Post-Newtonian Freely Specifiable Initial Data for Binary Black Holes in Numerical Relativity

TL;DR

This paper develops astrophysically realistic initial data for binary black holes in numerical relativity, aligning with post-Newtonian results and improving upon previous conformally flat solutions.

Contribution

It introduces a new solution to the constraint equations that matches post-Newtonian metrics up to 2PN order, incorporating black hole motion and coordinate transformations.

Findings

Solution differs from standard post-Newtonian metric by a coordinate transformation.

The initial data matches post-Newtonian results up to 2PN order.

Differences from Bowen-York solutions appear at second post-Newtonian order.

Abstract

Construction of astrophysically realistic initial data remains a central problem when modelling the merger and eventual coalescence of binary black holes in numerical relativity. The objective of this paper is to provide astrophysically realistic freely specifiable initial data for binary black hole systems in numerical relativity, which are in agreement with post-Newtonian results. Following the approach taken by Blanchet, we propose a particular solution to the time-asymmetric constraint equations, which represent a system of two moving black holes, in the form of the standard conformal decomposition of the spatial metric and the extrinsic curvature. The solution for the spatial metric is given in symmetric tracefree form, as well as in Dirac coordinates. We show that the solution differs from the usual post-Newtonian metric up to the 2PN order by a coordinate transformation. In addition, the solutions, defined at every point of space, differ at second post-Newtonian order from the exact, conformally flat, Bowen-York solution of the constraints.