Detweiler's gauge-invariant redshift variable: analytic determination of the nine and nine-and-a-half post-Newtonian self-force contributions

TL;DR

This work addresses the problem of determining higher-order post-Newtonian corrections to Detweiler's gauge-invariant redshift for the first-order gravitational self-force. It advances the analytic PN program by deriving exact expressions for the 9PN and 9.5PN contributions to the redshift function $U_1^t(u)$ using Regge-Wheeler-Zerilli and MST techniques, including logarithmic and transcendental constants. The authors validate key coefficients against Shah, Friedman and Whiting's MST-based results, confirming agreement for essential terms and consistency with numerical estimates for others. Additionally, the results are translatable into the EOB formalism, updating the linear-in-mass-ratio coefficients of the radial potential $A(u;\nu)$ and thereby enhancing high-PN accuracy for gravitational two-body modeling and waveform predictions.

Abstract

Continuing our analytic computation of the first-order self-force contribution to Detweiler's redshift variable we provide the exact expressions of the ninth and ninth-and-a-half post-Newtonian terms.