High-Accuracy Comparison between the Post-Newtonian and Self-Force Dynamics of Black-Hole Binaries

TL;DR

This paper compares high-precision predictions of binary black-hole dynamics from post-Newtonian and self-force methods, validating their consistency and improving the understanding of gravitational interactions in strong fields.

Contribution

It provides a detailed, high-order comparison between PN and SF approaches, including new PN coefficients up to 7PN and validation of regularization techniques.

Findings

PN and SF results agree within high precision

New PN coefficients measured up to 7PN order

Validation of regularization methods in both formalisms

Abstract

The relativistic motion of a compact binary system moving in circular orbit is investigated using the post-Newtonian (PN) approximation and the perturbative self-force (SF) formalism. A particular gauge-invariant observable quantity is computed as a function of the binary's orbital frequency. The conservative effect induced by the gravitational SF is obtained numerically with high precision, and compared to the PN prediction developed to high order. The PN calculation involves the computation of the 3PN regularized metric at the location of the particle. Its divergent self-field is regularized by means of dimensional regularization. The poles proportional to 1/(d-3) which occur within dimensional regularization at the 3PN order disappear from the final gauge-invariant result. The leading 4PN and next-to-leading 5PN conservative logarithmic contributions originating from gravitational-wave tails are also obtained. Making use of these exact PN results, some previously unknown PN coefficients are measured up to the very high 7PN order by fitting to the numerical self-force data. Using just the 2PN and new logarithmic terms, the value of the 3PN coefficient is also confirmed numerically with very high precision. The consistency of this cross-cultural comparison provides a crucial test of the very different regularization methods used in both SF and PN formalisms, and illustrates the complementarity of these approximation schemes when modelling compact binary systems.