Pointwise derivative of J
ProvedExactVacuumAtlas.Variation.hasDerivAt_maxwellDiv_ofeinstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity
Pointwise derivative of J
Given the ε-derivatives of the Christoffel symbols (chrVar), of F^{ab} (raise2Var) and of ∂F^{ab} at y, the derivative of J^b at y is LJ^b.
Scope. Exactly the hypotheses printed in the formal statement.
中文摘要 J 的导数。
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 n} {F : ℝ → TwoForm n} {h : Metric n} {f : TwoForm n} {y : Pt n}
Formal statement
theorem ExactVacuumAtlas.Variation.hasDerivAt_maxwellDiv_of
(hΓ : ∀ a b c, HasDerivAt (fun ε => christoffel (g ε) (ginv ε) a b c y) (chrVar (g 0) (ginv 0) h a b c y) 0)
(hU : ∀ a b, HasDerivAt (fun ε => raise2 (ginv ε) (F ε) a b y) (raise2Var (ginv 0) h (F 0) f a b y) 0)
(hpU : ∀ a b, HasDerivAt (fun ε => pd a (raise2 (ginv ε) (F ε) a b) y)
(pd a (raise2Var (ginv 0) h (F 0) f a b) y) 0) (b : Fin n) :
HasDerivAt (fun ε => maxwellDiv (g ε) (ginv ε) (F ε) b y) (linMaxwell (g 0) (ginv 0) (F 0) h f b y) 0 := by sorrySource
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#L93-L100; 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
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.