Is Birkhoff's Theorem Valid in Einstein-Aether Theory?

We attempt to answer whether Birkhoff's theorem (BT) is valid in the Einstein-Aether (EA) theory. The BT states that any spherically symmetric solution of the vacuum field equations must be static, unique, and asymptotically flat. For a general spherically symmetric metric with metric functions $A(r,t)$ \& $B(r,t)$, and aether components $a(r,t)$ \& $b(r,t)$, we prove the conditions for the staticity of spacetime using two different methods. We point out that BT is valid in EA theory only for special values of $c_1+c_3$, $c_1+c_4$, and $c_2$, where we can show that all these special cases are asymptotically flat. In particular, when the aether has only a temporal component, i.e., $b(r,t)=0$ and the $c_{14} \neq 0$ case gives us spherically symmetric static black holes without horizons; that is, they have naked singularities, at least for special values of $c_{14}$. Thus, the cosmic censorship conjecture is violated for the case BT holds. However, when we have an aether vector with temporal and radial components, we only prove that the staticity and the flatness at infinity hold for a special metric and particular combination of the aether parameters. For this case, there exist universal horizons instead of naked singularities.
Comment: 19 pages, no figures