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

TL;DR

The authors develop a framework to obtain dynamical invariants for the conservative two-body problem at 3PN in general relativity by reducing the 3PN ADM Hamiltonian to an ordinary Hamiltonian via a contact transformation and then computing the radial action $i_r(E,j)$. They present the full set of invariants for generic bound orbits and also specialize to circular motion, deriving explicit expressions for energy, angular momentum, and frequency relations in terms of the Delaunay variables, while clarifying how the 3PN regularization ambiguities coalesce into a single combination $\sigma(\nu)$. The invariants are expressed through a detailed 3PN radial-action expansion with coefficients $i_1(\nu)$, $i_2(\nu)$, $i_3(\nu)$, and $i_4(\nu)$ and the ambiguity-dependent terms, enabling 3PN phasing and comparison with other coordinate schemes. The work also analyzes the impact of the Wilson-Mathews truncation and highlights the potential importance of the circular-orbit ambiguities for observable quantities such as the last stable circular orbit.

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 Schäfer 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.