Instability of the regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black hole

  • We studied the instability of the regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black holes under charged scalar perturbations. The unstable modes satisfy the superradiant condition, but not all modes satisfying the superradiant condition are unstable. The instability occurs when the cosmological constant is small and the black hole charge is not too large. The Gauss-Bonnet coupling constant makes the unstable black hole more unstable when both the black hole charge and cosmological constant are small, and makes the stable black hole more stable when the black hole charge is large.
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Peng Liu, Chao Niu and Cheng-Yong Zhang. Instability of the regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black hole[J]. Chinese Physics C.
Peng Liu, Chao Niu and Cheng-Yong Zhang. Instability of the regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black hole[J]. Chinese Physics C. shu
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Instability of the regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black hole

    Corresponding author: Cheng-Yong Zhang, zhangcy@email.jnu.edu.cn
  • Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China

Abstract: We studied the instability of the regularized 4D charged Einstein-Gauss-Bonnet de-Sitter black holes under charged scalar perturbations. The unstable modes satisfy the superradiant condition, but not all modes satisfying the superradiant condition are unstable. The instability occurs when the cosmological constant is small and the black hole charge is not too large. The Gauss-Bonnet coupling constant makes the unstable black hole more unstable when both the black hole charge and cosmological constant are small, and makes the stable black hole more stable when the black hole charge is large.

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    I.   INTRODUCTION
    • It is known that general relativity should be modified from both the viewpoint of theory and observation. For examples the general relativity can not be renormalized and can not explain the dark side of the Universe. On the other hand, the Lovelock theorem states that in four dimensional vacuum spacetime, the general relativity with a cosmological constant is the unique metric theory of gravity with second order equations of motion and covariant divergence-free [1]. Beyond general relativity, one must go to higher dimensional spacetime, or add extra fields, or allow higher order derivative of metric or even abandon the Riemannian geometry. Various modified gravity theories have been proposed [2].

      By rescaling the GB coupling constant in a special way, a regularized four dimensional GB black hole solution was found recently [3]. This work provides a new classical four dimensional gravity theory and has inspired many studies, including the regularized black hole solutions [46], perturbations [710], shadow and geodesics [11], thermodynamics [12], and other aspects [13, 14]. It should be mentioned that the 4D EGB gravity was found inconsistent based on the theory proposed in [3]. Fortunately, some proposals have been raised to circumvent the issues of the 4D EGB gravity, including adding an extra degree of freedom to the theory [6], or breaking the temporal diffeomorphism invariance [16], where a well-defined theory was formulated. On the other hand, while the 4D EGB gravity formulated in [3] may run into trouble at the level of action or equations of motion, the spherically symmetric black hole solution derived in [3] can be obtained from conformal anomaly and quantum corrections [15] and those from Horndeski theory [6], which means the spherically symmetric black hole solution itself is meaningful and worthy of study.

      The regularized black hole solution has some remarkable properties. Its singularity at the center is timelike. The gravitational force near the center is repulsive and the free infalling particles can not reach the singularity [3]. One may expect that the regularized black hole solution would also show some new properties under perturbations, and some related works have been done [7, 9]. The study of the stability of black hole is an active area in black hole physics. It can be used to extract the parameters of the black hole such as its mass, charge and angular momentum. The stability of black hole is also related to gravitational wave, black hole thermodynamics, information paradox and holography, etc. [17]. Among these studies, the stability of black hole in asymptotic de Sitter (dS) spacetime is intriguing. For examples, the spontaneous scalarization of Kerr-dS black hole in scalar-tensor theory behaves very differently from those in asymptotically flat spacetime under perturbations [18]. The four dimensional Reissner-Nordström-de Sitter (RN-dS) may violate the strong cosmic censorship [19, 20]. The higher dimensional RN-dS and Gauss-Bonnet-de Sitter (GB-dS) black holes are unstable [21, 22].

      A quite surprising and still not very well understood result was discovered in [2325], where they showed that RN-dS black hole is unstable under charged scalar perturbations. Such instability satisfies superradiance condition [26]. However, only the monopole $ l = 0 $ suffers from this instability. Higher multipoles are stable. This is distinct from superradiance. To understand the precise mechanism, one may need the nonlinear studies, which was partially answered in recent works [20]. In this paper, we consider the instability of the regularized 4D charged EGB black hole in asymptotic dS spacetime under charged scalar perturbations. We will see that the behavior is very different with the case in asymptotic flat spacetime which has been done in [9]. The Gauss-Bonnet coupling constant plays a more subtle role here.

