Prove2Me

Explicit lowered Riemann tensor

Proved
ExactVacuumAtlas.EinsteinMaxwell.NP.riemLow_expl

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

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

Explicit lowered Riemann tensor

R_{abcd} = ½(∂c∂b g{ad} − ∂c∂a g{db} − ∂d∂b g{ac} + ∂d∂a g{cb}) + Σ g^{ef}(Γ{e,da}Γ{f,cb} − Γ{e,ca}Γ{f,db}).

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 下指标 Riemann 的显式公式。

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_Riemann
import Definitions.Def_Atlas_Transport_Connection
import Definitions.Def_Atlas_Transport_MatrixCalc
open scoped ContDiff
open Matrix
open ExactVacuumAtlas
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas.EinsteinMaxwell.NP
variable {n : ℕ}
variable {g gi : Metric n} {U : Set (Pt n)}
Formal statement
theorem ExactVacuumAtlas.EinsteinMaxwell.NP.riemLow_expl (hm : MetricData g gi U) {y : Pt n} (hy : y ∈ U) (a b c d : Fin n) :
    riemLow g gi a b c d y = (hessG g c b a d y - hessG g c a d b y - hessG g d b a c y
      + hessG g d a c b y) / 2 + quadQ g gi a b c d y := 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/Riemann.lean#L219-L235; SHA-256 dfe20b65e1328fa8bd4849f7fe94465fa58439b24537be187d13ea5b0e6928df. 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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