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Electromagnetic conservation in source-free curved spacetime

Physical scope

A local model deduction for the smooth zero-matter-action Einstein-Maxwell branch on a Lorentzian four-manifold. It asserts covariant conservation of electromagnetic stress on stationary source-free configurations only; it does not assert existence, global energy conservation, or empirical validity.

Statement

Adopt the zero-matter specialization of Interior stationary Einstein–Maxwell–matter action: the matter functional is identically zero on a singleton matter configuration space, so its current and Hilbert tensor vanish. Let $(M, g, N, o)$ be a smooth, time-oriented, oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , and let $F$ be the smooth physical curvature two-form of a connection in a fixed normalized circle-bundle sector, with the local $F = κ_{0} d a_{i}$ convention of Local potentials and smooth circle-bundle connection data. If the adopted compact-variation action is stationary, then its electromagnetic stress tensor

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

obeys

$\nabla^{a} T_{a b}^{E M} = 0.$ This is a local conditional deduction for the stated classical model. It does not assert a global conserved energy, existence of a stationary configuration, or empirical validity.

Facts & Assumptions

Given: The smooth geometry, fixed-sector connection, adopted action, and zero-matter specialization in the Statement.

[F1]

Interior stationary Einstein–Maxwell–matter action: stationarity is adopted for independent compactly supported metric, fixed-sector connection, and matter variations. The selected current and stress are the matter first-variation coefficients.

[F2]

Einstein–Maxwell equations from compact action stationarity: for that action and compact variations, stationarity is equivalent to its Einstein, Maxwell, and matter Euler–Lagrange equations. The connection equation is $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , and $d F = 0$ because $F$ is a connection curvature.

[F3]

Maxwell stress algebra and covariant conservation: for a smooth field satisfying $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} J^{b}$ , $\nabla^{a} T_{a b}^{E M} = - F_{b c} J^{c}$ ; its proof derives the identity by contracting the homogeneous Maxwell equation and applying the product rule. This supplier is a current draft with Step-1 readiness and verification.precheck: pending; its exact identity is used conditionally.

[F4]

Electrovacuum tensor Cauchy system and Einstein–Maxwell primitive quantities and SI conventions: $T^{E M}$ has the displayed SI normalization on the adopted Lorentzian signature.

[F5]

Local potentials and smooth circle-bundle connection data and Circle gluing, curvature and connection variations: the normalized local representatives glue in one fixed circle-bundle sector, and their physical curvature is $F = κ_{0} d a_{i}$ .

[A1]

The matter functional is $S_{m} \equiv 0$ on a singleton configuration space. Hence every matter variation vanishes and the two coefficients $T^{m}, j$ in [F1] are zero. This choice introduces no empirical premise.

Proof

Given: The data and assumptions in the Statement.

1.1A1F1algebra

Since $S_{m}$ is the zero functional, its first variation is zero for every admissible compactly supported metric, connection, or matter variation. Consequently the matter stress and current defined by [F1] satisfy $T_{a b}^{m} = 0$ and $j^{a} = 0$ .

2.1step 1.1F2F3F4F5algebra

Apply [F2] to the stationary configuration. Its connection equation and the value $j = 0$ from step 1.1 give $\nabla_{a} F^{a b} = 0$ ; the homogeneous equation $d F = 0$ also holds because $F$ is the physical curvature of a smooth connection with the normalization in [F5]. These are exactly the source-free Maxwell hypotheses of [F3].

3.1step 2.1F3algebra∎

Apply [F3] with $J = 0$ . Its pointwise identity reduces to $\nabla^{a} T_{a b}^{E M} = 0$ , as claimed. This is covariant local conservation, not a statement about a global energy charge.

Electromagnetic conservation in source-free curved spacetime

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources