Stability of Reissner-Nordström black hole in de Sitter background under charged scalar perturbation
Zhiying Zhu, Shao-Jun Zhang, C. E. Pellicer, Bin Wang, Elcio Abdalla
aa r X i v : . [ h e p - t h ] A ug Stability of Reissner-Nordstr¨om black hole in de Sitter background undercharged scalar perturbation
Zhiying Zhu , , Shao-Jun Zhang , C. E. Pellicer , Bin Wang , Elcio Abdalla Department of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China Department of Physics and Electronic Science,Changsha University of Science and Technology, Changsha, Hunan 410076, China Escola de Ciˆencias e Tecnologia, Universidade Federal do Rio Grande do Norte,Caixa Postal 1524, 59072-970, Natal, Rio Grande do Norte, Brazil Instituto de F´ısica, Universidade de S˜ao Paulo, CEP 05315-970, S˜ao Paulo, Brazil
We find a new instability in the four-dimensional Reissner-Nordstr¨om-de Sitter black holesagainst charged scalar perturbations with vanishing angular momentum, l = 0. We show thatsuch an instability is caused by superradiance. The instability does not occur for a largerangular index, as explicitly proved for l = 1. Our results are obtained from a numericalinvestigation of the time domain-profiles of the perturbations. PACS numbers: 04.70.Bw, 42.50.Nn, 04.30.Nk
I. INTRODUCTION
Perturbations around black holes have been intriguing objects of discussions for decades. One ofthe main reasons is their astrophysical interest. Real black holes always have interactions with therespective astrophysical environment through absorption or evaporation processes. Starting fromanalyses of the perturbations around black holes, we can inquire about their stability. If black holesare unstable under small perturbations, they will inevitably be destabilized and disappear or trans-form dynamically into another object, thus they simply cannot exist in nature as it is originallydefined. For the (3+1)-dimensional asymptotically flat black holes, such as Schwarzschild, Reissner-Nordstr¨om (RN) and Kerr black holes, the corresponding stability has been checked thoroughlyagainst different kinds of perturbations including neutral scalar, electromagnetic and gravitationalperturbations and they were found stable. Analyses of these perturbations and proofs of stabilityof (3+1)-dimensional Schwarzschild-de Sitter (dS), RN-dS, and Kerr-dS black holes have been re-ported. Recently, motivated by the discovery of the correspondence between physics in the anti-deSitter (AdS) spacetime and conformal field theory (CFT) on its boundary (AdS/CFT), the per-turbations around four-dimensional AdS black holes have been examined and (3+1)-dimensionalAdS black holes were found stable under neutral scalar, electromagnetic and gravitational pertur-bations. It was concluded that all of the considered four-dimensional black holes tested for stabilityare stable, except the string theory generalization of Kerr-Newman black holes whose stabilitieshave not been tested due to the difficulty in decoupling the angular variables in their perturbationequations. For a review on this topic, see [1, 2] and references therein.The physics in higher dimensions is much richer. In contrast to the four-dimensional results,various instabilities have been found. In a wide class of D ≥ l , while the first few lowest multipoles are stable [5, 6].Recently, in the high-dimensional RN-dS black hole background, numerical investigations have un-covered the surprising result that the RN black holes in dS backgrounds are unstable for spacetimedimensions larger than six [7, 8].The dynamical perturbations of a black hole background can usually be described by a sin-gle wavelike equation, where a growing mode of the perturbation indicates the instability of theblack hole. Besides such a procedure, there also exists a superradiant instability in the black holespacetime. Considering the classical scattering problem for a perturbation field in a black holebackground, we can have superradiance, a phenomenon where the reflected wave has larger ampli-tude than the incident wave. If the effective potential contains an extra local minimum out of theblack hole in addition to the local maximum, the superradiance will get successively amplified inthe valley of the local potential near the black hole, which will result in the superradiant instabilitydestabilizing the black hole. Reviews of the superradiance stability can be found in Refs. [1, 9].Most investigations