key: cord-0294690-0j3u2ck5
authors: Sakalli, .Izzet; Kanzi, Sara
title: Topical Review: Greybody Factors and Quasinormal Modes for Black Holes in Various Theories -- Fingerprints of Invisibles
date: 2022-05-03
journal: nan
DOI: 10.55730/1300-0101.2691
sha: eb755d267bc8cf1a7c94bdf9111844cff33480c9
doc_id: 294690
cord_uid: 0j3u2ck5

We give a pedagogical introduction to black holes (BHs) greybody factors (GFs) and quasinormal modes (QNMs) and share the recent developments on those subjects. In this study, our primary focus will be on the bosonic and fermionic GFs and QNMs of various BH and brane geometries and reveal the fingerprints of the invisibles with the radiation spectra to be obtained by the WKB approximation and bounding the Bogoliubov coefficients (together with the Miller-Good transformation) methods. (*Due to the notification of arXiv"The Abstract field cannot be longer than 1,920 characters", the appeared Abstract is shortened. For the full Abstract, please download the Article.)

BH thermal radiation or the so-called Hawking radiation (HR) exists at the intersection of classical general relativity (GR) and quantum field theory (QFT), and it may hold the key to unraveling the puzzles of a quantum gravity theory. In general relativity, classical macroscopic BHs obey rules that are similar to thermodynamic laws [1, 2] . When it comes to describing quantum fields in a BH geometry, around the event horizon, the similarity to the ordinary thermodynamics becomes exact since the BH has a temperature and entropy [3, 4] .

Namely, the Hawking radiation is nothing but the black-body radiation at the event horizon.

However, before reaching an observer (detected at spatial infinity in an asymptotically flat spacetime) and being observed, this thermal radiation must travel through a curved spacetime geometry. As a result, the surrounding spacetime acts as a potential radiation barrier, causing a variation from the black-body radiation spectrum as observed by the asymptotic observer. In summary, the GF is the difference between the spectrum of asymptotic and the black-body radiations.

The classic Hawking radiation calculation [3] employs a semi-classical approximation and thus it demonstrates that BHs have a thermal spectrum:

in which Ξ(ω) indicates the expectation value for the number of particles of a particular species released in a mode with frequency ω, T H is the Hawking temperature, the minus/plus sign describes bosons/fermions, and Γ(ω) represents the GF, which is the probability for an outgoing wave to reach spatial infinity or the observer [2] . This corresponds to the probability of an incoming wave to be absorbed by the BH (the absorption probability). The overall BH emission rate is obtained by integrating Eq. (1.1) across all spectra. Note that the BH emission spectrum would be identical to that of a black-body if Γ(ω) was a constant.

The computations of the GFs date back to 1976s [5, 6] . Although the computation setup is straightforward, yet retrieving accurate results might be problematic. The particle perturbation to the BH geometry is given by linearized wave equations, which explain the scattering of particles off BHs (the reader is referred to [7] and references therein). The applications of linear perturbation theory to 4-dimensional and higher dimensional BH geometries were investigated in [8] [9] [10] and [11, 12] , respectively. The resultant radial equation obtained for the perturbations in BH geometries is generally reduced to a 1-dimensional Schrödinger equation in which the 1-dimensional coordinate is created by the tortoise coordinate (representing the geometry of spacetime outside of the BH) and the effective potential identifies the BH geometry as well as the sort of perturbation that is being addressed. Normally, the potentials are enormously complicated and finding a definite solution to the Schrödinger equation (by required boundary conditions (BCs) for the scattering problem) could be impossible; thus one must rely on numerical works or approximation techniques like the Wentzel-Kramers-Brillouin (WKB) approximation described in [5, 6] . When studying the perturbation of BHs obtained in some particular theories like string, dilaton, axion etc., the exact solutions to the wave equations can be found in terms of Mathieu, hypergeometric, and Heun functions [13] [14] [15] [16] .

The absorption probability Γ(ω) can be stated as the square of the transmission coefficient for the related Schrödinger problem of the examined BH geometry. In other words, the tunneling possibility for the barrier defined by the particular potential is known as the GF. One of the main purposes of this review article is to show the computation of Γ(ω) in a variety of BH spacetimes. Parameter σ(ω) denotes absorption cross-section whıch is strongly connected to the GF, which is derived from the optical theorem [17] as follows:

σ(ω) = Γ(ω)|Ψ(ω)| 2 in which Ψ(ω) denotes the asymptotic plane wave projection of the incoming spherical wave function, and thus |Ψ(ω)| 2 can be normalized to unity. So, the answer to the question of why Γ(ω) and σ(ω) are considered as equivalent in many studies becomes clear for a reader. Moreover, it is worth noting that the above result is valid in the case of both incoming and outgoing asymptotic states are determined. The latter remark is mostly possible in asymptotic flat spacetimes, but the vice-versa cases are also possible, albeit to a lesser extent [18, 19] .

In this paper, we shall review the computation of GFs Γ(ω) or σ(ω) and later on T (ω), whose details will be given in the next sections for static, stationary, and spherically symmetric BHs (with and without charge/cosmological constant) even in higher dimensions.

To this end, bosonic, fermionic, and gravitational [163] perturbations are considered, and the transmission and reflection coefficients for the regarded Schrödinger equatıon are then computed [11, 12] . When having analytical solutions are impossible, we consider specific approximation methods whose details will be given in the associated sections for the perturbed fields [6, 22] . One of the important information that we want to say at this point is that there is a general consequence for the asymptotically flat BHs in the low frequency approximation, ω T H and ωr H 1 where r H represents the radius of the event horizon:

σ(ω) for the s-wave of a minimally coupled massless scalar field is equal to the area of the event horizon: σ(ω) = A H [23] . After taking into account the scattering's sub-leading contributions (the higher partial-waves with the angular momentum component > 0), σ(ω) always modifies, although the universality is maintained [23] .

GFs or absorption cross-sections, which are natural outcomes of the investigations made in the BH scattering theory, have a large literature (the reader is referred to [7, 24] for topical reviews). Let us take a quick look at some recent researches that are directly relevant to the current review on the GFs. At this point, it is useful to focus in particular on the most classical BHs belonging to the Schwarzschild family, whose GF investigations have been thoroughly conducted (mainly for bosonic emissions) [2] . In addition to the Schwarzschild family solutions, the so-called brane-world scenarios have also ignited the works on the GFs [25, 26] . In this regard, the numerous researches have focused on the GFs of the brane BHs (see for example [27, 28] ). Among them the notable works that we can show are Refs.

GF studies about the asymptotically flat BHs is undoubtedly the study of [48] . Moreover,

Ref. [49] demonstrated that the perturbation theory methods in quantum mechanics guide us to investigate the imaginary (asymptotic) frequency regimes. This methodology can also be applied to calculate the GFs of massless/massive bosonic, fermionic, and vector fields [2] . As a result, in-depth understanding of Hawking radiation from d-dimensional spherically symmetric BHs could be gained. Studies made in this direction for the BHs found in the string theory can be seen in [50, 51] . GFs of the AdS BHs in the circumstance of the AdS/CFT correspondence [52] are still attractive research area (see for example [53] ). In summary, the findings of [2] revealed the generic behaviors of the thermal correlation functions, which are dual to the GFs in the dual gauge theory, at both low and high frequencies.

