We perform the stability analysis on scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory by computing quasinormal mode spectrum. It is noted that the appearance of these black holes with scalar hair is closely related to the instability of Reissner-Nordström black holes without scalar hair in the EMS theory. The scalarized black hole solutions are classified by the node number of , where is called the fundamental branch and denote the excited branches. Here, we show that the excited black holes are unstable against against the -mode scalar perturbation, while the fundamental black hole is stable against all scalar-vector-tensor perturbations. This is consistent with other scalarized black holes without charge found in the Einstein-Scalar-Gauss-Bonnet theory. Hence, we may regard the instability of the black holes as the Gregory-Laframme instability arising from an effective mass term.
Quasinormal modes of scalarized black holes in the Einstein-Maxwell-Scalar theory
Yun Soo Myung***e-mail address: email@example.com and De-Cheng Zou†††e-mail address: firstname.lastname@example.org
Institute of Basic Sciences and Department of Computer Simulation, Inje University Gimhae 50834, Korea
Center for Gravitation and Cosmology and College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
Recently, a scalarization of the Reissner-Nordström (RN) black holes was investigated in the Einstein-Maxwell-scalar (EMS) theory which is a simpler theory than the Einstein-Scalar-Gauss-Bonnet-scalar (ESGB) theory . Here, may increase beyond unity, compared to for the RN black hole. The EMS theory is a second-order theory which includes three propagating modes of scalar, vector, and tensor. In this case, the instability of RN black hole was determined solely by the linearized scalar equation because the RN black hole is stable against tensor-vector perturbations theory [2, 3, 4, 5]. It is shown that the appearance of the scalarized charged black hole is closely associated with the Gregory-Laflamme (GL) instability of the RN black hole without scalar hair . A difference with the ESGB theory  is that there is no scalarization bands in the EMS theory, implying no upper bound on the coupling constant as the scalarized charged black holes.
The scalarized black holes without charge have been found from the ESGB theories [7, 8, 9]. It is emphasized that these black holes with scalar hair are connected to the appearance of instability for the Schwarzschild black hole without scalar hair. We note that the instability of Schwarzschild black hole in ESGB theory is considered as not the tachyonic instability but the GL instability  when comparing it with the GL instability of the Schwarzschild black hole in the Einstein-Weyl gravity . Here, the notion of the GL instability comes from the three observations [12, 13, 14, 15]: i) The instability is based on the -mode perturbation for either massive scalar or massive tensor. ii) The perturbed equation should include an effective mass term, so that the potential develops negative region near the horizon of black hole but it becomes positive just after crossing the -axis, leading to . Actually, this corresponds to a weaker condition than the sufficient condition of instability () including the tachyonic instability because the integral of potential may be positive. iii) The instability of a black hole without hair is closely related to the appearance of a newly black hole with hair where the hair is defined by either non-zero scalar or Ricci tensor outside the horizon.
Concerning the stability of scalarized black holes, it turns out that the black hole is stable against all perturbations, while black holes are unstable against the -mode) scalar perturbation in the Einstein-Born-Infeld-scalar theory  and the ESGB theory . The former was based on the scalar perturbation only, while the latter was based on the spherically symmetric tensor perturbations including the scalar perturbation. For the stability of scalarized charged black hole in the EMS theory, the black hole was mentioned within the scalar perturbation .
In this work, we wish to carry out the stability analysis on the scalarized charged black holes in the EMS theory by computing quasinormal mode spectrum. We wish to employ the full tensor-vector-scalar perturbations splitting into the axial and polar parts. Observing the potentials around the black holes with and together with computing quasinormal frequencies of the five physically propagating modes, we will find that the black hole is stable against all perturbations, while black holes are unstable against the -mode) scalar perturbation in the EMS theory.
2 Scalarized charged black holes
We start by mentioning the action of EMS theory without scalar potential 
where is a scalar field, is a Maxwell-scalar coupling constant as a mass-like parameter, and is the Maxwell kinetic term. In this work, we do not consider the Einstein-Maxwell-dilaton theory with a usual coupling of . The EMS theory describes three of a massive scalar, a massless vector, and a massless tensor which provide five (1+2+2=5) physically dynamical modes propagating on the scalarized charged black hole background.
First of all, we derive the Einstein equation from the action (1)
with and . The Maxwell equation takes the form
Importantly, the scalar equation is given by
For our purpose, we introduce the metric ansatz as 
with a metric function , in addition to potential and scalar . We would like to mention that the RN black hole solution () is defined, irrespective of any value of . However, a scalarized charged black hole is defined by restricting an allowable range for . The threshold of instability for a RN black hole is closely related to the appearance of the fundamental branch where the scalarized charged black hole appears. Also, the GL instability of a RN black hole without scalar hair is connected to the appearance of scalarized charged black holes.
where the prime () denotes differentiation with respect to its argument. From (8), one has a relation of . Accepting an outer horizon located at , one finds an approximate solution to four equations in the near-horizon
where the coefficients are determined by
which include the scalar charge and the electrostatic potential at the horizon .