      The paper is organized as follows. Section 2 describes the regularized 4D EGB-RN-dS black hole and gives the reasonable parameters region. Section 3 shows the charged scalar perturbation equations. Section 4 describes the numerical method we used and gives the results of the quasinormal modes. Section 5 is the summary and discussion.

    II.   THE 4D CHARGED EGB-DS BLACK HOLE
    • The four dimensional spherical symmetric EGB black hole in electrovacuum dS spacetime can be written as [4]

      $ ds^{2} = -f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}), $

      (1)

      where the metric function

      $ f(r) = 1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+4\alpha\left(\frac{M}{r^{3}}-\frac{Q^{2}}{r^{4}}+\frac{\Lambda}{3}\right)}\right), $

      (2)

      and the gauge potential

      $ A = -\frac{Q}{r}dt. $

      (3)

      Here M is the black hole mass, Q is the black hole charge and $ \Lambda $ the positive cosmological constant. When $ \alpha\to0 $, this solution returns to the RN-dS black hole. As $ r\to\infty $, it gives the asymptotic dS spacetime with an effective positive cosmological constant. Note that the solution (1) coincides formally with those obtained from Horndeski theory [6] and those from conformal anomaly and quantum corrections [15].

      The GB coupling constant $ \alpha $ can be either positive or negative here. In appropriate parameter region, the solution has three horizons: the inner horizon $ r_{-} $, the event horizon $ r_{+} $ and cosmological horizon $ r_{c} $. For negative $ \alpha $, the metric function may not be real in small r region. But since we are only interested in the region between $ r_{+} $ and $ r_{c} $, we allow negative $ \alpha $ in this work. For convenience, hereafter we fix the black hole event horizon $ r_{+} = 1 $. Then the mass parameter can be expressed as

      $ M = 1-\frac{\Lambda}{3}+Q^{2}+\alpha. $

      (4)

      Note that to ensure $ f(1) = 0 $, there must be $ \alpha>-\frac{1}{2} $. The parameter region where allows the black hole event horizon $ r_{+} $ and the cosmological horizon $ r_{c} $ can be determined by requiring $ f'(r_{+})>0 $, which implies the black hole temperature is positive. This leads to

      $ Q^{2}+\alpha+\Lambda<1. $

      (5)

      This formula is very similar to the neutral case [8]. The parameter region is shown in Fig. 1.

      Figure 1.  The parameter region that allows the event horizon $ r_{+} $ and cosmological horizon $ r_{c} $. The region is bounded by $ Q^{2}+\alpha+\Lambda<1, $ $ -0.5<\alpha $, $ \Lambda>0 $ and Q>0. As Q increases, the allowed region for $ (\Lambda,\alpha) $ shrinks. The extremal value of $ Q = \sqrt{3/2} $.

    III.   THE CHARGED SCALAR PERTURBATION
    • It is known that fluctuations of order $ {\cal{O}}(\epsilon) $ in the scalar field in a given background induce changes in the spacetime geometry of order $ {\cal{O}}(\epsilon^{2}) $ [26]. To leading order we can study the perturbations on a fixed background geometry. Let us consider a massless charged scalar field $ \psi $ on the background (1). Its equation of motion is

      $ 0 = D^{\mu}D_{\mu}\psi\equiv g^{\mu\nu}\left(\nabla_{\mu}-iqA_{\mu}\right)\left(\nabla_{\nu}-iqA_{\nu}\right)\psi, $

      (6)

      where q is the scalar charge and $ \nabla_{\mu} $ the covariant derivative. For generic background, we can take the following decomposition

      $ \psi = \sum\limits_{lm}\int d\omega e^{-i\omega t}\frac{\Psi(r)}{r}Y_{lm}(\theta,\phi). $

      (7)

      Here $ Y_{lm}(\theta,\phi) $ is the spherical harmonics on the two sphere $ S^{2}. $ The angular part and the radial part of the perturbation equation (6) decouple. What we are interested in is the radial part which can be written as the Schrödinger-like form

      $ 0 = \frac{\partial^{2}\Psi}{\partial r_{\ast}^{2}}+\left(\omega^{2}-\frac{2qQ}{r}\omega-V_{\rm{eff}}\right)\Psi, $

      (8)

      where the tortoise coordinate $ dr_{\ast} = dr/f $ is introduced. The effective potential reads

      $ V_{\rm{eff}} = -\frac{q^{2}Q^{2}}{r^{2}}+f\left(\frac{l(l+1)}{r^{2}}+\frac{\partial_{r}f}{r}\right). $

      (9)

      Unlike to the case in asymptotic flat spacetime where only one potential barrier appears between $ r_{+} $ and $ r_{c} $, here a negative effective potential well may appear between $ r_{+} $ and $ r_{c} $. We will see that this potential well plays an important role in the instability of charged EGB-dS black hole under perturbations.

