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

We obtain an exact slowly rotating Einstein-bumblebee black hole solution by solving the corresponding rr and tθ components of the gravitational field equations for both cases:A)b u=(0,b(r),0,0) and B) bu=(0,b(r),b(θ),0) .Then,we check the other gravitational field equations and the bumblebee field motion equations using this solution.We find that for case A,there indeed exists a slowly rotating black hole solution for an arbitrary LV(Lorentz violation)coupling constant l;however,for case B,this slowly rotating solution exists if and only if coupling constant l is as small as or smaller than angular momentum a.Thus far,no full rotating black hole solution has been published;hence,the Newman-Janis algorithm cannot be used to generate a rotating solution in the Einstein-bumblebee theory.This is similar to the Einstein-aether theory,wherein only some slowly rotating black hole solutions exist.To study the effects of this broken Lorentz symmetry,we consider the black hole greybody factor and find that,for angular index l=0 ,LV constant l decreases the effective potential and enhances the absorption probability,which is similar to the results for the non-minimal derivative coupling theory.