Sixth post-Newtonian nonlocal-in-time dynamics of binary systems

TL;DR

This paper advances the understanding of binary system dynamics at the sixth post-Newtonian order by deriving gauge-invariant measures of nonlocal-in-time effects, including scattering angles and Hamiltonians, with detailed expansions.

Contribution

It provides the first comprehensive gauge-invariant characterization of nonlocal-in-time dynamics at 6PN, including scattering angles and Hamiltonians, with novel expansions and the identification of a simple mass-ratio dependence.

Findings

Computed nonlocal part of scattering angle for hyperbolic motions.

Derived nonlocal part of averaged Hamiltonian for elliptic motions.

Identified a simple mass-ratio dependence in gravitational-wave energy loss.

Abstract

We complete our previous derivation, at the sixth post-Newtonian (6PN) accuracy, of the local-in-time dynamics of a gravitationally interacting two-body system by giving two gauge-invariant characterizations of its complementary nonlocal-in-time dynamics. On the one hand, we compute the nonlocal part of the scattering angle for hyberboliclike motions; and, on the other hand, we compute the nonlocal part of the averaged (Delaunay) Hamiltonian for ellipticlike motions. The former is computed as a large-angular-momentum expansion (given here to next-to-next-to-leading order), while the latter is given as a small-eccentricity expansion (given here to the tenth order). We note the appearance of $\zeta(3)$ in the nonlocal part of the scattering angle. The averaged Hamiltonian for ellipticlike motions then yields two more gauge-invariant observables: the energy and the periastron precession as functions of orbital frequencies. We point out the existence of a hidden simplicity in the mass-ratio dependence of the gravitational-wave energy loss of a two-body system.