\section{Introduction}

The standard model (SM) of particle physics and the general relativity (GR) cannot explain everything in the universe, such as dark energy and that what occurs in the vicinity of a black hole. Thus, on the very high energy scales, one reconsiders combining the SM with GR in a unified theory, i.e., "quantum gravity". The standard model extension (SME), proposed by Kostelesk\'{y} and collaborators \cite{Kostelecky1,Kostelecky2,Kostelecky3,Kostelecky4,Kostelecky5,Kostelecky6,Kostelecky7,Kostelecky8}, is this effective field theory combining GR and SM at low energy scales, and incorporates all possible background fields that violate the fundamental symmetries existent in nature, the Lorentz invariance, happening on the high energy scales (about Planck scale). Studying the Lorentz violation (LV) is a useful approach toward investigating the foundations of modern physics. Besides SME, string theory \cite{kostelecky198939}, noncommutative field theories \cite{carroll,carroll2,carroll3}, spacetime-varying fields \cite{bertolami69,bertolami692,bertolami693}, loop quantum gravity theory \cite{gambini,gambini2}, brane world scenarios \cite{burgess03,burgess032,burgess033}, massive gravity \cite{fernando} and Einstein-aether theory \cite{jacobson,jacobson2} are other proposals of Lorentz violation.

SME can be used to calculate a number of predictions which can be tested in modern high-precision experiments \cite{Kostelecky8,tasson}. The primary LV in the gravity sector of SME is the form of $s^{\mu\nu}R_{\mu\nu}$, where $s^{\mu\nu}$ is a tensor field possessing a nonzero background configuration and defines preferred frames, $R_{\mu\nu}$ is the Ricci tensor. The simplest form of $s^{\mu\nu}$ is the bumblebee field $B^\mu$, which is expected to produce new and maybe even eccentric phenomena through the coupling term $B^{\mu}B^{\nu}R_{\mu\nu}$. The name of "bumblebee" comes from the alleged remark that bumblebees should not be allowed to fly since we did not know how their wings produce lift. The surprising property of this bumblebee gravity is that, unlike the absence of $U(1)$ gauge symmetry, it does not forbid the propagation of massless vector modes \cite{bluhm2008}. So one expects to reveal a variety of physical relics which may be of interest in studies of dark energy and dark matter due to the appearance of Nambu-Goldstone and massive Higgs in theories with the spontaneous Lorentz symmetry breaking \cite{Kostelecky6,Kostelecky7,bluhm}.

In this Einstein-bumblebee gravity theory, Bertolami and P\'aramos studied the 4-dimensional vacuum solutions including the purely radial, the radial/temporal or the axial/temporal LV \cite{bertolami}. In recent years, Casana {\it et al} \cite{casana} obtained an exact Schwarzschild-like black hole solution; Xu {\it et al} studied more general static black hole solutions and their properties \cite{xu}. In 2020, we obtained an exact Kerr-like solution and studied its black hole shadow \cite{ding2020}. Though this solution does not seem to satisfy the bumblebee field equations, Liu {\it et al} found that it can still be satisfied under some certain conditions, i.e., considered as an approximate solution of the bumblebee field equations \cite{liu}. After that we derived a slowly rotating black hole solution which satisfies all field equations \cite{ding2021}. Lately, we derived a black hole solution and a cosmological solution of Einstein-Gauss-Bonnet gravity coupled to bumblebee field \cite{ding2022}, found that the Guass-Bonnet term and the bumblebee field can both act as a form of dark energy. In  a high dimensional spacetime, we obtained an exact  AdS-like black hole solution \cite{ding2023}, found that the conceptions of black hole horizon area/entropy and volume inside horizon should be renewed due to its anisotropy.

