Stability of scalarized charged black holes in the Einstein-Maxwell-Scalar theory

We analyze the stability of scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory with quadratic coupling. These black holes are labelled by the number of $n=0,1,2,\cdots$, where $n=0$ is called the fundamental black hole and $n=1,2,\cdots$ denote the $n$-excited black holes. We show that the $n=0$ black hole is stable against full perturbations, whereas the $n=1,2$ excited black holes are unstable against the $s(l=0)$-mode scalar perturbation. This is consistent with the EMS theory with exponential coupling, but it contrasts to the $n=0$ scalarized black hole in the Einstein-Gauss-Bonnet-Scalar theory with quadratic coupling. This implies that the endpoint of unstable Reissner-Nordstr\"{o}m black holes with $\alpha>8.019$ is the $n=0$ black hole with the same $q$.


Introduction
A scalarization of the Reissner-Nordström (RN) black holes was obtained in the Einstein-Maxwell-Scalar (EMS) theory [1]. The EMS theory is a simple second-order theory providing three kinds of propagating modes (scalar, vector, and tensor) around the black hole background. It is important to note that the appearance of the scalarized charged black holes is closely connected to the instability of the RN black hole with α > 8.019 for q = Q/M = 0.7 [2]. We note that these black holes are labelled as the n = 0(α ≥ 8.019), 1(α ≥ 40.84), 2(α ≥ 99.89), · · · black holes with α coupling constant.
All black hole solutions could be linearly tested to confirm that some solutions are selected as black holes in the curved spacetimes. Concerning the stability of scalarized black holes, it was shown that the n = 0 black hole is stable against l = 0(s-mode) scalar perturbation, while n = 1, 2, · · · black holes are unstable against the s-mode scalar perturbation in the Einstein-Born-Infeld-Scalar theory [3]. As was mention in [4], recently, a crucial difference between exponential and quadratic couplings in the Einstein-Gauss-Bonnet-Scalar (EGBS) theory states that the n = 0 black hole is stable against radial (spherically symmetric) perturbations for the exponential coupling, while it is unstable for the quadratic coupling. In the former case, the n = 0 black hole could be regarded as the endpoint of the evolution of unstable Schwarzschild black hole, whereas this is not the case for the latter. More recently, it is argued that the quadratic term controls the onset of the instability giving the n = 0 black hole, while the higher-order terms including the exponential coupling control the nonlinearities quenching the instability and thus, control the stability of the n = 0 black hole in the EGBS theory [5]. Very recently, the spontaneous scalarization of black holes and its stability in the EGBS theory were studied by including a massive and self-interacting term for different couplings [6,7].
For the stability of scalarized black holes in the EMS theory with exponential coupling [8], it is known that the n = 0 black hole is stable against full perturbations, while n = 1, 2 black holes are unstable against the s-mode scalar perturbation. Interestingly, the endpoint of unstable RN black holes may be the stable n = 0 black hole with the same q in the EMS theory with exponential coupling. Hence, it is very curious to know the stability issue of the n = 0, 1, 2 black holes in the EMS theory with quadratic coupling. It is noted that the n = 0 black hole is stable in the EMS theory with quadratic coupling by mentioning the positive potentials [9].
In this work, firstly we wish to show why the n = 0 black hole is stable (unstable) against radial perturbations for the exponential (quadratic) coupling in the EGBS theory by providing two kinds of potentials. It is known that the radial perturbations for l = 0, 1modes are equivalent to the full perturbations for the same modes. However, this is not true for higher modes of l = 2, 3, 4, · · · which are necessary to introduce the full perturbations.
