Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

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.

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Quasinormal modes of black holes and black branes

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.

Quasinormal modes of black holes: From astrophysics to string theory

Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically anti-de Sitter spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite temperature field theory. The quasinormal modes are defined for gauge-invariant quantities which obey an incoming-wave boundary condition at the horizon and a Dirichlet condition at the boundary. As an application, we explicitly find poles of retarded correlation functions of R-symmetry currents and the energy-momentum tensor in strongly coupled finite temperature N=4 supersymmetric SU(Nc) Yang-Mills theory in the limit of large Nc.

Quasinormal modes and holography

In 1982, the Belgian Pilots' Guild raised the question of what effect exposure to radar radiation--for example, that encountered in passing a pilot launch's radar--might have on the human body. Recapitulating investigations of this question, this article states that in 25 years of experience with radar, there have been no known incidents of pilots being affected by radar waves. In the future, however, involvement by some pilots with Vessel Traffic Service shore-based radar could affect pilots somewhat differently from limited exposure to pilot launch radar. Pilots who find themselves in new working conditions close to an emitting source should exercise care all times.

About heat waves

We provide a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in d dimensions This includes all possible types of gravitational perturbations (tensor, vector and scalar type) as described by the Ishibashi–Kodama master equations The frequencies for Schwarzschild are dimension independent, while for Reissner–Nordstrom are dimension dependent (the extremal Reissner–Nordstrom case must be considered separately from the non–extremal case) For Schwarzschild de Sitter, there is a dimension independent formula for the frequencies, except in dimension d = 5 where the formula is different For Reissner–Nordstrom de Sitter there is a dimension dependent formula for the frequencies, except in dimension d = 5 where the formula is different Schwarzschild and Reissner–Nordstrom Anti–de Sitter black hole spacetimes are simpler: the formulae for the frequencies will depend upon a parameter related to the tortoise coordinate at spatial infinity, and scalar type perturbations in dimension d = 5 lead to a continuous spectrum for the quasinormal frequencies We also address non–black hole spacetimes, such as pure de Sitter spacetime—where there are quasinormal modes only in odd dimensions—and pure Anti–de Sitter spacetime—where again scalar type perturbations in dimension d = 5 lead to a continuous spectrum for the normal frequencies Our results match previous numerical calculations with great accuracy Asymptotic quasinormal frequencies have also been applied in the framework of quantum gravity for black holes Our results show that it is only in the simple Schwarzschild case which is possible to obtain sensible results concerning area quantization or loop quantum gravity In an effort to keep this paper self–contained we also review earlier results in the literature

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2004/0008/0006/ATMP-2004-0008-0006-a004.pdf

On the classification of asymptotic quasinormal frequencies for $d$-dimensional black holes and quantum gravity

We provide a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in d dimensions. This includes all possible types of gravitational perturbations (tensor, vector and scalar type) as described by the Ishibashi-Kodama master equations. The frequencies for Schwarzschild are dimension independent, while for RN are dimension dependent (the extremal RN case must be considered separately from the non-extremal case). For Schwarzschild dS, there is a dimension independent formula for the frequencies, except in dimension d=5 where the formula is different. For RN dS there is a dimension dependent formula for the frequencies, except in dimension d=5 where the formula is different. Schwarzschild and RN AdS black hole spacetimes are simpler: the formulae for the frequencies will depend upon a parameter related to the tortoise coordinate at spatial infinity, and scalar type perturbations in dimension d=5 lead to a continuous spectrum for the quasinormal frequencies. We also address non-black hole spacetimes, such as pure dS spacetime--where there are quasinormal modes only in odd dimensions--and pure AdS spacetime--where again scalar type perturbations in dimension d=5 lead to a continuous spectrum for the normal frequencies. Our results match previous numerical calculations with great accuracy. Asymptotic quasinormal frequencies have also been applied in the framework of quantum gravity for black holes. Our results show that it is only in the simple Schwarzschild case which is possible to obtain sensible results concerning area quantization or loop quantum gravity. In an effort to keep this paper self-contained we also review earlier results in the literature.

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

Gravitational greybody factors are analytically computed for static, spherically symmetric black holes in d-dimensions, including black holes with charge and in the presence of a cosmological constant (where a proper definition of greybody factors for both asymptotically de Sitter and Anti-de Sitter spacetimes is provided). This calculation includes both the low-energy case—where the frequency of the scattered wave is small and real—and the asymptotic case—where the frequency of the scattered wave is very large along the imaginary axis—addressing gravitational perturbations as described by the Ishibashi-Kodama master equations, and yielding full transmission and reflection scattering coefficients for all considered spacetime geometries. At low frequencies a general method is developed, which can be employed for all three types of spacetime asymptotics, and which is independent of the details of the black hole. For asymptotically de Sitter black holes the greybody factor is different for even or odd spacetime dimension, and proportional to the ratio of the areas of the event and cosmological horizons. For asymptotically Anti-de Sitter black holes the greybody factor has a rich structure in which there are several critical frequencies where it equals either one (pure transmission) or zero (pure reflection, with these frequencies corresponding to the normal modes of pure Anti-de Sitter spacetime). At asymptotic frequencies the computation of the greybody factor uses a technique inspired by monodromy matching, and some universality is hidden in the transmission and reflection coefficients. For either charged or asymptotically de Sitter black holes the greybody factors are given by non-trivial functions, while for asymptotically Anti-de Sitter black holes the greybody factor precisely equals one (corresponding to pure blackbody emission).

https://www.math.tecnico.ulisboa.pt/~jnatar/talks/greytalk.pdf

Greybody Factors for d-Dimensional Black Holes

The exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-type equation to the complex plane and then performing a method of monodromy matching at several poles in the plane. While this method was successfully used in asymptotically flat space–time, as applied to both the Schwarzschild and Reissner–Nordstro/m solutions, its extension to nonasymptotically flat space–times has not been achieved yet. In this work it is shown how to extend the method to this case, with the explicit analysis of Schwarzschild–de Sitter and large Schwarzschild–anti–de Sitter black holes, both in four dimensions. We obtain, for the first time, analytic expressions for the asymptotic quasinormal frequencies of these black hole space–times, and our results match previous numerical calculations with great accuracy. We also list some results concerning the general classification of asymptotic quasinormal frequencies in d-dimensional space–times.

Asymptotic quasinormal frequencies for black holes in nonasymptotically flat space¿times

We prove that (possibly charged) test fields satisfying the null energy condition at the event horizon cannot overspin/overcharge extremal Kerr–Newman or Kerr–Newman–anti de Sitter black holes, that is, the weak cosmic censorship conjecture cannot be violated in the test field approximation. The argument relies on black hole thermodynamics (without assuming cosmic censorship), and does not depend on the precise nature of the fields. We also discuss generalizations of this result to other extremal black holes.

José Natário

Papers

On the classification of asymptotic quasinormal frequencies for $d$-dimensional black holes and quantum gravity

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

Greybody Factors for d-Dimensional Black Holes

Asymptotic quasinormal frequencies for black holes in nonasymptotically flat space¿times

Test fields cannot destroy extremal black holes