      The radial equation has following asymptotic behavior near the horizons.

      $ \Psi \to \left\{ {\begin{array}{*{20}{l}} {{e^{ - i\left( {\omega - \frac{{qQ}}{{{r_ + }}}} \right){r_ * }}}\sim {{\left( {r - {r_ + }} \right)}^{ - \frac{i}{{2{\kappa _ + }}}\left( {\omega - \frac{{qQ}}{{{r_ + }}}} \right)}},}&{r \to {r_ + },}\\ {{e^{i\left( {\omega - \frac{{qQ}}{{{r_c}}}} \right){r_ * }}}\sim {{\left( {r - {r_c}} \right)}^{ - \frac{i}{{2{\kappa _c}}}\left( {\omega - \frac{{qQ}}{{{r_c}}}} \right)}},}&{r \to {r_c}.} \end{array}} \right. $

      (10)

      Here $\kappa_{+} = \dfrac{1}{2f'(r_{+})}$ is the surface gravity on the event horizon and $\kappa_{c} = -\dfrac{1}{2f'(r_{c})}$ the surface gravity on the cosmological horizon. These asymptotic solution corresponds to ingoing boundary condition near the event horizon and outgoing boundary condition near the cosmological horizon. The system is dissipative and the frequency of the perturbations will be the composition of quasinormal modes (QNMs). With the specific boundary condition (10), the radial equation (8) can be solved as an eigenvalue problem. Only some discrete eigenfrequencies $ \omega $ would satisfy both the radial equation and boundary condition. One can write the eigenfrequency as $ \omega = \omega_{R}+i\omega_{I} $. When the imaginary part $ \omega_{I}>0 $, the amplitude of perturbation will grow exponentially and implies that the black hole is unstable at least in the linear perturbation level.

    IV.   THE INSTABILITY OF THE 4D CHARGED EGB-DS BLACK HOLE
    • The radial equation (8) with boundary conditions specified by (10) is generally hard to solve analytically, except a few exceptions such as the pure (A)dS spacetimes, Nariai spacetime and massless topological black holes [17]. In the general cases, approximations or numerical methods are required. Various numerical methods for QNMs calculations were developed, such as WKB method, shooting method, continued fraction method and Horowitz-Hubeny method [17]. Not all methods keeps high accuracy and efficiency in the charged case [27]. In this work, we adopt the asymptotic iteration method [28]. Also, we testify our results from the asymptotic iteration methods with time evolution.

    • A.   The asymptotic iteration method (AIM)

    • The asymptotic iteration method was used to solve the eigenvalues of the homogeneous second order ordinary derivative functions [29]. Later it was used to look for the quasinormal modes of black holes in asymptotic flat or (A)dS spacetime [28]. Let us first introduce an auxiliary variable

      $ \xi = \frac{r-r_{+}}{r_{c}-r_{+}}. $

      (11)

      It ranges from 0 to 1 as r runs from the event horizon to the cosmological horizon. The radial equation (8) then becomes

      $ \begin{aligned}[b] 0 =& \frac{\partial^{2}\Psi}{\partial\xi^{2}}\left(\frac{f}{r_{c}-r_{+}}\right)^{2}+\frac{\partial\Psi}{\partial\xi}\frac{f\partial_{\xi}f}{\left(r_{c}-r_{+}\right)^{2}} \\ \;\;\;\; &+\left[\left(\omega-\frac{qQ}{(r_{c}-r_{+})\xi+r_{+}}\right)^{2} -f\left(\frac{l(l+1)+\left(\xi+\frac{r_{+}}{r_{c}-r_{+}}\right)\partial_{\xi}f}{\left[(r_{c}-r_{+})\xi+r_{+}\right]^{2}}\right)\right]\Psi. \end{aligned} $

