Averaging Einstein's Equations: The Linearized Case

TL;DR

This paper presents a simple averaging method for linearized Einstein's equations that maintains tensorial properties and preserves FLRW metrics under averaging, with implications for cosmological perturbations.

Contribution

It introduces a straightforward averaging procedure for linearized Einstein equations that generalizes electrodynamics methods and maintains key geometric properties in cosmological models.

Findings

Yields approximately tensorial quantities in weak-field regimes

Preserves FLRW metrics under averaging to first order

Can be approximated by Zotov and Stoeger's method in certain cases

Abstract

We introduce a simple and straight-forward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology -- for weak-field and perturbed FLRW situations. In particular we demonstrate that it yields quantities which are approximately tensorial in these situations, and that its application to an exact FLRW metric yields another FLRW metric, to first-order in integrals over the local coordinates. Finally, we indicate some important limits of any linearized averaging procedure with respect to cosmological perturbations which are the result of averages over large amplitude small and intermediate scale inhomogeneities, and show our averaging procedure can be approximately implemented by that of Zotov and Stoeger in these cases.