We will perform the stability analysis on the n = 0, 1, 2 scalarized charged black holes in the EMS theory with quadratic coupling by observing the potentials and computing quasinormal mode spectrum. The full tensor-vector-scalar perturbations will be adopted for our analysis. Observing the potentials around the n = 0, 1, 2 black holes and together with computing quasinormal frequencies of the five physical modes, we show that the n = 0 black hole is still stable against full perturbations, while n = 1, 2 black holes are unstable against the l = 0-mode scalar perturbation in the EMS theory with quadratic coupling. This implies that the endpoint of unstable RN black holes with α > 8.019 is the n = 0(α ≥ 8.019) scalarized charged black hole with the same q. Furthermore, it states that there is no difference between exponential and quadratic couplings in the EMS theory, contrast to the EGBS theory.
2 n = 0, 1, 2, · · · black holes We consider the action of EMS theory with quadratic coupling [1] where α is a Maxwell-scalar coupling constant. If one considers a quadratic coupling of αφ 2 , one has to chooseφ = const to obtain the RN black hole with a different chargeQ 2 =φ 2 Q 2 .
In order to make the analysis clear, here, we choose an equivalent coupling of 1 + αφ 2 [9] withφ = 0 to give the same RN black hole.
From the action (1), the equations of motion are obtained as with G µν = R µν − (R/2)g µν and T µν = (1 + αφ 2 )(F µρ F ν ρ − F 2 g µν /4), and the Maxwell equation takes the form Here n represents the node number of ϕ(z) and it will denote the order number for labeling scalarized black holes.
The scalar equation is given by We consider the static scalar perturbed equation on the RN black hole background to identify the n = 0, 1, 2 black holes as where the RN (scalar-free) solutioñ is defined, irrespective of any value of α. Here (5) describes an eigenvalue problem: for given l = 0, requiring an asymptotically vanishing, smooth scalar selects a discrete set of the bifurcation points for scalar-free solution as α n (q = 0.7) = {8.019, 40.84, 99.89, · · · }.
In this case, the bifurcation points of the RN solution are the same as those of exponential coupling e αφ 2 [2,8] because the static scalar perturbed equation takes the same form as in (5). In Fig. 1, these solutions are classified by the node number n for ϕ(z) with z = r/(2M).
Furthermore, n will denote the order number for classifying different branches of scalarized black holes.
To obtain scalarized charged black holes, we have to introduce the spherically symmetric metric ansatz as with a metric function of N(r) = 1 − 2m(r)/r, in addition to electric potentialĀ 0 = v(r) and scalar fieldφ(r). We note that scalarized charged black holes are found by restricting an allowable range for α. The threshold of instability for the RN black hole is closely related to the appearance of the n = 0(α ≥ 8.019) black hole. Also, the static scalar perturbation around the RN black hole determines the appearance of n = 1, 2 · · · black holes. Plugging (7) into (2)-(4), one has the four equations where the prime ( ′ ) denotes differentiation with respect to its argument.
Considering the existence of a horizon located at r = r + , one finds an approximate solution to equations in the near horizon v(r) = v 1 (r − r + ) + . . . , where the four coefficients are given by This approximate solution involves two parameters of φ 0 = φ(r + ) and δ 0 = δ(r + ), which will be found when matching (12)-(15) with the asymptotic solutions in the far region On the other hand, we solve Eqs.(8)- (11) after replacing e αφ 2 with 1 + αφ 2 to obtain scalarized RN black holes in the EMS theory with exponential coupling. From Fig.   3, we find that the fundamental branch of exponential coupling is nearly the same as that of quadratic coupling. Here, both of the fundamental branches are defined from 0 to M α = 0.5/8.019 ≈ 0.062 where the RN black holes are unstable. For M/α > 0.062, the scalar hair (scalar charge Q s ) disappears and the branch merges with the stable RN branch.

Radial perturbations in the EGBS theory
It was known that the n = 0 black hole is stable against radial (spherically symmetric) perturbations for the exponential coupling, while it is unstable for the quadratic coupling by computing quasinormal modes of scalar perturbation [4]. They did not provide both of two potentials for exponential and quadratic couplings. Two potentials are necessary to see the stability of the n = 0 black hole with the different couplings intuitively. In this section, therefore, we will derive two potentials in the EGBS theory by carrying out the radial perturbations. It is clear that the radial perturbations for l = 0, 1-modes are equivalent to the full perturbations for the same modes. However, this is not true for the analysis of higher modes (l = 2, 3, 4, · · · ).
