Geodesic motion in Hořava-Lifshitz gravity

We propose an action for a free particle in Ho {r} {r}{}ava-Lifshitz gravity based on foliation preserving diffeomorphisms. The action reduces to the usual relativistic action in the low-energy limit and allows for subluminal and superluminal motions with upper and lower bounds on velocity, respectively. We find that deviation from general relativity is governed by a position-dependent coupling constant which also depends on the mass of the particle. As a result, lightlike geodesics are not affected whereas massive particles follow geodesics that become mass-dependent, hence the equivalence principle is violated. We make an exact study for geodesics in flat space and a qualitative analysis for those in a spherically symmetric curved background.