Commutator of frame derivatives
ProvedExactVacuumAtlas.EinsteinMaxwell.NP.fdir_commeinstein-maxwellexact-spacetime-atlas-vexact-vacuum-atlasgeneral-relativitynewman-penrose
Commutator of frame derivatives
For C^∞ f: [D_A, D_B] f = Σ_C c^C_{AB} D_C f.
Scope. Exactly the hypotheses printed in the formal statement.
中文摘要 标架导数对易子。
Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_NP_CCalc
import Definitions.Def_Atlas_EinsteinMaxwell_NP_FrameConn
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Riemann
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff Matrix
open Complex
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
variable {n : ℕ}
variable {g gi : Metric n} {E Θ : Fin n → Fin n → Pt n → ℂ} {U : Set (Pt n)}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.NP.fdir_comm (hF : FrameData g gi E Θ U) {x : Pt n} (hx : x ∈ U) (A B : Fin n) {f : Pt n → ℂ}
(hf : ContDiffOn ℝ ∞ f U) :
fdir E A (fdir E B f) x - fdir E B (fdir E A f) x = ∑ C, cfr E Θ C A B x * fdir E C f x := by sorrySource
He Wang, Exact Spacetime Atlas V: From Linearized Einstein–Maxwell Equations to Kerr–Newman Radiation Systems. Local source, no public repository; commit b8cd107a558e153ed41fb4972ac5315499d14778; Atlas/EinsteinMaxwell/NP/FrameConn.lean#L213-L231; SHA-256 2a10cd770a38610029f7546d603b6e7d5f8af619e7c052fbab673dadeffc4f32. Registered definitions: mission/PHYSICS_BOUNDARY.md (§K–§M); literature: P. Hintz, arXiv:2609.33661 (v1); Mark–Yang–Zimmerman–Chen, arXiv:1409.5800 (v1).
Human review
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.