on the superradiant instability were concentrated on the rotating black holes.Recently, the discussions of the superradiant stability were extended to the charged black holeswith charged scalar field perturbations [10, 11]. It was found that the superradiant instability canhappen more efficiently when one considers charged black holes and charged scalar fields [12–14].It would be of great interest to extend the study of the superradiant stability to black holes in thereal Universe, namely the black hole in the dS backgrounds.We will study small charged scalar field perturbations in the vicinity of a (3+1)-dimensionalRN black hole in a de Sitter background by doing numerical calculations. The stability of suchblack holes under charged scalar perturbations has never been examined previously. In the AdSspacetime, it was observed that the (3+1)-dimensional RN-AdS black hole can become unstabledue to the condensation of the charged scalar hair onto the black hole and finally the AdS blackhole can be destabilized under a small charged scalar field perturbation [15–17]. Does this observedinstability occur only for the AdS black holes because of their special spacetime properties? Willthe dynamical stability also happen in other four-dimensional black hole backgrounds? It is of greatinterest to test whether the (3+1)-dimensional RN-dS black hole can remain stable as other four-dimensional black holes under such charged scalar perturbing fields. This is the first motivationof the present paper. Besides, we would like to explore whether the stability against linearizeddynamical perturbations relates to the superradiant stability due to the superradiant amplificationof charged scalar waves.The organization of the paper is as follows. In Sec. II we review the (3+1)-dimensional RN-dSblack hole background and give the radial wave equation for a charged scalar field perturbation.In Sec. III we use the finite difference method to study the time-domain perturbation evolutionof the charged scalar field and test the stability against such perturbations. In Sec. IV, wederive the superradiant condition for the four-dimensional RN-dS black hole under a chargedscalar perturbation, and examine the relation between the dynamic stability and the superradiantstability. Finally, we provide conclusions in Sec. IV. We use natural units with G = ~ = c = 1. II. METRIC, PERTURBATION FIELDS AND EFFECTIVE POTENTIALS
We consider charged massive scalar field perturbation in the RN-dS black hole background withthe metric ds = − f ( r ) dt + 1 f ( r ) dr + r ( dθ + sin θdφ ) , (1)where f ( r ) = 1 − Mr + Q r − Λ r . (2)The integration constants M and Q are the black hole mass and electric charge, respectively. Λ isthe positive cosmological constant.The spacetime causal structure depends strongly on the zeros of f ( r ). Depending on the pa-rameters M, Q, and Λ, the function f ( r ) can have one to three or even no real positive zeros(for a negative cosmological constant). For the RN-dS cases we are interested in, f ( r ) has threereal, positive roots ( r c , r + , r − . ), and a real and negative root r = − ( r − + r + + r c ). The horizons r − , r + , r c are denoted as black hole Cauchy, event, and cosmological horizons respectively, whichsatisfy r − < r + < r c .The metric function f ( r ) can be expressed as, f ( r ) = Λ3 r ( r − r + )( r c − r )( r − r − )( r − r o ) . (3)Introducing the surface gravity κ i = | df /dr | r = r i associated with the horizon r = r i , we can writethe inverse of the metric function1 f = − κ − ( r − r − ) + 12 κ + ( r − r + ) + 12 κ c ( r c − r ) + 12 κ o ( r − r o ) . (4)Thus we can obtain the analytic form of the tortoise coordinate by calculating r ∗ = R f − ( r ) dr , r ∗ = − κ − ln (cid:18) rr − − (cid:19) + 12 κ + ln (cid:18) rr + − (cid:19) − κ c ln (cid:18) − rr c (cid:19) + 12 κ o ln (cid:18) − rr o (cid:19) . (5)It is easy to see that r ∗ → −∞ as r → r + and r ∗ → ∞ as r → r c .Consider now a charged scalar perturbation field ψ obeying the Klein-Gordon equation[( ∇ ν − iqA ν )( ∇ ν − iqA ν ) − µ ] ψ = 0 , (6)where q and µ are, respectively, the charge and mass of the perturbing field and A µ = − δ µ Q/r denotes the electromagnetic potential of the black hole. By adopting the usual separation of vari-ables in terms of a radial field and a spherical harmonic ψ lm ( t, r, θ, φ ) = Σ lm r Ψ lm ( t, r ) S lm ( θ ) e imφ ,the Schr¨odinger-type equations