Thermodynamics of Einstein-Aether Black Holes

TL;DR

This paper investigates various spherically symmetric vacuum solutions in Einstein-Aether theory, analyzing their horizon structures and thermodynamic properties, and provides explicit solutions for different parameter regimes.

Contribution

It classifies five classes of solutions in Einstein-Aether theory, derives explicit analytical forms, and studies their horizon thermodynamics.

Findings

Universal horizons are always inside Killing horizons.

All solutions are asymptotically flat with singularities at r=0.

Thermodynamic quantities like temperature and entropy are computed for universal horizons.

Abstract

We analyze several spherically symmetric exterior vacuum solutions allowed by the Einstein-Aether (EA) theory with a non static aether and study the thermodynamics of their Killing and universal horizons. We show that there are five classes of solutions corresponding to different values of a combination of the free parameters, $c_{2}$, $c_{13}=c_1+c_3$ and $c_{14}=c_1+c_4$, which are: (A) $c_2 \ne 0$ and $c_{13} \ne 0$ and $c_{14}=0$, (B) $c_2 \ne 0$ and $c_{13} = 0$ and $c_{14} = 0$, (C) $c_2 = 0$ and $c_{13} \ne 0$ and $c_{14} = 0$, (D) $c_2 = 0$, $c_{13} = 0$ and $c_{14} \ne 0$, and (E) $c_2 = - c_{13} \ne 0$ and $c_{14} \ne 0$. We present explicit analytical solutions for these five cases. All these cases have singularities at $r=0$ and are asymptotically flat spacetimes and possess both Killing and universal horizons with the universal horizons always being inside the Killing horizons. Finally, we compute the surface gravity, the temperature, the entropy and the first law of thermodynamics for the universal horizons.