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

TL;DR

This paper introduces a covariant averaging method for perturbed metrics and Einstein tensors that is explicit, requires no solving equations, and can be extended to static perturbations in four dimensions, enhancing the analysis of gravitational fields.

Contribution

It provides a lowest-order explicit covariant averaging formula involving an arbitrary smearing function and integrals, applicable to metrics and Einstein tensors without solving equations.

Findings

Covariant averaging formula in three dimensions for metrics and Einstein tensors.

Extension of the averaging method to static perturbations in four dimensions.

No need to solve differential equations for averaging process.

Abstract

We present an explicit averaging formula in lowest order. Besides an arbitrary smearing function it contains two integrals of this function. This is necessary in order to achieve covariance. There is no need to solve any equations. In three dimensions the same averaging formula yields a covariant averaging of the Einstein tensor and thus of the field equations. We also present a simple extension to static perturbations in four dimensions. Various further extensions of the formalism appear possible.