Modelling gravity on a hyper-cubic lattice

TL;DR

This paper introduces a lattice-based dynamical model related to General Relativity, capable of simulating classical and quantum gravity phenomena, with connections to weak-field gravity and the Einstein-Hilbert action.

Contribution

It presents a novel lattice model of gravity that approximates Einstein's theory and explores its classical and quantum implications.

Findings

Model reproduces weak-field gravity in harmonic gauge

Lattice action approximates Einstein-Hilbert action

Mean-field analysis reveals statistical properties

Abstract

We present an elegant and simple dynamical model of symmetric, non-degenerate (n x n) matrices of fixed signature defined on a n-dimensional hyper-cubic lattice with nearest-neighbor interactions. We show how this model is related to General Relativity, and discuss multiple ways in which it can be useful for studying gravity, both classical and quantum. In particular, we show that the dynamics of the model when all matrices are close to the identity corresponds exactly to a finite-difference discretization of weak-field gravity in harmonic gauge. We also show that the action which defines the full dynamics of the model corresponds to the Einstein-Hilbert action to leading order in the lattice spacing, and use this observation to define a lattice analogue of the Ricci scalar and Einstein tensor. Finally, we perform a mean-field analysis of the statistical mechanics of this model.