In this paper, we would seek a rotating black hole solution in 3-dimensional spacetime. The first black hole solution is BTZ(Ba\~{n}ados, Teitelboim and Zanelli) solution \cite{banados1,banados2}, which is asymptotically anti-de Sitter(AdS) and has on curvature singularity. However, it is a genuine black hole solution due to the presence of event horizon, Hawking radiation, entropy,  and playing a significant role to understand physical properties in higher dimensions by using many of toy models \cite{bengtsson1,bengtsson2}.  The black hole thermodynamics has raised some challenging questions: a statistical derivation of black hole entropy and an account of its microstates. A promised idea is the holographic principle: there is a nontrivial match between features of 2-dimensional conformal field theory (CFT) and features of black holes \cite{castro}. The microscopic degrees of freedom of the black hole are described by CFT living in the boundary. Quantum studies around BTZ black holes can help better understand AdS/CFT correspondence, T-duality and U-duality to classes of asymptotically flat black strings \cite{horowitz1,horowitz2}.

We will derive a 3-dimensional  rotating black hole solution and study the central charges of the dual CFT in the theory of Einstein gravity coupled to the bumblebee fields. The rest paper is organized as follows.   In Sec. II we give the background for the Einstein-bumblebee theory. In Sec. III, we give the  black hole solution by solving the gravitational and bumblebee field equations. In Sec. IV, we study its central charges  and find some effects of the Lorentz breaking constant $\ell$. Sec. V is for a summary.

\section{Einstein-bumblebee gravity in 3-dimensional spacetime}


In the bumblebee gravity model, one introduces the bumblebee vector field $B_{\mu}$ which has a nonzero vacuum expectation value, to lead a spontaneous Lorentz symmetry breaking in the gravitational sector via a given potential. In the three dimensional spacetime, the
 action of Einstein-bumblebee gravity is \cite{ding2021},
\begin{eqnarray}
\mathcal{S}=
\int d^3x\sqrt{-g}\Big[\frac{R-2\Lambda}{2\kappa}+\frac{\varrho}{2\kappa} B^{\mu}B^{\nu}R_{\mu\nu}-\frac{1}{4}B^{\mu\nu}B_{\mu\nu}
-V(B_\mu B^{\mu}\mp b^2)+\mathcal{L}_M\Big], \label{action}
\end{eqnarray}
where $R$ is Ricci scalar and $\Lambda$ is the cosmological constant. $\kappa=4\pi G/c^4$ for the three dimensions, where $G$ is the Newtonian constant.


The coupling constant $\varrho$ dominates the non-minimal gravity interaction to bumblebee field $B_\mu$. The term $\mathcal{L}_M$ represents possible interactions with matter or external currents.
The constant $b$ is a real positive constant. The potential $V(B_\mu B^{\mu}\mp b^2)$ triggers Lorentz and/or $CPT$ (charge, parity and time) violation. It gives a nonzero vacuum expectation value (VEV) for bumblebee field $B_{\mu}$ indicating that the vacuum of this model obtains a prior direction in the spacetime. This potential has a minimum at $B^{\mu}B_{\mu}\pm b^2=0$ and $V'(b_{\mu}b^{\mu})=0$ to ensure the destroying the $U(1)$ symmetry, where the field $B_{\mu}$ acquires a nonzero VEV, $\langle B^{\mu}\rangle= b^{\mu}$. Another vector $b^{\mu}$ is a function of the spacetime coordinates and has a constant value $b_{\mu}b^{\mu}=\mp b^2$, where $\pm$ signs mean that $b^{\mu}$ is timelike or spacelike, respectively.
The bumblebee field strength is
\begin{eqnarray}
B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.
\end{eqnarray}
This antisymmetry of $B_{\mu\nu}$ implies the constraint \cite{bluhm}
\begin{eqnarray}
\nabla ^\mu\nabla^\nu B_{\mu\nu}=0.
\end{eqnarray}