in the tortoise coordinate for each value of l read − ∂ Ψ ∂t + ∂ Ψ ∂r ∗ + 2 iq Φ ∂ Ψ ∂t − V ( r )Ψ = 0 , (7)where Φ = − Q/r and the effective potential V is given by V ( r ) = − q Φ ( r ) + f ( r ) (cid:18) l ( l + 1) r + µ + f ′ ( r ) r (cid:19) . (8)For a neutral massless scalar field perturbation, the effective potentials were described inRefs. [18–20], which vanish at the black hole and cosmological event horizons. However, for thecharged scalar field perturbation, it is easy to see that at both the black hole and cosmological eventhorizons, the effective potentials are negative. It is believed that the effective potential V describesthe scattering of ψ by the background curvature [21]. Usually, if the effective potential is negativein some region, growing perturbation, can appear in the spectrum, indicating an instability of thesystem under such perturbations. However, in Ref. [22] it was argued that this is not always true.They found that some potentials with a negative gap still do not imply instability. The criterionto determine whether a system is stable or not against linear perturbation is whether the time-domain profile for the evolution of the perturbation is decaying or not, or, in more general terms,the potential has to permit the existence of bound states [23]. Thus, in order to study the stabilityof the RN-dS black hole against charged scalar perturbation, we have to examine the evolution ofthe perturbation. In calculating the wave equation to obtain the evolution of the perturbation,usually the quasinormal boundary conditions should be employed by defining the solution with thepurely ingoing wave at the black hole event horizon and outgoing wave at the cosmological horizon. III. STABILITY ANALYSIS
We do not have analytic solutions to the time-dependent wave equation with the effectivepotentials considered here. We thus discretize the wave equation (7). Because of the appearanceof the term 2 iq Φ ∂ Ψ ∂t , the procedure used in Ref. [24] is not convenient here. One simple efficientdiscretization, used for example in Ref. [16], is to define Ψ( r ∗ , t ) = Ψ( j ∆ r ∗ , i ∆ t ) = Ψ j,i , V ( r ( r ∗ )) = V ( j ∆ r ∗ ) = V j and Φ( r ( r ∗ )) = Φ( j ∆ r ∗ ) = Φ j , and to write (7) as − (Ψ j,i +1 − j,i + Ψ j,i − )∆ t + 2 iq Φ j (Ψ j,i +1 − Ψ j,i − )2∆ t + (Ψ j +1 ,i − j,i + Ψ j − ,i )∆ r ∗ − V j Ψ j,i + O (∆ t ) + O (∆ r ∗ ) = 0 . (9)With the initial Gaussian distribution Ψ( r ∗ , t = 0) = exp[ − ( r ∗ − a ) b ] and Ψ( r ∗ , t <
0) = 0, we canderive the evolution of Ψ byΨ j,i +1 = − (1 + iq Φ j ∆ t )Ψ j,i − − iq Φ j ∆ t + ∆ t ∆ r ∗ Ψ j +1 ,i + Ψ j − ,i − iq Φ j ∆ t + (cid:18) − t ∆ r ∗ − ∆ t V j (cid:19) Ψ j,i − iq Φ j ∆ t . (10)In the following, we choose the parameters a = 10 and b = 3 in the Gaussian profile. Since the vonNeumann stability conditions usually require that ∆ t ∆ r ∗ <
1, we use ∆ t ∆ r ∗ = 0 . ϕ = Ψ /r in the following.It is necessary to point out that the time-domain profiles of perturbations include contributionsfrom all modes. It was argued that this method is based on the scattering of the Gaussian waveon the potential barrier and is independent of the boundary conditions at the black hole event andcosmological horizons [25]. Therefore it includes all possible instabilities due to different boundaryconditions.In order to extract dominant frequency from the time-domain profile of the perturbation, wewill use the Prony method. We can fit the profile data by superposition of damping exponents [26]Ψ( r, t ) ≃ p X i =1 C i e − iω i ( t − t ) . (11)We consider a late time period, which starts at t and ends at t = N ∆ t + t , where N is an integerand N ≥ p −
1. Then Eq. (11) is valid for each value from the profile data: x n ≡ Ψ( r, n ∆ t + t ) = p X j =1 C j e − iω j n ∆ t = p X j =1 C j z nj . (12)The Prony method allows us to find z i in terms of the known x n and, since ∆ t is also known tocalculate the quasinormal frequencies ω i . A. Neutral scalar perturbation
