Stationary and slowly rotating spacetimes in Ho\v{r}ava-Lifshitz gravity

Abstract

Stationary, axisymmetric and slowly rotating vacuum spacetimes in the Ho\v{r}ava-Lifshitz (HL) gravity are studied, and shown that, for any given spherical static vacuum solution of the HL theory (of any model, including the ones with an additional U(1) symmetry), there always exists a corresponding slowly rotating, stationary and axisymmetric vacuum solution, which reduces to the former, when the rotation is switched off. The rotation is universal and only implicitly depends on the models of the HL theory and their coupling constants through the spherical seed solution. As a result, all asymptotically flat slowly rotating vacuum solutions are asymptotically identical to the slowly rotating Kerr solution. This is in contrast to the claim of Barausse and Sotiriou, Phys. Rev. Lett. {\bf 109}, 181101 (2012), in which slowly rotating black holes were reported (incorrectly) not to exist in the infrared limit of the non-projectable HL theory.