Point-particle solution and the asymptotic flatness in 2+1D Hořava gravity

Abstract

We show that the solution corresponding to the gravitational field of a point particle at rest in $2+1$ nonprojectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava is exactly the same as $2+1$ general relativity with the same source. In general relativity this solution is well known, it is a flat cone whose deficit angle is proportional to the mass of the particle. To establish the system we couple the Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory to a point particle with relativistic action. Motivated by this solution, we postulate the condition of asymptotic flatness exactly in the same way of $2+1$ general relativity. A remarkable feature of this condition is that the dominant mode is not fixed, but affected by the mass of the configuration. In this scheme, another important coincidence with $2+1$ general relativity under asymptotic flatness is that the energy is the same (except for some coupling constants involved), the $z=1$ term with the derivative of the lapse function does not contribute.