Particle Motion with Ho\v{r}ava -- Lifshitz type Dispersion Relations

Using super Hamiltonian formalism, we study the motion of particles whose dispersion relations are modified to incorporate Ho\v{r}ava – Lifshitz type anisotropic scaling symmetry. We find the following as consequences of this modified dispersion relation: (i) The speed of a charged particle under a constant electric field grows without bound and diverges. (ii) The speed of a particle falling towards the horizon also grows without bound and diverges as the particle approaches the horizon. (iii) This particle reaches the horizon in a finite coordinate time, in contrast to the standard case where it requires infinite time.