Resonances of Schwarzschild Black Holes

Bachelot, Alain; Motet-Bachelot, A.

doi:10.1007/978-3-322-87871-7_5

Cited by 42 publications

(74 citation statements)

References 6 publications

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“…Furthermore, Leaver's general method has been discussed briefly and it has been shown that the infinite fraction approach breaks down for the special mode; this could presumably be the source of the discrepancy in the assignment of mode number to the special QNM solution. This issue as well as a detailed comparison of our results with those of other authors [12][13][14][15][16][17][18][19][20][21][22] would require further investigation.…”

Section: Discussionmentioning

confidence: 69%

“…The QNMs correspond to the poles of the amplitude for scattering of waves from black-hole barrier potentials. It has not been possible, in general, to find explicit analytic expressions for the QNM eigenfrequencies in closed form; therefore, various methods have been developed [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] for the approximate determination of these eigenfrequencies. These methods are generally based on the imposition of QNM boundary conditions on the solution of the wave equation such that the waves are outgoing at infinity and ingoing at the horizon; in this way, results have been obtained that are in general agreement with each other.…”

Section: Discussionmentioning

confidence: 99%

“…The inverted black-hole potential has an infinite number of bound states for j > 0; these states are further discussed in appendix A. It has been shown [21] that the number of QNMs is countably infinite as well. The mode number n = 0, 1, 2, .…”

Section: H Liu and B Mashhoonmentioning

confidence: 99%

“…Theoretical studies of black-hole formation using simplified models have revealed that in the late stages of collapse, the gravitational radiation emitted by the collapsing configuration would contain quasi-normal mode (QNM) oscillations characteristic of the black hole that is being formed [6]. The recognition of the intrinsic significance of QNMs following these studies has led to the development of numerical [7] and analytic [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] methods for the determination of QNM frequencies. Let us note, for the sake of concreteness, that the least-damped quadrupolar vibrations of a Schwarzschild black hole of mass M result in gravitational radiation of frequency ω 0 /2π ≈ 10 4 (M /M) Hz and damping constant −1 ≈ 6 × 10 −5 (M/M ) s.…”

Section: Introductionmentioning

confidence: 99%

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On the spectrum of oscillations of a Schwarzschild black hole

Liu

Mashhoon²

1996

Class. Quantum Grav.

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Abstract. The spectral properties of the resonant oscillations of a Schwarzschild black hole are studied. The analogy with the Coulomb wave equation is developed, and it is shown that the high-overtone quasi-normal modes are intimately connected with the long-range nature of the gravitational interaction. This is followed by an explicit construction of the quasi-normal eigenfunctions in analogy with the bound states of the inverted black-hole potentials. The special gravitational quasi-normal mode is discussed and its relationship with the algebraically special solution of the gravitational perturbations is clarified.

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Section: Discussionmentioning

confidence: 69%

Section: Discussionmentioning

confidence: 99%

Section: H Liu and B Mashhoonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the spectrum of oscillations of a Schwarzschild black hole

Liu

Mashhoon²

1996

Class. Quantum Grav.

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“…. ., characterizing oscillations with decreasing relaxation times (increasing imaginary part) [9,10]. On the other hand, the real part of the frequencies approaches an asymptotic constant value.…”

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confidence: 99%

Kerr black-hole quasinormal frequencies

Hod

2003

Phys. Rev. D

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Black-hole quasinormal modes (QNM) have been the subject of much recent attention, with the hope that these oscillation frequencies may shed some light on the elusive theory of quantum gravity. We compare numerical results for the QNM spectrum of the (rotating) Kerr black hole with an exact formula Reω → TBH ln 3 + Ωm, which is based on Bohr's correspondence principle. We find a close agreement between the two. Possible implications of this result to the area spectrum of quantum black holes are discussed.Gravitational waves emitted by a perturbed black hole are dominated by 'quasinormal ringing', damped oscillations with a discrete spectrum [1]. At late times, all perturbations are radiated away in a manner reminiscent of the last pure dying tones of a ringing bell [2][3][4][5]. The quasinormal mode frequencies (ringing frequencies) are the characteristic 'sound' of the black hole itself, depending on its parameters (mass, charge, and angular momentum).The free oscillations of a black hole are governed by the well-known Regge-Wheeler equation [6] in the case of a Schwarzschild black hole, and by the Teukolsky equation [7] for the (rotating) Kerr black hole. The black hole QNM correspond to solutions of the wave equations with the physical boundary conditions of purely outgoing waves at spatial infinity and purely ingoing waves crossing the event horizon [8]. Such boundary conditions single out discrete solutions ω (assuming a time dependence of the form e iωt ).The ringing frequencies are located in the complex frequency plane characterized by Imω > 0. It turns out that for a given angular harmonic index l there exist an infinite number of quasinormal modes, for n = 0, 1, 2, . . ., characterizing oscillations with decreasing relaxation times (increasing imaginary part) [9,10]. On the other hand, the real part of the frequencies approaches an asymptotic constant value.The QNM frequencies, being a signature of the blackhole spacetime are of great importance from the astrophysical point of view. They allow a direct way of identifying the spacetime parameters (especially, the mass and angular momentum of the central black hole). This has motivated a flurry of activity with the aim of computing the spectrum of oscillations (see e.g., [1] for a detailed review).Recently, the quasinormal frequencies of black holes have acquired a different importance [11][12][13][14][15][16][17] in the context of Loop Quantum Gravity, a viable approach to the quantization of General Relativity (see e.g., [18,19] and references therein). These recent studies are motivated by an earlier work of Hod [20]. Few years ago I proposed to use Bohr's correspondence principle in order to determine the value of the fundamental area unit in a quantum theory of gravity.To understand the original argument it is useful to recall that in the early development of quantum mechanics, Bohr suggested a correspondence between classical and quantum properties of the Hydrogen atom, namely that "transition frequencies at large quantum numbers should equal class...

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Analytic Calculation of Quasi-Normal Modes

Siopsis

Physics of Black Holes

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We discuss the analytic calculation of quasi-normal modes of various types of perturbations of black holes both in asymptotically flat and anti-de Sitter spaces. We obtain asymptotic expressions and also show how corrections can be calculated perturbatively. We pay special attention to lowfrequency modes in anti-de Sitter space because they govern the hydrodynamic properties of a gauge theory fluid according to the AdS/CFT correspondence. The latter may have experimental consequencies for the quark-gluon plasma formed in heavy ion collisions.

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Resonances of Schwarzschild Black Holes

Cited by 42 publications

References 6 publications

On the spectrum of oscillations of a Schwarzschild black hole

On the spectrum of oscillations of a Schwarzschild black hole

Kerr black-hole quasinormal frequencies

Analytic Calculation of Quasi-Normal Modes

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