Let us start with the EGBS theory where λ is the Gauss-Bonnet coupling constant and f (φ) is the coupling function. The Gauss-Bonnet term is given by R 2 GB = R 2 − 4R µν r µν + R µνρσ R µνρσ . Varying the action (18) leads to equations where Γ µν appeared in [10]. Plugging (7) into (19)-(20), we obtain tt and rr components of Einstein equation and scalar equation In order to obtain scalarized black holes first, we need two linearized equations which describe h µν and δφ propagating around the Schwarzschild black hole with metric function N(r) = 1 − 2M/r. This process is independent of coupling. They are derived by linearizing When solving (25) and (26) with static ansatz, one obtains a discrete spectrum of parameter λ as M/λ = {0.587, 0.226, 0.140 . . .}, which describes the n = 0, 1, 2, · · · scalarized black holes [11]. Now let us derive the scalarized black holes by choosing quadratic and exponential couplings as Considering the outer horizon located at r = r + , one finds an approximate solution to three equations in the near-horizon with δ 0 = δ(r + , λ) and φ 0 =φ(r + , λ). Performing an expansion of the reduced field equations at the horizon and requiring regularity of the metric functions and scalar field, one finds that scalarized black hole solutions exist only when the condition is satisfied. Moreover, two parameters of δ 0 and φ 0 could be determined by matching with the scalar charge Q s . After solving three equations (22)-(24) together with boundary conditions, we could read off the fundamental branch (n = 0) from the graphs of scalar charge vs mass. From Fig. 4, we observe that the fundamental branch of n = 0 black hole is a finite region of 0 < M/λ < 0.587 in the exponential coupling, while it is just a band with bandwidth of 0.587 < M/λ < 0.636 for quadratic coupling [12,4]. It is important to note that the latter exists in the stable Schwarzschild black hole bound (beyond the fundamental branch for exponential coupling). This is one of differences between exponential and quadratic couplings in the EGBS theory.
We perform radial (spherically symmetric) perturbations by choosing three perturbations of [H 0 (t, r), H 1 (t, r), δφ(t, r)] as where ǫ is a control parameter of perturbations. A decoupling process makes a single second order equation for scalar perturbation [4,13] g(r) 2 ∂ 2 δφ ∂t 2 − where g(r), C 1 (r) and U(r) are functions of N(r), δ(r) andφ(r) and similar expressions are given in the Appendix of [4]. Here we do not display these functions for simplicity.
Considering a further separation of perturbed scalar we obtain the Schrödinger equation for scalar perturbation where r * is the tortoise coordinate and a redefined scalar perturbation Z(r) read as Further, C 0 (r) satisfies Importantly, the potential is given by which takes a lengthly form. In Fig. 5, we plot the potentials V 0 (r, λ) for l = 0-mode scalar around the n = 0 black hole in the EGBS theory with exponential and quadratic couplings.
It is obvious that the potential for exponential coupling is positive outside the horizon, while the potential for quadratic coupling develops negative-positive-negative regions outside the horizon, leading to ∞ r + V 0 (r)g(r)dr < 0 [sufficient condition for instability]. This is one of differences between exponential and quadratic couplings. Thus, the endpoint of unstable Schwarzschild black holes may be the stable n = 0 black hole in the EGBS theory with exponential coupling only.

Full linearized theory
We consider the full perturbed fields around the background quantities Plugging (40) into Eqs.(2)-(4) leads to complicated linearized equations. Considering ten degrees of freedom for h µν , four for a µ , and one for δφ initially, the EMS theory describing a massive scalar and massless vector-tensor propagations provides five (1+2+2=5) physically propagating modes on the scalarized 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 (7), the perturbations can be decomposed into spherical harmonics Y lm (θ, ϕ) with multipole index l and azimuthal number m. This decomposition splits the tensor-vector perturbations into "axial (A)" which acquires a factor (−1) l+1 under parity inversion and "polar (P)" which acquires a factor (−1) l .
