Relative entropy as a measure of inhomogeneity in general relativity

TL;DR

The paper introduces relative volume entropy, based on Kullback-Leibler divergence, as a quantitative measure of inhomogeneity between two spacetimes in general relativity, useful for comparing their geometric differences.

Contribution

It proposes a novel inhomogeneity measure in general relativity using relative volume entropy derived from Kullback-Leibler divergence.

Findings

Applicable to gravitational wave spacetimes

Provides a lower bound on bits needed for metric description

Effective for comparing spacetime inhomogeneity

Abstract

We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.