Instability of Reissner-Nordstr\"{o}m black hole in Einstein-Maxwell-scalar theory

The scalarization of Reissner-Nordstr\"{o}m black holes was recently proposed in the Einstein-Maxwell-scalar theory. Here, we show that the appearance of the scalarized Reissner-Nordstr\"{o}m black hole is closely related to the Gregory-Laflamme instability of the Reissner-Nordstr\"{o}m black hole without scalar hair.


Introduction
Section 6 is devoted to obtaining a scalarized RN black hole by solving the four equations (41)-(44) numerically. It indicates that the appearance of the scalarized RN black hole is closely related to the GL instability of the RN black hole without scalar hair. Also, we obtain scalarized RN black holes for the quadratic coupling of αφ 2 for comparison to the exponential coupling e αφ 2 . Finally, we will describe our main results in section 7.

EMS and its linearized theory
The EMS theory is given by [9] S EMS = 1 16π where φ is the scalar field with a potential V φ , α is a positive coupling constant, and F 2 = F µν F µν is the Maxwell kinetic term. Here we choose V φ = 0 for simplicity. This theory implies that three of scalar, vector, and tensor are physically dynamical fields. It is noted that a different dilaton coupling of e −2α 0 φ was introduced for the Einstein-Maxwelldilaton theory originating from a low-energy limit of string theory [16,17]. Moreover, a quadratic coupling of αφ 2 will be considered as the other model to reveal scalarized charged black holes in section 7.
Now, let us derive the Einstein equation from the action (1) where G µν = R µν − (R/2)g µν is the Einstein tensor and T µν = F µρ F ν ρ − F 2 g µν /4 is the Maxwell energy-momentum tensor. The Maxwell equation is given by The scalar equation takes the form Consideringφ = 0 and electrically chargedĀ t = Q/r, one finds the RN solution from (2) and (3) with the metric function Here, the outer horizon is located at r = r + = M + M 2 − Q 2 = M(1 + 1 − q 2 ) with q = Q/M, while the inner horizon is at r = r − = M(1 − 1 − q 2 ). It is worth noting that (5) dictates a charged black hole solution without scalar hair. We stress that the RN solution (5) is a black hole solution to the EMS theory for any value of α. Hereafter we are interested in the outer horizon.
In order to explore the stability analysis, one has to obtain the linearized theory which describes the metric perturbation h µν , vector perturbation a µ and scalar perturbation ϕ propagating around the RN background (5) denoting by¯(overbar). By linearizing (2), (3), and (4), we find three linearized equations as ∇ µ f µν = 0, where the linearized Einstein tensor δG µν , the linearized energy-momentum tensor δT µν , and the linearized Maxwell tensor f µν are given by We note that an effective mass term of −αQ 2 /r 4 in (9) is replaced by −2λ 2 M 2 /r 6 in the ESGB theory [4]. Here the scalar coupling constant 'α > 0' plays the role of a mass-like parameter.

Instability of RN black hole
In analyzing the stability of the RN black hole in the EMS theory, we first consider the two linearized equations (7) and (8) [12,13]. Also, the even-parity perturbations with two physical degrees of freedom (DOF) were studies in [14,15]. It turns out that the RN black hole is stable against these perturbations. In this case, a massless spin-2 mode starts with l = 2, while a massless spin-1 mode begins with l = 1. The EMS theory provides 5(=2+2+1) DOF propagating around the RN background. Now, we focus on the linearized scalar equation (9) which determines the stability of the RN black hole in the EMS theory. Introducing and considering a tortoise coordinate r * defined by dr * = dr/f (r), a radial equation of (9) leads to the Schrödinger-type equation where the scalar potential V (r) is given by In Fig. 1, we find the α-dependent potentials for given l = 0, M = 1.1 and q = M/Q = 0.418 (a non-extremal RN black hole). The s(l = 0)-mode is allowed for the scalar perturbation and it is regarded as an important mode to test the stability of the RN black hole.
Hereafter, we consider this mode only.
A sufficient condition of ∞ r + drV (r)/f (r) < 0 for instability [18,19] leads to the bound as The first term 3/q 2 was found in analyzing the black hole dynamics in Einstein-Maxwelldilaton theory [16].
On the other hand, by observing the potential (15) carefully, the positive definite potential without negative region could be implemented by imposing the bound which guarantees a stable RN black hole. This is called the sufficient condition for stability.
We note that (16) is not a necessary and sufficient condition for the instability. Observing At this stage, we would like to mention that such potentials exist around neutral black holes (black holes without charge) in higher dimensions and the S-deformation has been used to confirm the stability of neutral black holes in higher dimensions [20]. We conjecture that the GL instability may occur for α th > 19.83, but the threshold of instability α th should be determined explicitly by the numerical computations. We expect that α th is located at the shaded region between α in and α po . Usually, if the potential V derived from physically propagating modes is negative in some region, a growing perturbation can appear It suggests that the RN black hole would be unstable for α > α th = 29.47 with q = 0.418, while it is stable for 19.83 < α < 29.47 showing negative region near the horizon. In the latter case, the S-deformation method could provide a complementary result to support the stability of such black holes by finding the deformed potential [21,22].
To determine the threshold of instability explicitly, one has to solve the second-order differential equation numerically which allows an exponentially growing mode of e Ωt (ω = iΩ) as an unstable mode. Here  Figure 3: Three α-curves as function of q. The upper blue curve represents α in (q) in (16) and the middle green curve indicates α th (q), while the lower red one denotes α po (q) (17).
Consequently, the GL instability bound for the RN black hole is given by which is considered as one of our main results. However, we could not determine an explicit form of α th (q) as function of q like as α in (q) in (16). In addition, the small unstable black appears when the bound satisfies r + < r c (q = 0.7) = 1.714 (21) at α = 8.019.

Static scalar perturbation
Here, it is worth checking the instability bound (20) again because the precise value of α th (q) determines scalarized RN black holes. This can be achieved by obtaining the static perturbed solutions to the linearized equation (14) with ω = 0 on the RN background. For a given l = 0 and q, requiring an asymptotically vanishing condition (ϕ ∞ → 0) leads to the fact that the existence of a smooth scalar determines a discrete set for α. In addition, it determines n = 0, 1, 2, · · · branches of scalarized black holes. Introducing a static condition (ω = 0) and a new coordinate of z = r/2M, the equation for u(r) reduces to where Here we wish to find a numerical solution even though an analytic solution is available for l = 0 case [9]. For this purpose, we first propose the near-horizon expansion for u(z) as This expression can be used to set data outside the outer horizon for a numerical integration from z = z + to z = ∞. Here the coefficients u ′ + and u ′′ + could be determined in terms of a free parameter u + as An asymptotic form of u(z) near the infinity of z = ∞ is given by where two relations are expressed in terms of u ∞ as At this stage, it is worth noting that we search for bound state scalar solution to (22) in the RN spacetime. We are free to choose the value of scalar field u + at the horizon because (22) is a linear differential equation and then, we choose u ∞ = 1 at infinity [2].
Actually, a numerical solution could be obtained by connecting the near-horizon form (23) to the asymptotic form (25)  these solutions are classified by the order number n = 0, 1, 2, · · · which is identified with the number of nodes for ϕ(z) = u(z)/z. We find that the n = 0 scalar mode without zero crossing represents the fundamental branch of scalarized black holes, while the n = 1, 2 scalar modes with zero crossings denote n = 1, 2 higher branches of scalarized black holes.
Actually, this corresponds to finding the l = 0 bifurcation points from the RN black hole with q = Q/M. Finally, we confirm that for given q, α n=0 (q) = α min (q) [underline value] recovers the threshold of instability α th (q) exactly.

GL instability
The instability of the RN black hole may be regarded as the GL instability since this instability is based on the s(l = 0) mode of a perturbed scalar and its linearized equation includes an effective mass term (not tachyonic mass of m 2 t < 0 presicely) which develops negative potential near the horizon from the Maxwell kinetic term. In this section, we wish to clarify the similarity and difference between the GL instability (modal instability) and tachyonic instability because the instability of RN black hole is closely related to appearance of scalarized RN black holes.
Let us first introduce the tachyon propagation with mass squared m 2 which provides a Schrödinger equation for radial part As is shown Fig. 5, the potential V t (r) for l = 0 tachyonic mode develops a positive region near horizon, while it approaches −0.01 as r → ∞ for m 2 t = −0.01. This shows clearly the tachyonic instability of RN black hole because the sufficient condition for instability ( ∞ r + drV t (r)/f (r) = −∞ < 0) is always satisfied with any mass m 2 t = −const < 0. We wish to mention that V t differs from V (r) in (15) in the sense that the latter is negative near horizon and becomes positive after crossing the r-axis. We regard '−αQ 2 /r 4 ' in V (r) as an effective mass term which can be made sufficiently negative by choosing α, making the scalar potential sufficiently negative in the near horizon. However, its role is limited to small r, because it approaches zero as r → ∞. Such a r-dependent mass term is necessary to have the scalarized RN black holes. Now we consider the stability of Schwarzschild black hole in Einstein-Weyl gravity whose action takes the form [6,8] where the Lichnerowicz operator is given by We note here that the condition of non-tachyonic mass requires m 2 2 > 0 because the Lichnerowicz operator contains −¯ . Before we proceed, we would like to mention the GL instability. For this purpose, we consider the perturbations around the 5D black string with ds 2 5 = ds 2 4 + dz 2 where ds 2 4 denotes the Schwarzschild line element, where the z-dependence is assumed to be of the form e ikz and the time-dependence takes the form of e Ωt . Actually, Eq.(31) takes the same form as the linearized black string equation for h (4) µν with the transverse-traceless gauge [3] (∆ L + k 2 )h (4) µν = 0 (33) except that the mass m 2 of the Ricci tensor is replaced by the wave number k along z direction. The GL instability states that the 5D black string is unstable against the metric perturbation for k < k th = 0.876/r + (long wavelength perturbation). The GL instability is an s(l = 0)-wave spherically symmetric instability from the four-dimensional perspective.
In addition, it is interesting to note that the dRGT massive gravity having a Schwarzschild solution when formulated in a diagonal bimetric form, has the same linearized equation as (31) except replacing δR µν by h µν [23,24].
As is shown in Fig. 6, all potentials develop negative region near the horizon, whereas their asymptotic limits are nonzero constants (V Z → m 2 2 , r → ∞). The former is similar to V (r) in (15), while the latter is different from V → 0 as r → ∞. This may imply that the structure of scalarized black holes differs from that of non-Schwarzschild black hole (Schwarzschild black hole with Ricci-tensor hair). Solving Eq.(34) with boundary conditions, one finds unstable tensor modes from Fig. 7. From Fig. 7, the GL instability mass bound for s(l = 0)-mode is given by where m th represents the threshold of GL instability.
On the other hand, we confirm the precise value of m th by solving the static Lichnerowicz-Ricci tensor equation as [7] where the eigenvalue λ should be determined by requiring the existence of a normalizable eigenfunction ψ µν . This amounts to seeking a negative eigenvalue λ for which the exponen- to the edge of the zone of Schwarzschild instability and the existence of non-Schwarzschild black holes. Importantly, this process is very similar to Section 4 for determining α th in the EMS theory. The difference is that many branches of α = {8.019, 40.84, 99.89, · · · } for q = 0.7 exist in the EMS theory, while a single branch of m 2 2 = 0.7677 exists for the EW gravity. This may be so because their asymptotic forms of potentials are different (V → 0 versus V Z → m 2 2 as r → ∞). Hence, the boundary condition at infinity is an asymptotically vanishing scalar (ϕ ∞ → 0) in the EMS theory, while it is a normalizable mode in the EM gravity. An actual correspondence would be met if one includes a mass term of V (φ) = 2αφ 2 in (1), leading to the s(l = 0)-mode potential which has similar asymptote (V cp → α as r → ∞) to V Z (r) in (35).