      (12)

      The above equation is generally hard to solve analytically. We turn to the numerical method. In terms of $ \xi $, the asymptotic solution near the horizons can be written as

      $ \Psi \to {\rm{ }}\left\{ {\begin{array}{*{20}{l}} {{\xi ^{ - \frac{i}{{2{\kappa _ + }}}\left( {\omega - \frac{{qQ}}{{{r_ + }}}} \right)}},}&{\xi \to 0,}\\ {{{\left( {\xi - 1} \right)}^{ - \frac{i}{{2{\kappa _c}}}\left( {\omega - \frac{{qQ}}{{{r_c}}}} \right)}},}&{\xi \to 1.} \end{array}} \right. $

      (13)

      Then we can write the full solution of (12) satisfying the asymptotic behavior (13) in the following form

      $ \Psi = \xi^{-\frac{i}{2\kappa_{+}}\left(\omega-\frac{qQ}{r_{+}}\right)}\left(\xi-1\right)^{\frac{i}{2\kappa_{c}}\left(\omega-\frac{qQ}{r_{c}}\right)}\chi(\xi). $

      (14)

      Here $ \chi(\xi) $ is a regular function of $ \xi $ in range $ (0,1) $ and obeys following homogeneous second order differential equation

      $ \frac{\partial^{2}\chi}{\partial\xi^{2}} = \lambda_{0}(\xi)\frac{\partial\chi}{\partial\xi}+s_{0}(\xi)\chi, $

      (15)

      in which the coefficients

      $ -\lambda_{0}(\xi) = \frac{i\left(\omega-\frac{qQ}{r_{c}}\right)}{(\xi-1)\kappa_{c}}-\frac{i\left(\omega-\frac{qQ}{r_{+}}\right)}{\kappa_{+}\xi}+\frac{f'(\xi)}{f(\xi)}, $

      (16)

      $ \begin{aligned}[b] -s_{0}(\xi) =& -\frac{\left(r_{c}-r_{+}\right)\left((\xi r_{c}\!-\!\xi r_{+}+r_{+})f'(\xi)+l(l\!+\!1)\left(r_{c}-r_{+}\right)\right)}{f(\xi)\left((\xi-1)r_{+}-\xi r_{c}\right){}^{2}}-\frac{\left(\omega-\dfrac{qQ}{r_{c}}\right)\left(\dfrac{\omega r_{c}-qQ}{2\kappa_{c}r_{c}}+i\right)}{2(\xi-1)^{2}\kappa_{c}} \\ &+\frac{\left(\omega-\dfrac{qQ}{r_{+}}\right)\left(\dfrac{qQ-r_{+}\omega}{2\kappa_{+}r_{+}}+i\right)}{2\kappa_{+}\xi^{2}}+\frac{\left(\omega-\dfrac{qQ}{r_{+}}\right)\left(\omega-\dfrac{qQ}{r_{c}}\right)}{2\kappa_{+}(\xi-1)\xi\kappa_{c}} \\ &+\frac{if'(\xi)\left(\omega-\dfrac{qQ}{r_{c}}\right)}{2(\xi-1)\kappa_{c}f(\xi)}-\frac{if'(\xi)\left(\omega-\dfrac{qQ}{r_{+}}\right)}{2\kappa_{+}\xi f(\xi)}+\frac{\left(r_{c}-r_{+}\right){}^{2}\left(\omega-\dfrac{qQ}{\xi r_{c}-\xi r_{+}+r_{+}}\right){}^{2}}{f(\xi)^{2}}. \end{aligned} $

      (17)

      The coefficients $ \lambda_{0}(\xi) $ and $ s_{0}(\xi) $ are regular functions of $ \xi $ in the interval $ (0,1) $. Differentiating (15) with respect to $ \xi $ iteratively leads to an $ (n+2) $-th order differential equation

      $ \chi^{(n+2)} = \lambda_n (\xi) \chi'(\xi) +s_n(\xi)\chi(\xi), $

      (18)

      where the coefficients are determined iteratively:

      $ \begin{aligned}[b] \lambda_{n}(\xi) &= \lambda'_{n-1}(\xi)+s_{n-1}(\xi)+\lambda_{0}(\xi)\lambda_{n-1}(\xi), \\ s_{n}(\xi) &= s'_{n-1}(\xi)+s_{0}(\xi)\lambda_{n-1}(\xi). \end{aligned} $

