  In this work we study a modified theory of gravity that contains up to fourth
order spatial derivatives as a model for the Horava-Lifshitz gravity. The
propagator is evaluated and, as a result, it is obtained one extra pole
corresponding to a spin two nonrelativistic massless particle, an extra term
which jeopardizes renormalizability, besides the unexpected general relativity
unmodified propagator. Then, unitarity is proved at the tree-level, where the
general relativity pole has shown to have no dynamics, remaining only the two
degrees of freedom of the new pole. Next, the nonrelativistic effective
potential is determined from a scattering process of two identical massive
gravitationally interacting bosons. In this limit, Newton's potential is
obtained, together with a Darwin-like term that comes from the extra non-pole
term in the propagator. Regarding renormalizability, this extra term may be
harmful, by power counting, but it can be eliminated by adjusting the free
parameters of the model. This adjustment is in accord with the detailed balance
condition suggested in the literature and shows that the way in which extra
spatial derivative terms are added is of fundamental importance.