From (36), selecting m th = 1 for r + = r c = 0.876, one finds the bound for unstable (small) black holes It is worth noting that r + = r c corresponds to the bifurcation point which allow a new non-Schwarzschild black hole [25]. At this stage, we note that the appearance of non-Schwarzschild black hole is closely related to the threshold of instability for Schwarzschild black hole in the Einstein-Weyl gravity [6,8].
We summarize whole properties for instability happened in the EMS theory and Einstein-Weyl gravity in Table 1. It is emphasized that the role of s-mode scalars ϕ in the EMS theory is replaced by a s-mode Ricci tensor δR µν (φ 0 ) in the Einstein-Weyl gravity.
Theory Einstein-Maxwell-scalar theory Einstein-Weyl gravity (30) BH without hair RNBH withφ = 0 SBH withR µν = 0 Linearized equation scalar equation (9) LR-equation (31) GL instability mode s-mode of ϕ s-mode of δR µν Bifurcation points α = 8.019, 40.84, 99.89, · · · for q = 0.7 m 2 2 = 0.7677 Potential and its asymptotic form V (r) in (15) and Small unstable BH r + < r c = 1.714 with α th = 8.019(q = 0.7) r + < r c = 0.876 with m th = 1 BH with hair scalarized RN BH non-Schwarzschild BH 6 Scalarized RN black holes 6.1 Exponential coupling Before we proceed, we note that the RN black hole solution is allowed for any value of α, while a scalarized RN black hole solution may exist only for α ≥ α th . The threshold of instability for a RN black hole reflects the disappearance of zero crossings in the perturbed scalar profiles. We explore a close connection between the instability of a RN black hole without scalar hair and appearance of a scalarized RN black hole. As a concrete example, we wish to find a scalarized RN black hole which is closely related to the q = 0.7(M = 1, Q = 0.7) and α ≥ 8.019 case (n = 0 case).
For this purpose, let us introduce the metric ansatz as [9] ds 2 sRN = −N(r)e −2δ(r) dt 2 + where a metric function is defined by N(r) = 1 − 2m(r)/r with the mass function m(r).
Also, we consider the U(1) potential and the scalar as A = v(r)dt and φ(r). Substituting these into Eqs.
(2)-(4) leads to the four equations Assuming the existence of a horizon located at r = r + , one finds an approximate solution to equations in the near horizon 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 (45)-(48) with the asymptotic solutions in the far region where Q s and Φ denote the scalar charge and the electrostatic potential, in addition to the ADM mass M and the electric charge Q. For simplicity, we choose φ ∞ = 0 here.
The EMS theory admits the RN black hole solution for any α. However, it becomes an unstable black hole for α > α th (q) (20), while it is stable against the scalar perturbation for α < α th (q). We note that 'α = α th (q)' indicates the threshold of instability. One expects that a scalarized RN black hole is allowed for α ≥ α th (q) when q ≥ 0.7. This means that the scalarized RN black holes bifurcates from the RN black hole hole at α = α th (q), but q increases beyond unity for the fixed α, implying that the scalarized RN black hole could be overcharged [9].
For the RN black hole with φ 0 = 0, the outer horizon is located at r + = 1.714 and the charge-mass ratio is given by q = 0.7. In the Fig.8 (left), one observes that for given α = 8.019, the ratio of q for the n = 0 scalarized RN black hole increases beyond the extremal RN black hole (q = 1) as φ 0 increases. Moreover, in the Fig. 8 (right), the scalar at the horizon φ 0 increases as the horizon radius r + decreases. The scalar at the horizon is terminated at r + = 1.714, corresponding to the RN outer horizon. It is the starting point for a scalarized RN black hole, while from (21) it corresponds to the ending point for unstable RN black hole.
