Static Spherical Vacuum Solution to Bumblebee Gravity with Time-like VEVs

The static spherical vacuum solution in a bumblebee gravity model where the bumblebee field $B_\mu$ has a one-component time-like vacuum expectation value $b_\mu$ is studied. We show that in general curved space-time solutions are not allowed and only the Minkowski space-time exists. However, it is surprising that non-trivial solutions can be obtained so long as a unique condition for the vacuum expectation $b^2\equiv -b^\mu b_\mu=2/\kappa$, where $\kappa=8\pi G$, is satisfied. We argue that naturally these solutions are not stable since quantum corrections would invalidate the likely numerical coincidence, unless there are some unknown \emph{fine-tuning} mechanisms preventing any deviation from this condition. Nevertheless, the naked singularities and the photon sphere of these novel but peculiar solutions are discussed, and we show that the extremal Reissner-Nordstr{\"o}m solution is a limit of one of our solutions.