Prove2Me

Pull-back of the Cartesian isotropic metric under toCart

Proved
ExactVacuumAtlas.Schwarzschild.Iso.pullback_cart

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

exact-vacuum-atlasgeneral-relativitylean4mathlib

For every real M and every x with ρ = x 1 > 0: (jacCart x)ᵀ · gIsoCart M (toCart x) · jacCart x = gIso M (x). Only ρ > 0 is needed (θ, φ arbitrary).

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_Isotropic_CartMaps
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
Formal statement
theorem ExactVacuumAtlas.Schwarzschild.Iso.pullback_cart (M : ℝ) {x : Pt 4} (hx : 0 < x 1) :
    (jacCart x)ᵀ * metricMatrix (gIsoCart M) (toCart x) * jacCart 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/CartMaps.lean#L89-L104; SHA-256 19aa01acaa7bc462019d7acb24320f1f0b760ab3f42a9833c3f2e0404b8b12b1. 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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