Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms

TL;DR

The paper derives a conservative 3PN Lagrangian in harmonic coordinates for compact binary dynamics, and computes ten Noetherian conserved quantities, revealing that the energy depends on a physical parameter $\lambda$ while two gauge constants $r'_1$ and $r'_2$ are removable by a coordinate change. It then constructs a unique contact transformation, including a function $F$ and counter-term $X_A^i$, that converts the harmonic-coordinate dynamics to an ordinary ADM-type Lagrangian, thereby establishing exact equivalence with the ADM Hamiltonian formalism of Damour, Jaranowski, and Schäfer. The equivalence requires a specific relation between the two formalisms, $\omega_{\rm static} = -\tfrac{11}{3}\lambda - \tfrac{1987}{840}$, and fixes the kinetic-ambiguity $\omega_{\rm kinetic}=\tfrac{41}{24}$; the logarithmic terms are gauge and disappear in the ADM framework. Overall, the work reconciles the harmonic-coordinate and ADM approaches to 3PN binary dynamics and provides explicit expressions for use in high-precision gravitational-wave templates.

Abstract

A Lagrangian from which derive the third post-Newtonian (3PN) equations of motion of compact binaries (neglecting the radiation reaction damping) is obtained. The 3PN equations of motion were computed previously by Blanchet and Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate positions, velocities and accelerations of the two bodies. At the 3PN order, the appearance of one undetermined physical parameter λreflects an incompleteness of the point-mass regularization used when deriving the equations of motion. In addition the Lagrangian involves two unphysical (gauge-dependent) constants r'_1 and r'_2 parametrizing some logarithmic terms. The expressions of the ten Noetherian conserved quantities, associated with the invariance of the Lagrangian under the Poincaré group, are computed. By performing an infinitesimal ``contact'' transformation of the motion, we prove that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and Schäfer.