      (19)

      Expanding the coefficient functions around a regular point $ \xi = \xi_0 $,

      $ \begin{aligned}[b] \lambda_{n}(\xi) &= \sum\limits_{j = 0}^{\infty}c_{n}^{j}(\xi-\xi_{0})^{j}, \\s_{n}(\xi) &= \sum\limits_{j = 0}^{\infty}d_{n}^{j}(\xi-\xi_{0})^{j}, \end{aligned} $

      (20)

      in which the expansion coefficients $ c_n^j $ and $ d_n^j $ are functions of $ \omega $, the iterative equations give

      $ c_{n}^{j} = (j+1)c_{n-1}^{j+1}+d_{n-1}^{j}+\sum\limits_{k = 0}^{j}c_{0}^{k}c_{n-1}^{j-k}, $

      (21)

      $ d_{n}^{j} = (j+1)d_{n-1}^{i+1}+\sum\limits_{k = 0}^{j}d_{0}^{k}c_{n-1}^{j-k}. $

      (22)

      For large enough n, the cutoff of the iteration in AIM is given by

      $ \frac{s_{n}(\xi)}{\lambda_{n}(\xi)} = \frac{s_{n-1}(\xi)}{\lambda_{n-1}(\xi)}, $

      (23)

      which leads to in terms of the expansion coefficients

      $ d_{n}^{0}c_{n-1}^{0} = d_{n-1}^{0}c_{n}^{0}. $

      (24)

      The quasinormal modes $ \omega $ can be worked out by solving this equation. We vary the iteration times and the expansion point to ensure the reliability of the results. The results are also checked by using other auxiliary variables such as $\xi = 1-\dfrac{r_{+}}{r}$ or $\xi = \left(1-\dfrac{r_{+}}{r}\right)/\left(1-\dfrac{r_{+}}{r_{c}}\right)$. Except some extremal cases ($ \Lambda\to0 $ or the black hole becomes extremal), they coincides well. Hereafter we set $ q = 1 $ for convenience.

    • B.   The eigenfrequencies of the charged scalar perturbation

    • Let us first study the effects of the black hole charge Q on the fundamental modes of the QNMs. It is shown in Fig. 2. From the left panel, we see that $ \omega_{R} $ increases with Q almost linearly. The slope is larger for larger $ \Lambda $ and $ \alpha $. In the right panel, $ \omega_{I} $ increases with small Q and then decreases with larger Q. For large enough Q, the black hole becomes stable. When Q is small, $ \omega_{I} $ increases with increasing $ \alpha $. This behavior is similar to the case in asymptotic flat spacetime [9]. However, when Q is large, $ \omega_{I} $ decreases with increasing $ \omega_{I} $. This is contrast with the cases in asymptotic flat spacetime. It implies that the positive GB coupling constant can make the black hole more unstable when the black hole charge Q is small and more stable when Q is large. Note further that no matter what values $ \Lambda,\alpha $ take, when $ Q\to0 $, both $ \omega_{R} $ and $ \omega_{I} $ tend to 0 from above. This implies the weakly charged black hole in dS spacetime is always unstable. The existence of $ \alpha $ does not change this phenomenon qualitatively.

      Figure 2.  The real part (left) and imaginary part (right) of the fundamental modes of the QNMs when $ l = 0 $. Solid lines for $ \Lambda = 0.01 $, dashed lines for $ \Lambda = 0.1 $

      Now we study the effects of $ \alpha $ on the fundamental modes in detail. Fig. 3 shows the real part of the fundamental modes $ \omega_{R} $ when $ Q = 0.1 $ and $ Q = 0.4 $. For fixed Q, the real part $ \omega_{R} $ increases almost linearly with $ \alpha $ and $ \Lambda $. Combining the results from Fig. 2, we conclude that $ \omega_{R}\propto\alpha\Lambda Q $.

      Figure 3.  The real part of the fundamental modes of the QNMs when $ l = 0 $. Left panel for $ Q = 0.1 $, right panel for $ Q = 0.4 $.

