Scalar perturbations and quasinormal modes of a nonlinear magneticcharged black hole surrounded by quintessence
Abstract
We study scalar perturbations and quasinormal modes of a nonlinear magneticcharged black hole surrounded by quintessence. The time evolution of scalar perturbations is studied for different parameters associated with the black hole solution. We also study the reflection and transmission coefficients along with absorption cross section for the considered black hole spacetime. It was shown that the real part of the quasinormal frequency increases with increase in the nonlinear magnetic charge while the module of the imaginary part of the frequency decreases. The analysis of the perturbations with changing quintessential parameter c showed that perturbations with high values of c become unstable.
1 Introduction
The stability of black hole spacetimes is one of the most interesting questions in general relativity and provides us with many answers related to the black hole itself. The study of various types of perturbations such as scalar, electromagnetic and gravitational on a black hole background is an active area of research. Dynamical evolution of any kind of perturbations on a black hole background can be classified into three stages: the initial outburst of the waves, damped oscillation which is also known as quasinormal modes (QNMs), and the late time power law tail. The first stage completely depends upon the initial perturbing field and it does not give us much information as regards the stability. The second stage is extremely important for black hole stability analysis and it consists of complex frequencies, the real part of which represents the real frequency of the perturbation and the imaginary part represents the damping. These quasinormal modes also provide information about different black hole parameters such as mass, angular momentum, charge etc.
In pioneering work metric perturbations of the Schwarzschild black hole have been studied by Regge and Wheeler [1] and Zerilli [2]. The author of Ref. [3] analyzed numerically scattering of waves on the Schwarzschild black hole [4]. Later Chandrasekhar presented a monograph about perturbation theory of black holes [5]. The main equation of the perturbation theory is a Schrödingerlike equation and, usually, it can be solved using semiclassical or numerical methods. Many authors performed this type of perturbative investigations of black holes (see, e.g., Refs. [6, 7, 8] and the references therein).
The discovery of gravitational waves opened a new window for black hole perturbation physics [9, 10, 11]. It has been discussed that the precision of the observation/experiment of gravitational waves allows us to test alternative theories of gravity [12, 13]. Moreover, it was also shown that, regardless of the existence of horizons, waveforms can be formed [14, 15]. On the other hand, it was stated that gravastars cannot be formed in a binary black hole merger process [16]. There is other work related to the study of the ringdown process through perturbation of black holes [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].
The solutions of the Einstein equations for black holes have a singularity problem. The appearance of singularities can be considered as a defect of Einstein’s general relativity. However, some regular black hole solutions have been obtained by various authors introducing nonlinear electrodynamics into the background gravity [48, 49, 50, 51, 52, 53, 54]. Different properties of regular black holes have been studied in the literature [55, 56, 57, 58, 59, 60].
Another interesting subject is the vacuum energy or quintessence, the existence of which changes the structure of the spacetime at asymptotic infinity – it will be not flat anymore. Furthermore, there will be a cosmological horizon and behind that the geometry becomes dynamic. Particularly, the effects of a repulsive cosmological constant are widely discussed in [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83]. The study of quasinormal modes of these black holes surrounded by quintessence is an interesting topic. The effects of the quintessential parameter on quasinormal frequencies of different black hole spacetimes were studied in [84, 85, 86, 87, 88, 89, 90, 91, 92]. Recently, the solution has been obtained of a regular black hole in the presence of quintessence [93]. In this work, we study the stability of this solution and the quasinormal modes of perturbations. We also study the reflection coefficient, graybody factor, and absorption coefficient of scalar perturbations.
The paper is organized as follows. In Sect. 2, we review the regular black hole solution surrounded by quintessence. In Sect. 3, we study the basic equations of scalar perturbations and, in Sect. 4, we give the numerical results of evolution of scalar perturbations and quasinormal modes. Section 5 is devoted to a study of the graybody factor and absorption coefficient. We summarize our results in Sect. 6.
Throughout the paper we use geometrized units where \(G=c=1\) and a metric with signature \((,+,+,+)\).
2 Nonlinear magneticcharged black hole surrounded by quintessence
3 Massless scalar perturbation
In this section, we shall briefly write the equations of scalar perturbations of a nonlinear magneticcharged black hole surrounded by quintessence (12). It was pointed out in [8] that, if there is no backreaction on the background, the perturbations of black hole spacetimes can be studied not only by adding the perturbation terms to the spacetime metric, but also by introducing fields into the spacetime metric.
The effective potential V(r) is plotted in Fig. 1 to see how it changes with the charge Q, the mass M and the spherical harmonic index l. The first panel shows the behavior of the potential for different values of the charge parameter. We can see that the height of the potential increases with the magnetic charge and this represents the suppression of emission modes for higher charge values. The middle panel shows the change in the potential when mass is varied and we can see that the potential decreases for an increase in mass. The right panel shows the increase of the peak of potential when we increase the spherical harmonic index. This again represents the suppression of scalar emission modes for higher values of l.
4 Numerical results
4.1 Evolution of scalar perturbations
4.2 Quasinormal modes
One of our main goals in this paper is to study the quasinormal modes (QNMs) and the stability of the perturbations in a nonlinear magneticcharged black hole spacetime surrounded by quintessence. We shall focus on massless scalar perturbations here.
Fundamental quasinormal frequencies (\(n = 0\)) for scalar perturbations of a nonlinear magneticcharged black hole surrounded by quintessence. Here \( \omega _{q} = 2/3 \), \( l = 2 \)
Q  c  \(\hbox {Re}(\omega )\)  \(\hbox {Im}(\omega )\) 

