Post-Newtonian expansion of gravitational waves from a particle in circular orbit around a Schwarzschild black hole

TL;DR

This paper analytically derives the post-Newtonian expansion of gravitational waves emitted by a test particle in circular orbit around a Schwarzschild black hole, providing precise coefficients up to high order for gravitational wave modeling.

Contribution

It presents the first analytical calculation of gravitational waveforms and luminosity coefficients up to 8th order beyond Newtonian for a test particle in Schwarzschild spacetime, including exact logarithmic term coefficients.

Findings

Derived gravitational waveforms up to v^8 order.

Confirmed analytical coefficients match previous numerical results.

Provides analytical benchmarks for gravitational wave physics of binary systems.

Abstract

Based upon the formalism recently developed by one of us (MS), we analytically perform the post-Newtonian expansion of gravitational waves from a test particle in circular orbit of radius $r_0$ around a Schwarzschild black hole of mass $M$. We calculate gravitational wave forms and luminosity up to $v^8$ order beyond Newtonian, where $v=(M/r_0)^{1/2}$. In particular, we give the exact analytical values of the coefficients of $\ln v$ terms at $v^6$ and $v^8$ orders in the luminosity and confirm the numerical values obtained previously by the other of us (HT) and Nakamura. Our result is valid in the small mass limit of one body and gives an important guideline for the gravitational wave physics of coalescing compact binaries.