On the other hand, one of the promising GF computation methods for obtaining the GF is the method described in Ref. [54] , which defines the general semi-analytic bounds for the GFs:

where σ (ω) is the GF which relies on radiated particles' frequency ω and parameter (angular momentum quantum number) and ℘ is defined by

in which V ef f is the effective potential of the 1-dimensional Schrödinger like wave equation and the prime symbol indicates the radial derivative: d dr . There are two constrains for the explicit positive function h: 1) h (r ) > 0 and 2) h(−∞) = h(∞) = ω. Without loss of generality, one can always set h = ω [54, 55] . Besides, there are other methodologies to determine the GFs. The most used of these methods are the WKB approximation, which is without a doubt the most widely utilized method [56] [57] [58] , the rigorous bound method [59, 60] , and the matching technique [61, 62] . Apart from those, the reader may also learn the additional approaches for computing the GFs by looking at those Refs. [63, 64] .

The distinctive ringing of BHs are called QNMs, which are the damped oscillations with a discrete set of complex frequencies. More technically speaking, the eigenmodes in dissipative systems are known as QNMs. Those eigenmodes are naturally produced by perturbations of classical gravitational backgrounds involving BHs, branes, and even the wormholes [65] .

QNMs decay with time for any stable system, and thus provide a useful description of the relaxation process. They are calculated by solving the wave equation of the considered BH perturbed by an external field. A disrupted BH tries to re-establish its equilibrium by generating energy in the form of gravitational waves. As a result, QNMs play critical role on the gravitational investigations like LIGO detector, which is designed for detecting the gravitational waves [66] . Additionally, in the gauge/gravity dualities theories, there is a relationship between the QNMs and the poles of a propagator in the dual field theory, which physicists use it as a tool to work on strong coupled gauge theories (or holography) [67] [68] [69] . The QNMs frequencies are defined by perturbing the spacetime of the BH with suitable boundary conditions: while the waves of the QNMs should be in the form of fully ingoing waves near the event horizon, they should be completely outgoing waves at spatial infinity. One of the most difficult aspects of understanding the QNMs is that they admit a non-self-adjoint eigenvalue problem: the system is not conserved, so that the energy is lost at both spatial infinity and the event horizon. Thus, unlike the ordinary or classical normal modes, the amplitude of oscillation decays in time for the QNMs [70] . The QNMs can also be utilized to test the theorem of no-hair, which claims that BHs contain only mass, charge, and angular momentum and for the BH quantization [16, [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] . Investigating QNMs can therefore provide new insights into physics beyond Einstein's general theory of relativity, which is an important instrumental for the (incomplete) theory of quantum gravity [86] . References [87] [88] [89] provide more detailed reviews of BH QNMs including their various computation methods, which are as follows: 1) Exact solutions [76, 84, 85, 90] 2) WKB approximation [91] [92] [93] [94] [100, 101] . On the other hand, the method that we will mainly focus on in this review will be the WKB approximation. We would like to briefly summarize this method now.

As noted above, QNMs can be represented as the solutions to a perturbation (wave) equation satisfying the certain boundary conditions. After separating the wave equation into the radial and angular parts, the radial part can be reduced to a 1-dimensional Schrödinger equation: 

one can define the tortoise coordinate as follows

The potential represented in terms of r * coordinates behaves in such a manner that when r * → −∞ (horizon) it vanishes and as r * → +∞ (spatial infinity) it might be a non-zero.

There is a particular location r * 0 that the potential reaches its maximum. In fact, the region considered for the WKB analysis is examined in three parts: Part I is from −∞ to r 1 , which is the initial turning point (where the potential vanishes), Part II covers the region from r 1 to r 2 , which is the second turning point, and Part III starts from r 2 to +∞. We apply the Taylor expansion around r * 0 in part II. In Parts I and III, a special exponential function can be used to approximate the solution as

After inserting Eq. (1.7) into Eq. (1.4), one gets S j as a function of V (r) and its derivative V (r). By considering the QNM boundary conditions:

Z ∼ e iωr * , r * → +∞, (1.8) we obtain the solutions of Parts I and III by matching them to Part II solution at the turning points r 1 and r 2 , respectively. In [91] , the WKB approach was effectively expanded to the sixth order. This permits the QNM frequencies to be calculated as a function of the potential and its derivatives at their maximum. Thus, one has the following QNM frequencies for the sixth order diagnosis:

(1.9)

Λ j expressions are also given in [91] . While Ref. [102] has some interesting current ideas on the convergence of the WKB series, Ref. [102] provides more information on the expansion parameter.

This review article is constructed as follows. In Section II, we supply a detailed review of GFs for the d-dimensional dS and AdS spacetimes. This section is nothing but a comprehensive overview of Ref. [2] and it is a useful starting point for learning the GFs in general. We discuss the black-body radiation, BH perturbation, reflection and transmission coefficients, GFs in the asympotic limit, flux, and universality. In Section III, we summarize the matching technique for computing the GFs and QNMs. Section IV is devoted to using of the WKB approximation for computing the GFs and QNMs. To assist the reader in conducting these computations, we provide specific applications. In this regard, the WKB approximation To this end, we consider various BH prototypes, which are the Schwarzschild-like BH in the BGM (SBHBGM), the Schwarzschild BH surrounded by quintessence field (SBHSQ), the brane-world BH solutions via a confining potential (BWBHSCP), and the charged (Taub)

NUT BHs (CTNBHs) to exemplify our viewpoint for computing the GFs and QNMs [164] .

Finally, we give our concluding remarks and comments on the GFs and QNMs in Section VI.

II. GFs FOR dS AND AdS SPACETIMES: BRIEF REVIEW OF REF. [2] For spin-0 waves of asymptotically flat BHs that are static and spherically symmetric, Harmark et al [2] presented a simple but generic derivation of the cross-section σ(ω). Their approach extends the results presented in [23] , which are limited to Schwarzschild BHs. For dS and AdS BHs [23] , they [2] showed that the cross-section is σ(ω) = A H , which is the BH area. In particular, they considered foremost s-waves ( = 0) to derive the GFs of asymptotically dS/AdS spacetimes. In fact, the literature concerning the GFs for BHs in NAF spacetimes, like dS or AdS, is preferably barren that there are less number of studies about this subject in the literature (see for example [16] [17] [18] [19] ). The most prominent feature of Ref. [2] is that the authors developed a general computation methodology that may be utilized to tackle different types of spacetime asymptotic.

For dS BHs, the GFs in the case of the low frequencies ω T H and ωr H 1 were analytically derived [2] . That was done in the case of small dS BHs, namely when r H is much shorter than the distance-scale adjusted by the cosmological horizon. The separation of the near horizon area from the distant region is the basis for their methodology. After some straightforward computations, they got the following non-trivial result: have a more richer structure. Moreover,ω d−2 ∼Â H admits a particular frequency, which corresponds to no reflection or full transmission Γ(ω) = 1 . Conversely, in the case of ω = 2n+d−1 together with n ∈ {0, 1, 2, · · · }, the pure reflection for the radiation Γ(ω) = 0 was determined. On the other hand, those frequencies, interestingly enough, exactly match the usual frequencies of scalar wave perturbations in pure AdS spacetime [46] . They also found other critical points, (2n+d−1−ω) 2 ∼Â H , that result in full transmission, Γ(ω) = 1.

The AdS/CFT correspondence, which links AdS BH events to thermal gauge theory, is particularly noteworthy because of the multifarious structure revealed by the GF for the AdS BHs (see for instance [52] and references therein).

