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Constant-radius charged product model

Physical scope

Conditional source-free SI model on either specified open Bertotti–Robinson product chart, with constant sphere radius L>0 and the specified smooth nonzero two-form. No maximal extension, asymptotic-flatness, source at a boundary, observation, or empirical realization is asserted.

Statement

Let $L > 0$ be a length. Choose either the product chart with dimensionless coordinates $τ, z$ , $z \in J$ for a nonempty open $J \subset (0, \infty)$ and $f (z) = z^{2}$ , or the chart with $z \in J$ for a nonempty open $J \subset R$ and $f (z) = 1 + z^{2}$ . Take the metric and geometric field of Constant-radius Bertotti-Robinson product data: $g_{L, f} = L^{2} (- f (z) d τ^{2} + f (z)^{- 1} d z^{2} + γ_{S^{2}}), F_{L} = - L d τ \land d z .$ For $G_{N}, c, μ_{0} > 0$ , set $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ and $F_{L} = α^{- 1} F_{L}$ . Under the adopted postulate Equation-based Einstein–Maxwell model with $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ , the SI field $F_{L}$ and metric $g_{L, f}$ satisfy $d F_{L} = 0, \nabla_{a}^{g} F_{L}^{a b} = 0, G_{a b} = κ_{E} T_{a b}^{E M} .$ The sphere factor has radius $L$ at each base point. This is a deduction for the stated open charts only.

Facts & Assumptions

Given: $L > 0$ , one of the two open product domains and functions $f$ , positive SI constants, the geometric product data, and the zero-source branch of the adopted postulate.

[F1]

Constant-radius Bertotti-Robinson product data: $g = L^{2} (- f d τ^{2} + f^{- 1} d z^{2} + γ_{S^{2}})$ and $F_{L} = - L d τ \land d z$ on the two stated open chart types.

[F2]

Product electrovacuum equations from full factor Ricci: for each such datum, $d F_{L} = d (⋆_{g} F_{L}) = 0$ and $Ric = 2 Q (F_{L})$ ; the full factor Ricci tensors are computed in its current proof. This mathematical theorem is draft with a complete proof and Step 1 authoring-ready record; its result is used conditionally, not as independent approval.

[F3]

Geometric electrovac data solve the selected SI equations: a source-free geometric electrovac datum scales by $F = α^{- 1} F$ to solve the selected SI equation 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: $α^{2} = 4 π G_{N} / (μ_{0} c^{4})$ and the geometric field has units $m^{- 1}$ while the SI two-form has units T in length coordinates.

[F5]

Equation-based Einstein–Maxwell model: the equation-based law is an adopted physical assumption; this item specializes it to $Λ = 0$ , $j = 0$ , and $T^{m} = 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric background convention is ${A C}_{ω}$ ; this pointwise product calculation makes no choice.

Proof

Given: The exact domains, field, metric, and current supplier statements in [F1]–[F5] and [A1].

1.1givenF1F2

For either allowed $f$ and domain, [F1] defines smooth product data with $L > 0$ . The current complete conditional verification [F2] proves the geometric Maxwell equations and $Ric = 2 Q (F_{L})$ there; the proof treats the two product factors and all mixed Ricci components.

2.1givenF3F4F5A1step 1.1∎

With constant $α > 0$ and units as in [F4], apply [F3] to the geometric electrovac datum from step 1.1. It gives the SI field $F_{L} = α^{- 1} F_{L}$ , the zero-divergence Maxwell equation, and $G_{a b} = κ_{E} T_{a b}^{E M}$ under the specialization of [F5]. No field equation or extension is asserted at a chart boundary, and this pointwise conversion adds no choice beyond [A1].

Constant-radius charged product model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources