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

TL;DR

This paper develops a systematic post-Newtonian expansion method to analytically solve the Regge-Wheeler equation for gravitational waves in Schwarzschild spacetime, enabling precise calculations of waveforms and luminosity for small-mass binary systems.

Contribution

It introduces a new iterative post-Newtonian method to obtain the ingoing Regge-Wheeler function in closed form, accurate to high post-Newtonian orders, aiding gravitational wave analysis.

Findings

Derived the Regge-Wheeler function to (post)$^{4}$-Newtonian order.

Provided formulas for gravitational wave luminosity including tail effects.

Validated the method's applicability to small-mass binary systems.

Abstract

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.