Mass ladder operators and quasinormal modes of the static BTZ-like black hole in the Einstein-bumblebee gravity

As the useful tools, the ladder operators can provide a deep insight into a system. In quantum mechanics, one find that the ladder operators relate families of solutions of the Schrödinger equation without detailed knowledge of the solutions. Extending the investigation to the massive Klein-Gordon equation in the curved spacetime, Cardoso et al. constructed generalized ladder operators for massive Klein-Gordon scalar fields in spacetimes with the conformal symmetry, and showed that these operators can map a solution of the massive Klein-Gordon equation into that with different mass squared [1, 2]. Thus, these operators are called the mass ladder operators. In Ref. [3], Mück used a scalar curvature term to replace the mass squared term and built the generalized ladder operators for the Klein-Gordon equation with a scalar curvature term. More recently, Katagiri and Kimura applied the ladder operators to investigate the quasinormal modes (QNMs) of a massive Klein-Gordon field in the Bañados-Teitelboim-Zanelli (BTZ) spacetime, which shows that the ladder operators can change indices of QNM overtones in different boundary conditions [4].

The aforementioned works on the ladder operators only focus on the Einstein gravity. It is of great interest to generalize the investigation to the Einstein-bumblebee gravity since it is one of simple and effective theories with the Lorentz violation [5]. In the bumblebee model, the Lorentz violation arises from the dynamics of the bumblebee field, i.e., a single vector or axial-vector field $ B_\mu $. In 2018, Casana and Cavalcante got an exact Schwarzschild-like black hole solution in this Einstein-bumblebee gravity, and investigated some classical tests, including the advance of the perihelion, bending of light and Shapiro's time-delay [6]. In 2020, Ding et al. obtained an exact Kerr-like solution and investigated the shadow of this Kerr-like black hole [7]. For a high dimensional spacetime, the authors of Ref. [8] got an exact AdS-like black hole solution, and investigated the thermodynamics and phase transitions of this high dimensional Schwarzschild-AdS-like black hole. For three-dimensional spacetime, Ding et al. gained an exact rotating BTZ-like black hole solution in the Einstein-bumblebee gravity theory, and obtained the central charges of the dual conformal field theory (CFT) on the boundary by using the thermodynamic method [9]. Then, Chen et al. investigated the quasinormal modes of a scalar perturbation around this rotating BTZ-like black hole in the Einstein-bumblebee gravity, and found that the Lorentz symmetry breaking parameter imprints only in the imaginary parts of the quasinormal frequencies [10]. Along this line, the Einstein-bumblebee gravity has been explored extensively on various aspects, including the exact solutions [11−19], the geodesics and gravitational lensing [20−24], the black hole shadow [25−29], the quasinormal modes and stability [30−37], and the energy extraction [38].

In this work, we construct the mass ladder operators for the static BTZ-like black hole obtained in Ref. [9] from the Einstein-bumblebee gravity, and probe the QNM frequencies of the mapped modes by mass ladder operators for the scalar perturbation under the Dirichlet and Neumann boundary conditions. We will find that the mass ladder operators, which depend on the Lorentz symmetry breaking parameter s in the Einstein-bumblebee gravity, can map a solution of the massive Klein-Gordon equation for the BTZ-like black hole into that with different mass squared and change indices of QNM overtones. Particularly, we find that there exists a threshold value $ s_c $ for all the modes, i.e., the quasinormal frequency has an imaginary part only if the parameter $ s<s_c $.

This work is organized as follows. In Sec. II, we first review the static BTZ-like black hole in the Einstein-bumblebee gravity, and then calculate the QNMs and frequencies for a massive perturbation of the static BTZ-like black hole with the Dirichlet and Neumann boundary conditions. In Sec. III, we construct the mass ladder operators in the static BTZ-like spacetime, discuss the relationship between the mass ladder operators and the QNMs, and analyze the effect of the overtone number n and the Lorentz symmetry breaking parameter s on the mapped modes. Finally, in Sec. IV, we summarize our work.

