# Quasinormal modes for nh-stu black holes

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## Abstract

We compute and investigate the behavior of scalar quasinormal frequencies for static, asymptotically anti de Sitter nh-stu black hole solution of \({\mathcal {N}}=2, D=4\) supergravity. We report their exact expression when the horizon topology is flat. The modes are purely imaginary unless they carry high angular momentum number. These modes for spherical horizon topology were numerically computed by using continued fraction method and found to be of two kinds. One such family consists of purely negative imaginary modes. The other family consists of complex modes with their imaginary part always negative, depicting stability of black holes against scalar perturbations. The complex modes show symmetry about imaginary axis, linear relationship with inverse temperature for large horizon radius and increased oscillation frequency for higher harmonics. Many of these properties are common to a family of black holes which asymptotes to anti de Sitter space, thus pointing to common features of thermalization dynamics of a class of dual holographic theories.

## 1 Introduction

Due to unique combination of presence of high curvatures in their geometry and simplicity of many of the known solutions, black holes have remained promising candidates to learn about novel quantum aspects of gravity. One useful such property is given by quasinormal modes in the black hole background. When one perturbs a black hole or its surrounding geometry, the perturbation oscillates like a normal mode of a closed system. These perturbations always decay with the corresponding frequencies usually being complex. A class of them either radiate out to infinity (for asymptotic flat or de Sitter case) or vanish at the boundary (for asymptotic anti de Sitter case). Such black hole oscillations are known as “quasi-normal modes” and they encode important properties about the dynamics of the black hole. The negative imaginary part of the complex frequencies is inversely proportional to the temperature of black hole. Quasinormal modes dominate the late time behaviour of the dynamics of the black hole akin to hydrodynamic modes. In the later stages of black hole formation, the gravitational waves include certain quasinormal modes that dominate the emission.

Quasinormal modes of black holes have been an interesting topic of discussion because they are those characteristics of black holes which do not depend on initial perturbations and are functions of black hole parameters only. This means that quasinormal modes encode unique characteristic features which hopefully can lead to the direct identification of the black hole existence. Recent interest in quasinormal modes of black holes arose since these quasi-normal frequencies are relevant to experiments for detecting gravitational waves such as the recent detection by LIGO of these waves emanating from colliding black holes [1]. The gravitational waves can also be emitted from supernovae or coalescence of binary neutron stars, which are thought to eventually form a black hole. Such waves, at late times, can have frequencies or features similar to those calculated for quasinormal modes. In general, quasinormal modes are important in black holes dynamics and appear in processes such as birth of black holes, collisions of two black holes, decay of different fields in a black hole background, etc [2, 3, 4].

Another source of interest in quasinormal modes is due to existence of a correspondence between asymptotic anti de Sitter gravity and quantum field theory in flat space-time, widely known as gauge/gravity correspondence [5]. According to this correspondence, a large static black hole in asymptotically AdS spacetime corresponds to an approximately thermal state of conformal field theory. The perturbation in the black hole corresponds to perturbation in thermal state, and the decay of the perturbation describes the return to the thermal equilibrium. So we can obtain a prediction for thermalization time scale in the strongly coupled field theory by introducing a perturbation in holographically dual AdS black hole solution which will ultimately evolve according to the quasi-normal frequencies. Thus, the quasinormal mode of an AdS black hole has an interpretation as a time scale for the approach to thermal equilibrium. Moreover, these modes are also related to Green’s function of appropriate operator corresponding to the perturbation in holographically dual field theories. There, the poles of the Green’s function are the quasi-normal frequencies. Thus, quasinormal modes using this duality has led to important progress in our understanding of the physics of a class of gauge theories [6, 7, 8, 9].

An asymptotic anti de Sitter black hole in gauged supergravity theories provides a dual description of certain strongly coupled non abelian quantum field theory at finite temperature. Due to inherent symmetries, these theories are more amenable to calculations and have been well studied in literature as a representative of black holes in presence of various matter. For more richer situations, we choose in this article to calculate quasi-normal modes of scalar perturbations in the background of BPS black hole solution in \({\mathcal {N}}=2, D=4\) Fayet-Iliopoulos gauged supergravity theories which contain certain scalar potential and a non-homogenous special Kahler manifold parameterized by the vector multiplet scalars [10]. The solution is referred as a non-homogeneous deformation of stu model (nh-stu). The remainder of the paper is organized as follows. In Sect. 2, we briefly review the nh-stu black hole solution of \({\mathcal {N}}=2,D=4\) supergravity. In Sect. 3, we drive the scalar field equation in this black hole background. We solve it to get exact quasinormal modes for flat horizon topology in Sect. 4. In Sect. 5, we analyze the field equation for the case with spherical horizon topology to express an implicit equation for quasinormal modes using modified version of continued fraction method [11, 12, 13, 14]. Numerical results from computation of quasinormal modes and their behavior are discussed in Sect. 5.1. The concluding Sect. 6 contains some outlook and scope for future studies.