First, we use the finite difference method proposed in Ref. [16] to reexamine the neutral masslessscalar field perturbations in RN-dS spacetime, which was first reported in Ref. [18] and laterconfirmed in Ref. [20]. For l = 0 , l = 0 mode there exists a negative potential well following a potential barrier near the cosmologicalevent horizon. But for the l = 1 mode the potential well does not exist. The time-domain evolutionof the perturbation in Fig. 1(b) shows that when l = 0, the field rapidly settles down to a constantvalue after some quasinormal oscillations. When l = 1, the perturbation field falls off exponentiallyafter the quasinormal oscillations. These results show very good agreement with those observedin Refs. [18, 20], which gives us confidence in the numerical method we employed. The neutralmassless scalar field perturbations do not grow in the time-domain profiles, although there is anegative potential well when l = 0. This shows that the RN-dS black hole is stable against theneutral massless scalar perturbation. But when the multipole index l = 0, the late time tail of theneutral scalar perturbation settles down to a constant value, instead of a decay, implying that the l = 0 mode is prone to instability. B. Charged scalar perturbation
It is of great interest to extend the study to the charged scalar field perturbation. One expectsthat with the new term in (9), the potential will become more negative when the scalar field ischarged. Whether this can lead to a deeper potential well and destabilize the RN-dS black holeconfiguration is a question to be answered.We first concentrate on the perturbation with the angular index l = 0 for the charged masslessscalar field. In Fig. 2(a) we exhibit the potential behavior with the increase of the charge of themassless scalar field. It is clear that with the increase of the charge of the scalar field, the potential (a) (b) FIG. 1: (a) The potential and (b) the time-domain profiles of neutral massless scalar perturbations for M = 1, Q = 0 .
5, Λ = 10 − , l = 0 (red line) and l = 1 (blue line). The potential V ( r ) is obtained from theevent horizon r + = 1 .
866 to the cosmological horizon r c = 172 . r = 100. has wider and deeper negative well and the potential barrier becomes smaller. For big enoughcharge q , i.e. q ≥
2, the potential barrier disappears and the potential falls continuously from asmall negative value at the cosmological event horizon to a very negative constant at the blackhole event horizon. Our main concern are regions where the effective potentials contain negativevalues, since possible instabilities are usually indicated in such regions.The response of a stable RN-dS black holes to external charged scalar perturbation is determinedby the late time evolution of the perturbation. In Fig. 2(b) we show the time-domain profiles forthe evolution of charged scalar perturbations. When the scalar field is charged, the late time tailof the perturbation will grow in the end. With the increase of the charge of the scalar field, thegrowth will become stronger. This can be determined by the slopes of the late time tails. Thegrowth of the charged scalar perturbation can be attributed to the negative potential well, whichcan trap and accumulate the perturbation and finally destabilize the background RN-dS black holespacetime. But when the charge of the scalar field is large enough, its tail decays instead of growingin the end. This can be understood from the corresponding monotonic behavior in the potential,which cannot hinder the perturbation from falling into the black hole.Above, we focused on the massless charged scalar field. It is of interest to explore the influenceof the scalar field mass on the stability of the black hole. For the massive charged scalar field, weobserve the influence on the potential by different values of the mass of the scalar field in Fig. 3.The dashed lines are for the neutral massive scalar fields, while the solid lines are for the chargedscalar fields with different masses. We see that, when the scalar fields become more massive, the (a) (b)
FIG. 2: (a) The potential and (b) the time-domain profiles of charged massless scalar perturbations for M = 1, Q = 0 .
5, Λ = 10 − , l = 0 and q = 0 . , . , ,
2. The potential V ( r ) is obtained from the eventhorizon r + = 1 .
866 to the cosmological horizon r c = 172 . r = 100. potential wells become narrower and less negative. When the mass of the scalar field is largeenough, the potential will have only a positive barrier between the black hole and cosmologicalevent horizons.The evolution of perturbations is shown in Fig. 4. The dashed lines are for the neutral scalarfield. We see that only for the massless scalar perturbation, which contains a constant tail, allmassive perturbations exhibit a decay behavior at late time. When the mass of the scalar fieldis large enough, i.e. µ ≥ .
