Equivalence between the ADM-Hamiltonian and the harmonic-coordinates approaches to the third post-Newtonian dynamics of compact binaries

TL;DR

This work establishes the physical equivalence of the conservative 3PN two-body dynamics derived in ADM Hamiltonian formalism and in harmonic coordinates by constructing an explicit mapping between the variable sets. It shows that the two results coincide up to a single physical ambiguity, relating the ADM static parameter $\omega_s$ to the harmonic parameter $\lambda$ via $\lambda = -\frac{3}{11}\omega_s - \frac{1987}{3080}$, and confirms the known kinetic ambiguity $\omega_k = 41/24$. The authors transfer the ADM 3PN Lagrangian and the conserved Poincaré quantities into the harmonic framework and derive a generalized Lagrangian linear in accelerations for the harmonic description. They also discuss the origin of regularization ambiguities, the role of logarithmic terms, and the persistent challenge of the static ambiguity in modeling compact bodies with point masses.

Abstract

The third post-Newtonian approximation to the general relativistic dynamics of two point-mass systems has been recently derived by two independent groups, using different approaches, and different coordinate systems. By explicitly exhibiting the map between the variables used in the two approaches we prove their physical equivalence. Our map allows one to transfer all the known results of the Arnowitt-Deser-Misner (ADM) approach to the harmonic-coordinates one: in particular, it gives the value of the harmonic-coordinates Lagrangian, and the expression of the ten conserved quantities associated to global Poincaré invariance.