Dynamical invariants for general relativistic two-body systems at the third post-Newtonian approximation

TL;DR

This paper derives all dynamical invariants for two-body systems in general relativity at the third post-Newtonian order, providing tools for analyzing orbital dynamics and stability in relativistic regimes.

Contribution

It introduces a method to extract invariants from the 3PN Hamiltonian using contact transformations and radial action variables, advancing the understanding of relativistic two-body dynamics.

Findings

Derived invariants for general orbits in 3PN approximation

Analyzed the special case of circular orbits

Discussed ambiguities in Hamiltonian regularization

Abstract

We extract all the invariants (i.e. all the functions which do not depend on the choice of phase-space coordinates) of the dynamics of two point-masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by showing how a contact transformation can be used to reduce the 3PN higher-order Hamiltonian derived by Jaranowski and Sch\"afer to an ordinary Hamiltonian. The dynamical invariants for general orbits (considered in the center-of-mass frame) are then extracted by computing the radial action variable $\oint{p_r}dr$ as a function of energy and angular momentum. The important case of circular orbits is given special consideration. We discuss in detail the plausible ranges of values of the two quantities $\oms$, $\omk$ which parametrize the existence of ambiguities in the regularization of some of the divergent integrals making up the Hamiltonian. The physical applications of the invariant functions derived here (e.g. to the determination of the location of the last stable circular orbit) are left to subsequent work.