Prove2Me

Reduced Maxwell equations: five components ⇔ two radial equations

Proved
ExactVacuumAtlas.EinsteinMaxwell.ES5.Generic.maxwell_iff

by He Wang · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-viexact-vacuum-atlasfive-dimensional-gravitygeneral-relativity

Reduced Maxwell equations: five components ⇔ two radial equations

Under ESData R B D A H W Φ Ψ, for x ∈ esDomain R: (∀ b, maxwellDiv (esMetric B D A H W) (esInverse B D A H W) (esFaraday Φ Ψ) b x = 0) ⇔ redJ_t = 0 ∧ redJ_σ = 0 at r = x 1.

Scope. All radial data (B, D, A, H, W, Φ, Ψ : ℝ → ℝ) and every R : Set ℝ with ESData R B D A H W Φ Ψ (R open, the seven functions C^∞ on R, B, D, A, H ≠ 0 on R); domain esDomain R = {x | x 1 ∈ R ∧ 0 < x 2 ∧ x 2 < 1}.

中文摘要 ESData 下：5 个 Maxwell 散度分量为零 ⇔ 两个约化 Maxwell 方程 redJ_t、redJ_σ 在 r = x 1 处为零。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_D_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Data
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Reduced
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Generic_Tables
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.D
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
open ExactVacuumAtlas.EinsteinMaxwell.ES5.Generic
set_option maxHeartbeats 8000000
set_option maxRecDepth 100000
set_option linter.unusedVariables false
set_option linter.unusedSimpArgs false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false
variable {R : Set ℝ} {B D A H W Φ Ψ : ℝ → ℝ}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.Generic.maxwell_iff (hE : ESData R B D A H W Φ Ψ) (x : Pt 5) (hx : x ∈ (esDomain R)) :
    (∀ b, maxwellDiv (esMetric B D A H W) (esInverse B D A H W) (esFaraday Φ Ψ) b x = 0) ↔ redJ_t B D A H W Φ Ψ (x 1) = 0 ∧ redJ_σ B D A H W Φ Ψ (x 1) = 0 := by sorry
Source
He Wang, Exact Spacetime Atlas VI: Five-Dimensional Einstein–Maxwell Certificates and Equal-Spin Radial Reduction. Local source, no public repository; commit 4a4048a; Atlas/EinsteinMaxwell/ES5/Generic/Reduced.lean#L512-L521; SHA-256 9b826fa37b3eb742414809da83be125f760708c6b7efb905716682ad3dda2820. Registered definitions: mission/PHYSICS_BOUNDARY.md (§N–§O); literature: R. C. Myers, M. J. Perry, Ann. Phys. 172 (1986) 304; F. R. Tangherlini, Nuovo Cimento 27 (1963) 636.
Human review

  • Endorsed by Shuze Chen · Oct 29, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 29, 2026

    Confirmed by the mission captain (proposal self-audit).

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