Averaging Spacetime: Where do we go from here?

TL;DR

This paper reviews the challenges of creating a covariant averaging procedure in General Relativity and discusses potential approaches, including geodesic parallel transport and Weitzenböck connection, to address the averaging problem.

Contribution

It provides a comparative review of existing averaging methods and proposes potential directions for developing a well-defined averaging procedure in gravity.

Findings

Identifies strengths and weaknesses of current averaging approaches.

Suggests parallel transport along geodesics or using Weitzenböck connection as promising methods.

Highlights open questions for future research in gravitational averaging.

Abstract

The construction of an averaged theory of gravity based on Einstein's General Relativity is very difficult due to the non-linear nature of the gravitational field equations. This problem is further exacerbated by the difficulty in defining a mathematically precise covariant averaging procedure for tensor fields over differentiable manifolds. Together, these two ideas have been called the averaging problem for General Relativity. In the first part of the talk, an attempt to review some the various approaches to this problem will be given, highlighting strengths, weaknesses, and commonalities between them. In the second part of the talk, an argument will be made, that if one wishes to develop a well-defined averaging procedure, one may choose to parallel transport along geodesics with respect to the Levi-Cevita connection or, use the Weitzenb\"ock connection and ensure the transportation is independent of path. The talk concludes with some open questions to generate further discussion.