Varying the action (\ref{action}) with respect to the metric yields the gravitational field equations
\begin{eqnarray}\label{einstein0}
G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}^B+\kappa T_{\mu\nu}^M,
\end{eqnarray}
where $G_{\mu\nu}=R_{\mu\nu}-g_{\mu\nu}R/2$, and the bumblebee energy momentum tensor $T_{\mu\nu}^B$ is
\begin{eqnarray}\label{momentum}
&&T_{\mu\nu}^B=B_{\mu\alpha}B^{\alpha}_{\;\nu}-\frac{1}{4}g_{\mu\nu} B^{\alpha\beta}B_{\alpha\beta}- g_{\mu\nu}V+
2B_{\mu}B_{\nu}V'\nonumber\\
&&+\frac{\varrho}{\kappa}\Big[\frac{1}{2}g_{\mu\nu}B^{\alpha}B^{\beta}R_{\alpha\beta}
-B_{\mu}B^{\alpha}R_{\alpha\nu}-B_{\nu}B^{\alpha}R_{\alpha\mu}\nonumber\\
&&+\frac{1}{2}\nabla_{\alpha}\nabla_{\mu}(B^{\alpha}B_{\nu})
+\frac{1}{2}\nabla_{\alpha}\nabla_{\nu}(B^{\alpha}B_{\mu})
-\frac{1}{2}\nabla^2(B^{\mu}B_{\nu})-\frac{1}{2}
g_{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(B^{\alpha}B^{\beta})\Big].
\end{eqnarray}
The prime denotes differentiation with respect to the argument,
\begin{eqnarray}
V'=\frac{\partial V(x)}{\partial x}\Big|_{x=B^{\mu}B_{\mu}\pm b^2}.
\end{eqnarray}
Varying instead with respect to the the bumblebee field generates the bumblebee equations of motion (supposing that there is no coupling between the bumblebee field and $\mathcal{L}_M$),
\begin{eqnarray}\label{motion}
\nabla ^{\mu}B_{\mu\nu}=2V'B_\nu-\frac{\varrho}{\kappa}B^{\mu}R_{\mu\nu}.
\end{eqnarray}

The contracted Bianchi identities ($\nabla ^\mu G_{\mu\nu}=0$) lead to conservation of the total energy-momentum tensor
\begin{eqnarray}\label{}
\nabla ^\mu T_{\mu\nu}=\nabla ^\mu\big( T^B_{\mu\nu}+T^M_{\mu\nu}\big)=0.
\end{eqnarray}




We suppose that there is no matter field and the bumblebee field is frosted at its VEV like in Refs \cite{casana,bertolami}, i.e., it is
\begin{eqnarray}\label{bbu}
B_\mu=b_\mu.
\end{eqnarray}
And the potential has a linear function form \begin{eqnarray}
V=\frac{\lambda}{2}(B_\mu B^\mu-b^2)^2,
\end{eqnarray}
where $\lambda$ is a constant. So this potential is $V=0$ and  its derivative $V'=0$ under the condition (\ref{bbu}).  Then the first two terms in Eq. (\ref{momentum}) are like those of the electromagnetic field, the only distinctive are the coupling items to Ricci tensor. Under this condition,  Eq. (\ref{einstein0}) leads to gravitational field equations \cite{ding2021}
\begin{eqnarray}\label{bar}
G_{\mu\nu}=\kappa (\lambda b_\mu b_\nu+b_{\mu\alpha}b^{\alpha}_{\;\nu}-\frac{1}{4}g_{\mu\nu} b^{\alpha\beta}b_{\alpha\beta})+\varrho\Big(\frac{1}{2}
g_{\mu\nu}b^{\alpha}b^{\beta}R_{\alpha\beta}- b_{\mu}b^{\alpha}R_{\alpha\nu}
-b_{\nu}b^{\alpha}R_{\alpha\mu}\Big)
+\bar B_{\mu\nu},
\end{eqnarray}
with
\begin{eqnarray}\label{barb}
&&\bar B_{\mu\nu}=\frac{\varrho}{2}\Big[
\nabla_{\alpha}\nabla_{\mu}(b^{\alpha}b_{\nu})
+\nabla_{\alpha}\nabla_{\nu}(b^{\alpha}b_{\mu})
-\nabla^2(b_{\mu}b_{\nu})-g_{\mu\nu}\nabla_\alpha\nabla_\beta(b^\alpha b^\beta)\Big].
\end{eqnarray}
\section{Rotating BTZ-like black hole solution}
The static spherically symmetric black hole metric in the four dimensional spacetime  has the form
\begin{eqnarray}\label{metric}
&&ds^2=-e^{2\Phi(r)}dt^2+e^{2\psi(r)}dr^2+r^2\big[\Omega(r) dt+d\phi\big]^2,
\end{eqnarray}
where

