Exact modifications on a vacuum spacetime due to a gradient bumblebee field at its vacuum expectation value

TL;DR

This paper develops an exact method to analyze how a gradient bumblebee field at its vacuum expectation value modifies vacuum spacetimes, providing new solutions and insights into Lorentz symmetry violation effects.

Contribution

It introduces a simplified, exact solution method for the bumblebee model with a gradient field, enabling explicit analysis of spacetime modifications and new solutions including rotating black holes.

Findings

Recovered known solutions like Schwarzschild and Kerr.

Derived new solutions with bumblebee field effects.

Showed the modified solutions retain key features of original spacetimes.

Abstract

This work belongs to the context of the standard-model extension, in which a Lorentz symmetry violation is induced by a bumblebee field as it acquires a nonzero vacuum expectation value. The mathematical formulation of a generic bumblebee model and its associated dynamical equations are presented. Then, these equations are considered for the vacuum and a substantial simplification is performed for the particular case of a gradient bumblebee field at its vacuum expectation value. After some further manipulation, a method to easily find solutions to the model is developed, in which the exact effect on the spacetime description due to the presence of this bumblebee field is explicitly provided. As some examples, the method is applied to determine the implications of the bumblebee field on the Schwarzschild spacetime and also on a rotating one. A previously published solution is recovered and some new ones are obtained. In the rotating situation, a simple solution is found which contains both the Kerr solution and the already published one as special cases. It is also shown its distinguished surfaces are still given by the same corresponding expressions for the Kerr solution. In conclusion, the mathematical improvement made is considered to be a significant contribution to the theory as a powerful tool to investigate its many aspects and consequences.