      The behavior of the imaginary part of the fundamental modes $ \omega_{I} $ is more interesting. In Fig. 4 we show the cases when $ Q = 0.1 $ and $ Q = 0.4 $. From the left and right panels, we see that $ \omega_{I} $ increases with small $ \Lambda $ and decreases with larger $ \Lambda $. The effects of $ \alpha $ on $ \omega_{I} $ is subtle and relevant to the $ \Lambda $ and Q. When both Q and $ \Lambda $ are small (left upper panel), $ \omega_{I} $ increases with $ \alpha $. For small Q and larger $ \Lambda $ (right upper panel), $ \omega_{I} $ increases with $ \alpha $ first and then decreases with $ \alpha $. For larger Q (lower panels), $ \omega_{I} $ decreases with $ \alpha $. This phenomenon is very different with the case in asymptotic flat spacetime [9], where $ \alpha $ roughly increases $ \omega_{I} $. This implies that the positive GB coupling constant $ \alpha $ can suppress the instability of black hole. As $ \alpha $ increases, it can even change the qualitative behavior of black hole under perturbations and render the unstable black hole stable.

      Figure 4.  The imaginary part of the fundamental modes of the QNMs when $ l = 0 $. The upper panel for $ Q = 0.1 $, lower panel for $ Q = 0.4 $. Left panel shows the cases for small $ \Lambda $, the right panel for larger $ \Lambda $.

      The instability we found here is very reminiscent of superradiance. However, the case is subtle here. With the similar method used in [9, 24], one can show that superradiance occurs only when

      $ \frac{qQ}{r_{+}}>\omega>\frac{qQ}{r_{c}}. $

      (25)

      Although the unstable modes encountered in our study indeed satisfy (25), it is observed that some oscillation frequencies corresponding to stable modes also lie in the range determined by (25). See Table 1 for evidence. In fact, as shown in [9, 24], the superradiant condition is the necessary but not sufficient condition for instability. The precise mechanism of the instability found here need more studies and will be addressed in section 5.

      $\alpha$$\frac{qQ}{r_{+}}$$\frac{qQ}{r_{c}}$$\omega$ (AIM)$\omega_I$ (time domain)

      0.50.10.02440420.0290311+0.0001514i0.0001515
      0.60.10.02485570.0297114+0.0000779i0.0000780
      0.650.10.02509870.0300692+0.0000191i0.0000192
      0.70.10.02535470.0304355-0.0000541i-0.0000547
      0.750.10.02562520.0308094-0.0001400i-0.0001390

      Table 1.  The fundamental modes when $ Q = 0.1,\Lambda = 0.12$ and $ l = 0$, which corresponds to the orange line in the upper right panel of Fig. 4. The last column is the imaginary part of the frequency extracted from the time evolution.

    • C.   Evolution of the perturbation field

    • We also directly compute the time-evolution of the perturbation field $ \psi $ to further reveal the instability of the 4D charged EGB-dS black hole. For time evolution, the Schrödinger-like equation becomes,

      $ -\frac{\partial^{2} \Psi}{\partial t^{2}} - \frac{2 i q Q}{r} \frac{\partial \Psi}{\partial t}+\frac{\partial^{2} \Psi}{\partial r_{*}^{2}}-V(r) \Psi = 0, $

      (26)

      In order to compute the time evolution of $ \psi $ we adopt the discretization introduced in [23]. The reliability of this numerical method can be verified by the convergence of computations when increasing the sampling density. We impose the following initial profile,

      $ \left\{ \begin{array}{l} \Psi ({r_*},t) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t < 0,\\ \Psi ({r_*},t) = \exp \left[ { - \dfrac{{{{({r_*} - a)}^2}}}{{2{b^2}}}} \right],\;\;\;\;\;\;t = 0. \end{array} \right. $

      (27)

      The discretization of equation (26) is implemented in $ (r_*,t) $ plane. Also, we fix $ \Delta t/\Delta r_* = 0.5 $ in order to satisfy the von Neumann stability conditions. Unlike some other analytical models where the $ r_* $ can be solved analytically, here the $ r_*(r) $ can only be obtained numerically. The $ r_*(r) $ diverges as $ \lim_{r\to r_+} \to -\infty $ and $ \lim_{r\to r_c} \to \infty $, hence the error of the numerical $ r_*(r) $ can be very large at the near horizon region. Therefore, we introduce a cutoff $ \epsilon $ solve the $ r_*(r) $ relation by solving the below system,

      $ r_*'(r) = 1/f(r),\,r_*(r_h + \epsilon) = 0, \, {\rm{with}}\, r\in [r_h + \epsilon,r_c - \epsilon], $

      (28)

      Usually the cutoff $ \epsilon $ should not be too small, otherwise it leads to significant error of the resultant $ r_*(r) $.