We expand the metric perturbations in tensor spherical harmonics under the Regge-Wheeler gauge, providing six degrees of freedom. The axial part h A µν (t, r, θ, ϕ) is composed of two radial modes h 0 (r) and h 1 (r) and the polar part h P µν (t, r, θ, ϕ) takes four radial modes [H 0 (r), H 1 (r), H 2 (r), K(r)] with time-dependence e −iωt . Similarly, we decompose the vector perturbations into the axial vector a A µ (t, r, θ, ϕ) with single mode u 4 (r) and the polar vector a P µ (t, r, θ, ϕ) with two modes u 1 (r) and u 2 (r), giving three degrees of freedom. Lastly, we have a polar scalar perturbation as We note that the linearized equations could be split into axial and polar parts.
In general, the axial part is composed of two coupled equations for MaxwellF (u 4 ) and Regge-WheelerK(h 0 , h 1 ), where the potentials are given by Here the tortoise coordinate r * ∈ (−∞, ∞) is defined by the relation of dr * /dr = e δ /N.
At this stage, it is worth noting that in the limits ofφ = δ = 0, V A FF (r), V A FK (r), and V A KK (r) recovers those for the RN black hole in the Einstein-Maxwell theory [14]. Here, we will derive the quasinormal modes propagating around n = 0, 1, 2 scalarized black holes by solving the two coupled equations directly.

Stability Analysis of n = 0, 1, 2 black holes
The stability analysis will be performed by getting quasinormal frequency of ω = ω r + iω i in e −iωt when solving the linearized equations with appropriate boundary conditions at the outer horizon: ingoing waves and at infinity: purely outgoing waves. We will compute the lowest quasinormal modes of the scalarized black holes by making use of a reasonable numerical background and the linearized equations (42)-(43) for axial part and the linearized equations (45)-(50) for polar part. To compute the quasinormal modes, we use a directintegration method [15].
Usually, a positive definite potential V (r) without any negative region guarantees the stability of black hole. On the other hand, a sufficient condition for instability is given by ∞ r + dr[e δ V (r)/N(r)] < 0 [16] in accordance with the existence of unstable modes. However, some potentials with negative region near the outer horizon whose integral is positive ( ∞ r + dr[e δ V (r)/N(r)] > 0) may not imply a definite instability. To determine the instability of the n = 0, 1, 2 black holes clearly, hence, 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 e −iωt of the perturbation is decaying or not. If ω i < 0(> 0), the black hole is stable (unstable), irrespective of any value of ω r . However, it is not an easy task to carry out the stability of scalarized charged black holes because these black holes comes out as not an analytic solution but a numerical solution. To have a reasonable numerical background, it needs hundreds of numerical solutions in the each branch. It is convenient to classify the linearized equations according to multiple index of l = 0, 1, 2, · · · because l determines number of physical fields at the axial and polar sectors.
where the potential V P 0 (r, α) is given by [9] V P 0 (r, α) = N e 2δ r 2 (N + r( which is the same form as that obtained by choosing radial perturbations [9]. We display three scalar potentials V P 0 (r, α) in Fig. 6 for l = 0 case around the n = 0 black hole. The whole potentials are positive definite except that the α = 8.65 case has negative region near the horizon. It does not represent instability because this is near the threshold of instability. Actually, the n = 0 black hole is stable against the l = 0 scalar perturbation.