In the asymptotic limit, the frequency through the imaginary axis is extremely high ω → +i∞ and thus |r H ω| 1 , there are no universal results for the GFs. Furthermore, it is notable that Γ(ω) for the Schwarzschild BH has the following expectation value for the radiated gravitons [2, 22] :

To comprehend the hypothesis outlined in [22] , one should check Refs. [26, 103] in which it was shown that, for a particular class of five-dimensional BHs, the GFs behave in such a way that at asymptotic infinity BH spectroscopy resembles the microscopic string's excitation spectrum. This suggests that GFs include information on the quantum structure of BHs.

One of the works made in [22] is based on the low-frequency results of [26] , which hypothesizes that by examining the aforementioned GFs one would be able to deduce on the BH's microscopical description at asymptotic frequencies. Whence, at asymptotic frequencies, BH microscopics would need new degrees of freedom (DoF) with somewhat unusual statistics, in particular, for the case of Schwarzschild BH (additional investigations along similar lines were later conducted in [104] ). On the other hand, the asymptotic calculations demonstrated that these additional microscopic DoF would have to include even more unusual statistics than in the Schwarzschild case for the charged or chargeless Reissner-Nordström or asymptotically dS BHs (for the relevant equations, the reader is referred to [2] ). Besides, for uncharged/charged AdS BHs and the asymptotically AdS case, Γ(ω) elicts the following expectation value for the number of radiated gravitons at asymptotic frequencies:

which is obviously the well-known black-body radiation.

To explore the generality of GFs in the asymptotic limit, one must first study the BH scattering. However, such a study reveals that some transmission and reflection coefficients hide the generality in the asymptotic limit. The characterization of gravitational perturbations can be considered by a Schrödinger equation in 1-dimension, with some potential V (r * ) related with the geometry of spacetime and characteristics of the perturbation, as previously stated. Let us deine the wave function Φ ω (r * ) as solution to the 1-dimensional Schrödinger-like wave equation:

which describes the scattering of an ingoing wave begun to propagate from spatial infinity r * = +∞ (or the cosmological horizon for asymptotically dS BHs). Thus, we have

where the reflection and transmission coefficients are denoted by R(ω) and T (ω), respectively. Although the both Φ ±ω admit the solution to the same wave equation, however one

which have other form of the reflection and transmission coefficients: (ω) and T (ω). Considering the following flux definition:

one can infer that the flux is independent of r * . For the entire region, r * → ±∞, after evaluating Eq. (2.8), we get

It is easy to check that when ω is real then Φ −ω = Φ * ω (here, * symbol means the complex conjugation) and whence = R * and T = T * . As a result, we rediscover one of the main outcomes of the quantum mechanics: | R| 2 + | T | 2 = 1. On the other hand, considering Φ ω as the Schrödinger solution with complex ω, which represents the dispersing of the outgoing wave coming from r * = −∞ (the outer event horizon), we get

in which T (ω) and (ω) have the same role as T (ω) and R(ω), respectively. As far as the space solutions of Eq. (2.5) possesses dimension 2, Φ ω should be a linear superposition of Φ ω and Φ −ω . So, one can set

and consequently, we have

Again, for real ω we have | | = |R|. However, it is not obligatory in the latter result to satisfy the complex frequencies ω. But, T = T always holds. Lastly, we would like to point out that Φ −ω continues to solve Eq. (2.5) by satisfying the following boundary conditions

for the other scattering coefficients (transmission/reflection): (ω) and T (ω). Similar to Eq. (2.11), Φ −ω is also a linear combination of Φ ω and Φ −ω :

So, we have double identities: T T = T T and R = . While the GF with complex frequency ω is defined as Γ(ω) = T (ω) T (ω), for the real frequencies it becomes Γ(ω) = |T (ω)| 2 . The GFs for the two scattering cases, in particular, turn out to be identical [2] .

Those findings are valid for either asymptotically flat or asymptotically dS geometries. For asymptotically AdS BHs, the r * coordinate spans the spacetime from −∞ (i.e., from event horizon) to a fix point, which can be chosen as zero, at spatial infinity. As shown in Ref. [46] , in the case of r * ∼ 0 the wave function behaves as 16) in which J ν is a first kind of Bessel function and j ∞ = d − 1, d − 3, d − 5 stand for the perturbations of tensor, vector, and scalar types, respectively. C ± are complex integration constants [46] . Therefore, when Re(ω) > 0 with r * −1, Eq. (2.16) admits the following asymptotic behavior [2] . The transmission and reflection coefficients at spatial infinity may therefore be defined in terms of the coefficients of the Bessel functions J ν with the aid of Eq. (2.17). For instance, the incoming waves obey the following matrix equation:

In fact, we should consider the solution at spatial infinity as if a plane-wave for r * −1 [2] . As previously stated, some of the T (ω) and R(ω) conceal generality of the GFs in the asymptotic limit. For all considered BH spacetimes, if T = 1, then Γ(ω) = T (ω). Plus, for all the asymptotically flat spacetimes [2]

where the parameter j could be

for tensor or scalar-type gravitational perturbations of neutral (charged) BHs and j = 2 j = 3d−7 2d−5

represents the vector-type gravitational perturbations of neutral (charged) BHs. On the other hand, it is found that for all asymptotically flat cases, we have

which yields the generic asymptotic GF:

Although the asymptotically dS spacetimes do not reveal any generality for the GFs aside from the T = 1, asymptotically AdS geometries do. Namely, for all AdS BHs, it was shown that when T = 1 = T , it leads to a generic result for the GF: Γ = 1 [2] . Moreover, in the case of = 0, one gets

The resemblance to the asymptotically flat case has now become even more obvious. Future investigations to be made on those topics will definitely help the researchers to get a better understanding of the observed signs of the generality/universality.

In this section, we shall address the investigations on the matching technique for computing the GFs. This technique is based on solution of the field equations in the near-horizon and far-field regions, and then comparing the solutions, via a intermediate region, to set up a complete and straightforward solution for the radial part of the field. Getting inspired from his PhD thesis, Unruh [6] did, in fact, use this procedure in 1976 for the Schwarzschild BH. The transmission/reflection coefficients for not only the bosons but also for the massive fermions from small BHs were considered [6] , and the same results were independently obtained by Ford [105] .

In Ref. Another useful and insightful work about this technique was also done in [2] , which calculates the GFs in the static and spherically symmetric BHs in d-dimensions, as indicated earlier. In fact, the monodromy matching techniques were initially designed to calculate the QNMs [2] and then were developed by different studies [95, 106] . In [107] , the monodromy matching technique was applied to derive the QNM frequencies of the d > 4 Schwarzschild and d = 4 Reissner-Nordström BHs for the ultra-high damping case. The equations are calculated using the monodromy of the perturbation technique, which is then carried out analytically to the complex plane. The potential in the non-physical locality appeared around the BH singularity [107] is heavily weighted in the evaluation. This research includes the low-energy form of the scattered wave with real and low frequencies, and the asymptotic case, in which the scattered wave's frequencies are extremely high along the imaginary axis. The gravitational perturbations marked out with the master equations of Ishibashi-Kodama were explored in both cases [11, 12] . For further approaches in this framework, we refer the reader to [108, 109] , in which the evaporation of a (d ≥ 4)-dimensional rotating BH possessing scalar DoF on the brane was considered. The GFs (and thus the absorption probabilities and cross-sections) for those BH spacetimes were analytically obtained for both high and low-energy regimes [109] . Essentially, the procedure performed was to consider the scalar fields propagating in the gravitational background persuade on the brane and to analytically solve the scalar field equation by applying the matching technique on the low-energy regime and matching up the far-field and near-horizon parts of the solution. In general, the analytic expression obtained for the absorption probability has dependency on the angular quantum numbers (l, m). The analytical results obtained in [108, 109] are hold just for low values of the energy parameter (ωr h ) and rotation parameter (a), and they were in harmony with the numerical results [109] . In addition to this, for the large values of ωr h and a, the agreement on both qualitative and quantitative levels was also confirmed.

