On constructing purely affine theories with matter

TL;DR

This paper investigates the possibility of deriving spacetime metrics from purely affine theories without prior metric assumptions, highlighting the limitations of local theories and suggesting non-locality may be necessary.

Contribution

It analyzes local purely affine theories with matter coupling, demonstrating their limitations and arguing that viable theories likely require non-locality to recover spacetime metrics.

Findings

Local affine theories do not define lightcones via shock wave propagation.

Replacing the Ricci tensor with the metric yields limited solutions.

Non-local theories are likely necessary for viable affine gravity models.

Abstract

We explore ways to obtain the very existence of a space-time metric from an action principle that does not refer to it a priori. Although there are reasons to believe that only a non-local theory can viably achieve this goal, we investigate here local theories that start with Schroedinger's purely affine theory [21], where he gave reasons to set the metric proportional to the Ricci curvature aposteriori. When we leave the context of unified field theory, and we couple the non-gravitational matter using some weak equivalence principle, we can show that the propagation of shock waves does not define a lightcone when the purely affine theory is local and avoids the explicit use of the Ricci tensor in realizing the weak equivalence principle. When the Ricci tensor is substituted for the metric, the equations seem to have only a very limited set of solutions. This backs the conviction that viable purely affine theories have to be non-local.