Averaging Spacetime: Where do we go from here?

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.