The First Law of Binary Black Hole Mechanics in General Relativity and Post-Newtonian Theory

TL;DR

This paper derives a first law for binary point-mass systems in general relativity using post-Newtonian analysis, connecting it with black hole mechanics and providing high-precision results for PN coefficients.

Contribution

It introduces a novel first law for binary point-masses in general relativity derived from first principles, extending black hole mechanics concepts to PN binary systems.

Findings

Derived a first law for binary systems at 3PN order

Established a relation between binding energy and redshift observable

Determined high-order PN coefficients with high precision

Abstract

First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are explored for PN spacetimes. As one such consequence, a simple relation between the PN binding energy of the binary system and Detweiler's redshift observable is established. Through it, we are able to determine with high precision the numerical values of some previously unknown high order PN coefficients in the circular-orbit binding energy. Finally, we propose new gauge invariant notions for the energy and angular momentum of a particle in a binary system.