      In order to obtain the late time evolution of the perturbation, we need to solve a large range or $ r_* $. Since $ 1/f $ tends to diverge when $ r\to r_h $ and $ r\to r_c $, in near horizon region the $ r_*(r) $ can be extracted analytically. A direct solution is to expand $ 1/f(r) $ with respect to $ r_h $, because the horizon requires that $ 1/f(r) \sim 1/(r-r_h) $. However, this direct expansion can lead to very large numerical error. A better solution is to expand $ f(r) $ with respect to $ r_h $, and then obtain the expansion of $ 1/f $ in terms of the expansion coefficients of the $ f(r) $. After solving the analytical expansion coefficients, one may glue the analytical expansion in the near horizon region and the numerical solution of the $ r_*(r) $ in $r\in [r_h + \epsilon,r_c - \epsilon]$. In this way, one may obtain a very large range of $ r_* $ and a long-time evolution can be realized.

      We show two examples of $ |\psi(t)| $ in log plot in Fig. 5, from which we can see that $ \ln |\psi| $ has linear dependence on t for late time evolution. From the left panel of Fig. 5, when Q is small the system is unstable ($ \ln |\psi| $ linearly decreases with t), while for large values of Q the system becomes stable ($ \ln |\psi| $ linearly grows with t). This is in accordance with previous results of the frequency analysis (see the right panel of Fig. 2). From the right panel of Fig. 5 we see that when $ \Lambda $ is small the system is unstable ($ \ln |\psi| $ linearly increases with t), while for large values of $ \Lambda $ the system becomes stable ($ \ln |\psi| $ linearly decreases with t). This is in accordance with previous results of the frequency analysis (see the bottom right panel of Fig. 4).

      Figure 5.  Left panel: the time evolution of the $ |\psi(r_* = 88.4216,t)| $ at $ \alpha = 2/3,\,\Lambda = 1/10 $, where each curve corresponds to different values of Q. Right panel: the time evolution of the $ |\psi(r_* = 88.4216,t)| $ at $ \alpha = 1/10,\,Q = 2/5 $, where each curve corresponds to different values of $ \Lambda $. For both plots we fixed $ a = 88.4216,\,b = 1/10 $.

      It is also important to verify the validity of the AIM with the time evolution. There are comprehensive methods to extract the frequencies from the time-domain profile of the perturbation, such as the Prony method used in [23]. Here we extract $ \omega_I $ by computing $ \partial_t\ln(|\psi|) $ at late time and compare that with those from the AIM. This simple process can only extract the imaginary part of the dominant mode and may be inaccurate enough when the QNMs are close to each other. But it is good enough for our purpose here. We provide the comparison in Table 1 (see the last two columns), from which we can see that all the time evolution results perfectly matches with those of the AIM.

      Finally, we show the unstable region of the charged EGB-dS black hole under charged scalar perturbation in Fig. 6. The black hole is unstable only when $ \Lambda $ and Q are not too large. As $ \Lambda $ or Q increases, the black hole becomes less unstable. For positive $ \alpha $, the unstable region shrinks. For negative $ \alpha $, the unstable region enlarges. Although we do not show the results when $ \Lambda\to0 $ due to the limitation of our numerical method, we can expect that there should be sudden drops since we have found that there is no instability for asymptotic flat charged EGB black hole under charged scalar perturbation [9]. This phenomenon was also disclosed for the RN-dS black hole in [24].

      Figure 6.  The unstable region of the charged EGB-dS black hole under charged scalar perturbation. The shadows under constant $ \alpha $ lines are corresponding unstable regions. We fix $ q = 1 $ here.

    • D.   Effective potential

    • Now let us take a look at the effective potential when $ l = 0 $. In the left panel of Fig. 7, we see that there is a negative potential well between $ r_{+} $ and $ r_{c} $. This potential well is the key point for the occurrence of instability. However, the negative effective potential does not guarantee $ \omega_{I}>0 $. For example, the case with $ \alpha = 0.8 $ has a negative potential well, but the corresponding $ \omega_{I}<0 $. Thus the existence of a negative potential well can be view as the necessary but not sufficient condition for the instability [30]. In the right panel of Fig. 7, the potential well disappears for larger $ \alpha $. The perturbation wave can be easily absorbed by the black hole and the corresponding background becomes stable under charged scalar perturbations. Note that the positive cosmological constant is crucial here to creating the necessary potential well.