In the present study, we pay attention to that the bumblebee field has a radial vacuum energy expectation because that the spacetime curvature has a strong radial variation, on the contrary that the temporal changes are very slow. So the bumblebee field is supposed to be spacelike($b_\mu b^\mu=$ positive constant) as that
\begin{eqnarray}\label{bu}
b_\mu=\big(0,b_0e^{\psi(r)},0\big),
\end{eqnarray}
where $b_0$ is a positive constant.
Then the bumblebee field strength is
\begin{eqnarray}
b_{\mu\nu}=\partial_{\mu}b_{\nu}-\partial_{\nu}b_{\mu},
\end{eqnarray}
whose components are all zero. And their divergences are all zero, i.e.,
\begin{eqnarray}
\nabla^{\mu}b_{\mu\nu}=0.
\end{eqnarray}
From the equation of motion (\ref{motion}), we have
\begin{eqnarray}
b^{\mu}R_{\mu\nu}=0\label{motion2}.
\end{eqnarray}
The gravitational field equations (\ref{bar}) become
\begin{eqnarray}\label{}
G_{\mu\nu}+\Lambda g_{\mu\nu}=\bar B_{\mu\nu}.
\end{eqnarray}

 For the metric (\ref{metric}), the nonzero components of Einstein tensor $G_{\mu\nu}$ and the bumblebee tensor $\bar B_{\mu\nu}$ are shown in the appendix.
Substituting the equation $G_{22}+\Lambda g_{22}=\bar B_{22}$ into $G_{02}+\Lambda g_{02}=\bar B_{02}$, one can obtain
\begin{eqnarray}\label{2202}
(1+\ell)\big[(r\Omega''+3\Omega')-r\Omega'(\Phi'+\psi')\big]=0,
\end{eqnarray}
where $\ell=\varrho b_0^2$ and the symbol $\prime$ means the derivative with their argument.
By using the motion equation (\ref{motion2}), one can obtain
\begin{eqnarray}\label{r11}
R_{11}=\frac{\psi'}{r}-(\Phi''+\Phi'^2-\Phi'\psi')+\frac{1}{2}e^{-2\Phi}r^2\Omega'^2=0.\end{eqnarray}
Substituting the equations $G_{22}+\Lambda g_{22}=\bar B_{22}$ and (\ref{r11}) into $G_{00}+\Lambda g_{00}=\bar B_{00}$, one can obtain
\begin{eqnarray}\label{2200}
(1+\ell)\left[r\Omega(r\Omega''+3\Omega')+\frac{1}{4}r^2\Omega'^2
-e^{2\Phi}\frac{\psi'}{r}-r^2\Omega\Omega'(\Phi'+\psi')\right]+e^{2\Phi+2\psi}\Lambda=0.
\end{eqnarray}
Substituting the equation (\ref{r11}) into $G_{11}+\Lambda g_{11}=\bar B_{11}$, one can obtain
\begin{eqnarray}\label{r11rr}
(1+\ell)\left[e^{2\Phi}\frac{\Phi'}{r}+\frac{1}{4}r^2\Omega'^2\right]+e^{2\Phi+2\psi}\Lambda=0.
\end{eqnarray}
The above two equations (\ref{2200}) and (\ref{r11rr}) can give that
\begin{eqnarray}\label{1100}
r\Omega\big[(r\Omega''+3\Omega')-r\Omega'(\Phi'+\psi')\big]-\frac{1}{r}e^{2\Phi}(\Phi'+\psi')=0.
\end{eqnarray}
Then the Eqs. (\ref{2202}) and (\ref{1100}) can give the following equations
\begin{eqnarray}
&&\Phi'+\psi'=0,\label{pp}\\&&r\Omega''+3\Omega'=0\label{omega}.
\end{eqnarray}
Eq. (\ref{omega}) can give
\begin{eqnarray}
\Omega=-\frac{j}{2r^2},
