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

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