Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics
TL;DR: In this article, the authors considered the problem of a small particle falling in a Schwarzschild background ("black hole") and examined its spectrum in the high-frequency limit, in terms of the traceless transverse tensor harmonics called electric and magnetic by Mathews.
Abstract: We are concerned with the pulse of gravitational radiation given off when a star falls into a "black hole" near the center of our galaxy. We look at the problem of a small particle falling in a Schwarzschild background ("black hole") and examine its spectrum in the high-frequency limit. In formulating the problem it is essential to pose the correct boundary condition: gravitational radiation not only escaping to infinity but also disappearing down the hole. We have examined the problem in the approximation of linear perturbations from a Schwarzschild background geometry, utilizing the decomposition into the tensor spherical harmonics given by Regge and Wheeler (1957) and by Mathews (1962). The falling particle contributes a $\ensuremath{\delta}$-function source term (geodesic motion in the background Schwarzschild geometry) which is also decomposed into tensor harmonics, each of which "drives" the corresponding perturbation harmonic. The power spectrum radiated in infinity is given in the high-frequency approximation in terms of the traceless transverse tensor harmonics called "electric" and "magnetic" by Mathews.
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TL;DR: Quasinormal modes are eigenmodes of dissipative systems as discussed by the authors, and they serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, such as viscosity, conductivity and diffusion constants.
Abstract: Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra require solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.
1,592 citations
TL;DR: The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.
Abstract: Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.
1,569 citations
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TL;DR: In this paper, a review of recent achievements on various aspects of black hole perturbations are discussed such as decoupling of variables in the perturbation equations, quasinormal modes (with special emphasis on various numerical and analytical methods of calculations), late-time tails, gravitational stability, anti-de Sitter/conformal field theory interpretation, and holographic superconductors.
Abstract: Perturbations of black holes, initially considered in the context of possible observations of astrophysical effects, have been studied for the past 10 years in string theory, brane-world models, and quantum gravity. Through the famous gauge/gravity duality, proper oscillations of perturbed black holes, called quasinormal modes, allow for the description of the hydrodynamic regime in the dual finite temperature field theory at strong coupling, which can be used to predict the behavior of quark-gluon plasmas in the nonperturbative regime. On the other hand, the brane-world scenarios assume the existence of extra dimensions in nature, so that multidimensional black holes can be formed in a laboratory experiment. All this stimulated active research in the field of perturbations of higher-dimensional black holes and branes during recent years. In this review recent achievements on various aspects of black hole perturbations are discussed such as decoupling of variables in the perturbation equations, quasinormal modes (with special emphasis on various numerical and analytical methods of calculations), late-time tails, gravitational stability, anti--de Sitter/conformal field theory interpretation of quasinormal modes, and holographic superconductors. We also touch on state-of-the-art observational possibilities for detecting quasinormal modes of black holes.
1,070 citations
TL;DR: In this article, a review of the mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime is presented.
Abstract: This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.
910 citations
TL;DR: In this paper, the gravitational quasi-normal frequencies of both stationary and rotating black holes are calculated by constructing exact eigensolutions to the radiative boundary-value problem of Chandrasekhar and Detweiler.
Abstract: The gravitational quasi-normal frequencies of both stationary and rotating black holes are calculated by constructing exact eigensolutions to the radiative boundary-value problem of Chandrasekhar and Detweiler. The method is that employed by Jaffe in his determination of the electronic spectra of the hydrogen molecule ion in 1934, and analytic representations of the quasi-normal mode wavefunctions are presented here for the first time. Numerical solution of Jaffe’s characteristic equation indicates that for each l -pole there is an infinite number of damped Schwarzschild quasi-normal modes. The real parts of the corresponding frequencies are bounded, but the imaginary parts are not. Figures are presented that illustrate the trajectories the five least-damped of these frequencies trace in the complex frequency plane as the angular momentum of the black hole increases from zero to near the Kerr limit of maximum angular momentum per unit mass, a = M , where there is a coalescence of the more highly damped frequencies to the purely real value of the critical frequency for superradiant scattering.
894 citations
Cites methods from "Gravitational field of a particle f..."
...In this paper the problem of determining the gravitational quasi-normal frequencies is cast, after the manner of Zerilli (1970), Chandrasekhar & Detweiler (1975, 1976), and Detweiler (1977, 1980), in the form of a linearized boundary-value problem on a stationary black hole background....
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...Understanding of covariant wave equations obeyed by ψ has come from studies by Wheeler (1955), Regge & Wheeler (1957), Zerilli (1970), Bardeen & Press (1973), Chandrasekhar (1975), and Chandraskhar & Detweiler (1975)....
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