Stationary scalar clouds around a rotating BTZ-like black hole in the Einstein-bumblebee gravity

In 1971, Ruffini and Wheeler proposed the no-hair conjecture [1], which is one of the most important developments in black hole physics, stating that stationary black holes should be characterized by only three parameters: mass, charge and angular momentum [2, 3]. This pioneering work has led to many investigations concerning the new hair of black holes. Based on the superradiance mechanism, Hod first analyzed the dynamics of a test massive scalar field surrounding Kerr black holes [4, 5] and further extended to the case of Kerr-Newman black holes [6, 7], and found out that there exist the stationary and regular scalar field configurations surrounding the realistic rotating black holes which are coined as stationary scalar clouds. Herdeiro et al. presented a family of solutions of Einstein’s gravity minimally coupled to a complex massive scalar field, which describes asymptotically flat, spinning black holes with scalar hair [8], and made a thorough investigation of the scalar clouds, which are bound states at the threshold of the superradiant instability, due to neutral (charged) scalar fields around Kerr(-Newman) black holes [9, 10, 11]. Sampaio et al. considered the Proca fields on the background of Reissner-Nordström black holes and obtained the marginal Proca cloud configurations [12], and the authors of Ref. [13] completed the analysis of bosonic fields on Kerr-anti-de Sitter (Kerr-AdS) black holes by studying Maxwell perturbations and constructed stationary Maxwell clouds with Robin boundary conditions. Extending the investigation to the excited states, Wang et al. found that, in contrast to Kerr black holes with ground state scalar hair, the first excited Kerr black holes with scalar hair have two types of nodes, including radial $n_{r}=1$ and angular $n_{\theta}=1$ nodes [14]. For a simple, geometrically elegant, black hole solution of three dimensional general relativity, Ferreira et al. showed that the BTZ black hole can support stationary scalar clouds of a massive scalar field by using appropriate Robin boundary conditions [15, 16]. Along this line, there have been accumulated interest to study various cloud configurations around black holes in the Einstein gravity [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] and modified gravity [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].

In this work, we focus on the stationary scalar clouds around a black hole in the Einstein-bumblebee gravity, which is one of simple effective theories with the Lorentz violation [44]. In the Einstein-bumblebee model, the Lorentz violation arises from the dynamics of the bumblebee vector field $B_{\mu}$ [45, 46]. Casana et al. found the first black hole solution in the Einstein-bumblebee gravity, $i.e.$, the exact Schwarzschild-like black hole solution, and presented the effects of the spontaneous Lorentz symmetry breaking on some classic tests, including the advance of perihelion, the bending of light, and Shapiro’s time delay [47]. More recently, Ding et al. obtained a three-dimensional rotating BTZ-like black hole solution in the Einstein-bumblebee gravity theory, and found the effects of the Lorentz breaking constant on its thermodynamics and central charges [48]. Considering a scalar perturbation around such a rotating BTZ-like black hole in the Einstein-bumblebee gravity, Chen et al. investigated the quasinormal modes of this rotating BTZ-like black hole and pointed out that the Lorentz symmetry breaking parameter imprints only in the imaginary parts of the quasinormal frequencies for the right-moving and left-moving modes [49]. Moreover, Ge et al. constructed the mass ladder operators for the static BTZ-like black hole in the Einstein-bumblebee gravity, and probed the quasinormal frequencies of the mapped modes by the mass ladder operators for a scalar perturbation under the Dirichlet and Neumann boundary conditions [50]. Since the three-dimensional gravity certainly offers potential insights into quantum gravity, it is of great interest to generalize the investigation to the stationary scalar clouds around a rotating BTZ-like black hole in the Einstein-bumblebee gravity. We will examine the influence of the spontaneous Lorentz symmetry breaking on the stationary scalar clouds, and check whether it is possible to distinguish between the BTZ black hole in the Einstein gravity and BTZ-like black hole in the Einstein-bumblebee gravity based on the scalar field configurations.

This work is organized as follows. In Section II, we briefly review the rotating BTZ-like black holes in the Einstein-bumblebee gravity, and present the massive Klein-Gordon equation and the appropriate boundary conditions for the non-extremal BTZ-like black holes. In Section III, we analyze stationary scalar configurations around the rotating BTZ-like black hole in the Einstein-bumblebee gravity. In Section IV, we investigate quasinormal modes for a massive scalar field which is governed by the Klein-Gordon equation. We conclude in the last section with our main results. For completeness, in Appendix A, we study massive Klein-Gordon equation in the extremal BTZ-like black hole.