Using the technique of matched asymptotic expansions in order to analytically compute the GFs was considered in [110] , where the minimally-coupled scalar fields propagating on the background of rotating BHs were used in the higher (odd) dimensions. Besides, the nonexistence of the superradiant was shown to be a result of reflecting or transparent border conditions since a global timelike Killing vector exists (Ω h < l −1 ), (here Ω h denotes the horizon angular velocity) [110] . It is worth noting that for this purpose only the positive frequencies modes are considered, namely the frequencies having ω − mΩ h > 0 [110] .

As being stated above, the first significant study on matching technique [111] [112] [113] concerning the GFs of the bosons and fermions was dexterously made by [6] . In that study, the Dirac interesting result appeared, based on this work [6] , is that all (quantum and/or classical) particles with a velocity v = 1 (see [6] for details) observe the BH as having roughly the same size.

In Ref. [108] , the emission of fermions from the aforementioned projected BH background were also studied. It was considered that the emitted particle modes couple at best minimally to the gravitational background, thus, they hold the free equations of motion [108] . When the Newman-Penrose (NP) formalism [114] is applied, by further assuming the factorized ansatz with spin-weighted spheroidal harmonics, the free equations of motion for particles with spin s = 0, 1 2 can form a master equation on the brane. Then, the analytical results to the radial general equation are obtained for the two asymptotic regions of the BH horizon (r ≈ r h ) and in the far-field (r >> r h ), these two solutions are related at a middle radial region for constructing a complete solution, which is valid at all radial regimes, for a field with arbitrary spin s. The derived expressions of the absorption probabilities (valid for ω → 0) for gauge bosons show that thermal radiation of spin-0 particles vanishes faster than the fermionic ones, spin-1 2 . Furthermore, it is proven that the total absorption cross-section behaves differently for gauge bosons and fermions in the low-energy limit than it does in the non-rotating case, while in the high-energy regime it behaves universally, achieving a constant, spin-independent asymptotic value.

This section is addressed to the calculation of GFs and QNMs by applying the WKB approximation. This approximation method is nothing but a semi-analytic technique, which is mostly used for determining the mode frequencies of the perturbed systems. This method is also used to derive analytic expressions for the QNMs with real and imaginary frequency components in terms of the BH parameters and numbers (n = 0, 1, 2, ..), which label the modes; for instance n = 0 is the fundamental mode, n = 1 denotes the first overtone mode, and so on. The WKB approach has found itself many different application areas in the literature including the computations of the GFs. It is indeed an accurate approach, which can be carried to the higher orders. In this context, Schutz and Will conducted one of the foundation investigations for the Schwarzschild BH in 1985 [115] . Moreover, it was found that the agreement of WKB approach with other methods is excellent especially for low lying modes [116, 117] , which are carried the results to third order beyond (within the eikonal approximation) [118] . In the following, we will go over the detailed applications for both GFs and QNMs based on the WKB approximation for both bosons and fermions.

The WKB approximation for the GFs is mostly used to calculate the reflection and transmission probabilities. In fact, the transmission coefficient can also be interpreted as GF, which determines the spectrum of Hawking radiation [22] . There are several valuable studies along this line. Its applications do not only take place in quantum mechanics, but also in higher dimensional gravity, astrophysics, string theory [119] , wormhole physics [120] , magnetized BHs [121] , and 5-dimensional charged Bardeen BH spacetimes (by involving the perturbations of scalar and electromagnetic fields) [122] etc. The use of the WKB approximation in BH physics has gained momentum when extended to the sixth order. The pioneering works on this subject undoubtedly belong to Konoplya and his co-authors (see [91, 92, 119, 120] and references therein). Besides, the third order WKB approximation is also in the agenda literature for studying the GFs; for example, the QNMs of the Hayward BH surrounded by quintessence have recently been studied in [123] for some special cases of the quintessence parameter.

At this stage, let us look at our first practical application. To this end, we consider Ref. [43] , which is one of our (Kanzi and Sakallı) most recent works. Thus, we shall employ the WKB approximation for computing the transmission and reflection coefficients for scalar perturbation of the polytropic BH [43] whose line element is

in which

and

In Eq. (4.4), L 2 = − 3 Λ in which Λ is the cosmological constant. The case of negative Λ will be considered throughout this section. In other words, we assume that the empty space has positive pressure but negative energy density, such as the AdS space. To determine the scattering coefficients, one needs to derive first the effective potential from the 1-dimensional Schrödinger-like wave equation. The following massive (m 0 : mass of boson) Klein-Gordon (KG) equation can be used to determine the wave equation of the associated scalar field:

where ∇ ν denotes the covariant derivative. Equation (4.5) can be alternatively expressed as

Since the line-element under consideration is Eq. (4.1), one computes the square root of determinant of the metric as √ −g = ρ 2 sin θ. In the sequel, considering the symmetries of the metric, we can define the following ansatz for the wave function:

where, needless to say, ω is the frequency of wave and m denotes the well-known azimuthal quantum number. Thus, one can get the radial equation as follows ∆ d dr ∆ dR (r) dr + m 2 a 2 − 4af mω + ω 2 r 2 + a 2 2 − ω 2 a 2 + m 2 0 r 2 + λ lm ∆ R (r) = 0, (4.8)

which λ lm represents the angular solution eigenvalue whose differential equation is given by

here

On the other hand, we have

In addition, let us utilize a transformation as

to obtain the 1-dimensional Schrödinger equation:

in which the effective potential V ef f is obtained as

where = d dr . In Fig. (1) , we have demonstrated the influence of the rotational parameter a on V ef f , i.e., Eq. (4.13). As it is shown in Fig. (1) , the height of the V ef f barrier rises as the rotation a increases. Now, it is worth recalling a well-known scientific fact: the chance of a particle tunneling relies on the proceeding particle energy in relation with the height of the barrier [124] .

Let us consider a moving wave that approaches the BH (r * → −∞) from spatial infinity r * → ∞ to determine the scattering coefficients. Naturally, while some of the waves are transmitted, the others will be reflected, due to the effective potential. Considering the scattering boundary conditions (2.13) with the following normalization condition |T | 2 + |R| 2 = 1, (4.14)

one can define the GF as (see [55] and references therein)

The definitions of R (ω) and T (ω) are given by [43, 91] T (ω) = 1 + exp (2πiK) −2 , (4.16)

where

in which r 0 is the special radial location which lead the potential V (r) to reach its maximum (V 0 ), i.e., to its peak. Let us set V (r 0 ) = V 0 . Moreover,

and Kerr-like BH (KlBH) in bumblebee gravity model (BGM) [125] and conducted a thorough investigation of their GFs and QNMs [126] . But, after being discovered by [127] , it was understood that the metric of [125] that was already used by us in [126] is wrong. Then, we revised all of our studies and demonstrated [128] that problem can be corrected when the slowly rotating KlBH [129] is considered. In this context, the information that we shall serve in this section is based on Ref. [128] .

