Black holes in ultraviolet-complete Hořava gravity

Hořava gravity is a proposal for completing general relativity in the ultraviolet by interactions that violate Lorentz invariance at very high energies. We focus on ($2+1$)-dimensional projectable Hořava gravity, a theory which is renormalizable and perturbatively ultraviolet-complete, enjoying an asymptotically free ultraviolet fixed point. Adding a small cosmological constant to regulate the long-distance behavior of the metric, we search for all circularly symmetric stationary vacuum solutions with vanishing angular momentum and approaching the de Sitter metric with a possible angle deficit at infinity. We find a two-parameter family of such geometries. Apart from the cosmological de Sitter horizon, these solutions generally contain another Killing horizon and should therefore be interpreted as black holes from the viewpoint of the low-energy theory. Contrary to naive expectations, their central singularity is not resolved by the higher derivative terms present in the action. It is unknown at present if these solutions form as a result of gravitational collapse. The only solution regular everywhere is just the de Sitter metric devoid of any black hole horizon.

Figure 1

Metric functions integrated inward and the corresponding curvature invariants for $F_{\infty }=1$, $f_{(1,0)}=0$ and $\lambda =\frac{15}{14}$. The qualitative behavior of the solution is the same for other values of $\lambda$.