Post-Newtonian Expansion of the Ingoing-Wave Regge-Wheeler Function

We present a method of post-Newtonian expansion to solve the homogeneous Regge-Wheeler equation which describes gravitational waves on the Schwarzschild spacetime. The advantage of our method is that it allows a systematic iterative analysis of the solution. Then we obtain the Regge-Wheeler function which is purely ingoing at the horizon in closed analytic form, with accuracy required to determine the gravitational wave luminosity to (post)$^{4}$-Newtonian order (i.e., order $v^8$ beyond Newtonian) from a particle orbiting around a Schwarzschild black hole. Our result, valid in the small-mass limit of one body, gives an important guideline for the study of coalescing compact binaries. In particular, it provides basic formulas to analytically calculate detailed waveforms and luminosity, including the tail terms to (post)$^3$-Newtonian order, which should be reproduced in any other post-Newtonian calculations.