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

We present a method of post-Newtonian expansion to solve the homogeneousRegge-Wheeler equation which describes gravitational waves on the Schwarzschildspacetime. The advantage of our method is that it allows a systematic iterativeanalysis of the solution. Then we obtain the Regge-Wheeler function which ispurely ingoing at the horizon in closed analytic form, with accuracy requiredto determine the gravitational wave luminosity to (post)$^{4}$-Newtonian order(i.e., order $v^8$ beyond Newtonian) from a particle orbiting around aSchwarzschild black hole. Our result, valid in the small-mass limit of onebody, gives an important guideline for the study of coalescing compactbinaries. In particular, it provides basic formulas to analytically calculatedetailed waveforms and luminosity, including the tail terms to(post)$^3$-Newtonian order, which should be reproduced in any otherpost-Newtonian calculations.Comment: 31 pages, KUNS 124