Prove2Me

E = 0 with F = 0 gives Ricci = 0

Proved
ExactVacuumAtlas.EinsteinMaxwell.ricciOf_eq_zero_of_einsteinMaxwell

by He Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity

E = 0 with F = 0 gives Ricci = 0

At a point with symmetric g, true inverse, F = 0 and all E_ab = 0: all ricciOf components vanish (trace argument in 4D).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 F=0 且 E=0 推出 Ricci=0（4 维取迹）。

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
open scoped ContDiff
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.ricciOf_eq_zero_of_einsteinMaxwell {g ginv : Metric 4} {F : TwoForm 4} {x : Pt 4}
    (hs : ∀ i j, g i j x = g j i x) (hinv : metricMatrix ginv x * metricMatrix g x = 1)
    (hF : ∀ i j, F i j x = 0) (hE : ∀ i j, einsteinMaxwell g ginv F i j x = 0) (i j : Fin 4) :
    ricciOf g ginv i j x = 0 := by sorry
Source
He Wang, Exact Spacetime Atlas IV: Certified Einstein–Maxwell Backgrounds and First Variations. Local source, no public repository; commit 543904bfcfa3bb169f43fc4d9dbd569999b71886; Atlas/EinsteinMaxwell/Certificate.lean#L98-L119; SHA-256 96047b8402974e172119af74b269b2642fe03c56261d6475022bc193053e6c50. 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

  • Endorsed by Shuze Chen · Oct 21, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 21, 2026

    Confirmed by the mission captain (proposal self-audit).

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