01, neutral scalar perturbations have a long-lived ringing at the latetime. The solid lines are for the scalar fields with fixed charge q = 0 . | ϕ | ∝ e κt at late time, we can getthe decay rate of perturbations in Table I. It is shown that, for both the neutral and charged scalarfield perturbations, the decay of tails first becomes quicker, and then slows down with the increaseof the mass of the scalar field. The massive field exhibiting oscillations in the late time tail hashigher energy, so that it decays slower. The influence of the scalar mass on the behavior of decayin the late time tail is consistent with that reported in the Kerr-Newman background [25].In summary we find that for the l = 0 mode, when the scalar field is charged, the perturbationcan grow. But such growth can be avoided if we increase the mass of the scalar field. The RN-dSblack hole background can be destabilized by the charged scalar field perturbation. FIG. 3: The potential V ( r ) is described from the event horizon r + = 1 .
866 to the cosmological horizon r c = 172 .
197 with M = 1, Q = 0 .
5, Λ = 10 − , and l = 0. The charge of the scalar field are q = 0 (dashed)and q = 0 . µ = 0 (red line), µ = 0 .
005 (green line), µ = 0 .
01 (blue line),and µ = 0 .
03 (black line) from bottom to top. µ = 0 µ = 0 . µ = 0 . µ = 0 . q = 0 0 − . − . − . q = 0 . . − . − . − . ∝ e κt at late time for l = 0, M = 1, Q = 0 . − . Here we obtained the value of κ for different value of the charge and mass of the field. Now, we discuss the l = 1 perturbation mode. We first look at the massless but charged scalarfield. In Fig. 5(a), we plotted the potential with the change of the charge q . It is similar to theone we observed for the l = 0 mode. With the increase of the charge, the region of negativepotential becomes wider and deeper, while the potential barrier becomes lower. When the chargeof the field is over a certain critical value, the potential becomes negative everywhere and decreasesmonotonically from the cosmological horizon to the black hole event horizon.In contrast to the situation in the l = 0 mode, the potentials with negative wells here donot imply instability. The time-domain profiles of perturbation are shown in Fig. 5(b). We seethat, when the charge of the scalar field increases, the decay of the perturbation tails becomesslower.We calculated that even until q = 5, which is 10 times the charge of the black hole itself,we still have the decay behavior of the perturbation. For larger q , in the numerical computationwe require smaller ∆ r ∗ to ensure convergence, what takes a much longer computation time. Thus,0 (a) (b)(c) (d) FIG. 4: Time-domain profiles for the scalar perturbations at r = 100 with M = 1, Q = 0 .
5, Λ = 10 − , and l = 0. The solid lines and the dashed lines denote q = 0 . q = 0, respectively. The masses of the fieldare (a) µ = 0, (b) µ = 0 . µ = 0 .
01, and (d) µ = 0 . in contrast with the l = 0 case, the RN-dS black hole background remains stable against chargedscalar perturbations when the angular index is l = 1. This result supports the argument given inRef. [22], that the negative effective potential does not guarantee any growing mode in the time-domain profiles for the evolution of the perturbations. The negative potential well can be viewedas a necessary condition for the instability of the black hole, rather than a sufficient condition.Generally, a bound state structure is required for the equivalent Schr¨odinger problem [23].For the l = 1 mode, the influence of the mass of the scalar field on the perturbation is illustratedin Fig. 6 for both neutral and charged scalar fields with fixed q = 0 .
5. It is shown that with theincrease of the mass of the scalar field, no matter whether it is neutral or charged, the decay ofperturbations first becomes quicker, and then slows down. This result can be seen in Table II withthe decay rate of perturbations fitted by | ϕ | ∝ e κt at late time, and is consistent with the l = 0case. When the mass of the scalar field is large enough, say µ ≥ .
05, one can see that the late1 (a) (b)
FIG. 5: (a) The potential and (b) the time-domain profiles of charged massless scalar perturbations for M = 1, Q = 0 .
5, Λ = 10 − , l = 1, and q = 0 , . , , . , .
8. The potential V ( r ) is obtained from the eventhorizon r + = 1 .
866 to the cosmological horizon r c = 172 . r = 100. time tail of a neutral scalar perturbation is dominated by a long-lived ringing. But for the chargedscalar field, the ringing will be followed by an exponential tail. (a) (b) FIG. 6: Time-domain profiles for the scalar perturbations at r = 100 with M = 1, Q = 0 .
5, Λ = 10 − , and l = 1. We consider (a) a neutral field and (b) a charged field with q = 0 .
