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

Abstract

We obtain a 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 this solution indeed for arbitrary LV 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 \(a\). 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 breaking, 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.