We now give the computation details of the bosonic QNMs for slowly rotating KlBH spacetime via the six order WKB approximation [128] , in which the scalar QNMs are elaborated. The slowly rotating KlBH metric in the BGM is given by [129] ds 2 ≈ − 1 − 2M r dt 2 − 4Mã sin 2 θ r dtdϕ + (1 + L)r r − 2M dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 , (4.21) whereã = √ 1 + La. Following the same process as previously prescribed in Eqs. (4.6)-(4.13), one gets the effective potential of the scalar waves propagating in the BH geometry (4.21) as follows [91] ). Thus, one can compute the QNMs as:

where Λ and Ω are the same as Eqs. (4.19) and (4.20) . The results for bosonic QNMs are tabulated in Table I for m = 0, l = 2, and the initial overtone n = 0. It can be inferred from For example, in the case of L ≈ 0 − 1.2, we have a reduction with the increasing LSB effect in the damping mode first and then this harmonious trend vanishes and a random behaviors appear, which can be deduced from Fig. 4 .

Before we get into the details of the Dirac perturbation (see Section V B) to derive the QNMs and GFs by using the WKB method, it might be worthy to mention some more investigations about the subject. In Ref. [130] , to analyze the behavior of Dirac perturbations when vacuum fluctuations are taken into consideration, the QNMs of a quantum corrected Schwarzschild BH were explored. For 4-dimensional Kerr BH, the numerical analyses of the fermionic QNMs, which were produced by using the third and sixth order semi-analytic WKB(J) and advanced iteration methods, can be found in [131] [132] [133] . Similar works were made for the f (R) global monopole [134] and for the massive Dirac field in the Kerr background [135] .

Furthermore, in Refs. [126, 128] , we also investigated the Dirac QNMs of KlBH in the BGM. To proceed the evaluations of the Dirac fields in the KlBH geometry, we used the 4-dimensional Dirac equation formulated in the NP formalism [136] in order to compute the fermionic V ef f for the KlBH [126, 128] :

where ± symbols represent the spin-up/down fermions. To derive the fermionic QNMs, one can employ the method for computing the bosonic QNMs prescribed in the previous section. Therefore, we analyze the potentials seen in Eq. Table VI for different values of the LSB quantity together for a = 0.4 . As it is shown in the Table VI, The general bounds on the Bogoliubov coefficients approach to determining the transmission probability or GFs, proposed in [137] , are discussed in this section. For 1-dimensional potential scattering, some generic bounds for reflection and transmission coefficients were established and also shown how the bounds are extended with various approaches [59, 138, 139] .

Regardless of the various successful methods applied for getting the GFs and QNMs, what matters at the end of the day is to have the effective potential and analyze its behavior under the physical changes. However, based on the particles character spin-0, spin-(±1/2) , the paths for deriving the effective potentials could be disparate.

Before going into the details of the subject seen in the title of this subsection, it would be better if we continue to make more literature reviews about the GFs. Calculation of HR in the semi-analytic approach for the 5-dimensional BH conformally coupled with scalar field yielded that the GF can 'suppress' the total Hawking radiation power [140] . In addition, in the impacts of Hawking radiation, both conformally coupled and charge parameters play an opposing role. For a d−dimensional Schwarzschild-Tangherlini BH spacetime [141] , it was shown that the GFs bound in bulk and brane-localized scalar field reduces as the spacetime dimension d rises.

Let us now consider the SBHBGM spacetime [55, 142] , which is given by [142] 

Recall that L denotes the LSB parameter. In order to derive the effective potential, one can invoke the following ansatz for the massless scalar field equation (4.6) [55]

and r * coordinate for the SBHBGM is found by r = √ 1 + L dr f . Thus, after some straightforward computations, the scalar effective potential of the SBHBGM can be derived as

where λ represents the angular eigenvalue, f is the metric function, and = d dr . Figure (6) reveals that V ef f disappears both at the event horizon and spatial infinity of the SBHBGM geometry. This picture assists us to analytically compute the GFs of the scalar field emitted from the SBHBGM. For obtaining the GFs, we employ the following general semi-analytic bound formula (see [55] and references therein) 

We then use the potential (5.3) to obtain the scalar GFs in the SBHBGM background. Thus, scalar (s) GFs are found to be

GFs of the SBHBGM arising from the scalar potential are depicted in Fig. (7) . Since a BH is not a perfect black-body, therefore, the GF of the HR must be < 1. Thus our findings, as shown in Fig. (7) , are fully in agreement with the latter remark. Moreover, the peak of the GFs reduce with the L parameter. To sum up, LSB has a GF-reducing effect for the bosonic fields.

We also investigated the same approach for regular Bardeen BH surrounded by quintessence (BBHSQ) in [62] , in which the scalar GFs are given by

where ω q denotes the parameter of state, c represents a positive normalization factor, and r h stands for the event horizon. We depicted the GFs of the BBHSQ in Fig. (8) . Meanwhile, the state parameter ω q and event horizon are related with each other [62] . It is clear from Fig. (8) that GF characteristics strongly depend on the state parameter ω q . Furthermore, according to Fig. (8) , very similar emissions occur near the critical ω q values (-1 3 and -1). In addition, the above methodology can also be used in the slowly rotating KlBH (4.21).

Thus, one finds out [128] 

In Fig. 9 , the obtained bosonic GFs of the KlBH are demonstrated. As it is shown in the plots, σ clearly drops with the growth in the LSB parameter. Namely, the LSB parameter acts as a fortifier for GFs of the spin-0 particles.

However, it is worth noting that employing a single semi-analytical method in combination with certain other methods can occasionally be necessary to complement and enrich the GFs. Therefore, we use supplementary methods like series, Miller-Good transformation, and rigorous bound to determine and improve the accuracy of the GFs [28, 63, 143, 144] .

In study [63] , we concentrate on the Miller-Good transformation approach [143] Substituting U (r) = ψ(r) √ f into the Schrödinger equation (4.12), we obtain

Further, we set [63] ψ (r) = 1 and ψ (r) = ρ (r) ρ (r)Ψ ρρ (ρ (r)) − 1 2 ρ (r) ρ (r) ρ (r) Ψ (ρ (r)) + 3ρ 2 (r) 4ρ 2 (r) ρ (r) Ψ (ρ (r)) , (5.15) where Ψ ρ denotes the derivative with respect to ρ. Thus, the wave equation (5.11) recasts in Ψ ρρ (ρ (r)) + k 2 ρ 2 (r) + 3ρ 2 (r) 4ρ 4 (r) − ρ (r) 2ρ 3 (r) Ψ (ρ (r)) = 0, (5.16) which can be simplified to the following form by defining a new effective potential K 2 :

Ψ ρρ (ρ (r)) + K 2 Ψ (ρ (r)) = 0. (5.17) Thus, the Schrödinger equation (5.11) which was previously represented in terms of ψ (r) and k (r) is now turned into anew 1-dimensional wave equation in terms of ρ (r) and K(ρ (r)).

