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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.