Let us begin by briefly reviewing the static BTZ-like spacetime in the Einstein-bumblebee gravity obtained in [9]. The action in this theoretical model with a negative cosmological constant $ \Lambda = -\frac{1}{l^2} $ is given by [39−41]

Here R describes the Ricci scalar, and κ is a constant that has a relationship with the three-dimensional Newton's constant G, given by $ \kappa = 8\pi G $. $ B_{\mu\nu} = \partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu} $ is the strength of bumblebee field $ B_\mu $, and $ \varrho $ is a coupling constant with the dimension of $ M^{-1} $ in this model. In addition, we note that the potential V has a minimum at $ B^{\mu}B_{\mu}\pm b^2 = 0 $, where b is a real positive constant and the signs $ \pm $ determine whether the field $ b_\mu $ is timelike or spacelike. This minimum yields a nonzero vacuum value $ \langle B_{\mu}\rangle = b_\mu $ with $ b_\mu b^\mu = \mp b^2 $, which leads to the breaking of the $ U(1) $ symmetry. It should be noted that the nonzero vector background $ b_\mu $ leads to the violation of Lorentz symmetry [5, 39−41].

From the action (1), one can get the static BTZ-like black hole solution in the Einstein-bumblebee gravity [9]

with

where M denotes the mass of the black hole, and $ s = \xi b^2 $ represents the spontaneous breaking of Lorentz symmetry due to the Einstein-bumblebee vector field with the form $ b_{\mu} = (0,b\xi, 0, 0) $. Considering the determinant of the metric $ g = -(1+s)r^2 $, we find that the metric becomes degenerate at $ s = -1 $. In the following analysis, we will consider the constraint $ s>-1 $ and compare with the results for the case of $ s = 0 $, i.e., without the Lorentz symmetry breaking. The Hawking temperature of this BTZ-like black hole is

with the horizon $ r_{H} = \sqrt{M}l $. For convenience, we will scale $ l = 1 $ in the numerical calculation.

In the static BTZ-like spacetime, the massive scalar field evolves according to

where μ is the mass of the scalar field. Assuming that the massive scalar perturbation Ψ has the form

we can obtain the radial equation

which can be solved in terms of the hypergeometric functions [42]. We introduce a variable $ z = \dfrac{r^2-r_H^2}{r^2} $ and then the radial equation (7) can be rewritten as

where A, B and C have forms

Rewriting the radial function $ \psi(z) $ as the form

we find that the function $ F(z) $ satisfies the standard hypergeometric equation

with

where the scalar field mass must obey $ \mu^2\geqslant\mu^2_{BF} $ with the Breitenlohner-Freedman (BF) bound $ \mu_{BF}^2 = -\dfrac{1}{l^2(1+s)} $. Thus, the general solution of the radial equation (8) can be given by a linear combination of hypergeometric functions F, i.e.,

where A and B are two constants of integration. By imposing the ingoing-wave condition at the black hole horizon, we obtain $ B = 0 $ and then the solution (15) has a more simple form

To probe the asymptotic behavior of $ \psi(z) $ at infinity as $ r\rightarrow\infty $ (i.e., $ z\rightarrow 1 $), we can expand the hypergeometric functions at $ 1-z $ [42], i.e.,

Thus, the general solution (16) can be expressed as

where the constants $ A_{\rm{I}} $ and $ A_{\rm{II}} $ are

The functions $ \psi^{(D)}(r) $ and $ \psi^{(N)}(r) $ respectively denote the forms of the solution (16) satisfied the Dirichlet boundary condition and Neumann boundary condition at infinity [4]. The forms of $ \psi^{(D)}(r) $ and $ \psi^{(N)}(r) $ are

With the Dirichlet boundary condition, we find $ a = -n $ or $ b = -n $, and then obtain the quasinormal frequencies for the massive scalar perturbation

The signs $ + $ and $ - $ respectively denote the left-moving and the right-moving modes. As the Lorentz symmetry breaking parameter s increases, the absolute value of the imaginary part of $ \omega^{(D)} $ decreases for different overtone number n [10], which is also shown in Fig. 1(a). Moreover, we also note that all the imaginary parts of $ \omega^{(D)} $ are negative, which means that the black hole is stable under the scalar perturbation with the Dirichlet boundary condition. And the corresponding wave-function $ \Psi^{(D)} $ at infinity has the asymptotic behavior

Figure 1.  (color online)The relationship between the imaginary part of the QNM frequencies and the Lorentz symmetry breaking parameter s with different overtone numbers n in different boundary conditions, where we have taken $ r_H = 1 $ and $ \mu^2 = -0.75 $.

