Full Classification of Static Spherical Vacuum Solutions to Bumblebee Gravity with General VEVs

TL;DR

This paper classifies all static spherical vacuum solutions in bumblebee gravity with general vacuum expectation values, revealing unique solutions and degeneracies not present in general relativity.

Contribution

It provides a comprehensive classification of solutions in bumblebee gravity with arbitrary VEVs, including conditions for degeneracy and the existence of Schwarzschild solutions.

Findings

Model degenerates for specific parameter sets, with solutions characterized by arbitrary functions.

Bumblebee gravity admits Schwarzschild solutions with certain matter distributions, unlike in general relativity.

The non-minimally coupled massless vector tensor theory is ill-defined at a specific coupling value.

Abstract

The static spherical vacuum solution in a bumblebee gravity model where the bumblebee field $B_\mu$ has a two-component space-like, light-like, and time-like vacuum expectation value $b_\mu$ is studied. Based on the results, we present a comprehensive classification of the static spherical vacuum solutions in bumblebee gravity with general vacuum expectation values. We find that the model becomes degenerate for a specific set of parameter combinations, where the solution can be characterized by an arbitrary function, which indicates that the non-minimally coupled massless vector tensor theory is ill-defined when $\xi=\kappa/2$. We also find that contrary to the situation in general relativity, the bumblebee gravity admits the exact Schwarzschild solution with non-zero matter distributions of certain forms. The implications of this result are discussed, suggesting that the experimental constraints within the solar system would be invalid.