Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation

TL;DR

This paper analytically determines the main radial potential for two-body gravitational interactions at the 4th post-Newtonian level, resolving divergencies and providing new insights into the potential's dependence on mass ratio.

Contribution

It provides the first complete analytical calculation of the 4PN radial potential, including the nu-dependent coefficient, using gravitational self-force methods and recent results.

Findings

The nu-independent part of the potential is obtained analytically.

The nu-dependent part of the potential is significantly negative.

Infrared divergencies are resolved through self-force calculations.

Abstract

We complete the analytical determination, at the 4th post-Newtonian approximation, of the main radial potential describing the gravitational interaction of two bodies within the effective one-body formalism. The (non logarithmic) coefficient a_5 (nu) measuring this 4th post-Newtonian interaction potential is found to be linear in the symmetric mass ratio nu. Its nu-independent part a_5 (0) is obtained by an analytical gravitational self-force calculation that unambiguously resolves the formal infrared divergencies which currently impede its direct post-Newtonian calculation. Its nu-linear part a_5 (nu) - a_5 (0) is deduced from recent results of Jaranowski and Sch\"afer, and is found to be significantly negative.