Thermodynamics of Einstein-Aether black holes

Abstract

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 2 , $$c_{13}=c_1+c_3$$ c 13 = c 1 + c 3 and $$c_{14}=c_1+c_4$$ c 14 = c 1 + c 4 , which are: (A) $$c_2 \ne 0$$ c 2 ≠ 0 and $$c_{13} \ne 0$$ c 13 ≠ 0 and $$c_{14}=0$$ c 14 = 0 , (B) $$c_2 \ne 0$$ c 2 ≠ 0 and $$c_{13} = 0$$ c 13 = 0 and $$c_{14} = 0$$ c 14 = 0 , (C) $$c_2 = 0$$ c 2 = 0 and $$c_{13} \ne 0$$ c 13 ≠ 0 and $$c_{14} = 0$$ c 14 = 0 , (D) $$c_2 = 0$$ c 2 = 0 , $$c_{13} = 0$$ c 13 = 0 and $$c_{14} \ne 0$$ c 14 ≠ 0 , and (E) $$c_2 = - c_{13} \ne 0$$ c 2 = - c 13 ≠ 0 and $$c_{14} \ne 0$$ c 14 ≠ 0 . We present explicit analytical solutions for these five cases. All these cases have singularities at $$r=0$$ 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.