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Field equations and balances of the selected scalar coupling

Physical scope

Conditional equations for one smooth classical charged-scalar functional coupled to the adopted smooth Einstein–Maxwell action. Current and stress balances hold on the scalar matter equation. No quantum interpretation, empirical matter claim, or coupled PDE solution is asserted.

Statement

Use the specified model of Specified classical charged complex-scalar model and the adopted total action of Interior stationary Einstein–Maxwell–matter action. Write $P_{a} : = D_{a} φ$ , $L_{φ} : = g^{a b} \overline{P_{a}} P_{b} + V (∣ φ ∣^{2})$ and

$E_{φ} : = D_{a} P^{a} - V^{'} (∣ φ ∣^{2}) φ, j^{a} : = K_{φ} λ Im (\overline{φ} P^{a}),$

$T_{a b}^{φ} : = K_{φ} Re (\overline{P_{a}} P_{b}) - \frac{K_{φ}}{2} g_{a b} L_{φ} .$

For smooth fields and the compact fixed-sector variations, stationarity of the selected scalar matter action in every scalar direction is equivalent to $E_{φ} = 0$ . Stationarity of the full action in all fields then gives

$\nabla_{a} F^{a b} = - μ_{0} j^{b}, G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{φ}),$

with $d F = 0$ for the connection curvature. On the scalar matter equation,

$\nabla_{a} j^{a} = 0, \nabla_{a} (T^{φ})^{a}_{b} = F_{b a} j^{a} .$

These are conditional field equations and local balance identities for the specified classical branch. They assert neither existence of a smooth solution nor empirical or quantum validity.

Facts & Assumptions

Given: The smooth scalar model and adopted total action, a fixed bundle sector, smooth $g, F, φ$ , constants and potential satisfying Specified classical charged complex-scalar model, compactly supported variations, and the indexed countable-choice convention [A2].

[F1]

Specified classical charged complex-scalar model: the selected matter action is exactly the classical functional $S_{φ}$ on the stated bundle and parameter branch.

[F2]

Classical charged-scalar action, current and stress variation: its scalar variation is $δ_{φ} S_{φ} = (K_{φ} / c) Re \int \overline{δ φ} E_{φ} {v o l}_{g}$ ; its connection and metric variations define the displayed $j^{a}$ and $T_{a b}^{φ}$ ; on shell it gives $\nabla_{a} j^{a} = 0$ and $\nabla_{a} (T^{φ})^{a}_{b} = F_{b a} j^{a}$ .

[F3]

Interior stationary Einstein–Maxwell–matter action: full admissibility is stationarity under independent compact variations of the metric, connection, and selected matter fields.

[F4]

Einstein–Maxwell equations from compact action stationarity: for this action, those independent stationarity conditions are equivalent to the sourced Maxwell equation, the Einstein equation with the Maxwell and selected-matter Hilbert tensors, and the selected matter Euler–Lagrange condition.

[A1]

$K_{φ} > 0$ , $λ$ is in the permitted representation branch, $V$ is real and smooth near all attained $∣ φ ∣^{2}$ , and all fields are smooth. The conclusion concerns only this smooth configuration domain.

[A2]

The compact-integration suppliers use the indexed countable-choice convention The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function. No full Axiom of Choice is used.

Proof

Proof technique: identify the scalar Euler–Lagrange equation and its current/stress variations, then apply the total-action stationarity theorem.

1.1A1A2F1F2

By [F1] and [F2], the scalar first variation vanishes for every compact complex-section variation exactly when $E_{φ} = 0$ . The same supplier gives the connection variation with current $j^{a}$ and metric variation with $T_{a b}^{φ}$ as displayed.

2.1A2F1F3F4step 1.1

Apply [F4] with the selected action and its matter, current, and stress variations from step 1.1. Full stationarity from [F3] therefore yields the sourced Maxwell and Einstein equations in the Statement, together with $E_{φ} = 0$ . The homogeneous identity $d F = 0$ is already part of the fixed-connection curvature interface.

3.1A2F2step 2.1algebra∎

Substitute $E_{φ} = 0$ from step 2.1 into the two off-shell balance identities in [F2]. Their right sides reduce to zero in the current identity and to $F_{b a} j^{a}$ in the stress identity, giving both on-shell balances. This uses no existence or regularity theorem for the coupled field equations.

Field equations and balances of the selected scalar coupling

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources