Gravity Discrete Vacuum Einstein Zero Deficit Of Critical Of Variation Formula O

The discrete vacuum equation

In the discrete geometry of Regge calculus, spacetime is approximated by a triangulated 3-manifold, a network of tetrahedra glued along triangular faces. The Regge action is a sum over the hinges, the edges of the triangulation, of the deficit angle times the edge length. The deficit angle measures how much the flat space around a hinge fails to close up; zero deficit means the geometry is locally flat there. The vacuum Einstein equation in this setting is the statement that the action is critical, that its first variation vanishes, and this is equivalent to the deficit being zero at every hinge.

The machine-checked theorem zero_deficit_of_critical_of_variationFormula_of_separating proves one direction of this equivalence under precise hypotheses. It assumes a first-variation formula that expresses the derivative of the action as a pairing of the deficit vector with conformal edge-length directions, and it assumes an incidence separation condition: if a deficit vector has zero pairing with every such direction, then the deficit vector itself is zero. Under those two assumptions, the theorem derives that if the action is critical at the flat potential, then the deficit is zero at every edge. The flat potential is the configuration where all edge lengths take their flat-space values, the natural background about which the action is varied.

The theorem is a formal statement in the framework's machine-checked library of formal theorems. It is not a proof of the full vacuum Einstein equivalence, because the reverse direction, from zero deficit to criticality, is recorded as a separate named input rather than derived. The separation condition is a real condition on the triangulation, not a consequence of local tetrahedron nondegeneracy; it is the rank condition that makes the pairing invertible. The theorem also does not claim that any particular triangulation satisfies the separation condition, nor that the first-variation formula holds for all geometries; those are hypotheses of the statement.

What the theorem establishes is a clean logical bridge: given the right variation formula and a separating incidence structure, the criticality of the action forces the local flatness of the geometry. This is the discrete analogue of the continuum fact that the Einstein equations follow from varying the Einstein-Hilbert action. The value of the theorem is that it isolates exactly which additional input, the incidence rank condition, is needed to close the gap between the variation formula and the vacuum equation.

theorem zero_deficit_of_critical_of_variationFormula_of_separating
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hFormula : ReggeFirstVariationFormula K hK)
    (hSep : IncidenceDeficitSeparating K)
    (hCrit : CriticalAtFlat K hK) :
    ZeroDeficitAtFlat K := by
  unfold CriticalAtFlat ReggeActionCriticalAtZero at hCrit
  unfold ZeroDeficitAtFlat
  have hdelta :
      (fun e : Fin K.nE => deficitAngle K (zeroPotential K) e) = 0 := by
    apply hSep
    intro η
    have happly := congrArg (fun L : VertexPotential K →L[ℝ] ℝ => L η) hCrit
    have hzero :
        (fderiv ℝ (reggeAction K hK) (zeroPotential K)) η = 0 := by
      simpa using happly
    have hformula := hFormula.variation_formula η
    rw [hformula] at hzero
    simpa using hzero
  intro e
  exact congrFun hdelta e

/-- First-variation formula before imposing zero deficit: the derivative of
the action pairs the deficit vector with conformal edge-length directions. -/
structure ReggeFirstVariationFormula
    (K : Triangulation3D) (hK : IncidenceConsistent K) where
  variation_formula :
    ∀ η : VertexPotential K,
      fderiv ℝ (reggeAction K hK) (zeroPotential K) η =
        ∑ e : Fin K.nE,
          deficitAngle K (zeroPotential K) e *
            directionalLengthCoefficient K η e

/-- Incidence rank/separation condition: a deficit vector whose pairing with
every conformal edge-length direction vanishes is zero.  This is a real
condition on the triangulation, not a consequence of local tetrahedron
nondegeneracy. -/
def IncidenceDeficitSeparating (K : Triangulation3D) : Prop :=
  ∀ δ : Fin K.nE → ℝ,
    (∀ η : VertexPotential K,
      ∑ e : Fin K.nE, δ e * directionalLengthCoefficient K η e = 0) →
    δ = 0

What this page does not claim

The theorem does not prove the full equivalence between criticality and zero deficit, only the forward direction under hypotheses. It does not assert that any specific triangulation satisfies the incidence separation condition. It does not claim that the first-variation formula holds for all geometries without the unit-edge normalization assumption.

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

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