On an expansion method for black hole quasinormal modes and Regge poles
TL;DR: In this paper, a new method for determining the frequencies and wavefunctions of black hole quasinormal modes (QNMs) and Regge poles was proposed. But the method is not suitable for the case of spacetimes of arbitrary spatial dimension.
Abstract: We present a new method for determining the frequencies and wavefunctions of black hole quasinormal modes (QNMs) and Regge poles. The key idea is a novel ansatz for the wavefunction, which relates the high-l wavefunctions to null geodesics which start at infinity and end in perpetual orbit on the photon sphere. Our ansatz leads naturally to the expansion of QNMs in inverse powers of L = l + 1/2 (in 4D), and to the expansion of Regge poles in inverse powers of ω. The expansions can be taken to high orders. We begin by applying the method to the Schwarzschild spacetime, and validate our results against existing numerical and Wentzel–Kramers–Brillouin methods. Next, we generalize the method to treat static spherically symmetric spacetimes of arbitrary spatial dimension. We confirm that, at lowest order, the real and imaginary components of the QNM frequency are related to the orbital frequency and the Lyapunov exponent for geodesics at the unstable orbit. We apply the method to five spacetimes of current interest, and conclude with a discussion of the advantages and limitations of the new approach, and its practical applications.
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TL;DR: In this paper, a review of 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, anti-de Sitter/conformal field theory interpretation, and holographic superconductors.
Abstract: Perturbations of black holes, initially considered in the context of possible observations of astrophysical effects, have been studied for the past 10 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, 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, anti--de Sitter/conformal field theory interpretation of quasinormal modes, and holographic superconductors. We also touch on state-of-the-art observational possibilities for detecting quasinormal modes of black holes.
1,070 citations
TL;DR: In this paper, the authors derived a geometric correspondence between the quasinormal-mode frequencies of Kerr black holes of arbitrary (astrophysical) spins and general spherical photon orbits, analogous to the relationship for slowly rotating holes.
Abstract: There is a well-known, intuitive geometric correspondence between high-frequency quasinormal modes of Schwarzschild black holes and null geodesics that reside on the light ring (often called spherical photon orbits): the real part of the mode’s frequency relates to the geodesic’s orbital frequency, and the imaginary part of the frequency corresponds to the Lyapunov exponent of the orbit. For slowly rotating black holes, the quasinormal mode’s real frequency is a linear combination of the orbit’s precessional and orbital frequencies, but the correspondence is otherwise unchanged. In this paper, we find a relationship between the quasinormal-mode frequencies of Kerr black holes of arbitrary (astrophysical) spins and general spherical photon orbits, which is analogous to the relationship for slowly rotating holes. To derive this result, we first use the Wentzel-Kramers-Brillouin approximation to compute accurate algebraic expressions for large-l quasinormal-mode frequencies. Comparing our Wentzel-Kramers-Brillouin calculation to the leading-order, geometric-optics approximation to scalar-wave propagation in the Kerr spacetime, we then draw a correspondence between the real parts of the parameters of a quasinormal mode and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the quasinormal-mode parameters to coefficients that modify the amplitude of the scalar wave. With this correspondence, we find a geometric interpretation of two features of the quasinormal-mode spectrum of Kerr black holes: First, for Kerr holes rotating near the maximal rate, a large number of modes have nearly zero damping; we connect this characteristic to the fact that a large number of spherical photon orbits approach the horizon in this limit. Second, for black holes of any spins, the frequencies of specific sets of modes are degenerate; we find that this feature arises when the spherical photon orbits corresponding to these modes form closed (as opposed to ergodically winding) curves.
175 citations
TL;DR: The purpose of the current Letter is to give some relations between gravitational lensing in the strong-deflection limit and the frequencies of the quasinormal modes of spherically symmetric, asymptotically flat black holes.
Abstract: The purpose of the current Letter is to give some relations between gravitational lensing in the strong-deflection limit and the frequencies of the quasinormal modes of spherically symmetric, asymptotically flat black holes. On the one side, the relations obtained can give a physical interpretation of the strong-deflection limit parameters. On the other side, they also give an alternative method for the measurement of the frequencies of the quasinormal modes of spherically symmetric, asymptotically flat black holes. They could be applied to the localization of the sources of gravitational waves and could tell us what frequencies of the gravitational waves we could expect from a black hole acting simultaneously as a gravitational lens and a source of gravitational waves.
129 citations
TL;DR: In this article, a thorough analysis of the quasinormal modes of the Bardeen black hole due to scalar perturbations was performed by varying the charge and mass of the field.
Abstract: The purpose of this paper is to study quasinormal modes (QNM) of the Bardeen black hole due to scalar perturbations. We have done a thorough analysis of the QNM frequencies by varying the charge $q$, mass $M$ and the spherical harmonic index $l$. The unstable null geodesics are used to compute the QNM's in the eikonal limit. Furthermore, massive scalar field modes are also studied by varying the mass of the field. Comparisons are done with the QNM frequencies of the Reissner-Nordstrom black hole.
121 citations
TL;DR: In this paper, the Asymptotic Iteration Method (AIM) was used to find radial QNMs for Schwarzschild, Reissner-Nordstrom (RN), and Kerr black holes.
Abstract: We discuss how to obtain black hole quasinormal modes (QNMs) using the asymptotic iteration method (AIM), initially developed to solve second-order ordinary differential equations. We introduce the standard version of this method and present an improvement more suitable for numerical implementation. We demonstrate that the AIM can be used to find radial QNMs for Schwarzschild, Reissner-Nordstrom (RN), and Kerr black holes in a unified way. We discuss some advantages of the AIM over the continued fractions method (CFM). This paper presents for the first time the spin 0, 1/2 and 2 QNMs of a Kerr black hole and the gravitational and electromagnetic QNMs of the RN black hole calculated via the AIM and confirms results previously obtained using the CFM. We also present some new results comparing the AIM to the WKB method. Finally we emphasize that the AIM is well suited to higher-dimensional generalizations and we give an example of doubly rotating black holes.
107 citations
References
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Book•
01 Jan 1978
TL;DR: A self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations is given in this paper.
Abstract: This book gives a self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations. Parts and chapter titles are as follows: fundamentals - ordinary differential equations, difference equations; local analysis - approximate solution of linear differential equations, approximate solution of nonlinear differential equations, approximate solution of difference equations, asymptotic expansion of integrals; perturbation methods - perturbation series, summation series; and global analysis - boundary layer theory, WKB theory, multiple-scale analysis. An appendix of useful formulas is included. 147 figures, 43 tables. (RWR)
4,776 citations
Book•
01 Jan 1983TL;DR: In a course of lectures on the underlying mathematical structures of classical gravitation theory given in 1978, Brandon Carter as discussed by the authors began with the statement ‘If I had been asked five years ago to prepare a course for recent developments in classical gravity theory, I would not have hesitated on the classical theory of black holes as a central topic of discussion. But I am grateful to them for their courtesy in assigning to me this privilege.
Abstract: In a course of lectures on the ‘underlying mathematical structures of classical gravitation theory’ given in 1978, Brandon Carter began with the statement ‘If I had been asked five years ago to prepare a course of lectures on recent developments in classical gravitation theory, I would not have hesitated on the classical theory of black holes as a central topic of discussion. However, the most important developments in gravitational theory during the last three or four years have not been in the classical domain at all…’ Carter is undoubtedly right in his assessment that the mathematical theory of black holes has not been in the mainstream of research in relativity since 1973. I therefore find it difficult to understand why the organizers of this meeting should have chosen precisely this topic for the opening talk of this meeting. But I am grateful to them for their courtesy in assigning to me this privilege.
4,165 citations
Book•
01 Jan 1966
TL;DR: In this paper, the authors present a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect, and the general approach to multiparticle reaction theory.
Abstract: Much progress has been made in scattering theory since the publication of the first edition of this book fifteen years ago, and it is time to update it. Needless to say, it was impossible to incorporate all areas of new develop- ment. Since among the newer books on scattering theory there are three excellent volumes that treat the subject from a much more abstract mathe- matical point of view (Lax and Phillips on electromagnetic scattering, Amrein, Jauch and Sinha, and Reed and Simon on quantum scattering), I have refrained from adding material concerning the abundant new mathe- matical results on time-dependent formulations of scattering theory. The only exception is Dollard's beautiful "scattering into cones" method that connects the physically intuitive and mathematically clean wave-packet description to experimentally accessible scattering rates in a much more satisfactory manner than the older procedure. Areas that have been substantially augmented are the analysis of the three-dimensional Schrodinger equation for non central potentials (in Chapter 10), the general approach to multiparticle reaction theory (in Chapter 16), the specific treatment of three-particle scattering (in Chapter 17), and inverse scattering (in Chapter 20). The additions to Chapter 16 include an introduction to the two-Hilbert space approach, as well as a derivation of general scattering-rate formulas. Chapter 17 now contains a survey of various approaches to the solution of three-particle problems, as well as a discussion of the Efimov effect.
4,044 citations
TL;DR: Quasinormal modes are eigenmodes of dissipative systems as discussed by the authors, and they serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, such as viscosity, conductivity and diffusion constants.
Abstract: Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra require solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.
1,592 citations
TL;DR: The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.
Abstract: Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.
1,569 citations