3 Post-Newtonian Expansion of the Regge–Wheeler Equation

In this section, we review a post-Newtonian expansion method for the Schwarzschild background, based on the Regge–Wheeler equation. We focus on the gravitational waves emitted to infinity, but not on those absorbed by the black hole. The black hole absorption is deferred to Section 4, in which we review the Mano–Suzuki–Takasugi method for solving the Teukolsky equation.

Since we are interested in the waves emitted to infinity, as seen from Equation (25View Equation), what we need is a method to evaluate the ingoing wave Teukolsky function in R ℓm ω, or its counterpart in the Regge–Wheeler equation, in X ℓmω, which are related by Equation (59View Equation). In addition, we assume ω > 0 whenever it is necessary throughout this section. Formulae and equations for ω < 0 are obtained from the symmetry X¯inℓmω = Xinℓ−m −ω.

 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 𝒪(𝜖2)

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