The binary black-hole problem at the third post-Newtonian approximation in the orbital motion: Static part

TL;DR

This work investigates the binary black-hole problem at the $3$PN level by refining PN expansions of time-symmetric BL and ML initial-data solutions and their static Hamiltonians. The authors demonstrate that the BL and ML static Hamiltonians differ at $3PN$ but can be reconciled by shifting black-hole centers to eliminate dipole moments, yielding a unique static Hamiltonian under this condition. They also explore static binary point-mass regularization schemes, showing consistency with BL/ML results up to $1/c^6$ and highlighting a static ambiguity at higher orders tied to regularization choices. The analysis clarifies the static part of the 3PN problem and emphasizes that a full, unique total Hamiltonian remains contingent on dynamical and radiative degrees of freedom beyond the static sector.

Abstract

Post-Newtonian expansions of the Brill-Lindquist and Misner-Lindquist solutions of the time-symmetric two-black-hole initial value problem are derived. The static Hamiltonians related to the expanded solutions, after identifying the bare masses in both solutions, are found to differ from each other at the third post-Newtonian approximation. By shifting the position variables of the black holes the post-Newtonian expansions of the three metrics can be made to coincide up to the fifth post-Newtonian order resulting in identical static Hamiltonians up the third post-Newtonian approximation. The calculations shed light on previously performed binary point-mass calculations at the third post-Newtonian approximation.