Static spherical vacuum solution to bumblebee gravity with time-like VEVs

Abstract The static spherical vacuum solution in a bumblebee gravity model where the bumblebee field $$B_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> has a one-component time-like vacuum expectation value $$b_\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> 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 two non-trivial solutions can be obtained so long as a unique condition for the vacuum expectation $$b^2\equiv -b^\mu b_\mu =2/\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>≡</mml:mo> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>b</mml:mi> <mml:mi>μ</mml:mi> </mml:msup> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mi>κ</mml:mi> </mml:mrow> </mml:math> , where $$\kappa =8\pi G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , is satisfied. One of the solutions contains a naked singularity, while the other exhibits features analogous to a confining potential. We argue that naturally these solutions are not stable since quantum corrections would invalidate the likely numerical coincidence, unless there are some unknown 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-Nordström solution is a limit of one of our solutions.