It is known that the scalarization bands exist for the ESGB theory [2]. A discrete set for η/M 2 obtained from static scalar perturbation corresponds to the right-end values of scalarization bands for a scalarized Schwarzschild black hole, while the left-end values are provided by the regularity constraint at the horizon (r 4 + ≥ 6η 2 φ 2 0 ). However, as is shown in Fig. 9, there are no scalarization bands in the EMS theory because we do not need to impose the regularity condition at the horizon. As a result, there is no upper bound on α as n = 0(α ≥ 8.019), 1(α ≥ 40.84), and 2(α ≥ 99.89).
Consequently, we obtain the scalarized RN black hole solution depicted in Fig. 10. The metric function N(r) has a different horizon at ln r = ln r + = 0.067 in compared to the RN horizon at ln r = ln r + = 0.539 and it approaches the RN metric function f (r) as ln r increases. Also, the scalar hair φ(r) starts with φ 0 = 0.44 at the horizon and it decreases as ln r increases, in compared to φ(r) = 0 for the RN black hole.

Quadratic coupling
Considering the quadratic coupling of αφ 2 , we have to chooseφ = const to obtain the RN black hole with different chargeQ 2 = αφ 2 Q 2 . In order to make the analysis simple, we may choose an equivalent coupling of 1 + αφ 2 withφ = 0 to give the RN black hole. In this case, the bifurcation points of the RN solution are the same as those of exponential coupling because the static scalar equation takes the same form as in (22  10, we observe that the quadratic coupling shows the similar properties to the exponential coupling.
As was mention in [26], however, the only difference between two coupling in the ESGB theory is that the n = 0 fundamental branch of scalarized black holes is stable for the exponential coupling, while the n = 0 fundamental branch is unstable for the quadratic coupling. Therefore, we expect that the similar thing will happen since the n = 0 scalarized RN black hole turned out to be unstable in the EMS theory with exponential coupling [27].

Discussions
First of all, we mention that scalarized RN black holes were found in the EMS theory. It is emphasized that the appearance of these black holes with scalar hair is closely related to the instability of the RN black hole without scalar hair in the EMS theory. Concerning the appearance scalarized RN black holes [9], it is very important to obtain the precise threshold α th of instability for the RN black hole in the EMS theory. In this work, we have obtained the GL instability bound (20) for the RN black hole in the EMS theory by considering s(l = 0)-mode scalar perturbation.
Roughly speaking, a shape of scalar potential V (r) in (15) determines the instability of RN black hole. The sufficient condition of ∞ r + dr[V (r)/f (r)] < 0 for instability [18,19] gives rises to an analytic bound (16), while the sufficient condition for stability is given by the other bound (17). Explicitly, for q = 0.418, the sufficient condition for instability takes the form of α > 30.74, whereas the sufficient condition for the stability is given by 0 < α ≤ 19.83. In the case of ∞ r + dr[V (r)/f (r)] > 0 with negative potential near the horizon, however, it is not easy to make a clear decision on the stability of the black hole.
Here it is still stable for 19.83 < α ≤ 29.47 with q = 0.418 even providing negative region near the horizon shown in Fig. 1. In this case, the S-deformation method might provide a complementary result to support the stability of such black holes by finding the deformed potential [21,22].
In general, the GL instability bound is not given by an analytic form. As was shown in Fig. 3 depending on q, it was determined by solving the linearized equation (9) numerically.
In the case of q = 0.418, the GL instability bound is α > 29.47 which is surely less than the sufficient condition for instability (α > 30.74). Importantly, this picture shows that the GL instability appeared in a simpler EMS theory than the ESGB theory and Einstein-Weyl gravity. For q = 0.7, we have obtained the GL instability bound of α > α th = 8.019. We have derived the precise value of threshold α th = 8.019 again by solving the static linearized equation numerically. Furthermore, we have obtained the n = 0(α ≥ 8.019) scalarized RN black hole by solving Eqs.(41)-(44) numerically for exponential and quadratic couplings.
Consequently, we have explored a clear connection between GL instability of RN black hole and scalarization of RN black hole.