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

TL;DR

This paper derives a third post-Newtonian Lagrangian for compact binaries, identifies conserved quantities, and proves its physical equivalence to the ADM-Hamiltonian formalism, clarifying the relationship between two approaches in gravitational dynamics.

Contribution

It provides a Lagrangian formulation at 3PN order, computes conserved quantities, and establishes the equivalence between harmonic-coordinate and ADM-Hamiltonian formalisms.

Findings

Derived the 3PN Lagrangian depending on positions, velocities, and accelerations.

Computed Noetherian conserved quantities associated with Poincaré invariance.

Proved the physical equivalence of harmonic-coordinate and ADM-Hamiltonian formalisms at 3PN.

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 \lambda reflects 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 Poincar\'e 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 Sch\"afer.