The rotating black hole in renormalizable quantum gravity: The three-dimensional Hořava gravity case

Abstract

Recently Hořava 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.

A remark on polar noncommutativity

“…(6) This was the relation eventually used in [7] for its convenience, and so despite the faulty starting point the results derived therein maintain their validity. However, it is important to point out that (6) is the fundamental commutation relation for polar noncommutative R 2 .…”

“…The question of noncommutativity in polar coordinates has been considered previously (for example, see [5] for a description of 'fuzzy' R 3 ). We wish to focus on work in [6] and [7], where a number of commutation relations were postulated for the fundamental commutator, [r, θ], in polar coordinates. These relations were determined and justified only by a dimensional argument, without any recourse to the validity of the association between the Hilbert space and the function space over which the Moyal -product acts.…”

Hořava gravity at a Lifshitz point: A progress report

“…carried out before the realization of the importance of the term a i a i that could play in the IR stability [95,96], so they all unfortunately belong to the branch of the non-projectable Hořava gravity that is plagued with the instability problem 16 . Lately, the case with β 0 = 0 were studied in [167][168][169][170]. In particular, it was claimed that slowly rotating black holes do not exist in this extension [169].…”

Rotating black holes in three-dimensional Hořava gravity

“…Reference[20] also includes an R 2 term in the action. We do not include this term and work fully within the infrared sector.…”