Canonical center and relative coordinates for compact binary systems through second post-Newtonian order

TL;DR

This paper derives explicit expressions for canonical coordinates in compact binary systems up to second post-Newtonian order, clarifying their relations and energy-momentum relations within Einstein's gravity.

Contribution

It provides the first explicit derivation of canonical center and relative coordinates and their inverse relations up to second post-Newtonian order for spinless binaries.

Findings

Explicit formulas for canonical coordinates up to 2PN order

Inverse relations between coordinates and canonical variables

Verification of Lorentz-invariant energy-momentum relation

Abstract

Based on a recent paper by Rothe and Sch\"afer on compact binary systems, explicit expressions for canonical center and relative coordinates in terms of standard canonical coordinates are derived for spinless objects up to second post-Newtonian approximation of Einstein's theory of gravity. The inverse relations, i.e. the dependence of the standard canonical coordinates on the canonical center and relative coordinates, are also given up to the second post-Newtonian approximation. The famous Pythagorean-theorem-type Lorentz-invariant relation between the system's total energy or Hamiltonian squared, the rest energy or mass squared - solely depending on relative coordinates -, and the total linear momentum squared are explicitly shown through second post-Newtonian approximation.