In order for the standard general relativistic (GR) model of the universe to match observation, investigators have found it necessary to introduce some fixes, including positing the existence of dark matter and dark energy. These hypotheses have had a profound effect on our understanding of the universe, but remain under debate, because dark matter has not been detected directly yet, and evidence for the existence of dark energy is mainly due to mismatches between observation and theory.
In a paper in the Journal of Mathematical Physics, Robert van den Hoogen argues that an alternative theory of gravity that is the result of averaging over sufficiently large scales might be an explanation for some, or even all, of the observed effects associated with dark matter and dark energy. An averaged theory of gravity, per se, is not derived, but the mathematical tools necessary to potentially derive an averaged theory are presented.
These tools include two major findings. First, to have a nontrivial covariant averaging procedure requires that the geometry of space-time not be restricted to being Riemannian, based on smooth manifolds, as in GR, but non-Riemannian with a flat affine connection. A second finding is that the averaging procedure used in previous work in development of “Macroscopic Gravity,” put forth by Zalaletdinov, is essentially that of parallel transport along arbitrary curves having a flat affine connection. This new observation adds significantly to the understanding of Zalaletdinov’s procedure.
The work, as presented, is more mathematical than physical, but physical interpretations are expected with future applications. A natural arena in which to apply these ideas is the teleparallel equivalent to GR, which attempts a unified theory based on a mathematical structure of distant parallelism.
Source: “Towards a covariant smoothing procedure for gravitational theories,” by R. J. van den Hoogen, Journal of Mathematical Physics (2017). The article can be accessed at https://doi.org/10.1063/1.4999065.
