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Adopted parallel-field initial state

Physical scope

An exact smooth-data example in the zero-matter-action Einstein-Maxwell model on a flat three-torus. It supplies compatible expanding initial data with nonzero parallel electric and magnetic fields and invokes only the conditional local-development theorem; it asserts no isotropy at later times or empirical observation.

Example

Let $L > 0$ be the physical side length of $T_{L}^{3} = (R / L Z)^{3}$ , with length coordinates $(X, Y, Z)$ and flat metric $h = d X^{2} + d Y^{2} + d Z^{2}$ . Give $T_{L}^{3}$ the orientation $d X \land d Y \land d Z$ . Fix the normalized circle-flux scale $κ_{0} > 0$ from Einstein–Maxwell primitive quantities and SI conventions, and set $B_{0} : = \frac{2 π κ_{0}}{L^{2}}, H : = \sqrt{\frac{κ}{3 μ_{0}}} B_{0} .$ Take $Λ = 0$ and define $K = H h$ , $e = B_{0} \partial_{Z}$ , and $b = B_{0} \partial_{Z}$ . The electric and magnetic observer fields are nonzero and parallel in this initial state.

The magnetic two-form is $β_{b} = B_{0} d X \land d Y$ . It is compatible with a nontrivial normalized circle bundle, not just a globally defined potential: on the universal cover $R^{3} \times U (1)$ , with $z \in U (1)$ , use the connection one-form $ω = - i z^{- 1} d z + \frac{2 π}{L^{2}} X d Y .$ Let the deck transformations be $T_{X} (X, Y, Z, z) = (X + L, Y, Z, e^{- 2 π i Y / L} z), T_{Y} (X, Y, Z, z) = (X, Y + L, Z, z), T_{Z} (X, Y, Z, z) = (X, Y, Z + L, z) .$ They commute because $e^{- 2 π i (Y + L) / L} = e^{- 2 π i Y / L}$ . The one-form $ω$ is invariant under these identifications, so it descends to a connection on the quotient bundle over $T_{L}^{3}$ . Its curvature is $d ω = \frac{2 π}{L^{2}} d X \land d Y,$ hence the physical curvature $κ_{0} d ω$ is exactly $β_{b}$ .

The diagram shows one fundamental square in the $X, Y$ directions. The opposite vertical edges are glued with the displayed $U (1)$ transition; the horizontal edges are glued without a fiber twist. It records the quotient convention but is not used in the verification.

Verification

Given: $L, κ_{0}, κ, μ_{0} > 0$ , $Λ = 0$ , the flat torus and fields in the Example, and the choice conventions required by Local evolution of the adopted electrovacuum model.

[F1] Constrained electrovacuum initial data: the four data constraints are $D_{i} e^{i} = D_{i} b^{i} = 0$ , $R (h) + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2} = 2 Λ + \frac{κ}{μ_{0}} (∣ e ∣_{h}^{2} + ∣ b ∣_{h}^{2})$ , and $D^{j} K_{j i} - D_{i} {tr}_{h} K = - \frac{κ}{μ_{0}} (e \times_{h} b)_{i}$ .

[F2] Circle gluing, curvature and connection variations: compatible normalized circle connection data glue to a fixed bundle sector whose physical curvature is $κ_{0}$ times the normalized curvature.

[F3] Local evolution of the adopted electrovacuum model: smooth constrained data whose magnetic two-form is the curvature in a supplied fixed normalized circle sector have a conditional local electrovacuum development; the theorem assumes ${A C}_{ω} + D C$ and makes no empirical claim. Its current draft has Step-1 authoring readiness and verification.precheck: pending; this example uses the theorem conditionally.

[A1] By Einstein–Maxwell primitive quantities and SI conventions, $κ_{0}$ has units Wb, the magnetic field has units T, and $K$ has units m $^{- 1}$ in length coordinates. The chosen value $B_{0} = 2 π κ_{0} / L^{2}$ has units T.

1.1givenF2A1algebraconstruct

The three deck transformations in the Example act by lattice translations on the base and their $U (1)$ multipliers commute: $T_{X} T_{Y} = T_{Y} T_{X}$ because the phase changes by $2 π$ over one $Y$ period; $T_{Z}$ commutes with both because it does not enter the phase. Thus they define a free proper deck action. Under $T_{X}$ , the fiber term $- i z^{- 1} d z$ changes by $- \frac{2 π}{L} d Y$ , while the base term changes by $+ \frac{2 π}{L} d Y$ ; under $T_{Y}, T_{Z}$ both terms are unchanged. Hence $ω$ descends. The right $U (1)$ action preserves it and its value on the vertical generator is $1$ , so it is a principal connection. Its curvature and physical flux are $κ_{0} d ω = B_{0} d X \land d Y$ and $\int_{{Z = 0}} β_{b} = B_{0} L^{2} = 2 π κ_{0}$ . This verifies the required normalized bundle sector without a global one-form potential on the torus.

1.2givenF1A1algebra

The fields are smooth and constant in the flat coordinates, so $D_{i} e^{i} = D_{i} b^{i} = 0$ . Also $R (h) = 0$ , ${tr}_{h} K = 3 H$ , $∣ K ∣_{h}^{2} = 3 H^{2}$ , and $e \times_{h} b = 0$ . By the definition of $H$ , $6 H^{2} = 2 κ B_{0}^{2} / μ_{0} = \frac{κ}{μ_{0}} (∣ e ∣_{h}^{2} + ∣ b ∣_{h}^{2})$ . The scalar constraint in [F1] follows with $Λ = 0$ ; the momentum constraint has both sides zero because $K$ is parallel and $e \times_{h} b = 0$ . Thus all four constraints hold.

2.1step 1.1step 1.2F3algebra∎

The bundle calculation in step 1.1 and constraint verification in step 1.2 satisfy every initial-data hypothesis of [F3]. Applying [F3] gives a smooth local development in the zero-matter-action electrovacuum model with the displayed nonzero parallel initial fields. This conclusion is conditional on the stated mathematical and model assumptions and reports no experiment.

Adopted parallel-field initial state

Example

Verification

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