Pointwise derivative of the Ricci expression
ProvedExactVacuumAtlas.Variation.hasDerivAt_ricci_ofeinstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity
Pointwise derivative of the Ricci expression
Given ε-derivatives of Γ and of ∂Γ at y, the derivative of the Ricci expression is ricciLin Γ C.
Scope. Exactly the hypotheses printed in the formal statement.
中文摘要 Ricci 表达式的导数。
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_ricci_of {Γ : ℝ → Fin n → Fin n → Fin n → Pt n → ℝ}
{C : Fin n → Fin n → Fin n → Pt n → ℝ}
(hΓ : ∀ a b c, HasDerivAt (fun ε => Γ ε a b c y) (C a b c y) 0)
(hpΓ : ∀ i a b c, HasDerivAt (fun ε => pd i (Γ ε a b c) y) (pd i (C a b c) y) 0) (b d : Fin n) :
HasDerivAt (fun ε => ricci (Γ ε) b d y) (ricciLin (Γ 0) C b d 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#L63-L70; 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.