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Exact coupled electromagnetic null-wave model

Physical scope

Exact source-free classical SI Einstein–Maxwell solutions on the specified plane-fronted null-wave chart, for arbitrary smooth electromagnetic profiles and smooth transverse harmonic gravitational profile. No asymptotic flatness, finite total energy, observation, or empirical validity is asserted.

Statement

Use length coordinates $(u, v, x, y)$ on $R^{4}$ . Let $a, b : R \to R$ be smooth profiles with units $m^{- 1}$ , and let $h : R^{3} \to R$ be smooth, dimensionless, and satisfy $h_{x x} + h_{y y} = 0$ . Set $P (u, x, y) = - (a (u)^{2} + b (u)^{2}) (x^{2} + y^{2}) + h (u, x, y),$ $g = - 2 d u d v + d x^{2} + d y^{2} + P d u^{2}, A = - (a (u) x + b (u) y) d u,$ $F = d A = a (u) d u \land d x + b (u) d u \land d y, F = α^{- 1} F,$ where $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ from SI and geometric solution parameter identifications. Use the orientation $d u \land d v \land d x \land d y$ and future unit field $N = \partial_{u} + (P + 1) \partial_{v} / 2$ , as in Smooth null electrovac wave candidate data. The metric has signature $(-, +, +, +)$ . Under the adopted postulate Equation-based Einstein–Maxwell model with no material stress or current and $Λ = 0$ , these data satisfy the source-free SI Einstein–Maxwell equations on all of $R^{4}$ . The claim is an exact model deduction; it asserts neither asymptotic flatness nor finite total wave energy.

Facts & Assumptions

Given: The length-coordinate null-wave data, smoothness and harmonicity hypotheses, positive SI constants, and the source-free specialization of the adopted postulate.

[F1]

Smooth null electrovac wave candidate data: $g = - 2 d u d v + d x^{2} + d y^{2} + P d u^{2}$ , $A = - (a x + b y) d u$ , and $F = a d u \land d x + b d u \land d y$ , with arbitrary smooth profiles and $h_{x x} + h_{y y} = 0$ .

[F2]

Exact electrovac plane-wave family: the defined data satisfy $d F = d (⋆_{g} F) = 0$ and $Ric = 2 Q (F)$ for the full stated profile class. This mathematical theorem is draft with a complete current proof and Step 1 authoring-ready record; it is used conditionally.

[F3]

Geometric electrovac data solve the selected SI equations: constant rescaling by $α^{- 1}$ turns any source-free geometric electrovac datum into a solution of the selected SI model. This local theorem is draft with its complete proof authored and is used conditionally, not as independent proof approval.

[F4]

SI and geometric solution parameter identifications: $α > 0$ is constant, $F = α F$ , and the two-form components have units $m^{- 1}$ geometrically and T in the SI convention.

[F5]

Equation-based Einstein–Maxwell model: the equation-based model is an adopted physical assumption, specialized here to $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometry and Hodge convention carries ${A C}_{ω}$ ; the local field rescaling makes no choice.

Proof

Given: The exact smooth null-wave data and supplier statements in [F1]–[F5] and [A1].

1.1givenF1F2

The data in the Statement are exactly the definition in [F1], with smooth $a, b, h$ and transverse Laplacian zero. The complete conditional verification [F2] therefore gives the geometric source-free Maxwell and Ricci equations on the whole coordinate domain $R^{4}$ .

2.1givenF3F4F5A1step 1.1∎

Apply the constant SI rescaling in [F3] to the datum of step 1.1. It yields $F = α^{- 1} F$ with the units and positive normalization in [F4]. Under the source-free adopted branch [F5], the resulting metric and field satisfy the SI equations at every point of $R^{4}$ ; the pointwise scaling makes no choice beyond [A1].

Exact coupled electromagnetic null-wave model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources