Superradiance of a Global Monopole in Reissner-Nordstr\"{o}m(-AdS) Space-time

In this article, the behavior of a charged and massive scalar field around a global monopole swallowed by a Reissner-Nordstr\"{o}m-Anti-de Sitter (RN-AdS) black hole is investigated by considering the Klein-Gordon equation in this geometry. The superradiance phenomenon and instability behavior of the black hole against charged scalar perturbations are studied for both AdS case and also for a RN black hole surrounded by a reflective mirror, i. e the black hole bomb case where in the latter we consider the case where the cosmological constant vanishes. The effects of the monopole on these cases are discussed analytically and also with the help of several graphs in detail. The monopole charge affects the superradiance threshold frequency and also effects the instability time scale for both cases. The existence of global monopole makes these black holes more stable against superradiance instability.


I. INTRODUCTION
Global monopoles are a special class of topological defects which may be produced in the early universe during the symmetry breaking phase transitions in General Unified Theories [1,2]. Various types of these defects such as monopoles, cosmic strings, domain walls or textures may be produced depending on the type of the broken symmetry. A possible mechanism for global monopoles to be formed is a spontaneously broken global O(3) symmetry to U (1) [3]. These monopoles have interesting gravitational properties such as the spacetime around a global monopole has a solid angle deficit and active gravitational mass of these monopoles vanishes [3]. They have also remarkable cosmological implications such as they are very dense objects and should dominate the early universe after being formed. This monopole problem can be avoided by the exponential expansion of the universe during the inflationary phase, which dilutes away the monopoles. For a review of topological defects and their astrophysical and cosmological implications, we refer to [4]. Understanding the gravitational effects of these defects and their interaction with the surrounding scalar, or electromagnetic fields are still an important topic in theoretical and observational cosmology.
In this article, we would like to investigate the dynamics of a massive, charged test scalar field in a cosmological spacetime where a global monopole is swallowed by a charged, massive black hole, namely, Reissner-Nordström-AdS(RN-AdS) black hole. We first investigate the Klein-Gordon equation(KGE) of a charged, massive scalar test particle. By using well known mathematical methods we can solve the KGE approximately which enables us to explore the instability conditions of superradiance phenomenon for our spacetime configuration.
Superradiance is basically a radiation amplification process that involves dissipative systems [5,6]. In the superradiant scattering, a scalar or electromagnetic wave sent far from the black hole is scattered by the black hole where the wave strength is enhanced by its interaction with the horizon of the black hole. The superradiant scattering occurs if the frequency of the wave is below a certain threshold frequency. The wave is enhanced because it gains rotational or electromagnetic energy from the black hole. Hence, it is a wave analogue of the Penrose process. The threshold frequency which determines whether the wave will be amplified when it is scattered depends on the angular velocity and electric potential of the horizon. One important property of superradiant scattering is that it may lead to instability if there exists a mechanism to feed the enhanced scattered waves back into the black hole until the scattered waves exceed the threshold frequency of this black hole. Hence this process may decrease the rotation velocity and horizon charge, lowering the threshold frequency, ending with a black hole with a smaller rotation frequency and electrical charge. By continuously feeding the waves below the threshold frequency, in the end, we may have a Schwarzschild black hole, which is known to be stable [7,8]. Moreover, with this mechanism, one could, in theory, extract energy from the black hole. In the classical domain, the first example of the phenomenon was presented by Zel'dovich [9], whose suggestion was to surround a rotating cylindrical absorbing object by a reflecting mirror, then by examining the case where the scalar waves strike upon it. One can also obtain a similar system for a Kerr black hole surrounded by a spherical reflective mirror. This black hole and mirror systems are called as "black hole bomb" by Press and Teukolsky [10], which has examined thoroughly in [11]. Hence, using a hypothetical reflective mirror one can observe the superradiant instability of a black hole. A more important question is that are there any natural mechanisms, for example, a potential barrier which continuously scatters back enhanced waves into the black hole. One such mechanism is the mass term of the scalar wave for Kerr black holes which behaves like a potential barrier [12][13][14][15]. However, the time scale of this instability for astrophysical black holes are greater than the Hubble time, hence this instability is ineffective for such black holes. However, this instability may be important for primordial black holes. Another candidate is the infinity of AdS black holes since for AdS black holes the infinity behaves like a reflective mirror. Despite this, large AdS black holes were shown to be stable [16]. However, the four dimensional rotating small [17] or charged [18] AdS black holes are unstable against superradiance instability of a scalar or charged scalar field. This observation is also present in rotating [19,20] and/or charged [21,22] AdS black holes in higher dimensions. For a more complete list of references to this topic and the other aspects of superradiance, we refer to the latest review [6].
As we have discussed in the previous paragraph, the superradiance phenomenon is not specific to rotational black holes, it can also be observed in the both RN and RN-AdS spacetimes, where now we have the Coulomb energy instead of rotational one. In this case, one needs to send a charged bosonic field to observe the phenomenon. However, here, superradiant scattering reveals itself for the frequencies bounded by the inequality ω < eΦ h , where e is the charge of the scalar field and Φ h is the electric potential of the horizon of the black hole sourced by its charge Q. In this article, we generalize the instability condition in the presence of a global monopole for both RN and RN-AdS spacetimes.
The article is organized as follows. In section 2 we present the spacetime corresponding to a global monopole swallowed by a black hole with mass M , and electrical charge Q in a cosmological background which is called RN-AdS global monopole spacetime and we obtain the Klein-Gordon equation for a scalar field by using an ansatz to separate the equation to its angular and radial parts. The quasinormal modes of RN-dS black hole with a global monopole are presented recently in [23]. Section 3 is devoted to the phenomenon of superradiance. We analytically investigate the stability properties of the spacetime configuration in two different cases under superradiance phenomena. In the first case, we have an AdS spacetime which behaves effectively as a reflecting box. In the second case, however, we consider the spacetime in the absence of the cosmological constant and surround the black hole by a reflecting box. For both cases, we discuss the effect of global monopole to superradiant instability by using analytical methods. We also present several graphs to visualize the effect of the monopole on the superradiant threshold frequency and also on the time scale of the instability for both AdS and black hole bomb cases. In section 4 we make a brief conclusion of the results we have found in this paper.

II. THE LINE ELEMENT AND THE KLEIN-GORDON EQUATION
In this section, we will present the line element for our spacetime configuration and also the Klein-Gordon equation for a charged scalar field. The behavior of the scalar field near the horizon and the radial infinity are derived. Using these results the superradiance condition is also derived and the effect of the monopole charge on the superradiant threshold frequency is discussed.

A. The Line Element
The spacetime line element around a global monopole swallowed by a Schwarzschild black hole is given in [3]. This solution is later generalized to a global monopole swallowed by a RN-(A)dS black hole implicitly in [25] with the line element with ∆ r is defined as, where M and Q are the total mass and the total charge, Λ is the cosmological constant of the black hole, and η is the contribution of the global monopole, which are the physical parameters of this space-time. One important note comes from the inspection of the pure global monopole configuration. The line element for such a configuration can be obtained by neglecting the black hole parameters, namely the mass M , the charge Q and Λ. Therefore the line element becomes [3,4], Rescaling the t and r variables by the transformations, we can rewrite the global monopole line element as, The line element (5) not only describes the asymptotic behavior of the global monopole outside the core but also states that the pure global monopole spacetime is not asymptotically flat, which describes a space with a solid deficit angle. Hence the area of a sphere of radius r is not 4πr 2 , but rather (1 − 8πη 2 ) 4πr 2 . Note that the space-time (5) also describes a "cloud of strings" solution [24], namely, a configuration where an ensemble of radially distributed straight cosmic strings, i. e. a Letellier spacetime, intersecting at a common point, which sometimes also called as the "string hedgehog configuration [25,26]". Note also that for positive values of (1 − 8πη 2 ), i.e (1 − 8πη 2 ) > 0, the equation (1) defines a spacetime such that ∆ r = 0 at a certain value of r. However for (1 − 8πη 2 ) < 0, r becomes a timelike variable and (5) can be interpreted as an anisotropic cosmological solution.

B. The Klein-Gordon Equation
The Klein-Gordon equation for a scalar field Φ that describes the dynamics of a massive electrically charged scalar particle of mass µ and charge e, in a curved spacetime is described by the equation, where g is the determinant of the metric tensor, with the value g = −r 4 sin 2 θ. The gauge differential operator D is defined as, where A µ is the vector potential.
It is straightforward to see that the Klein-Gordon equation is separable. Considering the usual separation ansatz, where m is the azimuthal quantum number and ω is the angular frequency of the scalar waves, and substituting (8) into (6) yields a separable equation on differential equation (6), which means we can write the total differential equation as distinct angular and radial equations separately. The angular part of the total differential equation leads to the associated Legendre differential equation where λ is the separation constant with well known expression whose solutions are given in terms of associated Legendre polynomials as whose values can be found by using the Rodrigues' formula [30]. The radial part of the Klein-Gordon is obtained as, Now, let us discuss the asymptotic behaviour of the scalar field near the horizon and the radial infinity for certain values of the parameters of the scalar field and the black hole.

C. The Asymptotic behaviour of the scalar field
The radial part of the Klein-Gordon differential equation is given by the equation (12), where ∆ r is now expressed for simplicity as, with the abbreviation term b and the AdS radius are defined as Consider now a tortoise coordinate transformation defined as, then the equation (12) takes the Schrödinger-like form, where the effective potential is defined as, D. Scalar field near the horizon Near the horizon, which corresponds to the largest root of ∆ r , r → r + where the coefficient of the radial equation (12) behaves as ∆ r → 0, the effective potential becomes, Here Φ h is the electric potential at near the event horizon defined by, Hence near the horizon we have, where r * is the tortoise coordinate given by the equation (15). Since our investigation is in the classical domain, we have to choose the negative sign in (20) which implies that there are only ingoing waves at the horizon, which implies that one must restrict the group velocity of the wave packet to a negative one. Classically speaking, no information can come out from a static black hole.

E. Scalar field at the infinity
At the radial infinity there are different asymptotic behaviour for the scalar field depending on the cosmological constant and mass of the scalar field. Hence we will discuss below these different cases, separately.

Nonvanishing Cosmological constant case
For nonvanishing cosmological constant, i. e = ∞, we have which implies that the boundary condition for the scalar field in this case is the following, due to the fact that AdS space behaves effectively as a reflecting mirror.

Vanishing Cosmological Constant case
For the vanishing cosmological constant, however, the behaviour of the scalar field is very different, since Hence, for vanishing cosmological constant, and if the scalar field is massive (µ = 0), then bound states that are decaying at infinity are possible for the scalar field if ω 2 < b 2 µ 2 with Hence, similar to RN or Kerr black holes, the mass of the scalar field can act as a potential barrier if it satisfies ω 2 < b 2 µ 2 . We see that the effect of the monopole term is to reduce the height of the potential barrier by a factor of b 2 = 1−8πη 2 < 1. However, it was shown in [15,27,28] that, unlike Kerr black holes, in the superradiant regime there are no metastable bound states for RN solution and RN black holes are stable against charged scalar perturbations. Hence we will not pursue the investigation of stability due to the mass of the scalar field in this paper. An open problem will be to investigate the stability of a global monopole swallowed by a charged and rotating black hole, a solution which awaits its discovery, against charged and massive scalar perturbations. However, this solution is not known as far as we know yet.
For the case where the mass of the scalar field vanishes or ω 2 ≥ b 2 µ 2 , then there is no bound state solutions and the field behaves as where For this case the superradiance scattering cannot lead to an instability unless one uses some artificial mechanisms such as surrounding the black hole with a reflective mirror as done in the black hole bomb mechanism.

F. Superradiance Condition
Here we derive superradiance condition for vanishing cosmological constant case. Let us consider a scattering experiment of a monochromatic scalar wave with frequency ω with a wave function of the form Φ =Re −iωt+imφ . When a scalar wave is sent from radial infinity with unit amplitude, and when we consider the black hole horizon as a one way membrane with no flux outside the horizon from the black hole, then the asymptotic form of the solution of the equation (16) can be written asR Here R and T are the amplitudes of the reflected and transmitted wawes, respectively. Note that the complex conjugate ofR, in which we will denote asR † , should be also a solution of the equation (16) since the potential V (r * ) is real and the solutions are invariant under t → −t, ω → −ω. ThenR andR † should be linearly independent and their Wronskian W =R∂r * R † −R † ∂r * R should be independent of r * . Calculating Wronskians near horizon and at the radial infinity and equating them one obtains Hence when the superradiant condition is satisfied, then the amplitude of the scattered wave becomes greater than it is sent. This phenomenon is called as the superradiant scattering. Note that for the AdS case, the condition for superradiance is also the same. This can be derived by the fact that the phase velocity of the waves flowing into the horizon changes sign relative to the group velocity of these waves. Now let us discuss the role of the monopole charge on the superradiant threshold frequency The monopole term, b 2 = 1 − 8πη 2 < 1, affect the superradiance threshold frequency since it changes the location of the outer horizon r + which is given by When the monopole is present, the location of the outer horizon increases relative to the case where the monopole is not present (b = 1). Hence the electric potential of the horizon decreases in the presence of the monopole term. Therefore, here we conclude that the presence of the monopole charge reduces the superradiant threshold frequency of the wave. A wave with frequency ω which may trigger the superradiant scattering when the monopole term is absent, may not trigger the superradiant scattering when the monopole term present.

III. SUPERRADIANCE INSTABILITY
Having obtained the Klein-Gordon equation for a charged and massive scalar field around a Reissner-Nordström-(A)dS global monopole black hole in the previous section, and also determined the role of the monopole term on the superradiant threshold frequency, we now investigate the phenomenon of superradiance against perturbations of charged and massive scalar field to understand the role of the monopole term on the superradiant instability of the black holes that we consider. The aim of this section is to find an instability condition for our space-time configuration for small mass and charge via solving the radial wave equation (12) in the low-frequency domain, i.e (r − r + ) << 1 ω , by exploiting the asymptotic matching technique. We separate our investigation into two cases where the first case corresponds to the anti-de Sitter spacetime where Λ < 0, that can be called natural superradiance since the infinity of the AdS spacetime behaves like a reflective mirror. In the second case, we will be interested in the superradiant instability in the absence of cosmological constant, i.e Λ = 0, using the method called as the black hole bomb, where the black hole is surrounded by a hypothetical reflective mirror [10]. In this section we consider the superradiance instability of a charged scalar field of a global monopole swallowed by a RN-AdS black hole. As we have said before, here we exploit the asymptotic matching technique where this technique divides the solution as near and far region solutions [29].

A -Near Region Solution
For small AdS black holes we have r + << , in the near region we assume (r − r + ) << 1 ω , Λ ∼ 0, r ∼ r + and ∆ r ∼ ∆, where we further assume that µ 2 r 2 + << 1 in the near region, since we are in the low frequency regime and the Compton wavelength of the perturbations must be large compared to the radius of the horizon. Now we will make a change of variable through the following definition, where z = 0 corresponds now the event horizon r = r + . Using (32) we have the following results, The radial equation (12) takes the form, where we have defined the so-called superradiant factor as, Now we can define an F-homotopic transformation of the following form, Substituting (36) to (34) we obtain, which is a hypergeometric differential equation with a general solution in the neighbourhood of z = 0 as F = az 1−γ F (1 + α − γ, β + 1 − γ; 2 − γ; z) + b F (α, β; γ; z) [30], where Therefore we can read of the solution of (34) as, Since we are in the classical limit, there will not be outgoing waves, therefore we have to set the coefficient B = 0. Now we analyse for the large values of r, i.e z → 1, the behaviour of the ingoing wave solution in the near region. To accomplish that, we will use hypergeometric transformation law z → 1 − z, which is given by [30], Since in the limit z → 1 ⇒ 1 − z → 0, we can use the property of the hypergeometric function F (α, β; γ; 0) = 1, to write the large r limit of the near region solution of the form

B -Far Region Solution
In the far region we assume r − r + >> M , such that the physical parameters of the black hole, namely the mass and the charge can be neglected, i.e M ∼ 0, Q ∼ 0. Hence the polynomial (2) now becomes, thus the radial part of Klein-Gordon equation (12) can be written as, Note that the equation (43) is the radial wave equation for AdS space-time with a global monopole. Moreover, we also observe that the monopole term b 2 in equation (43) does not vanish, which is adequate due to the fact that the monopole spacetime is not asymptotically flat. Hence we must keep the monopole term b 2 in the far region approximation. Let us start our calculation with a coordinate transformation defined as y = b 2 + r 2 2 , then we further transform that with y = b 2 x. With these transformations equation (43) takes the following form, Here we have setω 2 = ω 2 /b 2 , λ(λ + 1) = ν(ν + 1)/b 2 . Now lets use the following definitions, and the following ansatz, Substitution of (46) to (44) yields, where we have defined, such that, The equation (47) is in the form of hypergeometric differential equation and this equation admits a solution in the neighbourhood of x = ∞ as [30], x ), hence we can write a solution of (44) via (46) as, Taking the limit x → ∞ and using F (α, β; γ; 0) = 1, we see that the solution behaves as, However, at infinity, AdS spacetime behaves like a wall such that the scalar field Φ vanishes. This implies the restriction that the coefficient D must vanish.
To explore the equation (52) corresponding to the small values of r, i.e x → 1, we use the 1 x → 1 − x transformation law of the hypergeometric functions [30], which is given by, Note that, when x → 1 we have x − 1 → r 2 2 b 2 . Therefore the far region solution for small values of r is given by, We observe that the global monopole term, b 2 = 1 − 8πη 2 effects the far region solution (55) as a constant multiple of r, therefore we can safely apply the boundaries of the pure AdS space-time to analyse (55). When r → 0 the equation (55) diverges due to r −λ−1 → ∞. To obtain regular solutions we impose the condition as follows, Thus, the regularity condition (56) enables us to interpret m, which takes the values from the nonnegative integer numbers Z + , as a principal quantum number. Hence we obtain the discrete spectrum For the sake of abbreviation we have defined σ = λ/2 + 3/4 + 9 + 4µ 2 2 /4. Notice that the result (57) reduces to given in [17] when the mass of the field and monopole term vanishes, i.e µ = 0, b 2 = 1. Now, it is natural to assume that the condition (57) can be interpreted as the generator of the frequency spectrum of the normal modes at large distances, due to the fact that at infinity the structure of the RN-AdS black hole is similar to pure AdS background. In addition, one can still observe the effect of the global monopole in (57). Having said that, however, we should approach the current predicament more cautiously, since the inner boundaries of the pure AdS or RN-AdS black hole spacetimes are very different. For a pure AdS space-time, we have r = 0 as the inner boundary, on the other hand for the black hole case we have r = r + . Hence if one wishes to observe the effect of the black hole on the frequency spectrum, one must take into account of the possibility of tunneling of the wave through the potential located at r = r + , into the black hole and scattered back. Furthermore, the scattered amplitude of the wave may decrease or grows exponentially and may also cause the superradiant instability. To sum up, the quasinormal mode frequencies for the black hole case can be modified with additional complex frequencies as follows, where δ is possibly a small parameter signalling the effects of the charged black hole having the gravitational monopole. Exploiting the assumption (58), and using the Gamma function relations for small δ, we have where = (3 + 9 + 4µ 2 2 )/2, and Using (59) and (60) in the far region solution given by the equation (55) we obtain, Now if we want to be successful at the asymptotic matching procedure of the near region and the far region solutions we need a restriction on λ. The relation between λ and ν is given by, where b 2 = 1 − 8πη 2 . Taylor expansion of 1/b 2 yields, neglecting the O(η 2 ) term we have λ = ν. Therefore we can write the far region solution as, where we set the coefficients of r ν and r −ν−1 to A and B respectively. Matching the near region solution (41) with far region solution (64) yields then, Γ(m + 1 + 9 + 4µ 2 2 /2) Using the Gamma function property given by, we obtain, where we have defined ξ as, with ω = r 2 + /(r + − r − ) (ω − e Φ h ) and ω = (2b/ )(m + σ). Now we have, Hence the superradiance condition is The scalar field has dependence of ω as, Equation (71) implies with the condition (70) that the amplitude of the scalar field grows exponentially and causes instabilities. However one should bear in mind the effect of the global monopole term b 2 . The relevant physical choice of the global monopole term is b 2 = 1 − 8πη 2 > 0. Furthermore if we took η 2 as a positive number, i.e η 2 > 0, then 0 < b 2 < 1. To observe the net effect of the global monopole let us write an an explicit version of (70) as, FIG. 1: Graph of eΦ h / (ω QN ) − b 2 for m = 0, ν = 1, Λ = −3 × 10 −6 , Q = 0.8, µ = 0.1, e = 0.205 − 0.220 with different charge values of scalar field. As b 2 becomes smaller, which corresponds to larger values of η , we see that the condition δ > 0 starts not to hold. Instability condition only holds above the intersection points for these choosen parameter values.
As we have discussed in section (II F), we know that the monopole term causes an augmentation on the outer horizon which decreases the value of the electric potential and lower the threshold frequency. Moreover, inspection of equation (67) regarding the effect of global monopole shows a growth in δ therefore we observe a decrease in τ , since in superradiant instability, the time scale is given by In order to better understand the effect of the monopole, now, we will present several graphs. All of the graphs are plotted for unit black hole mass, i. e. M = 1. From figure (1), it is seen that the monopole term b 2 plays an important role in the superradiant threshold frequency. As we have said before, when the monopole term is present, the threshold frequency decreases. As a result, when the black hole contains a global monopole, the chances of having a superradiant scattering decreases with inceasing monopole term η. We also plot the graphs of changing of the time scale with monopole term b 2 for different parameters, namely, the mass of the scalar field µ, the black hole charge Q, the cosmological constant Λ and different mode values ν in figures (2a), (2b), (2c), (2d), respectively. We observe that for all of these four parameters, when the gravitational monopole term b 2 decreases the value of the time scale also decreases. In figure (2a), when the mass of the scalar field is absent, time scale differs drastically relative to massive scalar field. Furthermore, maximum change of the time scale with respect to gravitational monopole is observed in this case, as well. Figures (2b)  decrease, the effect of the monopole becomes more obvious. For a given µ, Q or , when monopole term b 2 decreases or equivalently η 2 increases, the instability time scale decreases, making the instability less effective for a monopole having more strength. The last figure (2d) is devoted to the dependence of the time scale to the modes ν of the scalar wave. The effect of the global monopole on the different modes of the perturbation term δ given in equation (67) and hence on the instability time scale τ comes from two different contributions. The first one is the explicit b −2ν term in δ, which comes from the coupling between AdS radius and monopole term b 2 . However, this term is not the only source of b 2 dependence for modes in δ and as we have said before, the location of the horizons are also contains b 2 dependence and the term (r + − r − ) ν also depends on mode number ν. It turns out that the combined effect of these two contributions determine the b 2 dependence of modes of instability time scale. The investigation of figure (2d) for first three modes shows that the main contribution on the time scale change for decreasing b 2 comes from the horizon terms since the global monopole effects and increases the values of them.
Hence, we conclude that in RN-AdS black holes having a global monopole, the onset of superradiant instability decreases with the monopole term b 2 . Nevertheless, if the instability occurs it will grow slower in comparison with the case when the monopole term is absent. In summary, we can conclude that the existence of global monopole makes the RN-AdS black holes more stable against superradiance instability. In this section, we discuss the instability condition in the absence of the cosmological constant Λ. As before, we will use the asymptotic matching technique to obtain the instability condition in addition with the so-called mirror condition which will become clear in the process of calculation. Inspection of the near region solution yields the same equation with AdS case since we have set the cosmological constant to zero in the case one for near region solution. Hence the near region solution of both cases are the same and we will use the same solution given in equation (39) and also we employ its far region limit given in equation (41). Hence all that remains is to find the far region solution.
In the far region, as before, we assume M ∼ 0, Q ∼ 0, where M and Q is the mass and the charge of the black hole. The polynomial ∆ r now becomes ∆ ∼ b 2 r 2 . Thus the radial part of the Klein-Gordon equation (12) is given by, where 2 = ω 2 /b 4 − µ 2 /b 2 and ν(ν + 1)/b 2 = λ(λ + 1). Equation (73) admits a general solution in terms of the Bessel functions [30] given by, and for small values of r it reduces to [30], Applying the similar mechanical steps that we have performed for the matching procedure in the previous case, we obtain the corresponding condition for the equations (75) and (41) given as, where b 2 = 1 − 8πη 2 and ω is the superradiant factor given by the equation (35). Notice that, we have used the approximation (62) in order the matching to work. In addition, we have found the coefficient of the near region solution A, as the following to obtain the (76). The main difference between the cases is the fact that in case I we have an AdS space-time which behaves effectively as a reflecting box. In case II to, however, we put a reflecting mirror by hand at the far region located at a radius r = r 0 , and as a result, the scalar field must vanish at the surface of the mirror. Hence, we have an additional condition between the amplitudes α and β due to the fact that equation (74) vanishes for r = r 0 . Therefore we have, and for small values of particle mass µ 2 << 1 it yields, frequency modes ν has similar graphs with differrent values, compared to the previous case. Figures (3c) and (2c) are also comparable since AdS space-time behaves as a reflecting box. The figure (3a) is different from its counterpart, namely, figure (1), since the coupling of the gravitational monopole is different in each case. Therefore we see that the threshold frequency is more sensitive in the changes of b 2 relative to the AdS case. Note that in figure (3a) we choose a small value of mirror radius r 0 , which means we put the mirror closer to the black hole compared to the AdS radius in the first case.
Hence the results that we have obtained for both cases are quite similar. The main difference lies in the fact that monopole term affects the real part of the frequency modes of the black hole bomb by a factor b 2 but for RN-AdS case the factor is b as the calculation procedure reveals, as a result, we may say that the chances of instability to occur is more likely in comparison with the AdS space. Another difference is the mode dependence of the instability time scale where in AdS case an explicit mode dependence exist with b −2ν term, whereas there is no such dependence in black hole bomb case as there is only b 2 term exist in this case. Hence the mode dependence of black hole bomb case only originates from the effect of the monopole on the horizons of the black hole.
We conclude that to obtain more accurate results concerning the comparison of superradiant instability in RN-AdS space-time with black hole bomb in RN spacetime, a numerical analysis is also needed.

IV. CONCLUSION
In this article, we have studied the dynamics of a massive, electrically charged scalar field in the background of a global monopole swallowed by a RN-AdS black hole space-time by investigating the charged and massive Klein-Gordon equation. Analyzing the asymptotic behavior of the scalar field near the horizon and far from it, we have discussed the effect of the monopole on the superradiance threshold frequency. We see that since the monopole term increases the location of the outer horizon and this frequency depends on the electric potential of the horizon, the existence of the monopole decreases the electric potential and hence the threshold frequency. Therefore, a wave which leads to superradiant scattering for RN(-AdS) spacetime may not lead to superradiant scattering in the presence of the monopole charge. Then we have exploited the asymptotic matching technique to inspect the stability conditions of both RN-AdS-monopole and RN-monopole black hole against charged scalar perturbations and found that global monopole effects the onset of instability in a negative way by coupling with the outer horizon of the black hole.Due to different couplings to b 2 terms for both cases, the onset is affected more for BH bomb case then RN-AdS case. The time scale of the scalar field is also affected by global monopole and causes the instability to grow slower both in the RN-AdS-monopole and RN-monopole space-times due to the effect of the gravitational monopole in the outer horizon. We have presented several figures to better see the effect of the monopole in these black holes. As a result of this paper, we conclude that, the existence of a global monopole makes these black holes more stable against superradiance instability.