We explore a way to obtain the metric from an action principle that does not refer to it a priori. We study local theories that start with Schr?dinger's purely affine theory, then we couple it to matter fields. We show that the propagation of shock waves does not define a lightcone 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.
展开▼