Using the so-called Schwarzian derivative:

K 2 can be rewritten as

In order to have positive ρ (r), one can set another parameter as j (r) = ρ (r) where j (r) > 0. Furthermore, by setting j (r) = J(r) −2 with J (r) > 0, one can obtain

For J = 1, it is clear that K 2 (ρ) = k 2 (r). As a result, the transmission amplitudes of both potentials are the same, and hence the transmission probability is the same. The second option is to discover a relationship between two parameters J and f that results in distinct potentials and hence the different transmission probabilities. To this end, Eq. (5.4)

is redefined as (see [63] and references therein) 

Moreover, by tuning J as being J 2 = f , we get a rather simpler integral and thus the transmission probability of the SBHSQ is obtained as follows:

which yields [63] T ≥ sec h 2 1 2ω

In summary, the obtained expression yields the GFs of the SBHSQ spacetime, which is Another efficient method for calculating GFs is to use the rigorous bound on the GFs, which was applied first by Boonserm and Visser [28] for the Schwarzschild BH. The

Regge-Wheeler equation that describes the motion of any sort of particle in a non-rotational spacetime can be used to calculate these bounds, which are valid for all frequencies. In the literature, so far, the rigorous bounds are considered for the GFs of the 4-dimensional

Reissner-Nordström BH, the higher dimensional Schwarzschild-Tangherlini BH, the charged dilatonic BH in (2 + 1) dimensions , the charged dilatonic BHs, the Kerr-Newman BH, and also for the BH and black string solutions in the dRGT massive gravity (see [42, [145] [146] [147] [148] [149] [150] and references therein). Now, we would like to draw the reader's attention to our yet another study [42] as an example of the use of rigorous bound on the GFs. In the framework of massive gravity theory coupled with non-linear electrodynamics (nem), the metric of the corresponding BH is given by [42] ds 2 = −f (r) dt 2 + dr 2 f (r) + r 2 dθ 2 + sin 2 θdϕ 2 , (5.26) which represents two metric function solutions of the charged dRGT BHs in the nem. The first metric function solution reads [42] f (r) = 1 − 2M r + 8r 2 3b 2 1 + qb r 2

where M is a constant of integration (mass parameter), b is a positive parameter, h 0 is a constant parameter, α = 1 + 3α 3 , and β = α 3 + 4α 4 . The second metric function solution is given by [42] f

To investigate the thermal radiation of the charged dRGT BHs in the nem, we consider the massless and charged KG equation:

where 30) and the electromagnetic potential is defined as

Let us consider the line-element (5.26) of the charged dRGT BH in the KG equation (5.29) with the following ansatz:

in which e iωt indicates the oscillating function and Y lm (Ω) denotes the spherical harmonics.

The tortoise coordinate can be easily obtained via the expression of dr * dr = 1 f (r) , which assists us to permute the radial wave equation that can be cast in 1-dimensional Schrödinger equation:

where

where f = df dr . Hereafter, we shall name the solutions by indices of 1 and 2. The V ef f of the first solution is found as

The V ef f of the second solution can be found out to be as

. By applying the Taylor expansion, we can derive the GF as

and

and

two parameters R h /r h denote upper/lower rigorous bounds, respectively, which are given by (5.46) and Substituting the V ef f of second solution, i.e., Eq. (5.37), in Eq. (5.5) to get (5.48) by using the Taylor expansion and integration, one can obtain

where W 2 = 2D + q 2 + ωbM q , (5.50)

and

Using Eq. (5.49), the rigorous bounds on the GFs for the second solution of the charged dRGT BH in the nem are plotted in Fig. (14) . For both GF solutions, the constant parameter b is set to be minimal. The charge parameter's increase first aids the GF in reaching its maximum value, but later the GF begins to decline. In both solutions, the aforementioned change can be seen after q = 4.5. As a result, we claim that in the nem, the charged GF cannot have a monotonous behavior.

After this stage, it will be appropriate to serve another interesting application for the GFs, which is about the BWBHSCP [44] . Before moving on to the GF computation details, let us take a look at the BWBHSCP metric structure. BWBHSCP has a static spherically symmetric line-element that can be described by the following metric ansatz: ds 2 = −e µ(r) dt 2 + e υ(r) dr 2 + r 2 dθ 2 + sin 2 θdϕ 2 .

(5.55)

There are two types of solutions [151] , the first solution reads for the lower bound, which causes this problem. In order to overcome this issue, one can replace the applied method with the one, which was prescribed in [28, 44] . To this end, we first set h = ω 2 − V ef f in Eq. (5.6). Since there are no restricted regions in this method [28] , one has 

This bound becomes pointless if ω 2 < V peak . Furthermore, we can compute the GF by using the transmission coefficient T l (ω) with the Miller-Good transformation technique [143] .

Thus, Eq. (5.59) is given by

To compute V peak , one should first define r peak . For instance, for an illustrative demonstration, if one sets B = 1, λ = 2, and α = 1, in the first solution, r peak can be computed The behaviors of the QNMs under the influence of the brane-world parameters α and β are shown in Table (III) for the first solution. The oscillation frequencies (Re(ω)) initially reduce with the rising α and β parameters then start to monotonically rise; the same characteristics can easily be observed for the damping rates (Im(ω)). As α parameter is increased, the QNMs of the second solution exhibit more regular behaviors: both Re(ω) and Im(ω) grow with the increasing α values.

For deriving the GFs of fermionic particles let us re-follow Ref. [55] for SBHBGM (5.1).

In order to compute the V ef f of the fermionic fields generating in the SBHBGM geometry, The physical quantities are chosen as B = 1 and λ = 6.

we consider the Chandrasekar-Dirac equations [114] :

where F 1,2 and G 1,2 are known as the Dirac spinors which indicate the components of the wave functions. The spin coefficients are represented by the Greek letters, moreover, the complex conjugate of any parameter is denoted by a over-bar symbol. The spin coefficients which are non-zero are found to be

4r . Moreover, D = a ∂ a , ∆ = n a ∂ a , and δ = m a ∂ a are the corresponding directional derivatives [55] . After a series of procedures whose details are given in [55] , the Dirac equations (5.62) are decomposed into the radial and angular equations. The radial parts can be reduced to the 1-dimensional Schrödinger equation with the following effective potential,

whose behaviors are depicted in Figs. (19) and (20) . After substituting the effective potential (5.64) in the semi-analytic GF equation (5.4) , one gets Additionally, for the slowly rotating KlBH case [128] , the expression for the fermionic GFs are defined as [128] 

in which Moreover, in line with the same thought, GFs of the CTNBHs have recently been studied in Refs. [152] and [153] , respectively. For the CTNBH [153] , the charged Dirac equations [154, 155] that govern the behavior of charged spin-1 2 particles are obtained with the NP formalism. Similarly, the charged KG equation has been employed in order to study the GF of bosons in the CTNBH geometry. For both type of charged particles, the corresponding Schrödinger like wave equations with the relevant effective potentials have been derived [153] . Thus, the effect of the NUT parameter (l) on the charged bosonic and fermionic GFs and QNMs have been thoroughly analyzed, which was the main aim of [153] . The result obtained in [153] for the bosonic GFs emitted from the CTNBH is as follows FIG. 21: σ + l (ω) versus ω graph (see Ref. [55] ).

where M , l, and q denote the mass, the NUT parameter, and the electric parameter of the BH, respectively. Moreover,q denotes the charge of the boson and r h = M ± M 2 + r 2 + q 2 is the event horizon of the CTNBH. Similarly, the fermionic GFs obtained from the Dirac perturbation of the CTNBH are as follows [153] σ ± l (ω) ≥∼ sec h 2 λ 2ω 2k 2 λl 2 qe (qer h + kl 2 ) 3 + in which e represents the charge of the fermion.

The impacts of the NUT parameter l and the CTNBH charge q on the GFs are illustrated in Figs. (24) and (25) . As can be easily seen from those figures, the GFs increase as the NUT parameter is increased. The latter remark is valid for the both bosonic and fermionic perturbations.

The influence of the NUT parameter l on the QNMs is also investigated in [153] . To this end, two sets of charge values are used: q =q = 0.1 and q =q = 0.2 for n = 0 and n = 1 when l = 1. Table V summarizes the findings, which show that increasing the NUT parameter reduces both oscillation and damping modes, in general. Moreover, for n = 1 state although the real part remains intact, but the damping modes raises until l = 3.3 then they start to decrease for both sets of the charges: q =q = 0.1 and q =q = 0.2.

To make the behaviors of bosonic QNMs more understandable and comprehensive, we have also plotted Fig. 26 for various NUT parameters l and charge q =q = 0.2. It is also worth noting that for other low charge values like q =q = 0.01, the figures plotted are almost identical to Fig. 26 that is why we have only depicted Fig. 26 .

For the Dirac perturbation of the CTNBH, the results obtained for the charged fermionic

QNMs [153] are tabulated in Table VI. The latter table is Table   VI indicates that the fermionic oscillation frequencies increase smoothly with the increasing NUT parameters l. However, such as in the bosonic case, this behavior is reversed for the damping modes. On the other hand, it is worth highlighting that the evolution (in the context of increase and decrease) of the QNM frequencies of the Dirac particles with the NUT parameter l is in harmony with their GF characteristics. Finally, to reveal the behaviors of the imaginary and real frequencies, we have depicted the Dirac QNMs in Fig.   27 . The both behaviors are in accordance not only with each other, but with the bosonic QNMs as well.

This study serves as an introduction to the analysis of QNMs and GFs of numerous curved spacetimes examined in astrophysics, higher-dimensional gravity, and string theory. Although currently there are some great reviews on the QNMs including many valuable studies, which are supported by the observational characteristics of the recently discovered gravitational waves [156, 157] , but the number of reviews written on the GFs is insignificant. Therefore, in this review article, we have mainly focused our efforts on the GFs rather than the QNMs. To this end, we have concentrated on certain approaches, topics, and computations that were either not reviewed or were briefly mentioned in the literature. As a result, instead of discussing general problems superficially, we have directed the reader to the more specialized literature. Thus, we believe that this review article can pave the way for the reader to learn and follow the methodologies given in this study for signing new works.

Finally, we would like to state that to humanity's surprise, many theoretical studies that were previously assumed to be difficult to test empirically and observationally have been confirmed by scientists employing and developing unique setups. Some of the most recent theory-experiment/observation pairings are as follows: 1) Higgs boson found by the Compact Muon Solenoid [158] and ATLAS [159] 2) Gravitational waves are observed [21] 3) First image of a BH [160] 4) The first room-temperature superconductor identified [161] 5) Observation stationary Hawking radiation in an analog BH [162] . All of those suggest that the GFs and QNMs may cease to be pure theoretical subjects and become measurable physical quantities much sooner than anticipated. In conclusion, the theoretical physics will continue to guide the experimental physics in this regard. 

The Four Laws of Black Hole Mechanics

Greybody Factors for d-Dimensional Black Holes

Particle Creation by Black Holes

Scalar QNMs plots for various values of the NUT

Black Holes and Thermodynamics

Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole

Absorption Cross Section of Small Black Holes

Scattering by Black Holes

Stability of a Schwarzschild Singularity

Gravitational Field of a Particle Falling in a Schwarzschild Geometry Analyzed in Tensor Harmonics

Perturbation Analysis for Gravitational and Electromagnetic Radiation in a Reissner-Nordström Geometry

A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions

Fermionic QNMs of the CTNBH for various combinations of charge and state values

Master Equations for Perturbations of Generalized Static Black Holes with Charge in Higher Dimensions

Exact Absorption Probabilities for the D3-Brane

Exact Absorption Probability in the Extremal Six-Dimensional Dyonic String Background

Greybody Factors of Holographic Superconductors with z = 2 Lifshitz Scaling

Analytical solutions in rotating linear dilaton black holes: resonant frequencies, quantization, greybody factor, and Hawking radiation

Can the Effective String see Higher Partial Waves?

Dirac QNMs plots for various values of the NUT parameter l based on Table VI. As the solid lines are for q

Absorption cross-section and decay rate of rotating linear dilaton black holes

Absorption Cross-Section and Decay Rate of Dilatonic Black Strings

Quantum Gravity in de Sitter Space

Observation of Gravitational Waves from a Binary Black Hole Merger

Greybody Factors at Large Imaginary Frequencies

Universality of Low Energy Absorption Cross-Sections for Black Holes

Greybody factors: Hawking radiation in disguise

Greybody factors for rotating black holes on codimension-2 branes

Black hole grey body factors and d-brane spectroscopy

Remarkable Analytic Relations among Greybody Parameters

Bounding the Greybody Factors for Schwarzschild Black Holes

Calculable Corrections to Brane Black Hole Decay I: The Scalar Case

Calculable Corrections to Brane Black Hole Decay II: Greybody Factors for Spin 1/2 and 1

Hawking Radiation from a (4 + n)-Dimensional Black Hole: Exact Results for the Schwarzschild Phase

Bulk versus Brane Emissivities of Photon Fields: For the Case of Higher-Dimensional Schwarzschild Phase

Graviton Emission from a Higher-Dimensional Black Hole

Black Hole Particle Emission in Higher-Dimensional Spacetimes

Hawking Emission of Gravitons in Higher Dimensions: Non-Rotating Black Holes

Graviton Emission in the Bulk from a Higher-Dimensional Schwarzschild Black Hole

Emissivities for the Various Graviton Modes in the Background of the Higher-Dimensional Black Hole

Evaporation of a Black Hole Off of a Tense Brane

Bulk versus Brane in the Hawking Radiation of Graviton: Black Holes Radiate Mainly into the Bulk when n ≥

Absorption and Emission Spectra of a Higher-Dimensional Reissner-Nordström Black Hole

Bulk and Brane Decay of a (4+n)-Dimensional Schwarzschild-de Sitter Black Hole: Scalar Radiation

Greybody factors of black holes in dRGT massive gravity coupled with nonlinear electrodynamics

GUP-modified Hawking radiation and transmission/reflection coefficients of rotating polytropic black hole

Physical properties of brane-world black hole solutions via a confining potential

Hawking Radiation from AdS Black Holes

On the Classification of Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes and Quantum Gravity

Asymptotic Quasinormal Frequencies for Black Holes in Non-Asymptotically Flat Spacetimes

Scattering of Scalar Waves from a Schwarzschild Black Hole

Perturbative Calculation of Quasinormal Modes of d-Dimensional Black Holes

Exact Results for Evaporating Black Holes in Curvature-Squared Lovelock Gravity: Gauss-Bonnet Greybody Factors

Higher-Derivative Corrected Black Holes: Perturbative Stability and Absorption Cross-Section in Heterotic String Theory

Large N Field Theories, String Theory and Gravity

Greybody factor and sparsity of Hawking radiation from a charged spherical black hole with scalar hair

Hawking Radiation of Five-Dimensional Charged Black Holes with Scalar Fields

GUP Modified Hawking Radiation in Bumblebee Gravity

Hawking Radiation As Tunneling

Quasinormal Modes, Scattering, and Hawking Radiation of Kerr-Newman Black Holes in a Magnetic Field

