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

TL;DR

This paper derives post-Newtonian expansions of black-hole initial data solutions, revealing differences at third order that can be reconciled by variable shifts, thus clarifying previous binary black-hole calculations.

Contribution

It provides the first detailed third post-Newtonian static Hamiltonian comparison for two black-hole solutions, highlighting how to align different initial data approaches.

Findings

Static Hamiltonians differ at third post-Newtonian order.

Shifting black hole positions aligns metrics up to fifth post-Newtonian order.

Clarifies the relation between different initial data solutions at third post-Newtonian approximation.

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.