Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems

We complete the analytical determination, at the 4th post-Newtonian (4PN) approximation, of the conservative dynamics of gravitationally interacting two-point-mass systems. This completion is obtained by resolving the infra-red ambiguity which had blocked a previous 4PN calculation [ P. Jaranowski and near-zone expansion due to infinite-range tail-transported temporal correlations found long ago [ L. Blanchet and T. Damour, Phys. Rev. D 37, 1410 ( 1988)]. This leads to a Poincare-invariant 4PN-accurate effective action for two masses, which mixes instantaneous interaction terms ( described by a usual Hamiltonian) with a ( time-symmetric) nonlocal-in-time interaction.