The Rotating Black Hole in Renormalizable Quantum Gravity: The Three-Dimensional Ho\v{r}ava Gravity Case

TL;DR

This paper presents the first exact rotating black hole solution in a three-dimensional Hořava gravity model, revealing unique singularity structures and thermodynamic properties influenced by Lorentz violation.

Contribution

It provides the first explicit rotating black hole solution in three-dimensional Hořava gravity, filling a significant gap in the understanding of such models.

Findings

The solution exhibits a ring curvature singularity inside the horizon.

A curvature singularity exists at the origin.

The standard first law of thermodynamics does not hold in this context.

Abstract

Recently Ho\v{r}ava proposed a renormalizable quantum gravity, without the ghost problem, by abandoning Einstein's equal-footing treatment of space and time through the anisotropic scaling dimensions. Since then various interesting aspects, including the exact black hole solutions have been studied but no "rotating" black hole solutions have been found yet, except some limiting cases. In order to fill the gap, I consider a simpler three-dimensional set-up with z=2 and obtain the exact rotating black hole solution. This solution has a ring curvature singularity inside the outer horizon, like the four- dimensional Kerr black hole in Einstein gravity, as well as a curvature singularity at the origin. The usual mass bound works also here but in a modified form. Moreover, it is shown that the conventional first law of thermodynamics with the usual Hawking temperature and chemical potential does not work, which seems to be the genuine effect of Lorentz-violating gravity due to lack of the absolute horizon.