Static Spherically Symmetric Solutions to modified Horava-Lifshitz Gravity with Projectability Condition

Abstract

In this paper we seek static spherically symmetric solutions of Horava-Lifshitz-like gravity with projectability condition. We consider the most general form of gravity action without detailed balance, and require the spacetime metric to respect the projectability condition. We find that for any value of $λ$, it may exists the solutions of topology $R\times M_3$, where $R$ is the time direction and $M_3$ is a three-dimensional maximally symmetric space depending on the value of cosmological constant and the potential of the action. Besides, in the UV region where $λ\neq 1$, we find Minkowski or de-Sitter space-time as the solution, while in the IR region where $λ=1$, we prove that (dS-)Schwarzschild solution is the only nontrivial solution. We also notice that the other static spherically symmetric solutions found in the literature do not satisfy the projectability condition and are not the solutions we get. Our study shows that in Horava-Lifshitz gravity with projectability condition, there is no novel correction to Einstein's general relativity in solar system tests.