\end{eqnarray}
where $j$ is an integral constant relating to its angular momentum.
From the Eq. (\ref{pp}), one can assume that $e^{2\Phi}=f(r)$ and  $e^{2\psi}=C/f(r)$, where $C$ is a constant to be determined. Substituting it into the equations $G_{11}+\Lambda g_{11}=\bar B_{11}$ and $G_{22}+\Lambda g_{22}=\bar B_{22}$, one can obtain that
\begin{eqnarray}
(1+\ell)^2\big(2f'+\frac{j^2}{r^3}\big)+4C\Lambda r=0,
\end{eqnarray}
which can give that
\begin{eqnarray}
f(r)= -m-\frac{C\Lambda r^2}{(1+\ell)^2}+\frac{j^2}{4r^2},
\end{eqnarray}
where $m$ is also an integral constant relating to its mass.
In order to get a BTZ-like solution, we choose the constant $C=(1+\ell)^2$, then the last rotating BTZ-like black hole solution is
\begin{eqnarray}\label{metricb}
ds^2=-f(r)dt^2+\frac{(1+\ell)^2}{f(r)}dr^2+r^2(d\phi-\frac{j}{2r^2}dt)^2,\qquad f(r)= -m-\Lambda r^2+\frac{j^2}{4r^2}.
\end{eqnarray}
This metric represents a purely radial LV black hole solution in a 3D spacetime. Its Kretschmann scalar is
\begin{eqnarray}
R_{\mu\nu\rho\tau}R^{\mu\nu\rho\tau}=\frac{12\Lambda^2}{(1+\ell)^4},
\end{eqnarray}
which is a finite constant in the whole spacetime as like as the original BTZ black hole, so there is also no curvature singularity at the origin. Its total energy-momentum tensor of the metric (\ref{metricb}) is
\begin{equation}\label{totalmom}
\!\!\!\!\!\!\!\! T_\nu^\mu=\frac{1}{\kappa}\left(
\begin{array}{ccc}
-\epsilon  & 0  & 0 \\
0  & p_r &0    \\
0  & 0  & p_t
\end{array}
\right),
\end{equation}
where $\epsilon$ is the energy density, $p_r$ is the radial pressure and $p_t$ is the tangential pressure, which read
\begin{eqnarray}
-\epsilon=p_r=p_t=\frac{-\Lambda}{(1+\ell)^2}.
\end{eqnarray}

It has two horizons: inner(Cauchy) horizon $r_+$ and outer(event) horizon $r_-$ which can be read from $f(r_{\pm})=0$,
\begin{eqnarray}
r_{\pm}=\sqrt{\frac{2}{-\Lambda}}\left(\sqrt{M+\sqrt{-\Lambda}J}\pm\sqrt{M-\sqrt{-\Lambda}J}\right),
\end{eqnarray}
where $M=m/8$ is its ADM mass and $J=j/8$ is its angular momentum. Like Kerr black hole, it has an ergosphere $r_{erg}$ which can be read from the time-time component $g_{tt}=0$,
\begin{eqnarray}
r_{erg}=\sqrt{r_+^2-r_-^2}.
\end{eqnarray}
Its surface gravity of the event horizon $\kappa$ is
\begin{eqnarray}
\kappa=-\frac{1}{1+\ell}\Big(\Lambda r_++\frac{16J^2}{r_+^3}\Big),
\end{eqnarray}
which can be read from the formula \cite{ding2020},
\begin{eqnarray}
\kappa=-\frac{1}{2}\lim_{r\rightarrow r_+}\sqrt{\frac{1}{g_{rr}X}}\frac{dX}{dr}, \qquad X=g_{tt}-\frac{g_{t\phi}^2}{g_{\phi\phi}}.
\end{eqnarray}
It is easy to prove that in the present case, the function $X=-f(r)$.



