Revisit Static Aether: Exact Vacuum Solution in Einstein-Aether Theory and Its Analytic Extension

We obtain an exact analytical vacuum solution of Einstein-Aether theory with a strictly static aether configuration and investigate its maximal extension. The solution depends only on the coupling $c_{14}$ and reduces to the Schwarzschild geometry in the limit $c_{14}=0$. We show that Schwarzschild is an isolated member of this family: for any nonzero $c_{14}$ the spacetime ceases to be a black hole and instead becomes either a naked singularity ($c_{14}<0$) or a wormhole-like geometry ($0<c_{14}<2$). By constructing the complete analytic extension, we demonstrate that the internal infinity of the wormhole corresponds to an extremal Killing horizon. Crossing this horizon leads to a new spacetime region where the causal roles of time and radial coordinates are exchanged, and the spacetime ultimately terminates at a spacelike singularity. The resulting global structure, summarized by the corresponding Carter-Penrose diagrams, reveals a previously unexplored causal completion of the static-aether vacuum spacetime.