As a concrete scalarized black hole solution with , we display the two numerical solutions [metric function only] with the coupling constant locating on the fundamental branch in Fig. 1. We emphasize that the non-zero scalar of case is not allowed for . Two graphs of in Fig. 2 are between and 45. It is worth noting that the excited branch solutions take the similar forms as the case. For simple notation, we call these scalarized charged black holes as the black holes. For different , we may find different scalarized charged black holes by classifying the black holes. We will perform the stability analysis on the black hole solutions with in the next section. Although the black holes exist, one expects to find similar features as the black holes.
3 Linearized equations
We start by considering the perturbed fields around the background quantities
Plugging (16)-(18) into Eqs.(2)-(4) leads to complicated linearized equations. Considering ten degrees of freedom for , four for , and one for initially, the EMS theory describing a massive scalar and massless vector-tensor propagations provides five (1+2+2=5) physically propagating modes on the black hole background. The stability analysis should be based on these physically propagating fields as the solutions to the linearized equations. In a spherically symmetric background (5), the perturbations can be decomposed into spherical harmonics with multipole index and azimuthal number . This decomposition splits the tensor-vector perturbations into “axial (A)” which acquires a factor under parity inversion and “polar (P)” which acquires a factor .
We expand the metric perturbations in tensor spherical harmonics in the Regge-Wheeler gauge. For the axial part with two modes and , the perturbed metric takes the form
where asterisks denote symmetrization. For polar perturbations with four modes (), we have
On the other hand, we decompose the vector perturbations into
where we gauge away. Based on the vector perturbations, we construct the perturbed Maxwell tensor which is relevant to linearized equations as
Lastly, we have a polar scalar perturbation as
The linearized equations could be split into axial and polar parts.
In general, the axial part is composed of two coupled Maxwell and Regge-Wheeler equations,
where the potentials are given by
The above may lead to a diagonalized equation as
Here the tortoise coordinate is defined by the relation of and a diagonalized potential is given by .
On the other hand, the polar part takes six coupled equations for Zerilli, Maxwell, and scalar equations
with and . Interestingly, these coupled equations describe three physically propagating modes.
4 Stability Analysis
The stability analysis will be performed by getting quasinormal frequency of when solving the linearized equations with appropriate boundary conditions at the outer horizon and infinity. Usually, a positive definite potential without any negative region guarantees the stability of black hole. On the other hand, a sufficient condition for instability is given by  in accordance with the existence of the unstable modes. However, some potentials with negative region near the outer horizon whose integral is positive () do not imply a definite instability. To determine the instability of the black holes clearly, one has to solve all linearized equations for physical perturbations numerically.
Accordingly, the criterion to determine whether a black hole is stable or not against the physical perturbations is whether the time evolution of the perturbation is decaying or not. If , the black hole is stable (unstable), irrespective of any value of . However, it is a nontrivial task to carry out the stability of a scalarized charged black hole because this black hole comes out as not an analytic solution but a numerical solution. In order to develop the stability analysis, it is convenient to classify the linearized equations according to multiple index because determines number of physical fields at the axial and polar sectors.
4.1 case: one DOF
For (-mode), the linearized equation obtained from the polar part is given entirely by a scalar equation ()
where the potential is given by 
We display four scalar potentials in Fig. 2 for case around the black hole. The whole potentials are positive definite except that the case having negative region near the horizon does not represent instability really because it is near the threshold of instability. Actually, the black hole is stable against the (-mode) scalar perturbation since the case corresponds to the threshold of instability satisfying the condition of . Although this condition does not rule out the possibility of unstable modes, one does not find any unstable modes. We confirm it from Fig. 3 that the imaginary frequency is negative for , implying a stable black hole. Importantly, we observe that is independent of . This is very similar to the stable RN black hole for in the EMS theory . From this observation, one finds a similarity and difference between and RN black holes in Table 1.
Now let us turn to the stability issue of the black holes. We observe from Fig. 4 that for the black hole, while the whole potentials are negative definite for the black hole. This implies that the black holes are unstable against the -mode) scalar perturbation. Clearly, the instability could be found from Fig. 5 because their imaginary frequencies are positive. Here the red curve denotes the instability (positive ) of RN black hole as a function of . Attaching (Right) Fig.3 on Fig. 5 indicates the negative imaginary frequency around the (stable) black hole. This instability may be regarded as the GL instability because it corresponds to the -mode instability.
