Lowest order covariant averaging of a perturbed metric and of the Einstein tensor II

TL;DR

This paper develops a covariant averaging method for perturbed metrics and Einstein tensors in three dimensions, with a simplified approach that respects covariance and explores extensions to four dimensions.

Contribution

It introduces the most general lowest-order covariant averaging formula in three dimensions and discusses its application and limitations in four dimensions.

Findings

A general averaging formula involving a bitensor with six basis tensors.

The averaging formula preserves the Einstein tensor under certain conditions in 3D.

Extension to 4D static perturbations is possible, but Einstein tensor preservation is not.

Abstract

We generalize and simplify an earlier approach. In three dimensions we present the most general averaging formula in lowest order which respects the requirements of covariance. It involves a bitensor, made up of a basis of six tensors, and contains three arbitrary functions, which are only restricted by their behavior near the origin. The averaging formula can also be applied to the Einstein tensor. If one of the functions is put equal to zero one has the pleasant property that the Einstein tensor of the averaged metric is identical to the averaged Einstein tensor. We also present a simple covariant extension to static perturbations in four dimensions. Unfortunately the result for the Einstein tensor cannot be extended to the four dimensional case.