  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.
