Discrete Gravitational Dimensions

TL;DR

This paper investigates a discrete model of gravitational extra dimensions using effective field theory, revealing non-local effects at non-linear levels and implications for constructing finite discrete gravity theories.

Contribution

It introduces a minimal discretization approach for gravitational extra dimensions and analyzes its non-linear behavior and UV/IR connections.

Findings

Linear theory recovers 5D continuum physics

Non-linear theory exhibits non-locality in the discrete dimension

Model remains valid as an effective theory above the compactification scale

Abstract

We study the physics of a single discrete gravitational extra dimension using the effective field theory for massive gravitons. We first consider a minimal discretization with 4D gravitons on the sites and nearest neighbor hopping terms. At the linear level, 5D continuum physics is recovered correctly, but at the non-linear level the theory becomes highly non-local in the discrete dimension. There is a peculiar UV/IR connection, where the scale of strong interactions at high energies is related to the radius of the dimension. These new effects formally vanish in the limit of zero lattice spacing, but do not do so quickly enough to reproduce the continuum physics consistently in an effective field theory up to the 5D Planck scale. Nevertheless, this model does make sense as an effective theory up to energies parametrically higher than the compactification scale. In order to have a discrete theory that appears local in the continuum limit, the lattice action must have interactions between distant sites. We speculate on the relevance of these observations to the construction of finite discrete theories of gravity in four dimensions.