In this section, we investigate stationary scalar configurations around the rotating BTZ-like black hole in the Einstein-bumblebee gravity. For this case, the resonance condition should be satisfied, $i.e.$, the field’s frequency equals to the critical frequency

which leads to $\alpha=0$ and $c=1$, $a=\beta-i\ell\sqrt{1+s}k/(2r_{+})$ in Eq. (15). Then the solutions in Eq. (18) reduce to

where $\beta$ is given in Eq. (15), and the boundary condition (22) for scalar clouds becomes

By scanning the parameter space of the system for given values of $s$, $\mu^{2}$, $k$, $r_{+}$ and $\zeta$, one may obtain scalar clouds. Here, for concreteness we take $\mu^{2}=-0.65$, $\zeta=0.9\pi$ and, without loss of generality, we take the AdS radius $\ell=1$ to measure all other physical quantities. One should further note that the boundary parameter $\zeta$ is bounded in the range $[\zeta_{*},\pi)$, where

satisfying $\zeta_{*}\in(\frac{\pi}{2},\pi)$. Since $\zeta_{*}$ increases as the Lorentz symmetry breaking parameter $s$ increases, the allowed parameter space decreases as the increase of $s$.

To explore the impact of the Lorentz symmetry breaking parameter $s$ on scalar clouds, in Fig. 1 we present scalar clouds for the case of $n=0$, $\zeta=0.9\pi$ and for various Lorentz symmetry breaking parameters $s$ ($s=-0.5,-0.1,0,0.1,0.5$) with fixed $k=1$ (left panel) and $k=2$ (right panel). It is shown that the clouds exist for smaller background mass with fixed background angular velocity by increasing $s$ with fixed $k$. In Fig. 2 we exhibit scalar clouds for the case of $n=0$, $\zeta=0.9\pi$ and for various quantum number $k$ ($k=1,2,3,4$) with fixed $s=-0.5$ (left panel) and $s=0.5$ (right panel). It is shown that the clouds exist for larger background mass with fixed background angular velocity by increasing $k$ with fixed $s$. The results shown in the above figures indicate the Lorentz symmetry breaking parameter $s$ and the quantum number $k$ have entirely opposite effects on the existence lines of nodeless scalar clouds. One should further notice that the Lorentz symmetry breaking parameter $s$ has no effect on the extremal lines (the dashed black curve in the figure), which is consistent with Eq. (6). This is completely different from the effect of the MOG parameter $\alpha$ on stationary scalar clouds in Kerr-MOG black holes, where the increase of parameter $\alpha$ makes the extremal lines lower [37].

Based on the above observation, in Fig. 3 we present scalar clouds in the parameter space by taking a few pairs of ($k,s$), including $(k=1,s=0)$, $(k=2,s=0.36794)$, $(k=3,s=0.42830)$ and $(k=4,s=0.45342)$. It is interesting to note that the existence lines for all the cases we considered here are the same, and we term such configurations as degenerate clouds. This indicates that infinite degenerate clouds may be constructed by increasing $k$ and the corresponding $s$ is bounded by $s=-(1+1/\mu^{2})$. The existence of degenerate clouds raises a question, $i.e.$ how to distinguish BTZ black holes and BTZ-like black holes. To solve this issue, we further explore the radial profile of degenerate clouds in Fig. 4, by taking the exactly the same parameters given in Fig. 3. It is shown clearly that the radial profiles of degenerate clouds $\phi$ are completely different, indicating that degenerate clouds may be discerned by the radial profile.

Before proceeding, we would like to stress that in the above we only find the nodeless scalar clouds ($n=0$), $i.e.$, the ground state stationary clouds, which is different from the Kerr case [8, 14]. Considering that the clouds only exist at the threshold of superradiance [69], we will study the quasinormal modes in the BTZ-like black hole and find out the corresponding reasons.

For a massive scalar field $\Phi$ satisfying the Klein-Gordon equation (8), from Eqs. (17) and (21) we can obtain quasinormal frequencies by solving the following equation

where the Robin parameter satisfies $\zeta\in[0,\pi)$. Obviously, if $\zeta=0$, $i.e.$, the so-called Dirichlet boundary condition, we can solve Eq. (27) exactly, and get the left-moving frequencies

and the right-moving frequencies

which reduce to the quasinormal frequencies given in Ref. [16] for the BTZ black hole if the Lorentz symmetry breaking parameter $s=0$. Similarly, if $\zeta=\pi/2$, $i.e.$, the so-called Neumann boundary condition, we have

For other values of Robin parameter $\zeta$, we have to solve Eq. (27) numerically and obtain the numerical results for the quasinormal frequencies.

