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Journal ArticleDOI

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

TL;DR: In this paper, the exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-like equation to the complex plane and then performing a method of monodromy matching at the several poles in the plane.
Abstract: The exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-like equation to the complex plane and then performing a method of monodromy matching at the several poles in the plane. While this method was successfully used in asymptotically flat spacetime, as applied to both the Schwarzschild and Reissner-Nordstrom solutions, its extension to non-asymptotically flat spacetimes 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 spacetimes, 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 spacetimes.
Citations
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Journal ArticleDOI
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

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this paper, resurgence and transseries are used to encode the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms with its Stokes constants, which are recast in equivalent physical languages: either a statistical mechanical language, as motions in chains and lattices; or a conformal field theoretical language, with underlying Virasoro-like algebraic structures.

211 citations

Journal ArticleDOI
TL;DR: In this article, a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in d dimensions is provided, including all possible types of gravitational perturbations (tensor, vector and scalar type) as described by the Ishibashi-Kodama master equations.
Abstract: 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

199 citations

Journal ArticleDOI
TL;DR: In this article, the singularity of a black hole in the AdS/CFT correspondence was studied and a relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators was established.
Abstract: We study black hole singularities in the AdS/CFT correspondence. These singularities show up in CFT in the behavior of finite-temperature correlation functions. We first establish a direct relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimensions. This allows us to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the exponential falloff of finite-temperature correlators at large imaginary frequency. We construct new gauge invariant observables whose divergences reflect the presence of the singularity. We also find a UV/UV connection that governs physics inside the horizon. Additionally, we comment on the possible resolution of the singularity.

163 citations

References
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Book
01 Jan 1983
TL;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

Journal ArticleDOI
TL;DR: In this article, it was shown that a Schwarzschild singularity, spherically symmetrical and endowed with mass, will undergo small vibrations about the spherical form and therefore remain stable if subjected to a small nonspherical perturbation.
Abstract: It is shown that a Schwarzschild singularity, spherically symmetrical and endowed with mass, will undergo small vibrations about the spherical form and will therefore remain stable if subjected to a small nonspherical perturbation.

2,105 citations

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this paper, the decay of a scalar field outside a Schwarzschild anti-de Sitter black hole was investigated by computing the complex frequencies associated with quasinormal modes.
Abstract: We investigate the decay of a scalar field outside a Schwarzschild anti--de Sitter black hole. This is determined by computing the complex frequencies associated with quasinormal modes. There are qualitative differences from the asymptotically flat case, even in the limit of small black holes. In particular, for a given angular dependence, the decay is always exponential---there are no power law tails at late times. In terms of the AdS-CFT correspondence, a large black hole corresponds to an approximately thermal state in the field theory, and the decay of the scalar field corresponds to the decay of a perturbation of this state. Thus one obtains the time scale for the approach to thermal equilibrium. We compute these time scales for the strongly coupled field theories in three, four, and six dimensions, which are dual to string theory in asymptotically AdS spacetimes.

988 citations


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Journal ArticleDOI
TL;DR: In this paper, the authors used Bohr's correspondence principle to provide a missing link between the Boltzmann-Einstein formula and the area-entropy thermodynamic relation for black holes.
Abstract: During the last twenty-five years evidence has been mounting that a black-hole surface area has a discrete spectrum. Moreover, it is widely believed that area eigenvalues are uniformly spaced. There is, however, no general agreement on the spacing of the levels. In this Letter we use Bohr's correspondence principle to provide this missing link. We conclude that the area spacing of a black hole is $4\ensuremath{\Elzxh}\mathrm{ln}3$. This is the unique spacing consistent both with the area-entropy thermodynamic relation for black holes, with the Boltzmann-Einstein formula in statistical physics, and with Bohr's correspondence principle.

695 citations