AVERAGING SPACE-TIME: WHERE DO WE GO FROM HERE?

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 Weitzenbock connection and ensure the transportation is independent of path. The talk concludes with some open questions to generate further discussion. 1. Smoothing of Space-time: What are the Problems? The current widely accepted, standard cosmological model is a Universe that is homogeneous and isotropic on the largest of length scales filled with a mixture of baryonic matter, electromagnetic radiation, neutrinos, dark matter and dark energy. The detailed measurements of the cosmic microwave background via the WMAP program and other observational programs lend credence to this model. Interestingly enough, the two dark components are estimated to make up approximately 95% of the matter content of the Universe today and yet we have no true understanding of their composition or nature. Actually, the dark components can only be observed through their gravitational effects on baryonic matter and photons and are not directly observed. Can there be an alternative description for these observational effects that does not assume the existence of these mysterious dark quantities? One possibility is that both dark matter and dark energy are artefacts of some effective averaged theory of gravitation. If one employs Einstein's theory of General Relativity as our theory of gravity, then the resulting cosmological model based on the assumptions of homogeneity and isotropy is described by a single function of time. This idealized model is mathematically elegant and consistent with the observations on cosmological scales (provided one adds sufficient quantities of dark matter and dark energy to the model). However, can one simply ignore the structure and inhomogeneity that is present on smaller scales? Voids, walls, filaments, clusters, and super clusters are all examples of such structures that can be observed. Indeed, the smaller the scale, the larger the inhomogeneity. Can one neglect the effects of these inhomogeneities on our smoothed out idealized model? If not, how does