Spherically symmetric analytic solutions and naked singularities in Einstein–Aether theory

We analyze all the possible spherically symmetric exterior vacuum solutions allowed by the Einstein–Aether theory with static aether. We show that there are three classes of solutions corresponding to different values of a combination of the free parameters, $$c_{14}=c_1+c_4$$ , which are: $$ 0< c_{14}<2$$ , $$c_{14} < 0$$ , and $$c_{14}=0$$ . We present explicit analytical solutions for $$c_{14}=3/2, 16/9, 48/25, -16$$ and 0. The first case has some pathological behavior, while the rest have all singularities at $$r=0$$ and are asymptotically flat spacetimes. For the solutions $$c_{14}=16/9, 48/25\, \mathrm {\, and \,}\, -16$$ we show that there exist no horizons, neither Killing horizon nor universal horizon, thus we have naked singularities. This characteristic is completely different from general relativity. We briefly discuss the thermodynamics for the case $$c_{14}=0$$ where the horizon exists.