5. The masses of the field are µ = 0(red line), µ = 0 .
005 (green line), µ = 0 .
05 (blue line), µ = 0 . µ = 0 . µ = 0 µ = 0 . µ = 0 . µ = 0 . µ = 0 . q = 0 − . − . − . − . − . q = 0 . − . − . − . − . − . ∝ e κt at late time for l = 1, M = 1, Q = 0 . − . Here we obtained the value of κ for different value of the charge and mass of the field. IV. DISCUSSION
The discussions presented in the previous section are focused on numerical calculations, but donot offer much physical insight. Could the instability of the charged scalar field perturbation for the l = 0 state be triggered by the superradiant instability? If we look at the effective potential, thereis a valley following the barrier. This is similar to the characteristic potential to accommodate thesuperradiant instability discussed in Refs. [10, 11, 14]. If there is superradiance, in the valley ofthe potential the reflection from the potential barrier could be amplified and its energy could beaccumulated continuously which could lead to instability.For the RN-dS case, to examine the superradiant instability, we have to consider the classicalscattering problem for the charged scalar field. The problem can be reduced to the wavelikeequation d Ψ dr ∗ + ( ω − V ( r ))Ψ = 0 , (13)with the effective potential V ( r ) = 2 qQωr − q Q r + f ( r ) (cid:18) l ( l + 1) r + µ + f ′ ( r ) r (cid:19) . (14)The effective potential has the following asymptotic behavior: when r → r + , V ≈ qQωr + − q Q r andwhen r → r c , V ≈ qQωr c − q Q r c respectively. In a scattering experiment, (13) has the followingasymptotic behavior Ψ ∼ Be − i ( ω − qQ/r + ) r ∗ , as r → r + ( r ∗ → −∞ ) , (15)Ψ ∼ e − i ( ω − qQ/r c ) r ∗ + Ae + i ( ω − qQ/r c ) r ∗ , as r → r c ( r ∗ → ∞ ) , (16)where A is called the amplitude of the reflected wave or the reflection coefficient, and B is the trans-mission coefficient. We adopted the boundary condition that the wave comes from the cosmological3horizon, partially passes through the potential barrier and falls inside the event horizon, while therest reflects back to the cosmological horizon [27, 28]. Using the constancy of the Wronskian, wecan show that 1 − | A | = ω − qQ/r + ω − qQ/r c | B | . (17)The reflected wave has larger amplitude than the incident one when qQr c < ω < qQr + . (18)Such an amplification of the incident wave is called superradiance. When the cosmological constantΛ approaches to zero, r c becomes infinite, and the superradiant condition Eq. (18) is equivalentto the one for the RN black hole in the presence of charged scalar perturbations, 0 < ω < qQ/r + [10, 11, 29]. Here, ω is the real oscillation frequency of the perturbation.Using the Prony method to fit the data in Fig. 2(b) at late time, we can obtain the dominantfrequencies of the modes in Table III for charged scalar perturbations and vanishing angular mo-mentum, l = 0. One can see that both the growing modes and decay mode satisfy the superradiantcondition (18). This means that the instability that we found is caused by the superradiance, butnot all the superradiant modes are unstable. This supports the discussion in a recent work [30]where it was argued numerically and analytically that the superradiance is a necessary condition,instead of a sufficient condition, for the instability. q ω qQ/r c qQ/r + . . . i . . . . . i . . . . i . . . − . i . . M = 1, Q = 0 .
