Covariance of the J residual
ProvedExactVacuumAtlas.Variation.maxwellDiv_transformeinstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity
Covariance of the J residual
Under CoordChange U V Φ J K with pulled-back data as in the transport card, the J residual transforms as a contravariant vector (acted on by the inverse Jacobian K); 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 J 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 n)} {Φ : Pt n → Pt n} {J K : Pt n → Matrix (Fin n) (Fin n) ℝ}
{gV giV : Metric n} {FV : TwoForm n} {g gi : Metric n} {F : TwoForm n}
Formal statement
theorem ExactVacuumAtlas.Variation.maxwellDiv_transform (hc : CoordChange U V Φ J K) (hm : MetricData gV giV V)
(hFs : ∀ i j, ContDiffOn ℝ ∞ (FV i j) 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 n} (hy : y ∈ U)
(b : Fin n) :
maxwellDiv g gi F b y = ∑ q, K y b q * maxwellDiv gV giV FV q (Φ y) := 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/Covariance.lean#L184-L221; 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
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.