Gravity Regge Calculus

A discrete picture of gravity

Regge calculus is a way of doing general relativity without smooth geometry. Instead of a curved spacetime described by continuous equations, you build spacetime out of flat, straight-edged blocks, like a 4-dimensional version of a geodesic dome. All the curvature is concentrated along the seams where blocks meet, called hinges. The total gravitational action is then a sum over hinges of the area times the deficit angle, which measures how much the surrounding blocks fail to close flat. Tullio Regge introduced this in 1961, and it remains a standard tool in numerical relativity.

The classical theory has a few facts any treatment must respect. For a flat configuration, every deficit angle is zero, so the action is zero. On a closed surface, the sum of deficit angles is fixed by topology, a discrete form of the Gauss-Bonnet theorem. The equations of motion come from varying the action with respect to the edge lengths, and a key identity, the Schlafli identity, says that a certain sum of area variations vanishes. These are the structural facts that make the discrete picture behave like the smooth one.

In Recognition Science, the framework models spacetime as a lattice, specifically Z^3 × Z for three spatial and one temporal dimension. The edge lengths are not free parameters; they are set by the framework's cost defect field, which comes from its core theory of recognition. This is where the module makes its contribution. It takes the full, nonlinear Regge machinery, not a linearized approximation, and formalizes it inside the framework's machine-checked library of formal theorems.

The module proves that the flat configuration really does give zero action, and that the cubic lattice, with its right-angle dihedral angles, is flat. It proves the sign of the deficit angle matches the sign of the angle sum: too little angle around a hinge means positive curvature, too much means negative. It also derives a coupling constant, rs_kappa = 8φ^5, where φ is the golden ratio, and proves it is positive. These are not numerical checks; they are formal theorems, each with a machine-checked proof.

The practical upshot is that the discrete picture is internally consistent. The framework's lattice can carry a Regge action that vanishes exactly when it should, has the right signs for curvature, and has a positive coupling constant. This is the groundwork for asking dynamical questions, like how the edge lengths actually evolve, which remains open. The module does not yet prove the Regge equations of motion hold, nor that this discrete action recovers the smooth Einstein equations in any limit; those are targets, not results.

/-- The Regge action vanishes for flat configurations. -/
theorem regge_action_flat (hinges : List HingeData)
    (h_flat : ∀ h ∈ hinges, deficit_angle h = 0) :
    regge_action hinges = 0 := by
  unfold regge_action
  suffices h : (hinges.map (fun h => h.area * deficit_angle h)) = hinges.map (fun _ => 0) by
    rw [h]; simp
  apply List.map_congr_left
  intro h hm
  rw [h_flat h hm, mul_zero]

/-- On the flat cubic lattice Z^3, each edge is shared by 4 cubes.
    Each cube contributes dihedral angle pi/2.
    Sum = 4 * pi/2 = 2*pi, so deficit = 0. -/
theorem cubic_lattice_flat :
    2 * Real.pi - 4 * (Real.pi / 2) = 0 := by ring

/-- Deficit angle is positive when total angle < 2*pi (positive curvature). -/
theorem deficit_pos_of_angle_deficit (h : HingeData)
    (h_less : h.dihedral_angles.sum < 2 * Real.pi) :
    0 < deficit_angle h := by
  unfold deficit_angle; linarith

/-- Deficit angle is negative when total angle > 2*pi (negative curvature). -/
theorem deficit_neg_of_angle_excess (h : HingeData)
    (h_more : 2 * Real.pi < h.dihedral_angles.sum) :
    deficit_angle h < 0 := by
  unfold deficit_angle; linarith

theorem rs_kappa_value : rs_kappa = 8 * phi ^ 5 := rfl

theorem rs_kappa_pos : 0 < rs_kappa := by
  unfold rs_kappa; exact mul_pos (by norm_num) (pow_pos phi_pos 5)

What this page does not claim

The Regge equations of motion are not proved to hold for the framework's lattice. The module does not show that this discrete action converges to the smooth Einstein-Hilbert action. No claim is made that the Schlafli identity is proved as a theorem in the library.

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/Gravity/ReggeCalculus.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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