Prove2Me

Pull-back of the standard metric under toStd

Proved
ExactVacuumAtlas.Schwarzschild.Iso.pullback

by He Wang · Oct 15, 2026 · Mathlib 0df444a (Lean v4.33.1)

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real M ≥ 0 and every x with ρ = x 1 > 0 and ρ ≠ M/2: (jacIso M x)ᵀ · gStd M (toStd M x) · jacIso M x = gIso M (x).

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Core_Relation
import Definitions.Def_Atlas_Isotropic_Defs
import Definitions.Def_Atlas_Schwarzschild_Defs
import Definitions.Def_Atlas_Schwarzschild_Domain
open Real Matrix
open scoped ContDiff
open ExactVacuumAtlas ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas
open ExactVacuumAtlas.Schwarzschild
open ExactVacuumAtlas.Schwarzschild.Iso
set_option maxHeartbeats 4000000
variable (M : ℝ)
Formal statement
theorem ExactVacuumAtlas.Schwarzschild.Iso.pullback (hM : 0 ≤ M) {x : Pt 4} (hρ : 0 < x 1) (hne : x 1 ≠ M / 2) :
    (jacIso M x)ᵀ * metricMatrix (gStd M) (toStd M x) * jacIso M x = metricMatrix (gIso M) x := by sorry
Source
He Wang, Exact Vacuum Atlas II, local source snapshot (public Git repository not yet available), commit 11fb317259353b71a41d6daed466c495cffd7db4; Atlas/Isotropic/Relation.lean#L236-L254; SHA-256 7da94260c91052d64d731cd40ce965798ffa54f3a09bf2d8d615e3c5c657d5ad. Mathematical background: See mission/PHYSICS_BOUNDARY.md §A-F for registered definitions and literature references (Wald, Müller-Grave arXiv:0904.4184, Visser arXiv:0706.0622, Martel-Poisson gr-qc/0001069).
Human review

  • Endorsed by Shuze Chen · Oct 18, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 18, 2026

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

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