Similarly, for the Neumann boundary condition, one has $ c-a = -n $ or $ c-b = -n $, and then the quasinormal frequencies for the scalar field are

As shown in Fig. 1(b), similar to the case of the Dirichlet boundary condition, the imaginary part of $ \omega^{(N)} $ increases for different overtone numbers n with the increase of the parameter s under the Neumann boundary condition, except the fundamental mode $ n = 0 $, i.e., the scalar field without nodes, where the imaginary part of $ \omega^{(N)} $ decreases slowly with the increase of s. It should be noted that the asymptotic behavior of the wave-function $ \Psi^{(N)} $ at infinity becomes

For the d-dimensional spacetime $ (M,g_{\mu\nu}) $ with the symmetry under the conformal transformation $ g_{\mu\nu}\rightarrow g'_{\mu\nu} = e^{2Q}g_{\mu\nu} $ and $ Q = \frac{1}{d}\nabla_\sigma\zeta^\sigma $, there exists a conformal Killing vector (CKV) $ \zeta^\mu $ satisfied the conformal Killing equation

Since the closed conformal Killing vector (CCKV) $ \zeta^{\mu} $ meets an extra condition

it satisfies the reduced Killing equation

Recent investigations show a CCKV $ \zeta^{\mu} $ is an eigenvector of Ricci tensor with a constant eigenvalue χ [1, 2], i.e.,

One can introduce a one-parameter family of operators

where $ {\cal{L}}_{\zeta} $ denotes the Lie derivative with respect to $ \zeta^\mu $ and k is a parameter. The commutation relation between the operator $ D_{k} $ and the d'Alembertian is given by

Acting the commutator on a scalar field ϕ, one can obtain

Obviously, the operator $ D_k $ maps a scalar field ϕ with mass squared $ \mu^2 = -\chi k(k+d-1) $ to another scalar field $ D_k\phi $ with a new mass squared $ \mu'^2 = -\chi(k-1)(k+d-2) $, so $ D_k $ can be regarded as a mass ladder operator, which shifts the parameter k to $ k-1 $ in the mass squared term. Thus, such kind of operators is called as the mass lowering operator $ D_{k_-} $. Similarly, one can introduce the mass raising operator $ D_{k_+} $, which shifts the parameter k to $ k+1 $ in the mass squared term.

For the BTZ-like spacetime in the Einstein-bumblebee gravity (2), we obtain four independent CCKVs

and the corresponding four mass ladder operators

Obviously, the mass ladder operators depend on the Lorentz symmetry breaking parameter s in the Einstein-bumblebee gravity. In an AdS$ _d $-like spacetime, the BF-bound $ \mu^2_{BF} $ for a scalar field can be expressed as [1, 4]

For the BTZ-like black hole spacetime in the Einstein-bumblebee gravity (2), we find that χ is exactly equal to the BF-bound $ \mu^2_{BF} = -\dfrac{1}{l^2(1+s)} $, and then the commutation relation (31) with the d'Alembertian becomes

This means that Eq. (32) can be further simplified as

As in [1, 2], $ D_{i,k} $ maps a solution Ψ to the Klein-Gordon equation with the mass squared $ \mu^2 = \dfrac{k (k+2)}{l^2(1+s)} $ into another solution $ D_{i,k}\Psi $ with the mass squared $ \mu^2 = \dfrac{(k-1)(k+1)}{l^2(1+s)} $, namely, shifts k to $ k-1 $ in the mass squared term.

To require that the characteristic exponents with the mass $ \Delta_{\pm} = \frac{1}{2}[1\pm\sqrt{1+\mu^{2}l^{2}(1+s)}] $ in (23) and (25) are real, we here only focus on the mass squared $ \mu^2\geqslant\mu^2_{BF} $. As in [4], one can introduce a parameter ν, which obeys

This means that $ \nu = -1\pm\sqrt{1+\mu^2l^2(1+s)} $. And the relationship between $ \mu^2l^2(1+s) $ and ν for different Lorentz symmetry breaking parameters s is presented in Fig. 2, which shows that the function $ \mu^2l^2(1+s) $ is symmetric about $ \nu = -1 $. Thus, without loss of generality, we only focus on the right part of the parabola where $ \nu = -1+\sqrt{1+\mu^2l^2(1+s)} $ with $ \nu\in[-1,\infty) $. For a given parameter ν, there are two values of k satisfied $ k(k+2) = \nu(\nu+2) $, i.e.,

Figure 2.  (color online)The term $ \mu^2l^2(1+s) $ as a function of ν for different Lorentz symmetry breaking parameters s.

