Prove2Me

Covariance of the E residual

Proved
ExactVacuumAtlas.Variation.einsteinMaxwell_transform

by He Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity

Covariance of the E residual

Under CoordChange U V Φ J K with pulled-back data as in the transport card, the E residual transforms as a covariant 2-tensor (Jᵀ · E · J); exact index form in the statement.

Scope. Nonlinear (not linearised) transformation law; used by the certificate transport and the gauge argument.

中文摘要 E 残差的坐标协变律。

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_Certificate
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff
open Matrix
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas
variable {n : ℕ}
variable {U V : Set (Pt 4)} {Φ : Pt 4 → Pt 4} {J K : Pt 4 → Matrix (Fin 4) (Fin 4) ℝ}
  {gV giV : Metric 4} {FV : TwoForm 4} {g gi : Metric 4} {F : TwoForm 4}
Formal statement
theorem ExactVacuumAtlas.Variation.einsteinMaxwell_transform (hc : CoordChange U V Φ J K) (hm : MetricData gV giV V)
    (hg : ∀ y ∈ U, metricMatrix g y = (J y)ᵀ * metricMatrix gV (Φ y) * J y)
    (hgi : ∀ y ∈ U, metricMatrix gi y = K y * metricMatrix giV (Φ y) * (K y)ᵀ)
    (hF : ∀ y ∈ U, metricMatrix F y = (J y)ᵀ * metricMatrix FV (Φ y) * J y) {y : Pt 4} (hy : y ∈ U) :
    Matrix.of (fun a b => einsteinMaxwell g gi F a b y) =
      (J y)ᵀ * Matrix.of (fun a b => einsteinMaxwell gV giV FV a b (Φ y)) * J y := by sorry
Source
He Wang, Exact Spacetime Atlas IV: Certified Einstein–Maxwell Backgrounds and First Variations. Local source, no public repository; commit 543904bfcfa3bb169f43fc4d9dbd569999b71886; Atlas/EinsteinMaxwell/Variation/Covariance.lean#L261-L270; SHA-256 9354896334506569e5c4491ef3a59340742907640cf5d1c465ba5a6e2e9ad40b. Registered definitions and literature: mission/PHYSICS_BOUNDARY.md (§H–§J); Müller–Grave arXiv:0904.4184; Newman et al., J. Math. Phys. 6 (1965) 918.
Human review

  • Endorsed by Shuze Chen · Oct 21, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 21, 2026

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

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