Quasinormal modes and AdS/CFT correspondence of a rotating BTZ-like black hole in the Einstein-bumblebee gravity

Abstract We obtain exact expressions for the quasinormal modes (QNMs) of the massive scalar, fermionic and vector perturbations around a rotating BTZ-like black hole in the Einstein-bumblebee gravity. We find that the Lorentz symmetry breaking (LSB) parameter $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> leaves its imprint only on the imaginary parts of the quasinormal frequencies and the corresponding perturbation field decays more slowly for a larger $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> , except for the left-moving quasinormal frequencies $$\omega _L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> with positive mass and the right-moving ones $$\omega _R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:math> with negative mass for the fundamental modes under the vector perturbation where the imaginary parts are independent of $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> . We also note that, regardless of the type of perturbations, the real parts depend only on the angular quantum number, which are the same as those in the standard BTZ black hole. Furthermore, we investigate the AdS/CFT correspondence from the QNMs and observe that the expected universal relation for the left and right conformal weights ( $$h_L,h_R$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> </mml:math> ) of the boundary operators dual to various fields still holds even for the BTZ-like black hole in the Einstein-bumblebee gravity. These results strongly support the AdS/CFT correspondence and could help us better understand the Einstein-bumblebee gravity with the Lorentz symmetry violation.