We obtain an exact slowly rotating Einstein-bumblebee black hole solution by solving the corresponding and components of the gravitational field equations for both cases: A) and B) . 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 ; however, for case B, this slowly rotating solution exists if and only if coupling constant 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 , LV constant decreases the effective potential and enhances the absorption probability, which is similar to the results for the non-minimal derivative coupling theory.

我们通过求解以下两种情况的引力场方程的对应和分量来获得精确的缓慢旋转的爱因斯坦-大黄蜂黑洞解：A)和 B) 。然后，我们使用该解检查其他引力场方程和大黄蜂场运动方程。我们发现对于情况 A，对于任意 LV（洛伦兹违反）耦合常数，确实存在一个缓慢旋转的黑洞解；然而，对于情况 B，当且仅当耦合常数等于或小于角动量a时，这种缓慢旋转的解才存在. 到目前为止，还没有发布完整的旋转黑洞解决方案；因此，Newman-Janis 算法不能用于生成爱因斯坦-大黄蜂理论中的旋转解。这类似于爱因斯坦-以太理论，其中仅存在一些缓慢旋转的黑洞解。为了研究这种破坏的洛伦兹对称性的影响，我们考虑了黑洞灰体因子，发现对于角指数，LV 常数降低了有效势并提高了吸收概率，这与非最小导数耦合的结果相似理论。