Slowly rotating Einstein-bumblebee black hole solution and its greybody factor in a Lorentz violation model

TL;DR

This paper derives a slowly rotating black hole solution in Einstein-bumblebee gravity with Lorentz violation and analyzes how Lorentz violation affects the black hole's greybody factor, revealing that Lorentz violation can enhance absorption probabilities.

Contribution

It provides the first exact slowly rotating Einstein-bumblebee black hole solution considering different Lorentz violation configurations and explores the impact on greybody factors.

Findings

Existence of slowly rotating solutions depends on Lorentz violation parameters.

Lorentz violation decreases effective potential and increases absorption probability for scalar fields.

No full rotating black hole solution is currently available in Einstein-bumblebee theory.

Abstract

We obtain an exact slowly rotating Einstein-bumblebee black hole solution by solving the corresponding $rr$ and $t\phi$ components of the gravitational field equations in both cases: A, $b_\mu=(0,b(r),0,0)$; B, $b_\mu=(0,b(r),\mathfrak{b}(\theta),0)$. Then we check the other gravitational field equations and the bumblebee field motion equations by using this solution. We find that in the case A, there exists a slowly rotating black hole solution indeed for arbitrary LV (Lorentz violation) coupling constant $\ell$; however as in the case B, there exists this slowly rotating solution if and only if the coupling constant $\ell$ is as small as or smaller than the angular momentum $a$. Till now there seems to be no full rotating black hole solution, so one can't use the Newman-Janis algorithm to generate a rotating solution in Einstein-bumblebee theory. It is similar as that in Einstein-aether theory where there exists only some slowly rotating black hole solutions. In order to study the effects of this Lorentz symmetry broken, we consider the black hole greybody factor and find that when angular index $l=0$, the LV constant $\ell$ decreases the effective potential and enhances the absorption probability, which is similar to that of the non-minimal derivative coupling theory.