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.
