First Law of Compact Binary Mechanics with Gravitational-Wave Tails

TL;DR

This paper extends the first law of binary mechanics to include the non-local tail effects that appear at 4PN order in the conservative two-body dynamics. It derives how tail-induced non-locality modifies the radial action and energy–angular momentum balance, and proves that the redshift z_a remains the physically meaningful quantity via a Fokker-Lagrangian approach. Using these results, it relates the periastron advance to the averaged redshift in the circular limit and confirms consistency with existing 4PN circular-orbit results through self-force data. The work provides a consistent framework linking tail effects, redshift, and orbital precession, with cross-checks against ADM and EOB formalisms and numerical results.

Abstract

We derive the first law of binary point-particle mechanics for generic bound (i.e. eccentric) orbits at the fourth post-Newtonian (4PN) order, accounting for the non-locality in time of the dynamics due to the occurence of a gravitational-wave tail effect at that order. Using this first law, we show how the periastron advance of the binary system can be related to the averaged redshift of one of the two bodies for a slightly non-circular orbit, in the limit where the eccentricity vanishes. Combining this expression with existing analytical self-force results for the averaged redshift, we recover the known 4PN expression for the circular-orbit periastron advance, to linear order in the mass ratio.