The stealth Kerr solution in the bumblebee gravity

TL;DR

This paper identifies a special vector-tensor gravity (bumblebee) model with $\xi_1=2\kappa$, $\xi_2=0$, $V=0$ that admits a stealth Kerr black hole: a Kerr metric accompanied by a nontrivial vector field whose energy–momentum tensor cancels, leaving the geometry indistinguishable from GR Kerr. The authors show the stealth Kerr can be obtained from the stealth Schwarzschild solution via the Newman–Janis algorithm, using both the tetrad and Giampieri formalisms; however, outside this exact coupling, the algorithm fails to generate rotating solutions. They also note that the stealth Kerr behaves like a charged rotating BH with $Q=-\sqrt{2\kappa}\,\lambda_0 M$ in this theory, drawing a Kerr–Newman analogy. The results illuminate how nonminimal vector couplings can yield elegant BH solutions and motivate future studies of perturbations, thermodynamics, and observational tests in modified gravity contexts.

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.