Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique

TL;DR

This paper introduces a unique generating technique for exact solutions in bumblebee gravity, extending known spacetimes like Kerr-Newman-(A)dS, and explores how geodesic choices influence solution properties.

Contribution

It presents a simple, unique method to construct exact bumblebee gravity solutions from vacuum solutions, including extensions to spacetimes with cosmological constant and electromagnetic fields.

Findings

The technique allows construction of bumblebee extensions of Kerr-Newman-(A)dS spacetime.

The bumblebee field is proportional to the tangential vector of a geodesic in the background spacetime.

The method's extension depends on the choice of geodesic, affecting solution properties.

Abstract

In this work, we show that if the bumblebee field in the Einstein-bumblebee theory is given by its vacuum expectation value ($B_{\mu}=b_{\mu}$) and it is not dynamical ($\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}=0$), then these conditions uniquely provide a generating technique, allowing us to construct exact solutions to bumblebee gravity from the vacuum solutions by adding a term $\sim b_{\mu}b_{\nu}$ to the metric tensor (thus proving the uniqueness of the method, presented in [Eur. Phys. J. C 82 (2022) 613]). Also, we show that the bumblebee field within this technique is proportional to the tangential vector of the (timelike or spacelike) geodesic curve in the background vacuum spacetime, and can be easily found knowing the solution to the Hamilton--Jacobi equation. Moreover, we prove that this technique can be extended to the case of any non-zero cosmological constant and the presence of the electromagnetic field. We apply this generating technique and obtain the bumblebee extension of the Kerr--Newman--Taub-NUT--(anti-)de Sitter spacetime. We show that this extension is not unique, as it depends on the exact geodesic curve one chooses to associate a bumblebee field with. Then, by considering various special cases of this generic solution, we demonstrate that the condition of the global reality of the bumblebee field limits the set of geodesics with which we can associate it.