Practical definition of averages of tensors in general relativity

TL;DR

This paper introduces a practical and natural method for averaging tensor fields on manifolds, including a derivative operation, with applications to general relativity and minimal geometric assumptions.

Contribution

It provides a new, straightforward definition of tensor averaging and derivatives applicable to general manifolds, facilitating the study of tensor differential equations.

Findings

Defines tensor averages using scalar integrations from a physical basis

Introduces covariant and Lie derivatives for averaged tensors

Applicable to general n-dimensional manifolds

Abstract

We present a definition of tensor fields which are average of tensors over a manifold, with a straightforward and natural definition of derivative for the averaged fields; which in turn makes a suitable and practical construction for the study of averages of tensor fields that satisfy differential equations. Although we have in mind applications to general relativity, our presentation is applicable to a general n-dimensional manifold. The definition is based on the integration of scalars constructed from a physically motivated basis, making use of the least amount of geometrical structure. We also present definitions of covariant derivative of the averaged tensors and Lie derivative.