Area-angle variables for general relativity

TL;DR

The paper addresses discretizing general relativity using area-type variables by proposing area-angle Regge calculus on triangulated 4-manifolds. It introduces variables $A_t$ and $\phi_e^\tau$ with local constraints, and an action $S[A_t,\phi_e^\tau,\lambda_t^\tau,\mu^\sigma_{ee'}]$ that combines a Regge-like term $\sum_t A_t \epsilon_t(\phi)$ with closure and gluing constraints, recovering the standard Regge action when solved. The main result is that the constrained area-angle data determine the complete geometry of each 4-simplex and its tetrahedra, providing a local, GR-consistent discretization that aligns with Plebanski BF theory and has potential links to spinfoam quantum gravity. This offers a promising route for classical lattice gravity computations and non-perturbative quantum gravity analyses, including possible perturbative studies on flat backgrounds and semiclassical limits of spinfoam models.

Abstract

We introduce a modified Regge calculus for general relativity on a triangulated four dimensional Riemannian manifold where the fundamental variables are areas and a certain class of angles. These variables satisfy constraints which are local in the triangulation. We expect the formulation to have applications to classical discrete gravity and non-perturbative approaches to quantum gravity.

Figures (1)

• Figure 1: The geometric meaning of equation (\ref{['aa']}): the 2d angle $\alpha_{ij,kl}$ belonging to the shaded triangle can be expressed in terms of 3d angles associated the thick edges of the tetrahedron $k$, or equivalenty of the tetrahedron $l$.