## 2 Review of \({\mathcal {N}}=2, D=4\) supergravity solution

We present a brief description in this section of a solution of \({\mathcal {N}}=2, D=4\) supergravity coupled to \(n_V\) Abelian vector multiplets. More details are available in original paper [10]. We represent the space time indices by greek letters. The bosonic field content consists of a veilbein \(e^a_{\mu }\), \(n_V +1\) vector potentials denoted as \(A^{\varLambda }_{\mu }\) with \(\varLambda =0,\ldots ,n_V\) and \(n_V\) complex scalars denoted as \(z^i\) with \(i=1,\ldots ,n_V\). The vector potential \(A^{0}_{\mu }\) denotes graviphoton while the rest of them are components of vector supermultiplets.

*a*can be set to unity by scaling radial variable. The above solution represents a black hole, with a horizon at \(r_0=\sqrt{c_1/a}\). Three charges and FI parameters together satisfy Dirac quantization condition given as \(g_0p^0-g^ip_i=-\kappa \). In terms of the parameters chosen here, this constraint assumes the form

*B*denotes \(\frac{c_4^2}{r_0^2}\).

## 3 The scalar field equation

*l*denotes the angular momentum number. The radial part of the field equation turns out to be,

*z*vanishes at the horizon and approaches unity at the boundary. We need to take into account horizon topology for further analysis.

## 4 Quasinormal modes for flat horizon topology

*B*in the differential equation reduces to unity. The scalar field equation for general mass simplifies to become

## 5 Quasinormal modes for spherical horizon topology

*l*.

2. Behavior near boundary,

*n*and the parameters (

*l*,

*B*and \(\omega \)) of the differential equation.

One possible way to test convergence of the power series in (28) at both the boundaries is to analyze the large-*n* behavior of \(\frac{a_{n+1}}{a_n}\).

*n*=0,

### 5.1 Numerical results

^{1}The elements of the infinite continued fraction are functions of \(\omega \) and we can get more number of quasi-normal frequencies as roots by increasing the number of terms in the continued fraction. The characteristic equation have an infinite number of terms, so we need to make an approximation by truncating it to certain number of terms before looking for the roots. So, for any desired accuracy, one needs to find the roots of the equation by increasing the number of terms in the truncated continued fraction. For a root with lowest imaginary value, we plot a graph between its imaginary part vs number of iterations.

*n*),

*c*, the faster will be the convergence, thus requiring lower number of terms for any given desired accuracy. We also list the values in Table 1.

Quasi-normal frequencies for B = 1/2

l | n | Re(\(\omega \)) | Im(\(\omega \)) | l | n | Re(\(\omega \)) | Im(\(\omega \)) |
---|---|---|---|---|---|---|---|

0 | 25 | 0.8573 | \(-\)3.1686 | 2 | 25 | 2.3539 | \(-\)2.6727 |

26 | 0.8640 | \(-\)3.1364 | 26 | 2.3541 | \(-\)2.6730 | ||

27 | 0.8609 | \(-\)3.1079 | 27 | 2.3544 | \(-\)2.6732 | ||

28 | 0.8503 | \(-\)3.0840 | 28 | 2.3547 | \(-\)2.6732 | ||

39 | 0.8337 | \(-\)3.0652 | 29 | 2.3548 | \(-\)2.6731 | ||

30 | 0.8120 | \(-\)3.0652 | 30 | 2.3550 | \(-\)2.6729 | ||

1 | 25 | 1.3909 | \(-\)2.8650 | 3 | 25 | 3.3729 | \(-\)2.5954 |

26 | 1.3984 | \(-\)2.8665 | 26 | 3.3729 | \(-\)2.5954 | ||

27 | 1.4045 | \(-\)2.8648 | 27 | 3.3729 | \(-\)2.5954 | ||

28 | 1.4086 | \(-\)2.8612 | 28 | 3.3729 | \(-\)2.5954 | ||

29 | 1.4105 | \(-\)2.8569 | 29 | 3.3729 | \(-\)2.5954 | ||

30 | 1.4106 | \(-\)2.8525 | 30 | 3.3729 | \(-\)2.5954 |

Imaginary quasi-normal frequencies for B = 1/2

l | Re(\(\omega \)) | Im(\(\omega \)) | l | Re(\(\omega \)) | Im(\(\omega \)) |
---|---|---|---|---|---|