      Figure 7.  The effective potential when $ l = 0 $. They correspond to the orange line in the upper right panel and orange line in the lower right panel of Fig 4, respectively.

      Now we consider the eigenfrequencies of the charged scalar perturbation when $ l = 1 $. We show the fundamental modes in Fig. 8. The left panel shows that there is still $ \omega_{R}\propto Q $ for higher l. But $ \alpha $ changes $ \omega_{R} $ little. All the fundamental modes has $\omega_{R} < \dfrac{qQ}{r_{c}}$ that live beyond the superradiant condition. The left panel is different from that in Fig. 2. Here $ \omega_{I} $ increases with Q and $ \alpha $ monotonically. Note that all the modes are stable now.

      Figure 8.  The real part (left) and imaginary part (right) of the fundamental modes when $ l = 1 $. We fix $ \Lambda = 0.05 $ here. The case with other $ \Lambda $ are similar.

      The stability of the higher l can be explained from the effective potential, as shown in Fig. 9. There is only one potential barrier and not potential well to accumulate the energy to trigger the instability.

      Figure 9.  The effective potential when $ l = 1 $. The cases with other $ Q,\Lambda $ are similar.

      We also provide the detailed time evolution for $ l \neq 0 $ in Fig. 10. From these two panels we see that the perturbations indeed decays in long time evolution and larger l leads to more significant decay of $ \psi $, and hence no instability occurs for higher values of l.

      Figure 10.  The time evolution of the $ |\psi(r_* = 88.4216,t)| $ in log plot at $ \alpha = 1/10, \, Q = 2/5 ,\, a = 88.4216, \, b = 1/10 $ and $ q = 1 $, where each curve in both panels corresponds to different values of $ \Lambda $. The left and the right panel corresponds to $ l = 1 $ and $ l = 2 $, respectively.

    V.   SUMMARY AND DISCUSSION
    • We studied the instability of charged 4D EGB-dS black hole under the charged massless scalar perturbation. This instability satisfies the superradiant condition. However, not all the modes satisfying the superradiant condition are unstable. The precise mechanism for the this instability is not well understood. But the positive cosmological constant $ \Lambda $ should play a crucial role. The instability occurs when the cosmological constant is small. This is reminiscent of the Gregory-Laflamme instability [31] since here exists hierarchy between the black hole event horizon and cosmological horizon. The instability here is different with the "$ \Lambda $ instability" found in [10, 21, 22] which occurs when the black hole charge and the cosmological constant are large.

      We analyzed this instability from the viewpoint of the effective potential. Higher l has only one potential barrier beyond the event horizon. The perturbation dissipates and leads no instability. The monopole $ l = 0 $ has a negative potential well between the event horizon and cosmological horizon, which can accumulate the energy to trigger the instability. But the negative potential well is just the necessary but not sufficient condition for the instability.

      Unlike the cases in asymptotic flat spacetime, the effects of the GB coupling constant $ \alpha $ on the perturbation is relevant to the black hole charge and cosmological constant. It makes the unstable black hole more unstable when both the black hole charge and cosmological constant are small, and makes the stable black hole more stable when the black hole charge is large. The weakly charged black hole in dS spacetime is always unstable. The existence of $ \alpha $ does not change this phenomenon qualitatively. However, $ \alpha $ can change the qualitative behavior when the black hole charge is large, and make the unstable black hole stable. We show that the unstable region of ($ \Lambda,Q $) shrinks with positive $ \alpha $ can enlarges with negative $ \alpha $. Unfortunately, we do not get the accurate enough results when $ \Lambda \to 0 $ due to the limitation of our numerical method. The case when black hole becomes extremal is also beyond the effectiveness of this method. The stability of extremal black hole may be very different with that of non-extremal black hole. There is "horizon instability" universally [32]. We leave them for further study.

      We point out several topics worthy of further investigations. The stability of massive perturbation should be explored in detail to reveal how mass term affects the stability of the charged perturbation. The stability of the 4D charged Einstein-Gauss-Bonnet anti de-Sitter are also definitely interesting to find out. We plan to explore these directions in near future.

    VI.   ACKNOWLEDGMENTS
    • We thanks Peng-Cheng Li, Minyong Guo and Shao-Jun Zhang for helpful discussions. Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work.

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