We examine the possibility that the gravitational contribution to the entro
py of a system can be identified with some measure of the Weyl curvature. I
n this paper we consider homothetically self-similar spacetimes. These are
believed to play an important role in describing the asymptotic properties
of more general models. By exploiting their symmetry properties we are able
to impose significant restrictions on measures of the Weyl curvature which
could reflect the gravitational entropy of a system. In particular, we are
able to show, by way of a more general relation, that the most widely used
"dimensionless" scalar is not a candidate for this measure along homotheti
c trajectories.