Black hole solutions with a linear equation of state in Ho\v{r}ava gravity and Einstein-Aether theory

Abstract

We provide a methodology to obtain black hole (BH) solutions in Ho\v{r}ava gravity (HG) and Einstein Aether (AE) theory for the spherically symmetric (SS) case with a static aether. This methodology consists of first specifying the form of the equation of state (EoS), rather than prescribing an energy density profile. The usual EoS for the static and SS case, $\rho = -p_r$, is no longer satisfied due to the presence of the HG AE terms. We study three linear EoS associated with: an analogue charged BH, a non-trivial extremal BH, and an ultra-relativistic stiff fluid, respectively. The HG AE terms lead to exotic behaviors, both in the physical properties of the solutions and in their thermodynamics. In Case I, the matter sources can be interpreted as an exotic anisotropic matter distribution, giving rise to an effective electric-potential term in the geometry. In Case II, we obtain a non trivial extremal BH solution for which the event horizon is $n_{\text{odd}}$ fold degenerate. In Case III, we find a solution with a non trivial repulsive potential, where the influence of the HG AE terms at short scales leads to the formation of a BH remnant whose horizon encloses a central singularity (instead of a de Sitter core as occurs in regular BHs).