Elsevier

Annals of Physics

Volume 434, November 2021, 168603
Annals of Physics

Quasinormal modes for non-minimally coupled scalar fields in regular black hole spacetimes: Grey-body factors, area spectrum and shadow radius

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Abstract

Black hole perturbation theory is a useful approach to study interactions between black holes and fundamental fields. A particular class of black hole solutions arising out of modification of Einstein’s general theory of relativity are regular black holes (RBHs) which can be constructed using a nonlinear electrodynamic Lagrangian. Because of their importance, we are interested in studying the behavior of three kinds of such RBHs under perturbations generated by an external field. Indeed, we investigate the quasinormal modes (QNMs) of a massive scalar field propagating near the RBHs which is non-minimally coupled to the Ricci scalar-tensor of background geometry. We will attempt to find the low-lying quasinormal frequencies of the perturbations by using WKB approximation. We shall also study the relationship between the QNMs and some characteristic properties of black holes such as grey-body factors, area quantization, and shadow radius.

Introduction

Recent observations [1], [2], [3], [4] have provided us with the strong evidences that the black holes do exist. In this regard, the behavior of the fields surrounding a black hole not only tells us about its presence but also helps us to determine its parameters. After perturbation, a black hole undergoes damped oscillations with complex frequencies. The modes of such oscillations are called quasinormal modes (QNMs) which correspond to solutions of the wave equation satisfying the boundary conditions, in general appropriate for purely ingoing waves at the horizon and purely outgoing waves at the asymptotic infinity. The study of QNMs of a black hole is an old and well established subject in physics [5], [6], [7]. Since there is a huge number of references on this subject, we only refer to some comprehensive reviews in Refs. [8], [9], [10].

On the other hand, an old problem in the black hole physics is the presence of spacetime singularity at the center of black holes [11], [12], [13]. In this paper, we study the black holes without singularity known as regular black holes (RBHs) or singularity free black holes. The first kind of RBH space–time in general relativity was proposed by Bardeen [14], and shortly after revived in Refs. [15], [16]. The Bardeen model satisfied the weak energy condition but it was not a vacuum solution of Einstein’s equations, so it was necessary to introduce some external form of matter or a modification of gravity. Ayon-Beato and Garcia (ABG) proposed a new nonlinear electrodynamics which, when coupled to gravity, produces an exact RBH solution that also satisfies the weak energy condition [17], [18], [19]. Subsequently, further analyses of singularity avoidance have been proposed in Refs. [20], [21], [22], [23], [24], [25], [26], [27].

Recently, there has been a revival of interest in alternative theories including RBHs. Studying more realistic configurations, which include coupling terms are very important [28]. This provides strong motivations for investigating the coupling of a scalar field with the Ricci scalar-tensor of RBH space–time geometry. There is a lot of interest in this kind of coupling in many research fields in physics, such as modified scalar-tensor theories (see e.g. [29] and Refs. therein), conformal gravity [30], cosmological models [31], [32], [33], [34]. In particular, we use WKB method to obtain the QNMs of perturbations of a scalar field with such a coupling around Bardeen, Hayward, and ABG black holes. The WKB method, initially proposed in Refs. [35], [36], [37] to obtain the QNMs of Schwarzschild background, provides a simple and powerful tool for studying properties of black holes. Further calculations for the Kerr and Reissner–Nordstrom black holes can be found in Refs. [38], [39], [40], [41]. There are other analytical and numerical methods than WKB to compute the QNMs of a particular black hole perturbations in Refs. [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56].

Using the WKB formula, the QNMs of neutral and charged minimal scalar field perturbations for RBHs have been studied in Ref. [57] and for wormholes and RBH using a phantom scalar field in Ref. [58]. Other calculations about the QNMs of RBHs in the case of minimal couplings could be found in Refs. [59], [60], [61], [62], [63] Perturbations of gravitational and Dirac fields in these backgrounds have also been studied in Refs. [64], [65], [66], [67], [68], [69]. However, these attempts are for theories with no coupling to the geometry, thus the main aim of this paper is to study the QNM spectra due to this kind of coupling. The QNMs are also important in determining some characteristic properties of black holes such as grey-body factors (GF), area quantization (AQ), and shadow radius (SR). These parameters are very important in analysis and verification of the obtained results from [1], [2], [3], [4]. This fact strongly motivates us to investigate the relationship between the QNMs and these quantities in the case of RBHs.

Very long ago, Hawking showed that a black hole can emit thermal radiation if the quantum effects are considered, known as Hawking radiation (HR) [70], [71], [72]. Because of the non-trivial spacetime geometry near the black hole, the initial radiation received by a distant observer will get modified by a coefficient called grey-body factor [73], [74], [75]. We will apply a numerical prescription employing WKB approximation based on [76], [77], [78] to study the behavior of GFs for RBHs in different regimes of parameters. The are also some efforts on the calculation of GFs in coupling of scalar theory with matter fields [79], four-dimensional Gauss–Bonnet gravity in [80], and 2+1-dimensional BTZ black holes in [81], [82], [83], [84].

It was conjectured by Bekenstein [85], [86] that the black hole entropy should be represented by a discrete spectrum in Planck units. That is, the area of a classical black hole behaves like an adiabatic invariant, and so, according to Ehrenfest’s theorem, the corresponding quantum operator must have a discrete spectrum. Moreover, Hod proposed [87] an equally spaced area spectrum and used the apparent existence of a unique QNM frequency in the large damping limit to uniquely fix the spacing. However, Maggiore [88] suggested that a black hole can be viewed as a damped harmonic oscillator whose physically relevant frequency is identical to the complex QNM frequencies having both real and imaginary parts. According to [88], for the highly excited QNMs the imaginary part is dominant over the real part and one can compute the AQ spectrum from imaginary part. We employ this prescription and a near horizon approximation to find equally spaced area spectrum in the case of RBHs in this paper.

Another important phenomenological feature of a black hole which is also closely related to the QNMs, is its SR [54]. The shadow image of the supermassive black hole in the center of M87 galaxy released by EHT [3], [4] greatly stimulated our enthusiasm for the research of RBH’s SR. More importantly, the research in this direction will open a new window to study the strong gravitational region near black hole horizon. The investigation of SR for several black holes have been done in Refs. [89], [90], [91], [92], [93], [94], [95]. There are also some attempts to compute SR of different non-singular black holes in Refs .[96], [97], [98]

The organization of the paper is as follows: In Section 2, we study the RBHs as solutions to the field equations of a non-linear electromagnetic model, which in the weak field limit are reduced to the standard Maxwell’s linear theory. In Section 3, we introduce the non-minimal coupling model and discuss the effective potential appeared in the Schrödinger-like equation of the scalar field dynamics. In Section 4, we use WKB method to derive the QNMs spectra and investigate the effects of physical parameters on the imaginary and real parts of frequencies. We provide a numerical verification on the GFs of HR for different values of RBH’s parameters. We also determine the AQ spectrum from the near horizon consideration and SR in the eikonal limit for this family of black holes. Finally, Section 5 is devoted to a brief summary and concluding remarks.

Section snippets

Regular black holes in non-linear electrodynamics

The Bardeen model [14], as the first RBH model in general relativity, is reinterpreted as the gravitational field of a non-linear magnetic monopole, i.e., as a magnetic solution of Einstein’s field equations coupled to a non-linear electrodynamics [17], [18], [19]. The model is described by the action S=d4xg116πGR14πL(F),where R is the scalar curvature and the Lagrangian of non-linear electrodynamics L(F) as a function of F=14FμνFμν, is given by L(F)=32αq22q2F1+2q2F52.Assume that G=1 and Fμν=

Propagation of a massive scalar field near RBHs

In order to gain insight into the quantum nature of RBHs, the kinematical properties provide relevant clues about their semiclassical aspects. From theoretical point of view, there are two different ways to initiate the perturbation of a black hole; one is by adding external test fields to the black hole geometry and the other is by perturbing the black hole metric itself or gravitational perturbation. The simplest way to study black hole perturbations due to external fields is to study the

QNMs of RBHs from WKB approximation

The WKB approximation is a promising technique for determining the QNM frequencies semi-analytically. It can be used for solving the scattering problem, which is necessary to find GFs of the black hole, and for the calculation of QNMs [105]. The main motivation for using this method is the similarity between the equation of perturbation theory for a particle and the one-dimensional Schrödinger equation for a potential barrier, for instance Eq. (3.6) that we obtained in the previous section. The

Conclusions

In this paper we performed a numerical study for QNMs of a neutral massive scalar field perturbations which is non-minimally coupled to the curvature of the spacetime geometry around some singularity free black holes known as RBHs. That is the case of great interest in the cosmological and modified gravity models. We have studied the most general examples of such spherically symmetric RBHs such as Bardeen, Hayward, and ABG black holes. We have also considered the GFs, AQ and SR as three

CRediT authorship contribution statement

Davood Mahdavian Yekta: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing – original draft, Writing – review & editing. Majid Karimabadi: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing – original draft, Writing – review & editing. S.A. Alavi: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing – original draft, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors, specially M.K. would like to thank R. Konoplya and V. Cardoso for useful discussion. We would also like to acknowledge J. Matyjasek and M. Opala [106] for sharing their Mathematica®notebook with higher order WKB corrections. All authors approved the version of the manuscript to be published.

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