Curvature profiles for quantum gravity

TL;DR

This paper introduces the curvature profile, a new global observable for curved metric spaces in quantum gravity, derived from quantum Ricci curvature, and explores its properties through examples like tetrahedra and spheres.

Contribution

It proposes the curvature profile as a novel, scale-dependent measure for quantum gravity and analyzes its behavior on regular polygons with conical singularities.

Findings

Curvature profile captures averaging properties of local curvature.

Distinct curvature profiles for tetrahedron and sphere confirm scale-dependent features.

Curvature profile reflects quantum geometric structures.

Abstract

Building on the recently introduced notion of quantum Ricci curvature and motivated by considerations in nonperturbative quantum gravity, we advocate a new, global observable for curved metric spaces, the curvature profile. It is obtained by integrating the scale-dependent, quasi-local quantum Ricci curvature, and therefore also depends on a coarse-graining scale. To understand how the distribution of local, Gaussian curvature is reflected in the curvature profile, we compute it on a class of regular polygons with isolated conical singularities. We focus on the case of the tetrahedron, for which we have a good computational control of its geodesics, and compare its curvature profile to that of a smooth sphere. The two are distinct, but qualitatively similar, which confirms that the curvature profile has averaging properties which are interesting from a quantum point of view.