On the accuracy of the post-Newtonian approximation

TL;DR

This paper uses post-Newtonian methods to accurately locate the innermost circular orbit of binary black holes, showing good convergence and agreement with numerical results, and questions some existing resummation techniques.

Contribution

It demonstrates the effectiveness of third post-Newtonian approximation in predicting the ICO and challenges assumptions about the black hole binary dynamics resembling test particles.

Findings

3PN approximation predicts ICO within 1% accuracy.

Good agreement between 3PN results and numerical calculations.

Post-Newtonian series converges well for comparable masses.

Abstract

We apply standard post-Newtonian methods in general relativity to locate the innermost circular orbit (ICO) of irrotational and corotational binary black-hole systems. We find that the post-Newtonian series converges well when the two masses are comparable. We argue that the result for the ICO which is predicted by the third post-Newtonian (3PN) approximation is likely to be very close to the ``exact'' solution, within 1% of fractional accuracy or better. The 3PN result is also in remarkable agreement with a numerical calculation of the ICO in the case of two corotating black holes moving on exactly circular orbits. The behaviour of the post-Newtonian series suggests that the gravitational dynamics of two bodies of comparable masses does not resemble that of a test particle on a Schwarzschild background. This leads us to question the validity of some post-Newtonian resummation techniques that are based on the idea that the field generated by two black holes is a deformation of the Schwarzschild space-time.