The quasi-normal modes of the Schwarzschild black hole

1975 ◽  
Vol 344 (1639) ◽  
pp. 441-452 ◽  
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.

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

1975 ◽  
Vol 345 (1641) ◽  
pp. 145-167 ◽  
Keyword(s):  
Black Hole ◽  
Wave Equation ◽  
Conservation Laws ◽  
Gravitational Waves ◽  
Kerr Black Hole ◽  
One Dimensional ◽  
Potential Barriers ◽  
Transmission Coefficients ◽  
Dimensional Wave ◽  
Axisymmetric Perturbations

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.

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

1977 ◽  
Vol 352 (1670) ◽  
pp. 325-338 ◽  
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.

On the metric perturbations of the Reissner–Nordström black hole

1979 ◽  
Vol 367 (1728) ◽  
pp. 1-14 ◽  
Keyword(s):  
Black Hole ◽  
Wave Equations ◽  
Schwarzschild Black Hole ◽  
One Dimensional ◽  
Dimensional Wave

The two pairs of one-dimensional wave equations which govern the odd and the even-parity perturbations of the Reissner–Nordström black hole are derived directly from a treatment of its metric perturbations. The treatment closely parallels the corresponding treatment in the context of the Schwarzschild black hole.

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

1976 ◽  
Vol 350 (1661) ◽  
pp. 165-174 ◽  
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.

Gravitational Waves

2018 ◽  
Author(s):  
Michele Maggiore
Keyword(s):  
Black Hole ◽  
Perturbation Theory ◽  
Gravitational Waves ◽  
Transfer Functions ◽  
Normal Modes ◽  
Cosmological Perturbation ◽  
Tests Of General Relativity ◽  
Pulsar Timing ◽  
Tensor Perturbations ◽  
Stochastic Backgrounds

A comprehensive and detailed account of the physics of gravitational waves and their role in astrophysics and cosmology. The part on astrophysical sources of gravitational waves includes chapters on GWs from supernovae, neutron stars (neutron star normal modes, CFS instability, r-modes), black-hole perturbation theory (Regge-Wheeler and Zerilli equations, Teukoslky equation for rotating BHs, quasi-normal modes) coalescing compact binaries (effective one-body formalism, numerical relativity), discovery of gravitational waves at the advanced LIGO interferometers (discoveries of GW150914, GW151226, tests of general relativity, astrophysical implications), supermassive black holes (supermassive black-hole binaries, EMRI, relevance for LISA and pulsar timing arrays). The part on gravitational waves and cosmology include discussions of FRW cosmology, cosmological perturbation theory (helicity decomposition, scalar and tensor perturbations, Bardeen variables, power spectra, transfer functions for scalar and tensor modes), the effects of GWs on the Cosmic Microwave Background (ISW effect, CMB polarization, E and B modes), inflation (amplification of vacuum fluctuations, quantum fields in curved space, generation of scalar and tensor perturbations, Mukhanov-Sasaki equation,reheating, preheating), stochastic backgrounds of cosmological origin (phase transitions, cosmic strings, alternatives to inflation, bounds on primordial GWs) and search of stochastic backgrounds with Pulsar Timing Arrays (PTA).

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

scholarly journals Black hole S-matrix for a scalar field

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Panos Betzios ◽  
Nava Gaddam ◽  
Olga Papadoulaki
Keyword(s):  
Black Hole ◽  
Normal Modes ◽  
Massless Scalar ◽  
Scattering Process ◽  
Schwarzschild Black Hole ◽  
Asymptotically Flat ◽  
Scalar Particles ◽  
Black Hole Background ◽  
S Matrix ◽  
Do So

Abstract We describe a unitary scattering process, as observed from spatial infinity, of massless scalar particles on an asymptotically flat Schwarzschild black hole background. In order to do so, we split the problem in two different regimes governing the dynamics of the scattering process. The first describes the evolution of the modes in the region away from the horizon and can be analysed in terms of the effective Regge-Wheeler potential. In the near horizon region, where the Regge-Wheeler potential becomes insignificant, the WKB geometric optics approximation of Hawking’s is replaced by the near-horizon gravitational scattering matrix that captures non-perturbative soft graviton exchanges near the horizon. We perform an appropriate matching for the scattering solutions of these two dynamical problems and compute the resulting Bogoliubov relations, that combines both dynamics. This allows us to formulate an S-matrix for the scattering process that is manifestly unitary. We discuss the analogue of the (quasi)-normal modes in this setup and the emergence of gravitational echoes that follow an original burst of radiation as the excited black hole relaxes to equilibrium.

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