The corresponding mass ladder operators $ D_{i,k_+} $ and $ D_{i,k_-} $ respectively shift ν to $ \nu+1 $ and $ \nu-1 $.

Now we are in a position to discuss the mapped solutions resulting from the mass ladder operators acting on the original solution Ψ. Since the operators $ D_{2,k\pm} $ and $ D_{3,k\pm} $ are non-global operators, they are not globally smooth in the φ direction. As a result, they change the quantum number m of the scalar field. However, the operators $ D_{0,k\pm} $ and $ D_{1,k\pm} $ can directly affect the frequencies of the scalar perturbation. Therefore, as in [4], we only focus on the effects of $ D_{0,k\pm} $ and $ D_{1,k\pm} $ on the general solution of the scalar field under different boundary conditions.

The general solution (16) of the scalar field can be rewritten as

The asymptotic behavior of the general solution near the horizon is

Under the Dirichlet or Neumann boundary conditions at infinity, the above general solution has the asymptotic behavior

Acting the mass ladder operators $ D_{0,k\pm} $ and $ D_{1,k\pm} $ on the exact solution (46), one can obtain the asymptotic behaviors of the mapped solutions near the horizon

where $ C_{0,k\pm} $ and $ C_{1,k\pm} $ are two constants

Under the Dirichlet boundary condition, the asymptotic behaviors of the mapped solutions acted by the mass ladder operators $ D_{0,k\pm} $ and $ D_{1,k\pm} $ at infinity can be expressed as

where

Combining with Eqs. (50), (51) and Eqs. (54)-(57), one can find that $ D_{0,k\pm} $ and $ D_{1,k\pm} $ respectively shift ω to $ \omega+i\dfrac{r_H}{l^2\sqrt{1+s}} $ and $ \omega-i\dfrac{r_H}{l^2\sqrt{1+s}} $, while they keep m unchanged. Thus, the quasinormal frequencies of the mapped solutions of the scalar field by the mass ladder operators $ D_{0,k\pm} $ and $ D_{1,k\pm} $ are

with

In particular, when the mass ladder operator $ D_{0,k+} $ acts on the fundamental mode, the mapped mode vanishes rather than becomes a “negative overtone mode” due to the restriction imposed by (58) as in [4]. Fig. 3 shows the change of quasinormal frequencies of the mapped modes $ D_{0,k\pm}\Psi^{(D)} $ and $ D_{1,k\pm}\Psi^{(D)} $ with the parameter s, which is similar to that of the original modes $ \Psi^{(D)} $. Notice that, for all the modes with $ \mu^2 = -0.75 $ and $ l = 1 $, there exists a threshold value $ s_{c} = \frac{1}{3} $, corresponding to a vertical dashed line in each panel, which means that all the modes stop increasing their imaginary parts when they reach this vertical line. As a matter of fact, the threshold value $ s_{c} = -\left(1+\dfrac{1}{\mu^2 l^2}\right) $ only depends on the mass of the scalar field, and increases with the increase of the mass squared term $ \mu^2 $, just as shown in Fig. 4 with $ \mu^2\leqslant0 $.

Figure 3.  (color online)The relationship between the imaginary part $ {\rm{Im}} $$ [\omega^{(D)}] $ and the Lorentz symmetry breaking parameter s respectively for $ \Psi^{(D)} $, $ D_{0,k\pm}\Psi^{(D)} $ and $ D_{1,k\pm}\Psi^{(D)} $ with different overtone numbers n, where we have taken $ r_H = 1 $ and $ \mu^2 = -0.75 $.

