ in step 2.1 and differentiate the tensor in [F2]. The product rule yields $μ_{0} \nabla_{a} T_{E M}^{a b} = (\nabla_{a} F^{a c}) F^{b}_{c} + F^{a c} \nabla_{a} F^{b}_{c} - \frac{1}{4} \nabla^{b} F^{2} = - μ_{0} j_{c} F^{b c} + \frac{1}{4} \nabla^{b} F^{2} - \frac{1}{4} \nabla^{b} F^{2}$ , where the sourced equation in [F1] gives the first term and step 2.1 gives the second.

4.1step 3.1algebra∎

The last two terms in step 3.1 cancel. Dividing by $μ_{0} > 0$ gives $\nabla_{a} T_{E M}^{a b} = - F^{b c} j_{c}$ , as claimed. The calculation uses no regularity beyond the stated smoothness and no physical or empirical premise.

Remarks

The sign is tied to $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ and to the Hilbert tensor in [F2]. Reversing the source convention reverses the exchange sign.
If the matter identity of Gauge/diffeomorphism matter identity and field exchange also applies, the two divergences cancel in the sum $T^{E M} + T^{m}$ .
The M03 current-form and Maxwell-action carriers are present as drafts and are used only at their inspected statement interfaces; this calculation does not approve their proofs. The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02; no independent acceptance is asserted: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.

Curved electromagnetic stress divergence

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources