# 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) $b_\mu= (0,b(r), \mathfrak{b}(\theta),0)$. 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 $\ell$; however, for case B, this slowly rotating solution exists if and only if coupling constant $\ell$ 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 $\ell$ decreases the effective potential and enhances the absorption probability, which is similar to the results for the non-minimal derivative coupling theory.
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Chikun Ding and Xiongwen Chen. Slowly rotating Einstein-bumblebee black hole solution and its greybody factor in a Lorentz violation model[J]. Chinese Physics C. doi: 10.1088/1674-1137/abce51
Chikun Ding and Xiongwen Chen. Slowly rotating Einstein-bumblebee black hole solution and its greybody factor in a Lorentz violation model[J]. Chinese Physics C.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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

###### Corresponding author: Xiongwen Chen, chenxiongwen@hhtc.edu.cn
• 1. Department of Physics, Huaihua University, Hunan, Huaihua 418008, China
• 2. Department of Physics, Hunan University of Humanities, Science and Technology, Hunan, Loudi 417000, China
• 3. Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China

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 for both cases: A) $b_\mu=(0,b(r),0,0)$ and B) $b_\mu= (0,b(r), \mathfrak{b}(\theta),0)$. 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 $\ell$; however, for case B, this slowly rotating solution exists if and only if coupling constant $\ell$ 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 $\ell$ decreases the effective potential and enhances the absorption probability, which is similar to the results for the non-minimal derivative coupling theory.

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