Figure 4.  (color online) The relationship between the imaginary part $ {\rm{Im}} $$ [\omega^{(D)}] $ and the Lorentz symmetry breaking parameter s respectively for $ \Psi^{(D)} $, $ D_{0,k_-}\Psi^{(D)} $ and $ D_{1,k\pm}\Psi^{(D)} $ with different masses $ \mu^2 $, where we have taken $ n = 0 $ and $ r_H = 1 $.

Under the Neumann boundary condition, we can obtain the asymptotic behaviors of the mapped solutions by mass ladder operators $ D_{0,k\pm} $ and $ D_{0,k\pm} $ at infinity

where

Combining with Eqs. (50), (51) and Eqs. (63)-(66), one can find that $ D_{0,k\pm} $ and $ D_{1,k\pm} $ respectively shift ω to $ \omega+i\dfrac{r_H}{l^2\sqrt{1+s}} $ and $ \omega-i\dfrac{r_H}{l^2\sqrt{1+s}} $, while they keep m unchanged. Thus, the quasinormal frequencies of the mapped solutions of the scalar field by the mass ladder operators $ D_{0,k\pm} $ and $ D_{1,k\pm} $ are

with

Similarly, under the Neumann boundary condition, there is also no “negative overtones” generated from the fundamental modes for $ D_{0,k_-} $ as in the case with the Dirichlet boundary condition. For the fundamental mode with $ \mu^2<0 $, the mass ladder operators do not change the BF-bound $ \mu^2_{BF} $. Furthermore, as in the Dirichlet boundary condition, we find that there also exists the same threshold value $ s_c = \frac{1}{3} $ for the mapped modes and the original modes. The threshold value $ s_c $ for all the mapped modes with the same mass squared term is a constant. Thus, the mass ladder operators keep the threshold value $ s_c $ for the mapped modes.

Moreover, as shown in Fig. 6, the threshold value $ s_c $ increases with the increase of $ \mu^2 $. It should be noted that the imaginary parts of the quasinormal frequencies of the mapped modes caused by $ D_{0,k_+}\Psi^{(N)} $ are all positive, which indicates that all the fundamental modes of $ D_{0,k_+}\Psi^{(N)} $ are unstable.

Figure 6.  (color online)The relationship between the imaginary part $ {\rm{Im}} $$ [\omega^{(N)}] $ and the Lorentz symmetry breaking parameter s respectively for $ \Psi^{(N)} $, $ D_{0,k_+}\Psi^{(N)} $ and $ D_{1,k\pm}\Psi^{(N)} $ with different masses $ \mu^2 $, where we have taken $ n = 0 $ and $ r_H = 1 $.

Figure 5.  (color online)The relationship between the imaginary part ${\rm{Im}}$$[\omega^{(N)}]$ and the Lorentz symmetry breaking parameter s respectively for $\Psi^{(N)}$, $D_{0,k\pm}\Psi^{(N)}$ and $D_{1,k\pm}\Psi^{(N)}$ with different overtone numbers n, where we have taken $r_H=1$ and $\mu^2=-0.75$.

In this work, we have investigated mass ladder operators constructed from the conformal symmetry of the static BTZ-like black hole in the Einstein-bumblebee gravity, and probed the QNM frequencies of the mapped modes by mass ladder operators for the scalar perturbation under the Dirichlet and Neumann boundary conditions. It is found that the mass ladder operators, which can change not only the mass squared but also the QNM frequencies, depend on the Lorentz symmetry breaking parameter s in the Einstein-bumblebee gravity. It is observed that the imaginary parts of the frequencies shifting by mass ladder operators increase with the increase of s. The changes of quasinormal frequencies of the mapped modes with the parameter s are similar to those of the original modes under two boundary conditions. However, under the Neumann boundary condition, the imaginary parts of the quasinormal frequencies of the mapped modes caused by the mass ladder operator $ D_{0,k_+} $ are all positive, which implies that all the corresponding fundamental modes are unstable. Moreover, it is also shown that mass ladder operators do not change the BF-bound $ \mu^2_{BF} $ as in the usual BTZ black hole, and the threshold value of the Lorentz symmetry breaking parameter $ s_{c} $, where the modes stop increasing their imaginary parts, increases with the increase of the mass squared $ \mu^2 $. These results could help us to further understand the conformal symmetry and Lorentz symmetry breaking of the static BTZ-like black hole in the Einstein-bumblebee gravity.