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

Abstract

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...

Approximative analytic study of fermions in magnetar’s crust; ultra-relativistic plane waves, Heun and Mathieu solutions and beyond

“…For example, the confluent Heun functions have been worked out in the context of the quasinormal modes for nonrotating black holes [Fiziev and Staicova, 2011] or as solutions to Schrödinger equation, for different rational potential functions, in thick braneworlds [Cunha and Christiansen, 2011].…”

“…Thus, because besides leptons, in dense nuclear matter, the nucleons interact among each other through three meson fields: the isoscalar-scalar meson σ, isoscalar vector meson ω and isovector-vector meson ρ, one may assume that currents and charge density produced by the single-particle spinors, as the ones in (22), are actually acting as sources for the Klein-Gordon Gordon equations for the time-and space-like meson fields ∆ − m 2 φ φ = g φ j p ∆ − m 2 φ φ 0 = g φ Q p (38) whereφ = { ω , ρ} and φ 0 = {ω 0 , ρ 0 }, g φ are the respective coupling constants, m φ are the meson masses and (see (21))…”

Heun Functions and Their Uses in Physics

“…o Fiziev is an expert in this topic. Two further articles by him and his collaborator is: Solving systems of transcendental equations involving the Heun functions, [72] and Application of the confluent Heun functions for finding the quasinormal modes of non rotating black holes [73].…”

Solving Systems of Transcendental Equations Involving the Heun Functions

“…This means that the use of the Heun functions is limited to the routines hidden in the kernel of Maple, which the user cannot change or improve-a situation that makes understanding the numerical problems or avoiding them adequately very difficult. On the positive side, those routines were found by the team to work well enough in many cases (see, for example, the match between theory and numerical results in [8], as well as the other applications of those functions in [9,10]). Yet, there are some peculiarities-there are values of the parameters where the routines break down leading to infinities or to numerical errors.…”

“…They occur in the problem of quasi-normal modes (QNM) of rotating and nonrotating black holes, which is to some extent the gravity analogue of the problem of the hydrogen atom. Finding the QNMs is critical to understanding observational data from gravitational wave detectors and proving or refuting the black holes existence ( [11,12] and also [8][9][10]). In this case, one has to solve a two-dimensional connected spectral problem with two complex equations in each of which one encounters the confluent Heun functions.…”