The General Spherically Symmetric Static Solutions in the Einstein-Aether Theory.

In the present work we have analyzed all the possible spherically symmetric vacuum solutions allowed by the Einstein-Aether (EA) theory. We show that there are solutions only for two specific values of $c_{14}$, zero and two. This result is completely general. For $c_{14}=0$ we have the Schwarzschild spacetime. On the other hand, for $c_{14}=2$ we have a family of new static solutions with an arbitrary function of the coordinate $r$. As a consequence of $c_{14}=2$ the EA coupling constant is zero, means that there is no coupling with the matter, which does not represent any problem to vacuum solutions. The problem arises when matching the vacuum exterior with some interior source matter since the EA parameter should be the same in the whole spacetime (interior and exterior). In this case the unique possibility would be $c_{14}=0$.