Living Reviews in Relativity

"Analytic Black Hole Perturbation Approach to Gravitational Radiation"
by
Misao Sasaki and Hideyuki Tagoshi  

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Abstract
1 Introduction
1.1 General
1.2 Post-Newtonian expansion of gravitational waves
1.3 Linear perturbation theory of black holes
1.4 Brief historical notes
2 Basic Formulae for the Black Hole Perturbation
2.1 Teukolsky formalism
2.2 Chandrasekhar–Sasaki–Nakamura transformation
3 Post-Newtonian Expansion of the Regge–Wheeler Equation
3.1 Basic assumptions
3.2 Horizon solution; z ≪ 1
3.3 Outer solution; 𝜖 ≪ 1
3.4 More on the inner boundary condition of the outer solution
3.5 Structure of the ingoing wave function to 2 𝒪 (𝜖 )
4 Analytic Solutions of the Homogeneous Teukolsky Equation by Means of the Series Expansion of Special Functions
4.1 Angular eigenvalue
4.2 Horizon solution in series of hypergeometric functions
4.3 Outer solution as a series of Coulomb wave functions
4.4 Matching of horizon and outer solutions
4.5 Low frequency expansion of the hypergeometric expansion
4.6 Property of ν
5 Gravitational Waves from a Particle Orbiting a Black Hole
5.1 Circular orbit around a Schwarzschild black hole
5.2 Circular orbit on the equatorial plane around a Kerr black hole
5.3 Waveforms in the case of circular orbit
5.4 Slightly eccentric orbit around a Schwarzschild black hole
5.5 Slightly eccentric orbit around a Kerr black hole
5.6 Circular orbit with a small inclination from the equatorial plane around a Kerr black hole
5.7 Absorption of gravitational waves by a black hole
5.8 Adiabatic evolution of Carter constant for orbit with small eccentricity and small inclination angle around a Kerr black hole
5.9 Adiabatic evolution of constants of motion for orbits with generic inclination angle and with small eccentricity around a Kerr black hole
6 Conclusion
7 Acknowledgements
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