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 t phi components of the gravitational field equations for both cases: A) b(mu) = (0, b(r), 0, 0) and B) b(mu) = (0, b(r), b(theta), 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.