Teukolsky-Starobinsky identities: A novel derivation and generalizations

Fiziev, P. P.

doi:10.1103/physrevd.80.124001

Cited by 52 publications

(38 citation statements)

References 55 publications

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“…Next, we rederive the algebraically special perturbations of Kerr (the TTMs) as examples of finding polynomial solutions of the confluent Heun equation. The fact that some Heun polynomial solutions of the radial equation correspond to the TTMs is not new [29], but we feel it is important to include these examples for completeness. Finally, we make use of these Heun polynomial solutions for the TTMs in the limit that a = 0 to reexamine the existence of QNMs and TTM R s at special frequencies on the NIA.…”

Section: Introductionmentioning

confidence: 99%

Gravitational perturbations of the Kerr geometry: High-accuracy study

2014

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We present results from a new code for computing gravitational perturbations of the Kerr geometry. This new code carefully maintains high precision to allow us to obtain high-accuracy solutions for the gravitational quasinormal modes of the Kerr space-time. Part of this new code is an implementation of a spectral method for solving the angular Teukolsky equation that, to our knowledge, has not been used before for determining quasinormal modes. We focus our attention on two main areas. First, we explore the behavior of these quasinormal modes in the extreme limit of Kerr, where the frequency of certain modes approaches accumulation points on the real axis. We compare our results with recent analytic predictions of the behavior of these modes near the accumulation points and find good agreement. Second, we explore the behavior of solutions of modes that approach the special frequency M ω = −2i in the Schwarzschild limit. Our high-accuracy methods allow us to more closely approach the Schwarzschild limit than was possible with previous numerical studies. Unlike previous work, we find excellent agreement with analytic predictions of the behavior near this special frequency. We include a detailed description of our methods, and make use of the theory of confluent Heun differential equations throughout. In particular, we make use of confluent Heun polynomials to help shed some light on the controversy of the existence, or not, of quasinormal and total-transmission modes at certain special frequencies in the Schwarzschild limit.

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Section: Introductionmentioning

confidence: 99%

Gravitational perturbations of the Kerr geometry: High-accuracy study

2014

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“…The values of the parameters when the BH mass is M = 1/2 and, if we choose |r ∞ | = 20 which turns out to be large enough to simulate numerically the actual infinity, are ( [16,19]):…”

Section: General Form Of the Equationsmentioning

confidence: 99%

“…The indirect approaches like the continued fractions method have some limitations and are not directly related with the physics of the problem. The RWE, the Zerilli equation and TRE, however, can be solved analytically in terms of confluent Heun functions, as done for the first time in [16][17][18][19]. Imposing the boundary conditions on those solutions directly (see [13,17]) one obtains a system of spectral equations (1) and (2) featuring the confluent Heun functions which can be solved numerically.…”

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confidence: 99%

Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes

Fiziev

Staicova

2011

Phys. Rev. D

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Although finding numerically the quasinormal modes of a nonrotating black hole is a well-studied question, the physics of the problem is often hidden behind complicated numerical procedures aimed at avoiding the direct solution of the spectral system in this case. In this article, we use the exact analytical solutions of the Regge-Wheeler equation and the Teukolsky radial equation, written in terms of confluent Heun functions. In both cases, we obtain the quasinormal modes numerically from spectral condition written in terms of the Heun functions. The frequencies are compared with ones already published by Andersson and other authors. A new method of studying the branch cuts in the solutions is presented -the epsilon-method. In particular, we prove that the mode n = 8 is not algebraically special and find its value with more than 6 firm figures of precision for the first time. The stability of that mode is explored using the ǫ method, and the results show that this new method provides a natural way of studying the behavior of the modes around the branch cut points. QUASI-NORMAL MODES OF BLACK HOLESThe study of quasinormal modes (QNMs) of a black hole (BH) has long history [1][2][3][4][5][6][7]. The reason behind this interest is that the QNMs offer a direct way of studying the key features of the physics of compact massive objects, without the complications of the full 3D general relativistic simulations. For example, by comparing the theoretically obtained gravitational QNMs with the frequencies of the gravitational waves, one can confirm or refute the nature of the central engines of many astrophysical objects, since those modes differ for the different types of objects -black holes, superspinars (naked singularities), neutron stars, black hole mimickers etc. [8][9][10][11][12][13].To find the QNMs, one needs to solve the second-order linear differential equations describing the linearized perturbations of the metric: the Regge-Wheeler equation (RWE) and the Zerilli equation for the Schwarzschild metric or the Teukolsky radial equation (TRE) for the Kerr metric and to impose the appropriate boundary conditions -the so-called black hole boundary conditions (waves going simultaneously into the horizon and into infinity) [1,3]. Additionally, one requires a regularity condition for the angular part of the solutions. And then, one needs to solve a connected problem with two complex spectral parameters -the frequency ω and the separation constant E (E = l(l + 1) -real for a nonrotating BH, with l the angular momentum of the perturbation). This Because of the complexity of the differential equations, until now, those equations were solved either approximately or numerically meeting an essential difficulty [1]. The indirect approaches like the continued fractions method have some limitations and are not directly related with the physics of the problem. The RWE, the Zerilli equation and TRE, however, can be solved analytically in terms of confluent Heun functions, as done for the first time in [16][17][18][19]. Imposin...

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“…They showed that in those cases, the new method indeed works better than the standard methods. Therefore, the new method can be readily applied to find the roots of the Regge-Wheeler equation [18], the Zerilli equation [28], the Teukolsky radial and angular equations [24], all of which are solved analytically in terms of confluent Heun functions. Using this algorithm, we were able to solve directly the problem of quasi-normal modes of a Schwarzschild ( [9]) and Kerr black hole ( [10]) with higher precision than that of the Broyden method.…”

Section: Discussionmentioning

confidence: 99%

Solving Systems of Transcendental Equations Involving the Heun Functions

Fiziev¹,

Staicova²

2012

AJCM

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The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Müller algorithm. The new algorithm is particularly useful in systems featuring the Heun functions and for them, the new algorithm gives distinctly better results than Newton's and Broyden's methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.

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Teukolsky-Starobinsky identities: A novel derivation and generalizations

Cited by 52 publications

References 55 publications

Gravitational perturbations of the Kerr geometry: High-accuracy study

Gravitational perturbations of the Kerr geometry: High-accuracy study

Application of the confluent Heun functions for finding the quasinormal modes of nonrotating black holes

Solving Systems of Transcendental Equations Involving the Heun Functions

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