\section{Central charge of the dual CFT}

According to the holographic theory \cite{witten}, one may expect that each sector of 3-dimensional  gravity which is either asymptotically anti-de Sitter(AdS) or AdS-like, there exists a dual 2-dimensional conformal field theory(CFT) that might live on the boundary of this AdS spacetime. From the black hole thermodynamics aspect, Yekta \cite{yekta} obtained the central charges of the CFT which can be constructed using the thermodynamics of the outer and inner horizons. He found the result was in complete agreement with that via the method of asymptotic symmetry group analysis \cite{liu1,liu2,liu3}. In this section, we study the central charges by using the thermodynamic method of black hole/CFT correspondence has been proposed in \cite{chen1,chen2,chen3}.

From the horizon equations $f(r_\pm)=0$, one can rewrite the black mass as
\begin{eqnarray}\label{}
M=-\frac{1}{8}\Lambda r_\pm^2+\frac{2J^2}{r_\pm^2}.
\end{eqnarray}
The temperature of outer horizon $T_+$ is defined by
\begin{eqnarray}\label{tp}
T_+=-\frac{1}{2\pi(1+\ell)}\Big(\Lambda r_++\frac{16J^2}{r_+^3}\Big)=\frac{-\Lambda}{2\pi(1+\ell)r_+}(r_+^2-r_-^2).
\end{eqnarray}
The temperature of the inner horizon $T_-$ should be a geometrical positive quantity and is constant over the inner horizon \cite{castro}, so it should be,
\begin{eqnarray}\label{tm} T_-=\frac{1}{2\pi(1+\ell)}\Big(\Lambda r_-+\frac{16J^2}{r_-^3}\Big)=\frac{-\Lambda}{2\pi(1+\ell)r_-}(r_+^2-r_-^2).
\end{eqnarray}
There are three thermodynamical quantities: entropy $S_\pm$ of the inner and outer horizons, angular momentum $J$ and pressure $P=-\Lambda/8\pi(1+\ell)^2$, i.e., mass $M$ can be expressed by the function of these quantities $M=M(S,J,P)$. The entropies are as following
\begin{eqnarray}\label{spm}
S_\pm=\int\frac{dM}{T_\pm}=\frac{1}{2}(1+\ell)\pi r_\pm,
\end{eqnarray}
the angular velocities $\Omega_\pm$ and the thermodynamical volumes $V_\pm$ are
\begin{eqnarray}\label{}
\Omega_\pm=\left(\frac{\partial M}{\partial J}\right)_{S,P}=\sqrt{8\pi P}\frac{r_\mp}{r_\pm},\qquad V_\pm=\left(\frac{\partial M}{\partial P}\right)_{S,J}=(1+\ell)^2\pi r_\pm^2.
\end{eqnarray}
So there exists the first law of thermodynamics and the Smarr formula for the outer horizon
\begin{eqnarray}\label{}
dM= T_+ dS_++\Omega_+ dJ+V_+ dP,\qquad 0\cdot M= T_+ S_+-2PV_++\Omega_+ J.
\end{eqnarray}
For the inner horizon, the Killing vector is spacelike inside the event horizon, then one should assign negative energy $-M$ to the inner horizon \cite{castro}, similar to the negative energies within the ergosphere. So  the first law of thermodynamics and the Smarr formula for the inner  horizon are
\begin{eqnarray}\label{}
dM=- T_- dS_-+\Omega_- dJ+V_- dP,\qquad 0\cdot M=- T_- S_--2PV_-+\Omega_- J.
\end{eqnarray}