5, Λ = 10 − and l = 0. V. CONCLUSION
In this paper, we have investigated the stability of the RN-dS black hole. In the four-dimensionalspacetime, the RN-dS black hole was found to be stable against the neutral scalar field perturbation[18–20]. However, recently, it has been reported that, when the spacetime dimensionality is
D >
6, the RN-dS black hole can become unstable if there are small gravitational perturbations of4scalar type [7, 8]. Here we found that, even in the four-dimensional spacetime, the instabilityof RN-dS black hole can appear under the perturbation of charged scalar field when the angularindex vanishes. But when the angular index is unit, the background four-dimensional RN-dSblack hole remains stable. We have noticed that the negative effective potential is not the onlycriterion to decide the stability of the background configuration; the practical tool for testingstability in spacetime backgrounds is the numerical investigation of the time-domain profiles of theperturbations. This supported the argument in Ref. [22] and the necessity of a bound state in theequivalent Schr¨odinger problem [23]. We have further explored the physical nature about why theinstability is triggered by the l = 0 mode in the four-dimensional RN-dS black hole. We found thatthis instability is caused by superradiance, but not all the superradiant modes are unstable. Acknowledgments
This work was supported in part by the National Natural Science Foundation of China. Z. Zwas also supported by China Postdoctoral Science Foundation under Grants No. 2011M500764and No. 2012T50414. E.A. and C.E.P. thank FAPESP and CNPq (Brazil) for support. [1] R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. , 793 (2011) [arXiv:1102.4014].[2] B. Wang, Braz.J.Phys. , 1029 (2005) [arXiv:gr-qc/0511133].[3] R. Gregory and R. Laflamme, Phys. Rev. Lett. , 2837 (1993) [arXiv:hep-th/9301052].[4] R. Gregory and R. Laflamme, Nucl. Phys. B , 399 (1994) [arXiv:hep-th/9404071].[5] R. A. Konoplya and A. Zhidenko, Phys. Rev. D , 104004 (2008) [arXiv:0802.0267].[6] M. Beroiz, G. Dotti and R. J. Gleiser, Phys. Rev. D , 024012 (2007) [arXiv:hep-th/0703074].[7] R. A. Konoplya and A. Zhidenko, Phys.Rev.Lett. , 161101 (2009) [arXiv:0809.2822].[8] V. Cardoso, M. Lemos and M. Marques, Phys. Rev. D , 127502 (2009) [arXiv:1001.0019].[9] V. Cardoso, arXiv:1307.0038.[10] S. Hod, Phys. Lett. B , 505 (2012).[11] S. Hod, Phys. Lett. B , 1489 (2013) [arXiv:1304.6474].[12] J. C. Degollado, C. A. R. Herdeiro and H. F. R´unarsson, Phys. Rev. D , 063003 (2013)[arXiv:1305.5513].[13] S. Hod, Phys. Rev. D , 064055 (2013) [arXiv:1310.6101].[14] S. -J. Zhang, B. Wang and E. Abdalla, arXiv:1306.0932.[15] X. He, B. Wang, R. -G. Cai and C. -Y. Lin, Phys. Lett. B , 230 (2010) [arXiv:1002.2679]. [16] E. Abdalla, C. E. Pellicer, J. de Oliveira and A. B. Pavan, Phys. Rev. D , 124033 (2010)[arXiv:1010.2806].[17] Y. Liu and B. Wang, Phys. Rev. D , 046011 (2012) [arXiv:1111.6729].[18] P. R. Brady, C. M. Chambers, W. Krivan and P. Laguna, Phys. Rev. D , 7538 (1997)[arXiv:gr-qc/9611056].[19] P. R. Brady, C. M. Chambers, W. G. Laarakkers and E. Poisson, Phys. Rev. D , 064003 (1999)[arXiv:gr-qc/9902010].[20] C. Molina, D. Giugno, E. Abdalla and A. Saa, Phys. Rev. D , 104013 (2004) [arXiv:gr-qc/0309079].[21] E. S. C. Ching, P. T. Leung, W. M. Suen and K. Young, Phys. Rev. Lett. , 4588 (1995)[arXiv:gr-qc/9408043].[22] K. A. Bronnikov, R. A. Konoplya and A. Zhidenko, Phys. Rev. D , 024028 (2012) [arXiv:1205.2224].[23] A. Bachelot and A. Motet-Bachelot, Ann. I. H. P.: Phys. Theor. , 3 (1993).[24] B. Wang, C. -Y. Lin and C. Molina, Phys. Rev. D , 064025 (2004) [arXiv:hep-tp/0407024].[25] R. A. Konoplya and A. Zhidenko, Phys. Rev. D , 024054 (2013) [arXiv:1307.1812].[26] E. Berti, V. Cardoso, J. A. Gonz´alez and U. Sperhake, Phys. Rev. D , 124017 (2007)[arXiv:gr-qc/0701086].[27] U. Khanal, Phys. Rev. D , 879 (1985).[28] T. Tachizawa and K. Maeda, Phys. Lett. A , 325 (1993).[29] J. D. Bekenstein, Phys. Rev. D7