Gravity Analysis Regge4 Dexact Action Symbol Phased Deficit Dot Resolved T12 Zer

A zero-momentum simplification

In numerical relativity, the Regge action is a discrete approximation of Einstein's theory of gravity, built from the angles and areas of a simplicial mesh. A key quantity is its Hessian, the second derivative of the action, which controls how small perturbations behave. Computing this Hessian exactly, especially for wave-like disturbances, is a difficult bookkeeping problem involving many terms. The declaration phasedDeficitDotResolvedT12_zeroMomentum is a machine-checked theorem that simplifies one piece of this Hessian for a special class of perturbations.

The theorem concerns a specific type of wave-like perturbation, called a plane-wave strain, which has a definite wavelength and direction. The declaration proves that when such a perturbation has zero net momentum, meaning its average spatial variation is zero, a complicated expression for the Hessian's cross-term simplifies dramatically. Instead of summing over many individual edges and hinges, the expression reduces to a sum over just four fundamental building blocks, each associated with a different way the perturbation can be transported across the simplicial mesh. This is a significant algebraic simplification, turning a potentially unwieldy sum into a compact and tractable form.

This result is a theorem, meaning it is formally proved within the machine-checked library of formal theorems. It is a structural lemma about the Hessian's behavior under a specific condition, not a claim about the full physical content of the theory. The theorem does not, for instance, establish that this simplified expression is the correct continuum limit of the Regge action, nor does it show that the full Hessian has any particular physical property. It is a precise algebraic identity that holds under the stated assumptions, providing a cleaner foundation for further analysis.

The practical value of this theorem is that it makes a difficult calculation more manageable. By proving that the zero-momentum case collapses to a sum over four terms, it provides a clear target for numerical checks and a stepping stone toward understanding the full Hessian. It is a piece of the larger puzzle of connecting the discrete Regge action to the smooth equations of general relativity, a connection that remains an open area of research.

theorem phasedDeficitDotResolvedT12_zeroMomentum (H : Mat4) (x : Wave4)
    (p : Fin 24) :
    phasedDeficitDotResolvedT12 H (fun _ => (0 : ℝ)) x p =
      ∑ μ : Fin 4,
        classDot
          (pushforwardClass
            (ReggeHinge4DStarKernel12.assembleStarMember μ) p) H := by
  unfold phasedDeficitDotResolvedT12
  refine Finset.sum_congr rfl fun μ _ => ?_
  simpa using
    phasedClassDot_zeroMomentum
      (pushforwardClass (ReggeHinge4DStarKernel12.assembleStarMember μ) p) H
      (fun i =>
        x i + transportOffset p (cubeOffsetT12 (starMemberCubeT12 μ)) i)

What this page does not claim

This theorem does not establish the correct continuum limit of the Regge action. It does not claim that the simplified expression holds for perturbations with non-zero momentum. It does not prove any specific physical property of the full Hessian, such as its signature or eigenvalues.

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/Analysis/Regge4DExactActionSymbol.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: