Exact solutions of Regge–Wheeler equation and quasi-normal modes of compact objects

Fiziev, P. P.

doi:10.1088/0264-9381/23/7/015

Cited by 98 publications

(157 citation statements)

References 29 publications

(34 reference statements)

Supporting

Mentioning

148

Contrasting

Order By: Relevance

“…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.…”

Section: Quasi-normal Modes Of Black Holesmentioning

confidence: 99%

“…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.In this article, for the first time we present finding l and ω directly in the case for gravitational perturbation s = −2 in a Schwarzschild metric, i.e. we solve the RWE and TRE analytically in terms of confluent Heun functions and we use a newly developed method (the two-dimensional generalization of the Müller method described in the internal technical report [20]) to solve the system of two transcendental equations with two complex variables.…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Fiziev

Staicova

2011

Phys. Rev. D

View full text Add to dashboard Cite

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...

show abstract

Section: Quasi-normal Modes Of Black Holesmentioning

confidence: 99%

mentioning

confidence: 99%

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

Fiziev

Staicova

2011

Phys. Rev. D

View full text Add to dashboard Cite

show abstract

“…o In a paper published in gr-qc/0603003, he studied the exact solutions of the Regge-Wheeler equation in the Schwarschild black hole interior [60]. o Fiziev is an expert in this topic.…”

Section: Some Examples Of the Heun Equation In Physical Applicationsmentioning

confidence: 99%

Heun Functions and Their Uses in Physics

Hortaçsu

2012

Mathematical Physics

View full text Add to dashboard Cite

Most of the theoretical physics known today is described using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2010. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. An integral transform solution using simpler functions also is not obtainable. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and few times updated version of the conference talk by the same author, published in

show abstract

“…The one-dimensional Müller algorithm ( [17]) is a quadratic generalization of the secant method, that works well in the case of a complex function of one variable. It has very good convergence for a large class of functions (~1.84) and it is very efficient when the starting point (the initial guess) is close to a root (for applications see [18] and [8]). It is also well convergent when working with special transcendental functions such as the Heun functions.…”

Section: Introductionmentioning

confidence: 99%

Solving Systems of Transcendental Equations Involving the Heun Functions

Fiziev¹,

Staicova²

2012

AJCM

View full text Add to dashboard Cite

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.

show abstract

Exact solutions of Regge–Wheeler equation and quasi-normal modes of compact objects

Cited by 98 publications

References 29 publications

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

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

Heun Functions and Their Uses in Physics

Solving Systems of Transcendental Equations Involving the Heun Functions

Contact Info

Product

Resources

About