[Paper Review] A look at area Regge calculus

Yasha Neiman|arXiv (Cornell University)|Aug 5, 2013

Noncommutative and Quantum Gravity Theories32 references3 citations

TL;DR

This paper investigates area Regge calculus, a discretized theory of gravity using triangle areas as dynamical variables instead of edge lengths. It shows that in the Euclidean and Lorentzian sectors with spacelike triangles and uniformly signed tetrahedra, non-trivial solutions inevitably yield a non-zero Ricci scalar, contradicting general relativity’s vacuum field equations, thus ruling out area Regge calculus as a viable discretization of GR in these sectors.

ABSTRACT

Area Regge calculus is a candidate theory of simplicial gravity, based on the Regge action with triangle areas as the dynamical variables. It is characterized by metric discontinuities and vanishing deficit angles. Area Regge calculus arises in the large-spin limit of the Barrett-Crane spinfoam model, but not in the newer EPRL/FK model. We address the viability of area Regge calculus as a discretization of General Relativity. We argue that when all triangles are spacelike and all tetrahedra have the same signature, non-trivial solutions of the area calculus are associated with a nonzero Ricci scalar. Our argument rests on a seemingly natural regularization of the metric discontinuities. It rules out the Euclidean area calculus, as well as the Lorentzian sector with all tetrahedra spacelike - the two setups usually considered in spinfoam models. On the other hand, we argue that the area calculus has attractive properties from the point of view of finite-region observables in quantum gravity.

Motivation & Objective

• To assess the viability of area Regge calculus as a discretization of classical General Relativity.
• To investigate whether area Regge calculus can reproduce vacuum GR solutions with zero Ricci scalar.
• To examine the implications of metric discontinuities and vanishing deficit angles in the area calculus framework.
• To evaluate the theory’s potential for finite-region observables in quantum gravity.

Proposed method

• The paper analyzes area Regge calculus using a regularization scheme for metric discontinuities, modeling them as distributions with finite jumps.
• It computes the Ricci scalar in the vicinity of triangles using a perturbative expansion around a discontinuous metric, focusing on the local curvature contribution.
• The analysis relies on the connection and holonomy transport in the normal plane of a spacelike triangle, showing that curvature arises solely from local metric discontinuities.
• It evaluates the Ricci scalar using the expression $ R = -\frac{1}{2} \sigma_{ij} \sigma^{ij} \cdot \text{sign}(r^2) $, where $ \sigma_{ij} $ quantifies the discontinuity.
• The theory is tested in both Euclidean and Lorentzian signatures, with special attention to spacelike triangles and tetrahedra of uniform signature.
• The paper compares results to standard Regge calculus and discusses implications for spinfoam models like Barrett-Crane and EPRL/FK.

Experimental results

Research questions

• RQ1 Can area Regge calculus reproduce vacuum General Relativity solutions with zero Ricci scalar in the Euclidean sector?
• RQ2 Does the absence of deficit angles in area Regge calculus lead to non-vanishing curvature in non-trivial configurations?
• RQ3 Is the non-zero Ricci scalar in area Regge calculus an artifact of regularization, or a fundamental inconsistency with GR?
• RQ4 Can area Regge calculus be consistently formulated in a Lorentzian spacetime with all spacelike triangles and uniformly signed tetrahedra?
• RQ5 Does the theory’s failure in standard spinfoam sectors imply the incompatibility of the Barrett-Crane model with GR?

Key findings

• In the Euclidean sector, all non-trivial solutions of area Regge calculus yield a positive, non-zero Ricci scalar, violating vacuum GR field equations.
• In the Lorentzian sector with spacelike triangles and uniformly signed tetrahedra, the Ricci scalar also remains non-zero and of uniform sign, contradicting GR.
• The non-vanishing Ricci scalar arises from metric discontinuities and is fully accounted for by the local curvature contribution, not holonomy effects.
• The result is robust under a seemingly natural regularization of metric discontinuities, suggesting it is not an artifact of the method.
• The theory cannot be reconciled with GR with a cosmological constant, as the curvature scale matches the discretization scale.
• The paper concludes that area Regge calculus is not a viable discretization of GR in the sectors relevant to spinfoam models, effectively ruling out the Barrett-Crane model.