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

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.