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The radial differential equations associated with separable perturbations of Kerr-Newman black holes are known to admit Liouvillian (closed-form) solutions for constrained frequencies and black hole parameters. In this paper, we show that the parameter constraints are satisfied exactly in the case of no rotation and thereby obtain a countable infinity of exact purely damped quasi-normal modes of fields on a Reissner-Nordstrom background at special values of the black hole charge-mass ratio. We show that with rotation the parameter constraints for Liouvillian quasi-normal modes are satisfied approximately in two distinct physical scenarios, where analytical approximations for angular eigenvalues are known. We arrive at functional expressions for quasi-normal frequencies and wave-functions in the case of near-extremal slow rotation and in a particular case of highly damped scalar modes of Kerr and Kerr-Newman. In the near-extremal case, our formulas extend a recent result of Hod to electromagnetic and gravitational perturbations. [ABSTRACT FROM AUTHOR]

We identify a countable infinity of new exact, closed-form, quasinormal mode perturbations of Reissner–Nordstrom black holes. We obtain a finite number of these modes explicitly, together with the values of the quasinormal frequency and the black hole charge for which the modes are valid. These modes are contained in the Liouvillian perturbations obtained from the application of Kovacic’s well-known algorithm to Chandrasekhar’s radial equations. Our results suggest that the set of quasinormal modes found in this paper, plus the known algebraically special perturbations, are the only Liouvillian quasinormal modes of Reissner–Nordstrom. [ABSTRACT FROM AUTHOR]

We apply the well-known Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of Reissner-Nordstrom, and specialized perturbations of the Kerr-Newman geometry. We also include the extreme geometries. We find all frequencies ω, in terms of black hole parameters and an integer n, which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closed-form perturbations of Kerr and Reissner-Nordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests. [ABSTRACT FROM AUTHOR]

Methods from the author’s previous work and the classes of solutions which they produce are extended to the wave equations which govern the massive scalar field and the massless spin- 1/2 field on the Kerr–Newman geometry and the massless fields of spin 1 and 2 on the Kerr geometry. The solutions found are exact and expressed in simple closed forms in terms of elementary functions, but they only exist when appropriate constraints hold on some of the black hole parameters and on the frequency of the field in some cases. The behavior on the horizon, at null infinity, and with respect to the angular variables is analyzed for some example solutions. For the examples studied, it is found that the ones having radial behavior of a normal mode are not free of angular singularities. An exact relation is established between the scalar wave equation on the extreme Kerr–Newman geometry and the Whittaker–Hill equation. [ABSTRACT FROM AUTHOR]

A recent paper by Gottlieb [J. Math. Phys. 29, 2434 (1988)] provides examples of acoustic wave equations, in various dimensions, that have nontrivial families of solutions that are progressing waves of order 1, and relates this to whether or not these equations satisfy Huygens’ principle. A statement made in that paper related to Huygens’ principle in one space dimension is clarified, and it is shown in this connection that, in general, the relationship between the possession of progressing wave solutions and the satisfaction of Huygens’ principle is more complex than the situation described by Gottlieb. In addition, the attractive properties of progressing waves of order 1 are retained by progressing waves of any finite order, and we use this to generalize in several ways Gottlieb’s results on ‘‘wake-free’’ solutions of the acoustic equation in three dimensions. [ABSTRACT FROM AUTHOR]