New method for exact results on quasinormal modes of black holes

In a recent series of papers, we have shown how the eikonal/geometrical optics approximation can be used to calculate analytically the fundamental quasinormal mode frequencies associated with coupled systems of wave equations, which arise, for instance, in the study of perturbations of black holes in gravity theories beyond General Relativity. As a continuation to this series, we focus here on the quasinormal modes of nonrotating black holes in scalar Gauss-Bonnet gravity assuming a small-coupling expansion. We show that the axial perturbations are purely tensorial and are described by a modified Regge-Wheeler equation, while the polar perturbations are of mixed scalar-tensor character and are described by a system of two coupled wave equations. When applied to these equations, the eikonal machinery leads to axial quasinormal modes that deviate from the general relativistic results at quadratic order in the Gauss-Bonnet coupling constant. We show that this result is in agreement with an analysis of unstable circular null orbits around black holes in this theory, allowing us to establish the geometrical optics--null geodesic correspondence for the axial quasinormal modes. For the polar quasinormal modes, the small-coupling approximation forces us to consider the ordering between eikonal and small-coupling perturbative parameters, one of which we show, by explicit comparison against numerical data, yields the correct identification of the quasinormal modes of the scalar-tensor coupled system of wave equations. These corrections lift the general relativistic degeneracy between scalar and tensorial eikonal quasinormal modes at quadratic order in Gauss-Bonnet coupling in a way reminiscent of the Zeeman effect. In general, our analytic, eikonal quasinormal mode frequencies (normalized to the General Relativity ones) agree with numerical results with an error of $\mathcal{O}(10%)$ in the regime of small coupling constant. Finally, we find that the analytical expressions for the quasinormal modes are common to a broad class of scalar-Gauss-Bonnet theories to leading eikonal order, showing a degeneracy between the quasinormal modes of nonrotating black holes in particular scalar-Gauss-Bonnet theories in the geometrical optics limit.

The quasinormal modes (QNMs) of a charged regular black hole are calculated in the eikonal approximation. In the eikonal limit, the QNMs of the black hole are determined by the parameters of unstable circular null geodesics. The behavior of the QNMs are compared with the QNMs of a Reisner–Nordström black hole by fixing some of the parameters that characterize the black holes and varying others. We observed that the parameter that is related to the effective cosmological constant at small distances determines the behavior of the QNMs of a regular charged black hole.

Much of our physical intuition about black hole quasinormal modes in general relativity comes from the eikonal/geometric optics approximation. According to the well-established eikonal model, the fundamental quasinormal mode represents wavepackets orbiting in the vicinity of the black hole's geodesic photon ring, slowly peeling off towards the event horizon and infinity. Besides its strength as a "visualisation" tool, the eikonal approximation also provides a simple quantitative method for calculating the mode frequency, in close agreement with rigorous numerical results. In this paper we move away from Einstein's theory and its garden-variety black holes and go on to consider spherically symmetric black holes in modified theories of gravity through the lens of the eikonal approximation. The quasinormal modes of such black holes are typically described by a set of coupled wave equations for the various field degrees of freedom. Considering a general, theory-agnostic, system of two equations for two perturbed fields, we derive eikonal formulae for the complex fundamental quasinormal mode frequency. In addition we show that the eikonal modes can be related to the extremum of an effective potential and its associated "photon ring". As an application of our results we consider a specific example of a modified theory of gravity with known black hole quasinormal modes and find that these are well approximated by the eikonal formulae.

The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schr\"odinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations, without resorting to a second-order reformulation. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.

Quasi-normal modes of black holes determine the damping of disturbances at intermediate times and are important in studying the dynamics of black holes and external fields around them. Currently, interest in quasi-normal modes is due to three of their features. The first one is connected with the possibility of observing quasi-normal modes and obtaining a "trace" of a black hole with the help of new-generation gravitational antennas under construction. Recently, collaborations between the Laser Interferometric Gravitational Wave Observatory (LIGO) and the French-Italian gravitational wave detector located at the European Gravitational Observatory (VIRGO) reported the observation of a gravitational wave signal corresponding to the spiral and merging of two black holes, resulting in the formation of a single black hole. It was shown that the observations agree with Einstein's theory of gravity with a high accuracy, limited mainly by a statistical error. The second concerns the anti-de Sitter / Conformal field theory correspondence, which implies that a large black hole in anti-de Sitter space corresponds roughly to thermal states in Conformal field theory. Thus, the damping of black hole perturbations can be associated with the return to thermal equilibrium of the perturbed state in the Conformal field theory. Note that in anti-de Sitter space, the quasi-normal modes of black holes control the decay of the field at later times, since there are no power-law tails and the decay is always exponential. The third feature is associated with the possible connection of quasinormal modes of black holes in some space-time geometries with the Choptyuk scaling. This paper investigates the low-lying frequencies of the quasi-normal modes of the Taub-NUT (Newman-Unti-Tamburino) black hole for scalar, electromagnetic, and Dirac perturbations using the Wenzel, Kramers, and Brillouin (WKB) approximation of the third order. The influence of the NUT metric parameter on the considered types of frequencies of quasi-normal modes is shown.

