The present article reveals important properties of the confluent Heun's functions. We derive a set of novel relations for confluent Heun's functions and their derivatives of arbitrary order. Specific new subclasses of confluent Heun's functions are introduced and studied. A new alternative derivation of confluent Heun's polynomials is presented.
The well-known Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in the linear approximation. From a mathematical point of view it presents a particular case of the confluent Heun equation and can be solved exactly, due to recent mathematical developments. We present the basic properties of its general solution. A novel analytical approach and numerical techniques for study the boundary problems which correspond to quasi-normal modes of black holes and other simple models of compact objects are developed.
The Teukolsky Master Equation is the basic tool for study of perturbations of the Kerr metric in linear approximation. It admits separation of variables, thus yielding the Teukolsky Radial Equation and the Teukolsky Angular Equation. We present here a unified description of all classes of exact solutions to these equations in terms of the confluent Heun functions. Large classes of new exact solutions are found and classified with respect to their characteristic properties. Special attention is paid to the polynomial solutions which are singular ones and introduce collimated one-way-running waves. It is shown that a proper linear combination of such solutions can present bounded one-wayrunning waves. This type of waves may be suitable as models of the observed astrophysical jets.(1.2b)
The Regge-Wheeler equation describes axial perturbations of Schwarzschild metric in linear approximation. Teukolsky Master Equation describes perturbations of Kerr metric in the same approximation. We present here unified description of all classes of exact solutions to these equations in terms of the confluent Heun's functions. Special attention is paid to the polynomial solutions, which yield novel applications of Teukolsky Master Equation for description of relativistic jets and astrophysical explosions. IntroductionAt present the study of different type of perturbations of the gravitational field of black holes (BH), neutron stars (NS) and other compact astrophysical objects is a very active field for analytical, numerical, experimental and astrophysical research. Ongoing and nearest future experiments, based on perturbative and/or numerical analysis of relativistic gravitational dynamics, are expected to provide critical tests of the existing theories of gravity [1].In the last five years the sensitivity of the operating detectors for gravitational waves LIGO, VIRGO, GEO, TAMMA has been improving at a formidable rate and one may expect the first direct observation of gravitational waves in the nearest future. Note that the existing observational results, collected in the last five years, give only limitations on the number of the BH-BH, BH-NS and NS-NS mergers. Since such mergers are still not observed, this number seems to be below the optimistic theoretical expectations, announced some eight years ago 1 . The number of real mergers is believed to be consistent with the recent theoretical and observational constraints [2]. The already started large projects like advanced LIGO and especially LISA, hopefully will bring into being the gravitational wave astronomy in the next decade. Thus we are expecting to discover new fundamental physics.Another outstanding physical problem is presented by the gamma ray bursts (GRB) -the most powerful explosions in our universe after the Big Bang and the relativistic jets, related with them, as well as with other astrophysical objects. Due to recent developments of gamma ray astronomy in space missions SWIFT, Chandra, Huble Space Telescope, Spitzer, HETTE-2, BeppoSAX, and AGILE, together with ground observations by ESA and many other observatories, we already have very good observational data, which is still waiting for adequate theoretical explanation. The recently started Fermi/GLAST mission will give us more complete and precise data in the nearest future. Concerning the theoretical situation one has to stress that the existing theoretical models of GRB do not give a clear and acceptable explanation of the observational facts [3]. Moreover, there is a kind of crisis in this area, since the observational data seem to contradict the old models of central engine of long GRB. In addition, the presence of BH in short GRB was recently refuted by the existing detectors of gravitational waves [4].New physical effects, due to the rotation of the gravitational field described by ge...
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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