Black holes and global structures of spherical spacetimes in
  Horava-Lifshitz theory
  We systematically study black holes in the Horava-Lifshitz (HL) theory by
following the kinematic approach, in which a horizon is defined as the surface
at which massless test particles are infinitely redshifted. Because of the
nonrelativistic dispersion relations, the speed of light is unlimited, and test
particles do not follow geodesics. As a result, there are significant
differences in causal structures and black holes between general relativity
(GR) and the HL theory. In particular, the horizon radii generically depend on
the energies of test particles. Applying them to the spherical static vacuum
solutions found recently in the nonrelativistic general covariant theory of
gravity, we find that, for test particles with sufficiently high energy, the
radius of the horizon can be made as small as desired, although the
singularities can be seen in principle only by observers with infinitely high
energy. In these studies, we pay particular attention to the global structure
of the solutions, and find that, because of the
foliation-preserving-diffeomorphism symmetry, ${Diff}(M,{\cal{F}})$, they are
quite different from the corresponding ones given in GR, even though the
solutions are the same. In particular, the ${Diff}(M,{\cal{F}})$ does not allow
Penrose diagrams. Among the vacuum solutions, some give rise to the structure
of the Einstein-Rosen bridge, in which two asymptotically flat regions are
connected by a throat with a finite non-zero radius. We also study slowly
rotating solutions in such a setup, and obtain all the solutions characterized
by an arbitrary function $A_{0}(r)$. The case $A_{0} = 0$ reduces to the slowly
rotating Kerr solution obtained in GR.