A slight difference is that the GL instability of RN black hole is determined by (stable for ) for given , whereas the sufficient condition for instability of RN black hole takes the form of . On the other hand, the is stable, while the black holes are unstable. However, these scalarized charged black holes are not defined for and as shown in Fig. 1 and Table 1.
4.2 case: three DOF
For case, the axial linearized equation takes a diagonalized form
where the potential is given by
We find from Figs. 6 and 7 that all potentials are positive definite for the black holes. This means that the black holes are stable against the axial vector perturbation. We confirm it from Fig. 8 that all are negative, indicating stable black holes. Moreover, the quasinormal frequency at coincides with that for the fundamental EM mode () around the RN black holes [22, 23].
Finally, we obtain the vector-led and scalar-led modes around the =0, 1, 2 black holes from the polar linearized equations (3)-(3). We find from Fig. 9 that all around the 0, 1, 2 are negative, implying stable black holes. We note that the fundamental frequency of vector-led mode takes the same value of for that around the RN black hole in the EM theory. Also, it is observed from Figs. 10 that all of scalar-led mode around the black holes are negative, implying stable black holes. However, it is noted that starting frequencies for the scalar-led mode depend on the 0, 1, 2 black holes: for , for , and for .
4.3 case: five DOF
First of all, we consider the axial part because of its simplicity. The axial linearized equations are given by two coupled equations for Regge-Wheeler-Maxwell system as
where three potentials are given by
Solving these coupled equation with boundary conditions leads to quasinormal frequencies for vector-led mode around the black holes as is shown in Fig. 11, implying stable black holes. Here we obtain the fundamental frequency of for vector-led mode around the RN black hole in the EM theory [22, 23], being independent of . We find from Fig. 12 that the black holes are stable against the gravitational-led mode. We note that the fundamental frequency of (for gravitational-led mode around the RN black hole in the EM theory) plays the role of a starting point for the black holes [22, 23].
Finally, the polar linearized equations are given by Eqs.(3)-(3) with . Here we have three modes: vector-led, gravitational-led, and scalar-led modes. We find from Figs.13, 14 and 15 that all are negative, implying the stable black holes. It is worth noting that the fundamental frequencies of vector-led and gravitational-led modes around the RN black hole in the EM theory take the same values as in the axial case: for vector-led mode and for gravitational-led mode. For the polar scalar-led mode, however, the quasinormal frequency starts differently from for , for , and for .
5 Summary and Discussions
In this work, we performed the stability analysis of the scalarized charged black holes in the EMS theory by computing quasinormal mode spectrum. This is a nontrivial task and completing it takes a long time because these black holes are found in numerically. In this case, we found quasinormal frequencies of physical modes around black holes, depending on the mass-like coupling . The (un)stable mode is found when the imaginary frequency is (positive) negative.
We have shown that the excited black holes are unstable against against the -mode scalar perturbation only, while the fundamental black hole is stable against all scalar-vector-tensor perturbations. Even though we have carried out the stability analysis on the , 1, 2 black holes, we expect to find from Fig. 5 that the 3, 4, 5, higher excited black holes are unstable against the -mode scalar perturbation. We could not find any unstable modes from the scalar-vector-tensor perturbations, as in the RN black hole . This is consistent with other scalarized black holes without charge found in the ESGB theory by making use of spherically symmetric perturbations . Here, the instability of the black holes is regarded as the Gregory-Laframme instability because it arose from the mode with an effective mass term.
Concerning connection between scalarized black holes and RN black hole in the EMS theory, the black hole (RN black hole) are stable against the -mode scalar perturbation for . We make a further connection between scalarized black holes in the EMS theory and RN black hole in the EM theory. The quasinomal frequencies of vector- and gravitational-led modes around the 0, 1, 2 black holes start from fundamental quasinormal frequencies of vector- and gravitational-led modes around the RN black hole in the EM theory [22, 23]. This provides the checking point for correctness for our computations. On the other hand, it is shown in Fig. 5 that quasinormal frequencies of scalar mode start differently from the 0, 1, 2 black holes. Similarly, it is indicated from Figs. 10 and 15 that quasinormal frequencies of scalar-led modes start differently from the 0, 1, 2 black holes. This implies that the scalar plays a key role in obtaining the 0, 1, 2 black holes and analyzing their stability in the EMS theory. The presence of the -mode scalar induces the GL instability on the black holes, whereas it does not induce any instability on the black hole.
Acknowledgments We are grateful to Yunqi Liu for useful discussions. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. NRF-2017R1A2B4002057).
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