Geometry Regge Action Second Variation Regge Action Second Variation Eq Canonica

Second variation of the Regge action

The Regge action is a way to do general relativity without a smooth metric. Instead of a continuous field, spacetime is approximated by a triangulation, a network of flat tetrahedra glued along their faces. Curvature lives on the edges, where the angles of the surrounding tetrahedra fail to add up to a full circle. The action assigns a number to each triangulation, and the classical equations of motion come from varying the edge lengths. This is the standard discrete approach to gravity, and its second variation is what the declaration examines.

In the framework's library, a machine-checked collection of formal theorems, the declaration reggeAction_secondVariation_eq_canonicalHessian states a precise relationship. It says that for any triangulation with a flat configuration, meaning one where the curvature is zero, the second derivative of the nonlinear Regge action along any conformal line through that flat point equals a quadratic form built from the canonical Hessian. The canonical Hessian is a specific matrix derived from the incidence structure of the triangulation, the record of which vertices connect to which edges and faces. The theorem asserts that the second variation of the full nonlinear action matches this canonical quadratic form at the flat point.

This is not a free-standing result. The theorem is conditional on a named input, ReggeActionSecondVariationInput, a structure that packages the assumption that the canonical second variation holds. The proof of the theorem is a one-line extraction: it simply unpacks the input and returns it. The heavy lifting, the actual calculation that the second variation of the nonlinear action equals the canonical Hessian, is not carried out in this declaration. It is stated as a requirement, a named input structure that must be supplied by a future, lower-level calculation involving the Cayley-Menger determinant and arccos chain-rule expansions.

What the declaration does establish is the shape of the claim and its logical role. It shows that the desired second-variation statement is a well-formed proposition, and that once the analytic facts are provided, the theorem follows trivially. It also sets up a companion statement about the cubic remainder, the error term beyond the quadratic approximation, which is likewise conditional on a separate input. The declaration does not prove that the second variation actually equals the canonical Hessian, nor does it show that the cubic remainder is bounded. Those remain targets for the lower-level calculation, not achievements of this declaration.

/-- Phase-D second-variation theorem, conditional on the named nonlinear
second-variation input. -/
theorem reggeAction_secondVariation_eq_canonicalHessian
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_second : ReggeActionSecondVariationInput K hK h_flat) :
    CanonicalHessianSecondVariationAtZero K hK :=
  h_second.canonical_secondVariation

/-- Phase-D second-variation theorem, conditional on the named nonlinear
second-variation input. -/
theorem reggeAction_secondVariation_eq_canonicalHessian
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_second : ReggeActionSecondVariationInput K hK h_flat) :
    CanonicalHessianSecondVariationAtZero K hK :=
  h_second.canonical_secondVariation

/-- Phase-D second-variation theorem, conditional on the named nonlinear
second-variation input. -/
theorem reggeAction_secondVariation_eq_canonicalHessian
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_second : ReggeActionSecondVariationInput K hK h_flat) :
    CanonicalHessianSecondVariationAtZero K hK :=
  h_second.canonical_secondVariation

What this page does not claim

The declaration does not prove the second variation equals the canonical Hessian; it only states the theorem conditional on an input. The declaration does not establish the cubic remainder bound for the nonlinear action. The declaration does not provide the explicit form of the canonical Hessian or the lower-level Cayley-Menger calculation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Geometry/ReggeActionSecondVariation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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