0.1  0.0  0.48366  \(\) 0.096758 
0.1  0.0001  0.483432  \(\) 0.096701 
0.1  0.001  0.48137  \(\) 0.096191 
0.5  0.0  0.485999  \(\) 0.095791 
0.5  0.0001  0.485769  \(\) 0.095736 
0.5  0.001  0.483698  \(\) 0.095235 
0.9  0.0  0.498703  \(\) 0.089525 
0.9  0.0001  0.498468  \(\) 0.089477 
0.9  0.001  0.496348  \(\) 0.089045 
Fundamental quasinormal frequencies (\(n = 0\)) for scalar perturbations of a nonlinear magneticcharged black hole surrounded by quintessence. Here \( \omega _{q} = 2/3 \), \( Q = 0.6 \)
l  c  \(\hbox {Re}(\omega )\)  \(\hbox {Im}(\omega )\) 

1  0.0  0.295528  \(\) 0.096078 
1  0.0001  0.295377  \(\) 0.096024 
1  0.001  0.294019  \(\) 0.095544 
2  0.0  0.487769  \(\) 0.095027 
2  0.0001  0.487538  \(\) 0.094973 
2  0.001  0.485461  \(\) 0.094480 
3  0.0  0.681045  \(\) 0.094792 
3  0.0001  0.68073  \(\) 0.094737 
3  0.001  0.677896  \(\) 0.094240 
In Tables 1 and 2, we list the fundamental quasinormal frequencies \( n = 0 \) of scalar perturbations of a nonlinear magneticcharged black hole surrounded by quintessence. Table 1 shows the variation with respect to the charge for different values of c. The real part of the frequency decreases with increasing c. With increasing charge the real part of the frequency increases but the magnitude of the imaginary part of the frequency decreases. In Table 2, we list the fundamental quasinormal frequencies with respect to smaller spherical harmonic index l for different values of c. We can see that the real part of the frequency decreases and the magnitude of the imaginary part of the frequency also decreases with increase in c for the same l. Here we have to note that the WKB method has low accuracy in the small l regime.
In Fig. 4, we plot the quasinormal frequencies with respect to spherical harmonic index l for \( c=0 \), \( c=0.0001 \) and \( c=0.001 \). The left panel shows the real frequencies and the right panel shows the imaginary frequencies. Here we consider \( M = 1 \), \( Q = 0.6 \) and \( \omega _{q}=2/3 \).
In Fig. 5, we plot the quasinormal frequencies with respect to the charge parameter Q for \( n=0 \), \( n=1 \). The left panel shows the real frequencies and the right panel shows the imaginary frequencies. Here we consider \( M = 1 \), \( c = 0.001 \), \( \omega _{q}=2/3 \) and \( l=2 \).
In Fig. 6, we plot the quasinormal frequencies with respect to spherical harmonic index for different values of quintessential parameter \( \omega _{q} \). The left panel shows the increase of real part of the frequency with l and we see that the change with respect to \( \omega _{q} \) is very small. The right panel shows the magnitude of imaginary part of quasinormal frequencies which decreases with increasing l.
In Fig. 7, we plot the quasinormal frequencies with respect to quintessential parameter \( \omega _{q} \). The left panel shows the increase of real part of the quasinormal frequency with the increase in quintessential parameter \( \omega _{q} \) and the right panel shows the decrease of imaginary part of quasinormal with the increase in quintessential parameter.
5 Graybody factors and absorption coefficient
5.1 Nature of graybody factors

\(\omega ^{2}<< V(r_{0}) \). Here the transmission coefficient is close to zero and the reflection coefficient is almost equal to one.

\(\omega ^{2}>> V(r_{0}) \). Here the transmission coefficient is close to one and the reflection coefficient is almost equal to zero.

\(\omega ^{2} \sim V(r_{0}) \). We shall consider this case because the WKB approximation has high value of accuracy for \( \omega ^{2} \sim V(r_{0}) \).
In these expressions \( b=n+\frac{1}{2} \), \(V^{n}(r_{0})=\mathrm{{d}}^{n}V/\mathrm{{d}}r_{*}^{n} \) at \( r = r_{0} \).
In the left and middle panel of Figs. 8, 9, 10 and 12, we have plotted the dependence of reflection and absorption coefficient on the frequency of scalar perturbations, respectively. Figure 8 shows the variation for different spherical harmonic indices l, Fig. 9 shows the variation for different charges Q, Fig. 10 shows the variation for different quintessential parameter values c and Fig. 12 shows the variation for different values of quintessential parameter \( \omega _{q}\).
When the spherical harmonic index is varied, the transmission coefficient \( T(\omega )^{2} \) becomes smaller and hence the reflection coefficient \( R(\omega )^{2} \) becomes larger as seen in the middle and left panel of Fig. 8. The transmission coefficient \( T(\omega )^{2} \) decreases with the increase in magnetic charge and hence the reflection coefficient \(R(\omega )^{2} \) increases as can be seen in the middle and left panel of Fig. 9. Similarly, \( T(\omega )^{2} \) increases with increase in quintessential parameter c and hence \( R(\omega )^{2} \) decreases as seen in the middle and left panel of Fig. 10. The change of transmission and reflection coefficient is very small when we vary the quintessential parameter \(\omega _{q} \) as can be seen from Fig. 11. In order to see the variation, we plot \( R(\omega )^{2} \), \( T(\omega )^{2} \) with higher resolution. We can see that transmission coefficient \(T(\omega )^{2} \) decreases for increase in quintessential parameter \( \omega _{q} \) and hence reflection coefficient \(R(\omega )^{2} \) increases.
5.2 Absorption cross section
6 Summary and conclusion
In this paper, we have focused on scalar perturbations of a nonlinear magneticcharged black hole surrounded by quintessence. First, we numerically calculated the time evolution of scalar perturbations around the considered black hole spacetime and we found that perturbations with higher spherical harmonic index l live longer. We also saw the behavior of perturbations with changing cosmological constant parameter c and we note that perturbations with the higher value of c becomes unstable. The behavior of the perturbation for different values of the charge parameter is also studied and reported.
We have used the sixthorder WKB method to calculate the quasinormal frequencies of scalar perturbations for a nonlinear magneticcharged black hole surrounded by quintessence. We studied the dependence of quasinormal frequencies on charge Q, spherical harmonic index l and cosmological constant parameter c. We see that the real part of quasinormal frequency increases with increase in charge Q for both fundamental (\( n = 0 \)) and first overtone mode (\( n = 1 \)) but the magnitude of the imaginary part of the frequency decreases. The magnitude of the imaginary part of the frequency decreases with increase in charge for different values of c. The real part of the quasinormal frequencies increase monotonically with respect to l but the variation with respect to c is extremely small and the lines overlap on the plot. The magnitude of the imaginary frequencies decreases with an increase in l and c.
We have further studied the graybody factors \( \gamma (\omega ) \) and the partial absorption cross section \( \sigma _{l} \) of scalar perturbations for a nonlinear magneticcharged black hole surrounded by quintessence. We studied the dependence of these quantities for different parameters of the black hole spacetime. The transmission coefficient \( T(\omega )^{2} \), or the graybody factor \(\gamma (\omega ) \), becomes smaller and hence reflection coefficient \(R(\omega )^{2} \) becomes larger with increase in l. The transmission coefficient \( T(\omega )^{2} \) decreases with the increase in charge and hence the reflection coefficient \(R(\omega )^{2} \) increases. Similarly, \( T(\omega )^{2}\) increases with increase in quintessential parameter c and hence \( R(\omega )^{2} \) decreases but the change is too small. We also investigated the effect of changing quintessential parameter \(\omega _{q} \) on transmission and reflection coefficient and partial absorption cross section and we found that the effect of \(\omega _{q}\) on these quantities are very small.
For future directions of research, it would be interesting to study the electromagnetic and gravitational perturbations for the nonlinear magneticcharged black hole surrounded by quintessence. We hope to understand the behavior of these perturbations with respect to the cosmological constant parameter c.
Notes
Acknowledgements
The authors thank Dimitry Ayzenberg and Askar Abdikamalov for helpful discussion. This work was supported by the National Natural Science Foundation of China (Grant no. U1531117) and Fudan University (Grant no. IDH1512060). H.C. also acknowledges support from the China Scholarship Council (CSC), Grant no. 2017GXZ019020. The research is supported in part by Grant no. VAFAF2008 and no. YFAFtech20188 of the Uzbekistan Ministry for Innovation Development, by the Abdus Salam International Centre for Theoretical Physics through Grant no. OEANT01 and by Erasmus+ exchange grant between Silesian University in Opava and National University of Uzbekistan. A.A. thanks the Nazarbayev University for the hospitality.
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