  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.
