Maxwell divergence depends only on data on an open set
ProvedExactVacuumAtlas.EinsteinMaxwell.ES5.maxwellDiv_congr_of_eqOneinstein-maxwellexact-spacetime-atlas-viexact-vacuum-atlasfive-dimensional-gravitygeneral-relativity
Maxwell divergence depends only on data on an open set
Componentwise EqOn of g, ginv, F on an open U gives equal maxwellDiv at every x ∈ U (statement as printed).
Scope. Exactly the hypotheses printed in the formal statement.
中文摘要 开集上数据相同则 Maxwell 散度相同。
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_D_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_ES5_Defs
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.EinsteinMaxwell ExactVacuumAtlas.EinsteinMaxwell.D
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.ES5
variable {U : Set (Pt 5)} {f : Pt 5 → ℝ} {F : ℝ → ℝ → ℝ} {x : Pt 5}
variable {g g' ginv ginv' : Metric 5} {Fm Fm' : TwoForm 5}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.ES5.maxwellDiv_congr_of_eqOn (hU : IsOpen U) (hg : ∀ i j, Set.EqOn (g i j) (g' i j) U)
(hgi : ∀ i j, Set.EqOn (ginv i j) (ginv' i j) U) (hF : ∀ i j, Set.EqOn (Fm i j) (Fm' i j) U)
(hx : x ∈ U) (b : Fin 5) : maxwellDiv g ginv Fm b x = maxwellDiv g' ginv' Fm' b x := by sorrySource
He Wang, Exact Spacetime Atlas VI: Five-Dimensional Einstein–Maxwell Certificates and Equal-Spin Radial Reduction. Local source, no public repository; commit 4a4048a; Atlas/EinsteinMaxwell/ES5/Tools.lean#L80-L89; SHA-256 8eb33cdee8c5956f9a82565c80aa58b049e59e7e5ed43a4f1176098b09966a0a. Registered definitions: mission/PHYSICS_BOUNDARY.md (§N–§O); literature: R. C. Myers, M. J. Perry, Ann. Phys. 172 (1986) 304; F. R. Tangherlini, Nuovo Cimento 27 (1963) 636.
Human review
Confirmed by the mission captain (proposal self-audit).
Confirmed by the moderator at approval.