Quasinormal modes of black holes with non-linear-electrodynamic sources in Rastall gravity

Dark matter density can be significantly enhanced by the supermassive black hole at the galactic center, leading to a structure called dark matter spike. Dark matter spike may change the spacetime properties of black holes that constitute deviations from GR black holes. Based on these interesting background, we construct a set of solutions of black holes in a dark matter spike under the Newtonian approximation and full relativity. Combining the mass model of M87, we study the quasinormal modes of black holes in the scalar field and axial gravitational perturbation, then compared them with Schwarzschild black hole. Besides, the impacts of dark matter on the quasinormal mode of black holes have been studied in depth. In particular, in the axial gravitational perturbation, our results show that the impacts of dark matter spike on the quasinormal mode of black holes can reach up to 10-4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-4}$$\\end{document}. These new features from quasinormal mode of black holes under the Newtonian approximation and full relativity may provide some help for the establishment of the final dark matter model, and provide a new thought for the indirect detection of dark matter.

We elucidate that a distinctive resonant excitation between quasinormal modes (QNMs) of black holes emerges as a universal phenomenon at an avoided crossing near the exceptional point through high-precision numerical analysis and theory of QNMs based on the framework of non-Hermitian physics. This resonant phenomenon not only allows us to decipher a long-standing mystery concerning the peculiar behaviors of QNMs but also stands as a novel beacon for characterizing black hole spacetime geometry. Our findings pave the way for rigorous examinations of black holes and the exploration of new physics in gravity.

Perturbations of black holes, initially considered in the context of possible observations of astrophysical effects, have been studied for the past ten years in string theory, brane-world models and quantum gravity. Through the famous gauge/gravity duality, proper oscillations of perturbed black holes, called quasinormal modes (QNMs), allow for the description of the hydrodynamic regime in the dual finite temperature field theory at strong coupling, which can be used to predict the behavior of quark-gluon plasmas in the nonperturbative regime. On the other hand, the brane-world scenarios assume the existence of extra dimensions in nature, so that multidimensional black holes can be formed in a laboratory experiment. All this stimulated active research in the field of perturbations of higher-dimensional black holes and branes during recent years. In this review recent achievements on various aspects of black hole perturbations are discussed such as decoupling of variables in the perturbation equations, quasinormal modes (with special emphasis on various numerical and analytical methods of calculations), late-time tails, gravitational stability, AdS/CFT interpretation of quasinormal modes, and holographic superconductors. We also touch on state-of-the-art observational possibilities for detecting quasinormal modes of black holes.

The physical interpretation of black hole's quasinormal modes is fundamental for realizing unitary quantum gravity theory as black holes are considered theoretical laboratories for testing models of such an ultimate theory and their quasinormal modes are natural candidates for an interpretation in terms of quantum levels. The spectrum of black hole's quasinormal modes can be re-analysed by introducing a black hole's effective temperature which takes into account the fact that, as shown by Parikh and Wilczek, the radiation spectrum cannot be strictly thermal. This issue changes in a fundamental way the physical understanding of such a spectrum and enables a re-examination of various results in the literature which realizes important modifies on quantum physics of black holes. In particular, the formula of the horizon's area quantization and the number of quanta of area result modified becoming functions of the quantum "overtone" number n. Consequently, the famous formula of Bekenstein-Hawking entropy, its sub-leading corrections and the number of microstates are also modified. Black hole's entropy results a function of the quantum overtone number too. We emphasize that this is the first time that black hole's entropy is directly connected with a quantum number. Previous results in the literature are re-obtained in the limit n \to \infty.

Using series solutions and time-domain evolutions, we probe the eikonal limit of the gravitational and scalar-field quasinormal modes of large black holes and black branes in anti-de Sitter backgrounds. These results are particularly relevant for the AdS/CFT correspondence, since the eikonal regime is characterized by the existence of long-lived modes which (presumably) dominate the decay time scale of the perturbations. We confirm all the main qualitative features of these slowly damped modes as predicted by Festuccia and Liu [G. Festuccia and H. Liu, arXiv:0811.1033.] for the scalar-field (tensor-type gravitational) fluctuations. However, quantitatively we find dimensional-dependent correction factors. We also investigate the dependence of the quasinormal mode frequencies on the horizon radius of the black hole (brane) and the angular momentum (wave number) of vector- and scalar-type gravitational perturbations.

The ringdown phase of gravitational waves emitted by a perturbed black hole is described by a superposition of exponentially decaying sinusoidal modes, called quasinormal modes (QNMs), whose frequencies depend only on the property of the black-hole geometry. The extraction of QNM frequencies of an isolated black hole would allow for testing how well the black hole is described by general relativity. However, astrophysical black holes are not isolated. It remains unclear whether the extra matter surrounding the black holes such as accretion disks would affect the validity of the black-hole spectroscopy when the gravitational effects of the disks are taken into account. In this paper, we study the QNMs of a Schwarzschild black hole superposed with a gravitating thin disk. Considering up to the first order of the mass ratio between the disk and the black hole, we find that the existence of the disk would decrease the oscillating frequency and the decay rate. In addition, within the parameter space where the disk model can be regarded as physical, there seems to be a universal relation that the QNM frequencies tend to obey. The relation, if it holds generically, would assist in disentangling the QNM shifts caused by the disk contributions from those induced by other putative effects beyond general relativity. The QNMs in the eikonal limit, as well as their correspondence with bound photon orbits in this model, are briefly discussed.

We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an "infrared" effect; (ii) the instability of all overtones under small-scale ("ultraviolet") perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert's pioneer work and Nollert and Price's analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the P\"oschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.

Quasinormal modes have played a prominent role in the discussion of perturbations of black holes, and the question arises whether they are as significant as normal modes are for self-adjoint systems, such as harmonic oscillators. They can be significant in two ways: Individual modes may dominate the time evolution of some perturbation, and a whole set of them could be used to completely describe this time evolution. It is known that quasinormal modes of black holes have the first property, but not the second. It has recently been suggested that a discontinuity in the underlying system would make the corresponding set of quasinormal modes complete. We therefore turn the Regge-Wheeler potential, which describes perturbations of Schwarzschild black holes, into a series of step potentials, hoping to obtain a set of quasinormal modes which shows both of the above properties. This hope proves to be futile, though: The resulting set of modes appears to be complete, but it no longer contains any individual mode which is directly obvious in the time evolution of initial data. Even worse, the quasinormal frequencies obtained in this way seem to be extremely sensitive to very small changes in the underlying potential. The question arises whether, and how, it is possible to make any definite statements about the significance of quasinormal modes of black holes at all, and whether it could be possible to obtain a set of quasinormal modes with the desired properties in another way. \textcopyright{} 1996 The American Physical Society.

Quasinormal modes are the modes of a wave propagating in the spacetime around the black hole. The definition is the wave solution that satisfies certain boundary conditions, i.e. only ingoing at the horizon and the outgoing at the infinity. According to these boundary conditions, the corresponding frequencies namely, quasinormal frequencies, are allowed to be a discrete set of complex num- ber. These yield damping modes to the wave solution. Practically, quasinormal frequencies can be numerically obtained by solving the SchrÄodinger-like equation under particular boundary conditions. However, many previous works have sug- gested the possibility to determine these frequencies analytically. Therefore in this thesis, we mainly aim to investigate quasinormal modes of black holes in various dimensions by using analytical method. For three dimensional cases, quasi- normal frequencies of BTZ and rotating BTZ solution are calculated. A large three dimensional AdS Schwarzschild black hole are explored and its quasinormal modes are also obtained. Then, a massive scalar perturbation on four dimensional Schwarzschild metric are numerically investigated. For five dimensions, first order perturbation is applied for the study of quasinormal modes of a five dimensional AdS Schwarzschild background. Ultimately, we have proposed semi-analytic calculation for the quasinormal frequencies of a rotating Kaluza-Klein black hole with squashed horizons.