  We consider a maximal extension of the Hilbert-Einstein action and analyze
several interesting features of the theory. More specifically, the motion is
non-geodesic and takes place in the presence of an extra force. These models
could lead to some major differences, as compared to the predictions of General
Relativity or other modified theories of gravity, in several problems of
current interest, such as cosmology, gravitational collapse or the generation
of gravitational waves. Thus, the study of these phenomena may also provide
some specific signatures and effects, which could distinguish and discriminate
between the various gravitational models.
