Uniqueness of static spherically symmetric vacuum solutions in the IR limit of Ho\v{r}ava-Lifshitz gravity

We investigate static spherically symmetric vacuum solutions in the IR limit of projectable nonrelativistic quantum gravity, including the renormalisable quantum gravity recently proposed by Ho\v{r}ava. It is found that the projectability condition plays an important role. Without the cosmological constant, the spacetime is uniquely given by the Schwarzschild solution. With the cosmological constant, the spacetime is uniquely given by the Kottler (Schwarzschild-(anti) de Sitter) solution for the entirely vacuum spacetime. However, in addition to the Kottler solution, the static spherical and hyperbolic universes are uniquely admissible for the locally empty region, for the positive and negative cosmological constants, respectively, if its nonvanishing contribution to the global Hamiltonian constraint can be compensated by that from the nonempty or nonstatic region. This implies that static spherically symmetric entirely vacuum solutions would not admit the freedom to reproduce the observed flat rotation curves of galaxies. On the other hand, the result for locally empty regions implies that the IR limit of nonrelativistic quantum gravity theories does not simply recover general relativity but includes it.