Prove2Me

IV's Ricci tensor in the KN frame

Proved
ExactVacuumAtlas.EinsteinMaxwell.NP.ricF_KN

by He Wang · Oct 23, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-vexact-vacuum-atlasgeneral-relativitynewman-penrose

IV's Ricci tensor in the KN frame

On domReg, the only potentially nonzero frame-Ricci entries are R(l,n) = R(n,l) = R(m,m̄) = R(m̄,m) = Q²/Σ²; all remaining entries are zero. The formula also covers Q = 0, in which case these four entries vanish as well.

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 KN 标架 Ricci：仅 (l,n)、(n,l)、(m,m̄)、(m̄,m) 四个位置可能非零，其值均为 Q²/Σ²，其余为零；Q=0 时这四项也为零。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Lift
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Certificate
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Defs
import Definitions.Def_Atlas_EinsteinMaxwell_KN_Domain
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_FrameNull
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNComm
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFaraday
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNFrame
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNSpin
import Definitions.Def_Atlas_EinsteinMaxwell_NP_KNWeyl
import Definitions.Def_Atlas_EinsteinMaxwell_NP_NPScalars
import Definitions.Def_Atlas_EinsteinMaxwell_NP_Riemann
import Definitions.Def_Atlas_Kerr_InAtoms
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
open Real Complex
open scoped ComplexConjugate
open ExactVacuumAtlas.EinsteinMaxwell.KN (Sig Del gKN giKN faradayKN domReg)
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
variable {M a Q : ℝ} {x : Pt 4}
local notation "OM" => omL (gKN M a Q) (giKN M a Q) (knE M a Q) (knTh M a Q)
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.NP.ricF_KN (hx : x ∈ domReg M a Q) (A B : Fin 4) :
    ricF (gKN M a Q) (giKN M a Q) (knE M a Q) A B x
      = (!![0, ((Q ^ 2 / Sig a (x 1) (x 2) ^ 2 : ℝ) : ℂ), 0, 0; ((Q ^ 2 / Sig a (x 1) (x 2) ^ 2 : ℝ) : ℂ), 0, 0, 0;
          0, 0, 0, ((Q ^ 2 / Sig a (x 1) (x 2) ^ 2 : ℝ) : ℂ); 0, 0, ((Q ^ 2 / Sig a (x 1) (x 2) ^ 2 : ℝ) : ℂ), 0] :
          Matrix (Fin 4) (Fin 4) ℂ) A B := by sorry
Source
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/KNWeyl.lean#L162-L173; SHA-256 13cac1a0822e84fed504493d4cce2dee2d08d9f7487c29671f1afdcefef95064. 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

  • Endorsed by Shuze Chen · Oct 24, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 24, 2026

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

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