Black hole solutions with a linear equation of state in Hořava gravity and Einstein-Aether theory

TL;DR

This work develops black hole solutions in the static, spherically symmetric sector of covariant Horava gravity and Einstein–Aether theory by prescribing linear equations of state rather than energy-density profiles. It derives three analytic cases: an analogue charged BH with a RN-like metric sourced by exotic anisotropic matter; an extremal BH with an $f(r)=(1-2M/r)^{n_{\mathrm{odd}}}$ horizon structure leading to zero temperature but area-law entropy; and a stiff-fluid case giving a Schwarzschild-like metric plus a short-distance repulsive term that yields inner and outer horizons and a remnant at small scales. The analysis shows that HG–AE terms modify horizon thermodynamics, yielding nontrivial temperature behavior, phase transitions in the heat capacity, and remnants that enclose a central singularity, while entropy continues to obey the area law via Wald’s approach. Collectively, the results illustrate how Lorentz-violating gravity theories alter BH structure and thermodynamics, with implications for UV modifications and horizon physics in beyond-GR contexts.

Abstract

We provide a methodology to obtain black hole (BH) solutions in Hořava 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, $ρ= -p_r$, 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).

Figures (1)

• Figure 1: First panel: Temperature for $C_2=1$ and for $n = 4\xi / \eta=3.5,4,4.5,5$. Second panel: Heat capacity for $C_2=1$ and for $n = 4\xi / \eta=3.5,5$