We confirm it from Fig. 7 that the imaginary frequency is negative for α ≥ 8.019, implying a stable n = 0 black hole. This is one of our main results. Hereafter, we will perform the stability analysis for higher multipoles on the n = 0 black hole only because the n = 1, 2 black holes turned out to be unstable against the l = 0 scalar perturbation. In other words, it seems meaningless to carry out a further stability analysis for the unstable n = 1, 2 black holes. 5.2 l = 1 case: n = 0 black hole In this case, we have three physical modes propagating around the n = 0 black hole. For l = 1 case, the axial linearized equation around the n = 0 black hole is given by where the potential takes the form We find that all potentials are positive definite for the n = 0(α ≥ 8.019) black hole. This means that the n = 0 black hole is stable against the axial l = 1 vector perturbation. We confirm it by showing that ω i is negative, indicating a stable black hole.
Finally, we obtain the vector-led and scalar-led modes propagating around the n=0 black hole by solving the polar l = 1 linearized equations (45)-(50). We find that ω i of vector-led mode around the n =0 is negative, implying a stable black hole. Also, it is found that ω i of scalar-led mode around the n = 0 black hole is negative, implying a stable black hole. we have three modes: vector-led, gravitational-led, and scalar-led modes. We find that all ω i of these modes are negative, implying the stable n = 0 black hole.

Summary and Discussions
It was shown in the EGBS theory that the n = 0 black hole is stable against radial perturbations for the exponential coupling, while it is unstable for the quadratic coupling. In the former case, the n = 0 black hole could be regarded as the endpoint of the evolution of unstable Schwarzschild black hole, whereas this is not the case for the latter. First of all, we discuss the differences between exponential and quadratic couplings for the n = 0 black hole (fundamental blanch) in the EGBS theory. We observe from Fig. 4 that the fundamental branch of n = 0 black hole is a finite region of 0 < M/λ < 0.587 in the exponential coupling, while it is just a band with bandwidth of 0.587 < M/λ < 0.636 for quadratic coupling where locates within the stable Schwarzschild black hole bound (beyond the fundamental branch for exponential coupling). This is one difference between exponential and quadratic couplings in the EGBS theory. Also, it is shown from Fig. 5 that the potential for exponential coupling is positive outside the horizon, while the potential for quadratic coupling develops negative-positive-negative regions outside the horizon, leading to ∞ r + V 0 (r)g(r)dr < 0 [sufficient condition for instability]. This is the other difference between exponential and quadratic couplings for the n = 0 black hole in the EGBS theory.
We have shown that the n = 1(α ≥ 40.84), 2(α ≥ 99.89) excited black holes are unstable against against the l = 0 scalar perturbation, while the n = 0(α ≥ 8.019) fundamental black hole is stable against all scalar-vector-tensor perturbations in the EMS theory with quadratic coupling. In the former case, the instability of the n = 1, 2, · · · black holes is regarded as the Gregory-Laframme instability because it arose from the s(l = 0) mode with an effective mass term. In the latter, we found all negative quasinormal frequencies (ω i < 0) of 9 = 1(l = 0) + 3(l = 1) + 5(l = 2) physical modes around the n = 0 black hole. In other words, we could not find any unstable modes from the l = 0, 1, 2 scalar-vector-tensor perturbations around the n = 0 black hole. Even though we have carried out the stability analysis on the n = 0, 1, 2 black holes, we expect from Fig. 7 that the other higher excited (n =3, 4, 5,· · · ) black holes are unstable against the s(l = 0)-mode scalar perturbation because their frequencies exist as branches along the unstable RN black holes. This is consistent with those for the EMS theory with exponential coupling [8], but it contrasts to the n = 0 scalarized black hole found in the ESGB theory with quadratic coupling when making use of radial (spherically symmetric) perturbations [4]. Actually, the n = 0 black hole found in the ESGB theory with exponential coupling has a similar property found in the EMS theory with exponential and quadratic couplings (See Figs. 3, 4). This implies that the endpoint of unstable RN black holes with α > 8.019 is the n = 0 scalarized black hole with the same q = 0.7 in the EMS theory with quadratic and exponential couplings.