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

TL;DR

The paper analyzes static spherical vacuum solutions in Bumblebee gravity with a time-like VEV $b_\mu$, showing that nontrivial curved solutions exist only for $b^2=2/\kappa$; two distinct non-Schwarzschild spacetimes emerge in this limit, both featuring naked singularities and lacking a Schwarzschild-like horizon. A reduced equation for the metric function $\alpha(r)$ yields constant-$\alpha$ and non-constant-$\alpha$ branches, with the constant case producing a naked singularity and the non-constant case connecting to extremal Reissner–Nordström in the limit $\ell\to 0$. The authors argue these solutions are generically unstable due to Hamiltonian boundedness and quantum corrections, implying no stable nontrivial time-like VEV solutions; nevertheless, the analysis illuminates how nonminimal coupling and Lorentz-violating vector fields reshape black-hole-like spacetimes, including their photon-sphere structure and potential gravitational lensing signatures. In the minimal-coupling regime ($\ell\ll1$), the singularities become less problematic, preserving some predictability, while photon-sphere analysis reveals one branch with no photon sphere and another with a single photon sphere at $r=(2-\ell)R_s$, suggesting observational discrimination through lensing could be possible in principle for certain branches. Overall, the work clarifies the existence, geometry, and instability of time-like VEV solutions in Bumblebee gravity and highlights avenues for future dynamical and observational studies, including quasinormal modes and shadows in naked-singularity spacetimes.

Abstract

The static spherical vacuum solution in a bumblebee gravity model where the bumblebee field \(B_μ\) has a one-component time-like vacuum expectation value \(b_μ\) 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 \(b^2\equiv -b^μb_μ=2/κ\), where \(κ=8πG\), 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{ö}m solution is a limit of one of our solutions.