In Fig. 5, we present the imaginary parts of some selected quasinormal frequencies as a function of $\zeta/\pi$ with different Lorentz symmetry breaking parameters for a given black hole configuration (the fixed $\ell$, $r_{+}$ and $r_{-}$) and scalar field configuration (the fixed $\mu^{2}$ and $k$). The red dots denote quasinormal frequencies corresponding to the Dirichlet boundary condition $(\zeta=0)$ and the red triangles correspond to those for the Neumann boundary condition $(\zeta=\frac{\pi}{2})$, which can be served as a check of the numerical computation. We can see clearly that, as a function of $\zeta$, the mode continuously connects the case of Dirichlet boundary condition with that of Neumann boundary condition for quasinormal frequencies. However, all the imaginary parts of quasinormal frequencies are negative except for the fundamental frequencies of type L, corresponding to the nodeless modes ($n=0$). For the nodeless modes, the imaginary parts of frequencies always change from negative to positive as the parameter $\zeta$ increases, which means that there exist the so-called superradiance zero-modes, occurring in between decaying modes and superradiantly amplified modes [15]. It should be noted that, regardless of the Lorentz symmetry breaking parameter $s$, the superradiance zero-modes only appear in the nodeless modes ($n=0$) of type L, which indicates that the stationary scalar clouds only exist in this ground state since the resonance condition corresponds precisely to the threshold of the superradiant instability [69]. This supports the observation shown in Figs. 1, 2 and 3 where we can only obtain the existence lines of nodeless scalar clouds ($n=0$).

In order to further analyze the superradiance zero-modes, we display the imaginary parts of some quasinormal frequencies of nodeless modes ($n=0$) as a function of $\zeta/\pi$ with different Lorentz symmetry breaking parameters for a given black hole configuration (the fixed $\ell$, $r_{+}$ and $r_{-}$) and scalar field configuration (the fixed $\mu^{2}$ and $k$) in Fig. 6. We observe that the threshold of the superradiant instability, $i.e.$, the critical value of $\zeta$ with Im$[\omega]=0$, increases as the Lorentz symmetry breaking parameter $s$ increases. In particular, we find this threshold of the superradiant instability is identically equal to the value of $\zeta$ given by Eq. (25) for the existence of the stationary scalar clouds. From Fig. 6, we note that the critical value of $\zeta\rightarrow 0.75\pi$ when $s=s_{app}$. Thus, we argue that, in the case of $\mu^{2}=-0.65$, $r_{+}=5$, $r_{-}=3$, $\ell=1$ and $k=1$, the threshold of the superradiant instability has a range $0.5\pi<\zeta<0.75\pi$, which suggests that the stationary scalar clouds only exist in this range. As a matter of fact, we only take $\mu^{2}=-0.65$ and $\ell=1$ in Figs. 1-4 with different valves of $r_{+}$, and so the superradiant instability can appear in the case of $\zeta=0.9\pi$.

We have investigated the Klein-Gordon equation for a massive scalar field on a rotating BTZ-like black hole in the Einstein-bumblebee gravity. By employing appropriate Robin boundary conditions at the AdS boundary, we have established the existence of stationary fundamental clouds. We found that, by increasing the Lorentz symmetry breaking parameter $s$ (the quantum number $k$), the clouds exist for smaller (larger) background mass with the same background angular velocity, which means that the parameters $s$ and $k$ have entirely opposite effects on the existence lines of scalar clouds. In particular, we obtained the degenerate clouds where different parameters $s$ and $k$ have the same existence line of scalar clouds, which has not ever been observed as far as we know. Interestingly, we pointed out that the radial profiles of degenerate clouds $\phi$ may be used to discriminate degenerate clouds. Moreover, we noticed that, below the limiting value $s=-(1+1/\mu^{2})$, there exist infinite degenerate clouds for any initial values of $s$ and $k$.

By analyzing quasinormal modes in BTZ-like black holes, we observed that, regardless of the Lorentz symmetry breaking parameter $s$, superradiant instabilities are only appeared in the nodeless mode ($n=0$) of type L, which illustrates the observation, shown in Figs. 1, 2 and 3, that only fundamental clouds are allowed. Moreover, we observed that the threshold of the superradiant instability, which determines scalar clouds, depends on the parameter $s$. Our results indicate that the Lorentz symmetry breaking parameter $s$ yields richer physics in the stationary clouds of massive scalar fields around black holes.