Discrete Gravity

TL;DR

This paper proposes a model of discrete space at the Planck scale, defining geometric properties like torsion and curvature, which recover classical continuous geometry in the limit of vanishing elementary volume.

Contribution

It introduces a discrete space framework with finite degrees of freedom per cell, characterizing geometry via displacement operators and spin connection, bridging to classical geometry.

Findings

Discrete space characterized by displacement operators and spin connection.

Torsion and curvature defined for discrete spaces.

Classical continuous geometry recovered in the limit of zero elementary volume.

Abstract

We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of freedom per elementary volume. In such discrete spaces, each elementary cell is completely characterized by displacement operators connecting a cell to the neighboring cells and by the spin connection. We define the torsion and curvature of the discrete spaces and show that in the limiting case of vanishing elementary volume the standard results for the continuous curved differentiable manifolds are completely reproduced.