Rigorous Bounds on Transmission, Reflection, and Bogoliubov Coefficients

Bounding the Bogoliubov Coefficients

Greybody Factors of Charged Dilaton Black Holes in 2 + 1 Dimensions

Greybody Factor and Hawking Radiation of Charged Dilatonic Black Holes

Solution of Dirac Equation and Greybody Radiation around a Regular Bardeen Black Hole Surrounded by Quintessence

Effect of Quintessence on Geodesics and Hawking Radiation of Schwarzschild Black Hole

The thermodynamics of black holes

Wormholes without exotic matter: quasinormal modes, echoes and shadows

GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral

Linear dilatons, NS five-branes and holography

Quasi-normal modes of warped black holes and warped AdS/CFT correspondence

Holographic Isotropisation in Gauss-Bonnet Gravity

Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown

The Physical interpretation of the spectrum of black hole quasinormal modes

Generalized second law of thermodynamics in black hole physics

Black hole entropy quantization

Quantization of Black Hole Entropy from Quasinormal Modes

Quasinormal spectrum and quantization of charged black holes

Analytical Solutions in a Cosmic String Born-Infeld-dilaton Black Hole Geometry: Quasinormal Modes and Quantization

Quantization of higher-dimensional linear dilaton black hole area/entropy from quasinormal modes

Quasinormal modes of charged dilaton black holes and their entropy spectra. Modern

Spectroscopy of rotating linear dilaton black holes from boxed quasinormal modes

Resonance Spectra of Caged Stringy Black Hole and Its Spectroscopy

Spectroscopy of Rindler modified Schwarzschild black hole

Quasinormal Modes of a Schwarzschild Black Hole Immersed in an Electromagnetic Universe

Quantization of rotating linear dilaton black holes

Quasinormal modes of charged fermions in linear dilaton black hole spacetime:exact frequencies

Spectroscopy of z = 0 Lifshitz Black Hole

A short review of loop quantum gravity

Quasinormal modes of stars and black holes

Quasinormal Modes of Black Holes and Black Branes

A New Approach to Black Hole Quasinormal Modes: A Review of the Asymptotic Iteration Method

Quasinormal modes of charged scalars around dilaton black holes in 2+1 dimensions: Exact frequencies

Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB approach

Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations

Quasinormal modes of tensor and vector type perturbation of Gauss Bonnet black hole using third order WKB approach

Quasinormal resonances of a charged scalar field in a charged Reissner-Nordström black-hole spacetime: A WKB analysis

On the classification of asymptotic quasinormal frequencies for ddimensional black holes and quantum gravity

The quasinormal modes of quasinormal modes

Astrophysical signatures of boson stars: quasinormal modes and inspiral resonances

Quasinormal frequencies of D-dimensional Schwarzschild black holes: Evaluation via continued fraction method

Black hole quasinormal modes using the asymptotic iteration method

Quasinormal modes of dS and AdS black holes: Feedforward neural network method

Investigating a New Approach to Quasinormal Modes: Physics-Informed Neural Networks

Quasinormal modes of black holes and Borel summation

World Volume Approach to Absorption by Non-Dilatonic Branes

Effective Stringy Description of Schwarzschild Black Holes

Quantization of a scalar field in the Kerr spacetime

Asymptotic quasinormal frequencies for black holes in non-asymptotically flat spacetimes

Asymptotic black hole quasinormal frequencies

Greybody factors in a rotating black-hole background. II. Fermions and gauge bosons

Greybody factors for brane scalar fields in a rotating black hole background

Greybody factors for rotating black holes in higher dimensions

Superradiant Instability of Five-Dimensional Rotating Charged AdS Black Holes

Superradiance and black hole bomb in five-dimensional minimal ungauged supergravity

Superradiance and instability of small rotating charged AdS black holes in all dimensions

The Mathematical Theory of Black Holes

Black Holes Normal Modes: A Semianalytic Approach

Black-hole normal modes: A WKB approach. II. Schwarzschild black holes

Low-lying quasinormal modes of topological AdS black holes and hydrodynamics

Black-hole normal modes: A WKB approach. I. Foundations and application of a higher-order WKB analysis of potential-barrier scattering

Quasinormal modes of black holes: From astrophysics to string theory

Passage of radiation through wormholes of arbitrary shape

Quasinormal modes of magnetized black hole

Quasinormal modes, shadow and greybody factors of 5D electrically charged Bardeen black holes

Quasinormal modes of the Hayward black hole surrounded by quintessence: scalar, electromagnetic and gravitational perturbations

Exact Kerr-like solution and its shadow in a gravity model with spontaneous Lorentz symmetry breaking

Greybody radiation and quasinormal modes of Kerr-like black hole in Bumblebee gravity model

Comment on "Greybody radiation and quasinormal modes of Kerr-like black hole in Bumblebee gravity model

Reply to "Comment on 'Greybody radiation and quasinormal modes of Kerr-like black hole in Bumblebee gravity model

Slowly rotating Einstein-bumblebee black hole solution and its greybody factor in a Lorentz violation model

Quasinormal modes of a quantum-corrected Schwarzschild black hole: gravitational and Dirac perturbations

Dirac quasinormal frequencies of the Kerr-Newman black hole

Fermion Quasi-normal modes of the Kerr Black-Hole

Quasi-normal modes and fermionic vacuum decay around a Kerr black hole

Scalar and spinor QNMs of a black hole with a global monopole in f(R) gravity

Quasinormal modes of massive fermions in Kerr spacetime: Longlived modes and the fine structure

An Approach to Gravitational Radiation by a Method of Spin Coefficients

Some general bounds for 1-D scattering

Temperature and entropy of Schwarzschild-de Sitter space-time

Analytic bounds on transmission probabilities

Hawking Radiation of Five-dimensional Charged Black Holes with Scalar Fields

The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes

Exact Schwarzschild-like solution in a bumblebee gravity model

Transmission probabilities and the Miller-Good transformation

Greybody factor and quasinormal modes of Regular Black Holes

Bounding the Greybody Factors for Non-rotating Black Holes

The Hawking effect and the bounds on greybody factor for higher dimensional Schwarzschild black holes

Bounding the greybody factors for scalar field excitations on the Kerr-Newman spacetime

Greybody factor for black holes in dRGT massive gravity

Greybody factor for massive fermion emitted by a black hole in de Rham-Gabadadze-Tolley massive gravity theory

Greybody factor for black string in dRGT massive gravity

Brane-World Black Hole Solutions via a

Greybody Radiation of scalar and Dirac perturbations of NUT Black Holes

Fermionic and Bosonic Greybody Factors as well as Quasinormal Modes for Charged Taub NUT Black Holes

Solution of the Dirac equation in the rotating Bertotti-Robinson spacetime

Dirac Equation on the Kerr-Newman Spacetime and Heun Functions

Detecting quasinormal modes of binary black hole mergers with second-generation gravitational-wave detectors

Constraints on quasinormal-mode frequencies with LIGO-Virgo binary-black-hole observations

Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC

Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC

First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole

Room-temperature superconductivity in a carbonaceous sulfur hydride

Observation of stationary spontaneous Hawking radiation and the time evolution of an analogue black hole

The interest on gravitational perturbations has gained momentum since the first indirect observation of gravitational waves, which was made on 14 September 2015 and was announced by the LIGO and Virgo collaborations on

Throughout this study, we shall usually refer to "QNMs" as frequencies of QNMs, rather than the corresponding amplitudes. The units = c = G = 1 will be used the most unless otherwise stated

Acknowledgments 55References 56

We would like to express our gratitude to the following colleagues who contributed to