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

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.

Fokker action of nonspinning compact binaries at the fourth post-Newtonian approximation

“…Part of the difference is due to the fact that we disagree with the treatment of the tail part of the Hamiltonian for circular orbits in Ref. [67]. However, even when using our own treatment of tails in their results, there still remains a discrepancy with the works [61][62][63]67] that we cannot resolve.…”

“…However, trying to relate our result to the result obtained from the ADM Hamiltonian approach [61][62][63]67], we find a difference with the latter works, occuring at orders G 4 and G 5 in the Hamiltonian. Part of the difference is due to the fact that we disagree with the treatment of the tail part of the Hamiltonian for circular orbits in Ref.…”

“…More recently, the important effect of gravitational wave tails at 4PN order [65,66] was included in the ADM Hamiltonian. This permitted to understand the IR divergencies in this calculation and to complete the 4PN dynamics [67] (see also [68]). Notice however that the latter work [67] did not perform a full consistent PN analysis but resorted to an auxiliary self-force calculation [69][70][71] to fix a last coefficient.…”

“…We show that the IR divergencies are perfectly consistent with the presence of the tail effect at 4PN order, which is incorporated consistently into the Fokker action. However, like in [67], we are obliged to introduce an arbitrary coefficient relating the IR cut-off scale to the a priori different scale present in the tail integral. This coefficient is determined by using a self-force calculation (both numerical [69,70] and analytical [71]), so that our end result for the energy of circular orbits at 4PN order has the correct self-force limit.…”

Radiation reaction for spinning bodies in effective field theory. II. Spin-spin effects

“…In a companion paper [4], we reported the resulting radiation-reaction equations of motion to linear order in the spins, which appear at fourth postNewtonian (4PN) order, using the effective field theory (EFT) formalism. At this order conservative contributions are also known using more traditional methods [5] as well as the EFT approach [6,7], encompassing spin-independent [8][9][10][11][12][13][14][15][16][17][18] (see also [19,20]) and spin-dependent terms. 1 See [5,55] for extensive reviews on the postNewtonian expansion, and [56][57][58][59][60][61] for reviews on the EFT approach to the binary problem.…”