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Timelike Geodesic Motion in Horava-Lifshitz Spacetime

Recently Ho$\breve{r}$ava proposed a non-relativistic renormalisable theory of gravitation. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory is expected to flow to the relativistic value $λ= 1$, and could therefore serve as a possible candidate for a UV completion of Einstein general relativity or an infrared modification thereof. In this paper under allowing the lapse function to depend on the spatial coordinates $x^i$ as well as $t$, we obtain the spherically symmetric solutions. And then by analyzing the behavior of the effective potential for the particle, we investigate the timelike geodesic motion of particle in the Ho$\breve{r}$ava-Lifshitz spacetime. We find that the nonradial particle falls from a finite distance to the center along the timelike geodesics when its energy is in an appropriate range. However, we find that it is complexity for radial particle along the timelike geodesics. There are follow different cases due to the energy of radial particle: 1) When the energy of radial particle is higher than a critical value $E_{C}$, the particle will fall from infinity to the singularity directly; 2) When the energy of radial particle equals to the critical value $E_{C}$, the particle orbit is unstable at $r=r_{C}$, i.e. the particle will escape from $r=r_{C}$ to the infinity or to the singularity, which is determined by the initial conditions of the particle; 3) When the energy of radial particle is in a proper range, the particle will rebound to the infinity or plunge to the singularity from a infinite distance, which is also determined by the initial conditions of the particle.