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

TL;DR

The paper resolves the IR ambiguities that blocked a complete $4$PN conservative two-body dynamics by identifying the tail-induced time-nonlocal correlations that spoil purely near-zone expansions. Using a reduced Fokker-type action derived via an ADM-based (ADMTT) formalism and carefully regularizing both UV and IR divergences, the authors obtain a Poincaré-invariant $4$PN Hamiltonian that splits into local and nonlocal-in-time tail terms. The nonlocal piece is matched with the near-zone contribution, and the remaining undetermined constant is fixed by comparing circular-orbit energetics to the effective-one-body radial potential, yielding a final, fully determined $H_{ ext{4PN}}$ with explicit local and nonlocal structure. This work clarifies the role of tail effects at $4$PN, provides a robust analytical foundation for high-precision binary dynamics, and informs future efforts in waveform modeling and EOB formulations.

Abstract

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 G.Schäfer, Phys. Rev. D 87, 081503(R) (2013)] by taking into account the 4PN breakdown of the usual 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 Poincaré-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.