Half-integral conservative post-Newtonian approximations in the redshift factor of black hole binaries

TL;DR

This work explains the emergence of half-integral conservative PN terms in the redshift factor for circular black-hole binaries by attributing them to hereditary tail-of-tail effects at $5.5$PN in the extreme mass ratio limit. It develops a multipolar-post-Minkowskian framework with near-zone matching to extract the conservative $5.5$PN metric pieces and isolates the tail contribution, finally producing the explicit SF correction to the redshift factor $u_{ ext{SF}}^T(y)= y\left[1 - \frac{13696}{525}\pi\,y^{11/2}\right]$ with $y=(G m_2 \Omega / c^3)^{2/3}$; this result agrees with Shah13 and reinforces the consistency between PN and SF approaches for circular orbits. By carefully separating conservative and dissipative parts using time-symmetry and performing a coordinate transformation to remove spurious quartic terms, the paper demonstrates that the leading half-integral effect is purely conservative and originates from non-local tail integrals. The methodology provides a robust path to higher-order hereditary effects and strengthens the cross-validation between analytical PN, SF, and perturbative methods in binary dynamics.

Abstract

Recent perturbative self-force computations (Shah, Friedman & Whiting, submitted to Phys. Rev. {\bf D}, arXiv:1312.1952 [gr-qc]), both numerical and analytical, have determined that half-integral post-Newtonian terms arise in the conservative dynamics of black-hole binaries moving on exactly circular orbits. We look at the possible origin of these terms within the post-Newtonian approximation, find that they essentially originate from non-linear "tail-of-tail" integrals and show that, as demonstrated in the previous paper, their first occurrence is at the 5.5PN order. The post-Newtonian method we use is based on a multipolar-post-Minkowskian treatment of the field outside a general matter source, which is re-expanded in the near zone and extended inside the source thanks to a matching argument. Applying the formula obtained for generic sources to compact binaries, we obtain the redshift factor of circular black hole binaries (without spins) at 5.5PN order in the extreme mass ratio limit. Our result fully agrees with the determination of the 5.5PN coefficient by means of perturbative self-force computations reported in the previously cited paper.