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

TL;DR

This paper demonstrates the physical equivalence between the ADM-Hamiltonian and harmonic-coordinates approaches to third post-Newtonian dynamics of compact binaries by explicitly mapping variables and transferring known results.

Contribution

It provides the explicit map between the two approaches, enabling transfer of results and confirming their physical equivalence at the third post-Newtonian order.

Findings

Established the explicit variable map between the two approaches

Transferred the ADM results to harmonic coordinates, including the Lagrangian and conserved quantities

Confirmed the physical equivalence of the two methods at 3PN order

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 Poincar\'e invariance.