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Einstein–Maxwell equations from compact action stationarity

Physical scope

Conditional equivalence between compact interior stationarity of the adopted smooth fixed-sector action and its metric, connection, and selected matter Euler–Lagrange equations. It asserts no existence, uniqueness, regularity improvement, empirical validity, or boundary variational principle.

Statement

Let $(M, g)$ , the fixed smooth connection sector, the constants $c, G_{N}, μ_{0} > 0$ , constant $Λ$ , and the selected matter functional $S_{m}$ be exactly those of Interior stationary Einstein–Maxwell–matter action. Write $F$ for the connection curvature, $j$ and $T^{m}$ for the current and Hilbert tensor in the selected matter variation, and define $G_{a b} : = {Ric}_{a b} - \frac{1}{2} R g_{a b}$ using the Ricci and scalar curvature contractions in the convention of Oriented Lorentzian fields, observers and coordinate conventions. Also set

$T_{a b}^{E M} : = \frac{1}{μ_{0}} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}), κ_{E} : = \frac{8 π G_{N}}{c^{4}} .$

The homogeneous identity $d F = 0$ holds because $F$ is the curvature of the smooth connection in the fixed sector. The adopted configuration is stationary under every admissible compactly supported metric, connection, and matter variation if and only if all of the following hold:

$\nabla_{a} F^{a b} = - μ_{0} j^{b}, G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m}), δ_{Ψ} S_{m} [ζ] = 0 for every admissible compactly supported ζ .$

The last clause is the selected matter model's Euler–Lagrange condition; an explicit local matter equation cannot be given until $S_{m}$ is specified. These are model deductions from the adopted action and smooth compact variation hypotheses, not evidence that a coupled solution exists or that the model describes nature.

Facts & Assumptions

Given: The total action and independent compact variation classes of Interior stationary Einstein–Maxwell–matter action, a fixed smooth bundle sector, and the selected matter action with its current and Hilbert tensor, under the indexed countable-choice convention [A2].

[F1]

Interior stationary Einstein–Maxwell–matter action: stationarity is adopted independently for compact metric, connection, and matter variations; the source and stress terms are the first variations of the selected $S_{m}$ .

[F2]

Compact potential and metric variation of the Maxwell action: for the fixed-sector Maxwell action, $δ_{A} S_{E M} = (μ_{0} c)^{- 1} \int (\nabla_{a} F^{a b}) α_{b} {v o l}_{g}$ and the matter variation is $c^{- 1} \int j^{b} α_{b} {v o l}_{g}$ ; their sum is stationary for all admissible local compact one-form variations exactly when $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ .

[F3]

Compact Einstein–Hilbert variation and SI coupling: for the stated compact interior metric variations, stationarity of the Einstein–Hilbert, Maxwell, and selected matter terms is equivalent to $G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m})$ .

[F4]

Circle gluing, curvature and connection variations: on the fixed principal $U (1)$ bundle, normalized local connection representatives $a_{i}$ give physical local potentials $A_{i} = κ_{0} a_{i}$ and a global physical curvature $F ∣_{U_{i}} = d A_{i} = κ_{0} d a_{i}$ , with $d F = 0$ . Differences of connections in this fixed sector are global base one-forms; for a compact physical-potential variation $α$ , $δ F = d α$ .

[A1]

Variations are compactly supported in the interior, or vanish on a neighbourhood of any physical boundary; the fixed bundle sector is held fixed.

[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: evaluate the independent first variations, then use their pointwise stationarity equivalences.

1.1A1A2F1F2algebra

By [F1], the connection variation of the total action is the sum of the Maxwell variation and the selected matter variation in [F2]. Their displayed integrals combine to $δ_{A} S_{t o t} = (μ_{0} c)^{- 1} \int_{M} (\nabla_{a} F^{a b} + μ_{0} j^{b}) α_{b} {v o l}_{g}$ . The local bump argument and its converse are exactly the conditional equivalence in [F2], so stationarity for every admissible $α$ is equivalent to the sourced Maxwell equation in the Statement.

1.2A1A2F1F3

Apply [F3] to the metric part of [F1]. For every admissible compact symmetric inverse-metric variation, its first variation vanishes exactly when $G_{a b} + Λ g_{a b} = κ_{E} (T_{a b}^{E M} + T_{a b}^{m})$ pointwise. The Maxwell tensor in [F3] is the tensor from the metric variation at fixed lower $F_{a b}$ ; it is not an independently guessed source.

1.3F1given

By the adopted stationarity clause in [F1], the matter-field variations vanish exactly under the condition $δ_{Ψ} S_{m} [ζ] = 0$ for every allowed compactly supported $ζ$ . This records the equations of the chosen functional without inventing a matter equation for an unspecified $S_{m}$ .

2.1A1F1F2F3F4step 1.1step 1.2step 1.3algebra∎

The three variation classes are independent. If the total action is stationary for all of them, steps 1.1, 1.2, and 1.3 give each displayed condition. Conversely, if all three conditions hold, each corresponding first variation vanishes by those steps, and their sum is the first variation of $S_{t o t}$ ; hence it vanishes for every allowed total variation. Finally, [F4] gives $d F = 0$ for every connection curvature in this model. No existence of a stationary configuration is used.

Einstein–Maxwell equations from compact action stationarity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources