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

TL;DR

This work delivers the first analytic 4PN determination of the conservative two-body gravitational interaction potential within the EOB framework. By connecting GSF results (via Detweiler’s redshift) and RWZ perturbation theory, the authors obtain an explicit ν-dependent expansion of the EOB radial potential A(u;ν), including a linear-in-ν term with precisely derived coefficients and a negative ν^2 contribution. They confirm a consistent 4PN structure, compare analytic results to recent numerical inferences, and discuss implications for infrared issues and the modeling of comparable-mass binaries. The results enhance the precision of analytical two-body dynamics and offer cross-validation for numerical relativity calibrations and future higher-PN efforts.

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 Schäfer, and is found to be significantly negative.