Analytic determination of the eight-and-a-half post-Newtonian self-force contributions to the two-body gravitational interaction potential

TL;DR

This work extends the analytic determination of the gauge-invariant two-body interaction potential in the EOB framework to 8.5PN order at linear order in the symmetric mass ratio $\nu$, by applying gravitational self-force theory to Detweiler's redshift variable. It delivers explicit analytic expressions for the $\nu$-linear PN coefficients $a_{1\rm SF}(u)$ at 7.5PN–9.5PN, reveals an increasing transcendental content including $\zeta(3)$ and $\gamma^2$, and provides a detailed mapping to Detweiler's function $u_1^t(u)$ via the coefficients $\alpha_n$, $\beta_n$, and $\gamma_n$, with several new analytic results and validation against Shah 2013 results. The paper also translates these analytic insights into higher-order PN terms for $a_{1\rm SF}(u)$ using Shah’s numerical data, and critically analyzes the convergence of the PN expansion in the strong-field regime by comparing with Akcay 2012ea results, highlighting the limitations of PN near the light ring. The findings underscore both the power and limits of analytic GSF methods for two-body dynamics and point to potential gains from second-order GSF analyses and from leveraging analytic structure to extract higher PN coefficients from numerical data.

Abstract

We {\it analytically} compute, to the eight-and-a-half post-Newtonian order, and to linear order in the mass ratio, the radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies, thereby extending previous analytic results. These results are obtained by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. We emphasize the increase in \lq\lq transcendentality" of the numbers entering the post-Newtonian expansion coefficients as the order increases, in particular we note the appearance of $ζ(3)$ (as well as the square of Euler's constant $γ$) starting at the seventh post-Newtonian order. We study the convergence of the post-Newtonian expansion as the expansion parameter $u=GM/(c^2r)$ leaves the weak-field domain $u\ll 1$ to enter the strong field domain $u=O(1)$.

Figures (1)

• Figure 1: The plot compares, on the interval $\frac{1}{6} \le u\le \frac{1}{3}$, the successive PN approximants of the quantity $\hat{a}_E(u)=a(u)/(2u^3 E(u))$ with $E(u)=(1-2u)/\sqrt{1-3u}$, to its "exact" numerical value. There are fifteen curves in the figure: fourteen PN approximants (from 3PN to 10.5 PN), and the fit model #14 of numerical relativity data in Ref. Akcay:2012ea. The curves are distinguishable from their values at $u=\frac{1}{3}$, where they are ordered from bottom to top as follows: 8PN, 5PN, 3PN, 5.5PN, 4PN, 9.5 PN, 8.5PN, 6.5PN, 9PN, 6PN, 7.5PN, 7PN, 10.5PN, model #14 (dotted curve), 10PN.