Regge calculus from a new angle

TL;DR

This work advances Regge calculus by replacing piecewise flat simplices with ones of constant curvature to incorporate a cosmological constant $\Lambda = 3\kappa$. It develops several angle based formulations, including a constraint-free first order action and dedicated 3d and 4d dihedral angle descriptions, with appropriate gluing constraints enforced by Lagrange multipliers. By leveraging the Schlaefli identity and Gram matrix relations, deficit angles and volumes are expressed as functions of angles, recovering the standard equations of motion on the constraint surface. These angle-centric approaches provide new avenues for quantum gravity models with a cosmological constant, potentially improving path integral measures and spin foam quantizations.

Abstract

In Regge calculus space time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show such a formulation allows to replace the length variables by 3d or 4d dihedral angles as basic variables. Moreover we will introduce a first order formulation, which in contrast to using flat simplices, does not require any constraints. These considerations could be useful for the construction of quantum gravity models with a cosmological constant.