0 | 0.0 | \(-\)0.23559 | 3 | 0.0 | \(-\)0.80073 |

0.0 | \(-\)0.80522 | 0.0 | \(-\)1.85155 | ||

0.0 | \(-\)1.77237 | 0.0 | \(-\)3.55538 | ||

0.0 | \(-\)4.40241 | 0.0 | \(-\)6.9749 | ||

0.0 | \(-\)26.8484 | 0.0 | \(-\)25.6236 | ||

0.0 | \(-\)36.5261 | 0.0 | \(-\)35.9239 | ||

0.0 | \(-\)54.2701 | 0.0 | \(-\)53.9987 | ||

0.0 | \(-\)45.2377 | 0.0 | \(-\)44.8469 | ||

1 | 0.0 | \(-\)0.35889 | 4 | 0.0 | \(-\)1.09163 |

0.0 | \(-\)1.04091 | 0.0 | \(-\)2.34747 | ||

0.0 | \(-\)2.23218 | 0.0 | \(-\)4.33724 | ||

0.0 | \(-\)4.90604 | 0.0 | \(-\)8.41743 | ||

0.0 | \(-\)26.656 | 0.0 | \(-\)24.6883 | ||

0.0 | \(-\)36.4274 | 0.0 | \(-\)35.5081 | ||

0.0 | \(-\)54.2251 | 0.0 | \(-\)53.8163 | ||

0.0 | \(-\)45.1731 | 0.0 | \(-\)44.5822 | ||

2 | 0.0 | \(-\)0.55516 | 5 | 0.0 | \(-\)1.42751 |

0.0 | \(-\)1.41471 | 0.0 | \(-\)2.9055 | ||

0.0 | \(-\)2.86867 | 0.0 | \(-\)5.22298 | ||

0.0 | \(-\)5.82491 | 0.0 | \(-\)10.285 | ||

0.0 | \(-\)26.2581 | 0.0 | \(-\)23.306 | ||

0.0 | \(-\)36.228 | 0.0 | \(-\)34.9706 | ||

0.0 | \(-\)54.1348 | 0.0 | \(-\)53.5865 | ||

0.0 | \(-\)45.0432 | 0.0 | \(-\)44.2464 |

*l*and

*B*. So we computed first ten lowest order quasi-normal modes from \(l=0\) to \(l=5\) with \(B=1/2\), which are shown explicitly in Fig. 2. Here, the size of the black hole is fixed to unity, i.e. \((r_0=1)\).

*B*. We plotted a graph between size of the black hole (\(r_0\)) and the quasi-normal frequencies in Fig. 4. This also depicts the variation of the mode with temperature which is inversely proportional to the size of the black hole horizon.

From the above figure, we can say that the imaginary part of the quasi-normal frequency of the large size black hole is the linear function of \(r_0\) and this linear relation doesn’t apply for the intermediate size of the black hole [20]. In this aspect, we find the black hole in our case behaves very similar to AdS black holes [6, 11, 12, 21, 22, 23, 24, 25].

## 6 Conclusion

In this article, we have calculated quasinormal modes for nh-stu black hole solution of \({\mathcal {N}}=2, D=4\) supergravity. The calculation is amenable for a particular constraint on charges and gauge couplings given as \(c_2=c_3\) in terms of parameters defined in Sect. 2. In this case, the scalar perturbation equation contains only 3 singularities. For the case of flat horizon topology, it can be exactly solved in terms of hypergeometric functions. These solutions lead to two classes of quasinormal modes, one being purely imaginary. Higher angular momentum numbers in comparison with given scalar mass and horizon radius can result in other class of purely real modes.

We also explored massless scalar perturbations for compact black hole solution of above kind with spherical horizon topology. We proved by adapting an argument for AdS black holes [6] that all quasinormal modes here will have non vanishing negative imaginary component. (See appendix A). So, they all can be interpreted as decaying, thus rendering the black hole stable against any massless scalar perturbation. Furthermore, the quasinormal modes obtained here can be divided into two types, one are purely imaginary negative modes. Higher angular momentum number contributes to faster decay of any such mode. The second kind of modes are complex frequencies with negative imaginary part. Here, the effect of increasing the angular momentum number contributes to shriller oscillations of the mode given by higher magnitude of real part of quasinormal frequencies. These complex modes come in pairs, i.e. for any given quasinormal mode, its negative conjugate is also another mode. Graphically, it leads to quasinormal modes placed symmetrically in the lower half of complex \(\omega \) plane. This complex conjugate symmetry of the roots is in consonance with the reflection principle in scattering, when quasinormal modes are interpreted as singularities of the scattering amplitude for an effective gravitational potential felt by traveling perturbations [26, 27, 28, 29]. When behavior of a quasinormal frequency with respect to horizon radius is investigated for given angular momentum number, we found a linear relationship for larger size of black holes. This relationship breaks down for intermediate size. Reducing horizon size leads to proportionate increase in the temperature of Hawking radiation. At intermediate sizes, one is likely to encounter a Hawking-Page transition [30], which depicts instability of anti de Sitter black holes preferring thermal gas in anti de Sitter space. Patterns of quasinormal modes do encode effects of phase transitions in many situations [31]. Most of the patterns obtained here for complex quasinormal frequencies are common with those obtained for simpler setting of AdS-Schwarzschild black holes, pointing to very general properties. This points towards general characteristics about dynamics of scalar hydrodynamic modes in holographically dual strongly coupled gauge theories [4]. Full spectrum of quasinormal modes also paves a way to find one loop partition function determinant in such theories [32, 33].

## Footnotes

- 1.
We implemented root finding algorithm using MATHEMATICA software.

## Notes

### Acknowledgements

This research is partly supported by DST-SERB grant no. SR/FTP/PS-149/2011.

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