# First Law of Compact Binary Mechanics with Gravitational-Wave Tails

## TL;DR

This paper derives a new first law for binary mechanics at 4PN order that includes gravitational-wave tail effects, linking periastron advance to redshift and confirming previous results for circular orbits.

## Contribution

It introduces the first law of binary mechanics at 4PN order with tail effects, connecting orbital parameters to redshift and validating with self-force data.

## Key findings

- Derived the 4PN first law including tail effects.
- Linked periastron advance to redshift in eccentric orbits.
- Confirmed circular orbit results using self-force data.

## 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.