It is obviously that the equality $T_+S_+=T_-S_-$ is true from the relations (\ref{tp}), (\ref{tm}) and (\ref{spm}). This means that the entropy product of the inner and outer horizon
\begin{eqnarray}\label{}
S_+S_-=(1+\ell)^2\frac{\pi^2J}{\sqrt{-\Lambda}},
\end{eqnarray}
is universal (mass-independent). So the central charges of the left- and right-moving sectors in the dual CFT must be the same,
 \begin{eqnarray}\label{}
c_L=c_R=6\frac{d}{dJ}\left(\frac{S_+S_-}{4\pi^2}\right)=\frac{3(1+\ell)^2}{2\sqrt{-\Lambda}}.
\end{eqnarray} These results of 2-dimensional CFT on the boundary can be used to describe 3-dimensional AdS gravity in the bulk.

Defining the left-moving temperature $T_L$ and right-moving temperature $T_R$ of the dual CFT,
 \begin{eqnarray}\label{}
T_L=\left(\frac{1}{T_+}+\frac{1}{T_-}\right)^{-1}=\frac{-\Lambda}{2\pi(1+\ell)}(r_+-r_-),\qquad T_R=\left(\frac{1}{T_+}-\frac{1}{T_-}\right)^{-1}=\frac{-\Lambda}{2\pi(1+\ell)}(r_++r_-),
\end{eqnarray}
then the entropies of the inner and outer horizon can be represented by
 \begin{eqnarray}\label{}
S_\pm=\frac{\pi^2}{3\sqrt{-\Lambda}}(c_LT_L\pm c_RT_R).
\end{eqnarray}
In addition, defining the energies of the left- and right-moving sectors of the dual CFT as
 \begin{eqnarray}\label{}
E_L=\frac{\pi^2}{6\sqrt{-\Lambda}}c_LT_L^2=\frac{M-\sqrt{-\Lambda}J}{2},\qquad E_R=\frac{\pi^2}{6\sqrt{-\Lambda}}c_RT_R^2=\frac{M+\sqrt{-\Lambda}J}{2},
\end{eqnarray}
then the black hole mass $M$ and angular momentum $J$ can be represented by
 \begin{eqnarray}\label{}
M=E_L+E_R, \qquad J=\frac{1}{\sqrt{-\Lambda}}(E_L-E_R).
\end{eqnarray}
In this way, we use the central charges on the boundary to reappear  the entropy, mass and angular momentum in the bulk.



\section{Summary}

In this paper, we have studied Einstein gravity coupled to a bumblebee field in a 3-dimensional spacetime. We obtain an exactly rotating BTZ-like black hole solution. The bumblebee field doesn't affect the locations of the black hole horizon and ergosphere.

This black hole is different from 4-dimensional Schwarzschild-like \cite{casana} or Kerr-like \cite{ding2020,ding2021} black holes that it cannot be asymptotically flat and has no curvature singularity at the origin. It is asymptotically AdS and has isotropic pressure, but is different from the 4- or higher dimensional AdS-like black holes \cite{maluf,ding2023} whose pressure are anisotropic.

We study the AdS/CFT correspondence of this black hole, find
that the entropy product of its inner and outer horizons is universal. So the central charges of the
dual CFT on the boundary can be obtained via the thermodynamic method, and they can reappear
black hole mass and angular momentum in the bulk.




\begin{acknowledgments} This work was supported by the Scientific Research Fund of the Hunan Provincial Education Department under No. 19A257, the National Natural Science Foundation (NNSFC)
of China (grant No. 11247013), Hunan Provincial Natural Science Foundation of China (grant No. 2015JJ2085 and No. 2020JJ4284).
\end{acknowledgments}

