Prove2Me

Pointwise derivative of E from its ingredients

Proved
ExactVacuumAtlas.Variation.hasDerivAt_einsteinMaxwell_of

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

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

Pointwise derivative of E from its ingredients

Given HasDerivAt at 0 of g, ricciOf, ricciScalar and emStress components at y, the ε-derivative of E_ab at y is LE_ab (product rule on the finite expression).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 E 的导数（由组成部分导数推出）。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Defs
import Definitions.Def_Atlas_Transport_MatrixCalc
open Filter Topology
open scoped ContDiff
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas
open ExactVacuumAtlas.Variation
variable {n : ℕ}
variable {g ginv : ℝ → Metric 4} {F : ℝ → TwoForm 4} {h : Metric 4} {f : TwoForm 4} {y : Pt 4}
Formal statement
theorem ExactVacuumAtlas.Variation.hasDerivAt_einsteinMaxwell_of (hg : ∀ i j, HasDerivAt (fun ε => g ε i j y) (h i j y) 0)
    (hRic : ∀ a b, HasDerivAt (fun ε => ricciOf (g ε) (ginv ε) a b y) (ricciVar (g 0) (ginv 0) h a b y) 0)
    (hR : HasDerivAt (fun ε => ricciScalar (g ε) (ginv ε) y) (scalarVar (g 0) (ginv 0) h y) 0)
    (hS : ∀ a b, HasDerivAt (fun ε => emStress (g ε) (ginv ε) (F ε) a b y)
      (emStressVar (g 0) (ginv 0) h (F 0) f a b y) 0) (a b : Fin 4) :
    HasDerivAt (fun ε => einsteinMaxwell (g ε) (ginv ε) (F ε) a b y)
      (linEinstein (g 0) (ginv 0) (F 0) h f a b y) 0 := 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/Bridge.lean#L125-L132; SHA-256 c0401e780ed4e7fdf93b3fceb547d092276cb438ae1a341a5d661c712a8a09c4. 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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