# 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 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 a slowly rotating black hole solution indeed for arbitrary LV (Lorentz violation) 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. Till now there seems to be no full rotating black hole solution, so one can't use the Newman-Janis algorithm to generate a rotating solution in Einstein-bumblebee theory. 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 Lorentz symmetry broken, 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.
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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.
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, Huaihua, 418008, P. R. China
• 2. Department of Physics, Hunan University of Humanities, Science and Technology, Loudi, Hunan 417000, P. R. 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, Hunan 410081, P. R. 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 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 a slowly rotating black hole solution indeed for arbitrary LV (Lorentz violation) 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. Till now there seems to be no full rotating black hole solution, so one can't use the Newman-Janis algorithm to generate a rotating solution in Einstein-bumblebee theory. 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 Lorentz symmetry broken, 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.

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