  Horava's proposal for non-relativistic quantum gravity introduces a preferred
time foliation of space-time which violates the local Lorentz invariance. The
foliation is encoded in a dynamical scalar field which we call `khronon'. The
dynamics of the khronon field is sensitive to the symmetries and other details
of the particular implementations of the proposal. In this paper we examine
several consistency issues present in three non-relativistic gravity theories:
Horava's projectable theory, the healthy non-projectable extension, and a new
extension related to ghost condensation. We find that the only model which is
free from instabilities and strong coupling is the non-projectable one. We
elaborate on the phenomenology of the latter model including a discussion of
the couplings of the khronon to matter. In particular, we obtain the parameters
of the post-Newtonian expansion in this model and show that they are compatible
with current observations.
