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

Abstract

Based on a recent paper by Rothe and Schäfer on compact binary systems (Rothe and Schäfer 2010 J. Math. Phys. 51 082501), explicit expressions for canonical center and relative coordinates in terms of standard canonical coordinates are derived for spinless objects up to second post-Newtonian (PN) approximation of Einstein’s theory of gravity (the third post-Newtonian order expressions are available in the form of supplementary data). 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 PN 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 PN approximation.