The stealth Kerr solution in the bumblebee gravity

Abstract

In this paper, we find Kerr solution accompanied with a nontrivial vector field as a solution to one of the simplest vector-tensor theories of gravity, namely the bumblebee model with an intriguing coupling constant between the Ricci curvature tensor and the vector field. We also demonstrate that the accompanied vector field can be generated via the Newman-Janis algorithm from a simple spherical vector field, which together with the Schwarzschild metric constitutes a solution to the same bumblebee model. It is probably the simplest example of a theory and its black-hole solutions for the Newman-Janis algorithm to hold except for general relativity.