Time-symmetric initial data for binary black holes in numerical relativity

TL;DR

This paper develops a new initial data solution for binary black holes in numerical relativity that aligns with post-Newtonian approximations up to second order, improving the physical realism of simulations.

Contribution

It introduces a conformal decomposition solution for time-symmetric initial data that matches post-Newtonian metrics up to 2PN order, extending previous models.

Findings

The solution is isometric to the post-Newtonian metric up to 2PN order.

The total ADM mass and individual masses are computed via surface integrals at infinity.

The binary's interacting mass-energy is well-defined at 2PN order and matches known post-Newtonian results.

Abstract

We look for physically realistic initial data in numerical relativity which are in agreement with post-Newtonian approximations. We propose a particular solution of the time-symmetric constraint equation, appropriate to two momentarily static black holes, in the form of a conformal decomposition of the spatial metric. This solution is isometric to the post-Newtonian metric up to the 2PN order. It represents a non-linear deformation of the solution of Brill and Lindquist, i.e. an asymptotically flat region is connected to two asymptotically flat (in a certain weak sense) sheets, that are the images of the two singularities through appropriate inversion transformations. The total ADM mass M as well as the individual masses m_1 and m_2 (when they exist) are computed by surface integrals performed at infinity. Using second order perturbation theory on the Brill-Lindquist background, we prove that the binary's interacting mass-energy M-m_1-m_2 is well-defined at the 2PN order and in agreement with the known post-Newtonian result.