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

TL;DR

This paper investigates static spherical vacuum solutions in bumblebee gravity with time-like VEVs, revealing that only Minkowski space is generally allowed unless a specific condition is met, leading to peculiar solutions with naked singularities.

Contribution

It demonstrates that non-trivial solutions exist under a unique vacuum condition, connecting these solutions to extremal Reissner-Nordström black holes and discussing their stability and singularities.

Findings

Solutions are generally Minkowski space unless a specific vacuum condition is satisfied.

Non-trivial solutions exhibit naked singularities and photon spheres.

The extremal Reissner-Nordström solution is a limit of these solutions.

Abstract

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.