First Law of Mechanics for Compact Binaries on Eccentric Orbits

TL;DR

This work generalizes the first law of binary mechanics to nonspinning compact binaries on generic eccentric orbits within the ADM-Hamiltonian framework, deriving it via orbital averaging and action-angle methods. It provides a robust set of consequences, including PDEs and a first-integral relation for $M$, $L$, $R$, and redshift data, and validates the results against 3PN PN calculations and circular-orbit limits. The paper then outlines concrete applications, notably using gravitational self-force data to inform PN and EOB models for eccentric binaries, including strong-field regimes, separatrix behavior, and noncircular EOB dynamics through constrained potentials $a(u)$, $\bar d(u)$, and $q(u)$, with key invariants like the redshift $\langle z_a \rangle$ guiding the connections. Overall, the framework strengthens the link between PN theory, GSF, and EOB for eccentricbinary systems, with implications for waveform modeling in gravitational-wave astronomy.

Abstract

Using the canonical Arnowitt-Deser-Misner Hamiltonian formalism, a "first law of mechanics" is established for binary systems of point masses moving along generic stable bound (eccentric) orbits. This relationship is checked to hold within the post-Newtonian approximation to general relativity, up to third (3PN) order. Several applications are discussed, including the use of gravitational self-force results to inform post-Newtonian theory and the effective one-body model for eccentric-orbit compact binaries.