On the equations governing the axisymmetric perturbations of the Kerr black hole

It is shown how Teukolsky’s equation, governing the perturbations of the Kerr black hole, can be reduced, in the axisymmetric case, to a one-dimensional wave equation with four possible potentials. The potentials are implicitly, dependent on the frequency; and besides, depending on circumstances, they can be complex. In all cases (i.e. whether or not the potentials are real or complex), the problem of the reflexion and the transmission of gravitational waves by the potential barriers can be formulated, consistently, with the known conservation laws. It is, further, shown that all four potentials lead to the same reflexion and transmission coefficients.

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On the reflexion and transmission of neutrino waves by a Kerr black hole

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1977.0002 ◽

1977 ◽

Vol 352 (1670) ◽

pp. 325-338 ◽

Cited By ~ 20

Keyword(s):

Black Hole ◽

Wave Equation ◽

Gravitational Waves ◽

Kerr Black Hole ◽

One Dimensional ◽

Potential Barriers ◽

Two Component ◽

Reflexion Coefficient ◽

Dimensional Wave

The equations governing the two-component neutrino are reduced to the form of a one-dimensional wave equation. And it is shown how the absence of super-radiance (i. e. a reflexion coefficient in excess of one) for incident neutrino waves and its manifestation for incident electromagnetic and gravitational waves (of suitable frequencies) emerge very naturally from the character of the respective potential barriers that surround the Kerr black hole.

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The quasi-normal modes of the Schwarzschild black hole

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1975.0112 ◽

1975 ◽

Vol 344 (1639) ◽

pp. 441-452 ◽

Cited By ~ 352

Keyword(s):

Black Hole ◽

Wave Equation ◽

Gravitational Waves ◽

Short Range ◽

Numerical Solutions ◽

Normal Modes ◽

Schwarzschild Black Hole ◽

One Dimensional ◽

Potential Barriers ◽

Dimensional Wave

The quasi-normal modes of a black hole represent solutions of the relevant perturbation equations which satisfy the boundary conditions appropriate for purely outgoing (gravitational) waves at infinity and purely ingoing waves at the horizon. For the Schwarzschild black hole the problem reduces to one of finding such solutions for a one-dimensional wave equation (Zerilli’s equation) for a potential which is positive everywhere and is of short-range. The notion of quasi-normal modes of such one-dimensional potential barriers is examined with two illustrative examples; and numerical solutions for Zerilli’s potential are obtained by integrating the associated Riccati equation.

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On the equations governing the gravitational perturbations of the Kerr black hole

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1976.0101 ◽

1976 ◽

Vol 350 (1661) ◽

pp. 165-174 ◽

Cited By ~ 41

Keyword(s):

Black Hole ◽

Wave Equation ◽

Kerr Black Hole ◽

Radial Equation ◽

One Dimensional ◽

Gravitational Perturbations ◽

Dimensional Wave

Teukolsky’s radial equation governing the general, non-axisymmetric, gravitational perturbations of the Kerr black hole is reduced to the form of a one-dimensional wave equation by making use of the transformation which enables the treatment of the non-axisymmetric modes in exactly the same way as the axisymmetric modes.

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On algebraically special perturbations of black holes

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1984.0021 ◽

1984 ◽

Vol 392 (1802) ◽

pp. 1-13 ◽

Cited By ~ 76

Keyword(s):

Black Holes ◽

Black Hole ◽

Gravitational Waves ◽

Special Class ◽

Mathematical Theory ◽

Kerr Black Hole ◽

Potential Barriers

Algebraically special perturbations of black holes excite gravitational waves that are either purely ingoing or purely outgoing. Solutions, appropriate to such perturbations of the Kerr, the Schwarzschild, and the Reissner-Nordström black-holes, are obtained in explicit forms by different methods. The different methods illustrate the remarkable inner relations among different facets of the mathematical theory. In the context of the Kerr black-hole they derive from the different ways in which the explicit value of the Starobinsky constant emerges, and in the context of the Schwarzschild and the Reissner-Nordström black-holes they derive from the potential barriers surrounding them belonging to a special class.

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