Improved initial data for black hole binaries by asymptotic matching of post-Newtonian and perturbed black hole solutions

TL;DR

This paper develops improved initial data for non-spinning black hole binaries by asymptotically matching post-Newtonian and perturbed Schwarzschild solutions, resulting in more accurate and horizon-penetrating coordinate systems for simulations.

Contribution

It introduces a novel matching procedure in isotropic coordinates that enhances the accuracy of initial data and provides horizon-penetrating coordinates for numerical relativity.

Findings

Better agreement between metrics in the buffer zone.

Global smooth 4-metric with errors comparable to approximation errors.

Horizon-penetrating coordinates with positive lapse everywhere.

Abstract

We construct approximate initial data for non-spinning black hole binary systems by asymptotically matching the 4-metrics of two tidally perturbed Schwarzschild solutions in isotropic coordinates to a resummed post-Newtonian 4-metric in ADMTT coordinates. The specific matching procedure used here closely follows the calculation in gr-qc/0503011, and is performed in the so called buffer zone where both the post-Newtonian and the perturbed Schwarzschild approximations hold. The result is that both metrics agree in the buffer zone, up to the errors in the approximations. However, since isotropic coordinates are very similar to ADMTT coordinates, matching yields better results than in the previous calculation, where harmonic coordinates were used for the post-Newtonian 4-metric. In particular, not only does matching improve in the buffer zone, but due to the similarity between ADMTT and isotropic coordinates the two metrics are also close to each other near the black hole horizons. With the help of a transition function we also obtain a global smooth 4-metric which has errors on the order of the error introduced by the more accurate of the two approximations we match. This global smoothed out 4-metric is obtained in ADMTT coordinates which are not horizon penetrating. In addition, we construct a further coordinate transformation that takes the 4-metric from global ADMTT coordinates to new coordinates which are similar to Kerr-Schild coordinates near each black hole, but which remain ADMTT further away from the black holes. These new coordinates are horizon penetrating and lead, for example, to a lapse which is everywhere positive on the t=0 slice. Such coordinates may be more useful in numerical simulations.