Numerical construction of initial data sets of binary black hole type using a parabolic-hyperbolic formulation of the vacuum constraint equations

Abstract

In this paper we investigate the parabolic-hyperbolic formulation of the vacuum constraint equations introduced by Rácz with a view to constructing multiple black hole initial data sets without spin. In order to respect the natural properties of this configuration, we foliate the spatial domain with 2-spheres. It is then a consequence of these equations that they must be solved as an initial value problem evolving outwards towards spacelike infinity. Choosing the free data and the "strong field boundary conditions" for these equations in a way which mimics asymptotically flat and asymptotically spherical binary black hole initial data sets, our focus in this paper is on the analysis of the asymptotics of the solutions. In agreement with our earlier results, our combination of analytical and numerical tools reveals that these solutions are in general not asymptotically flat, but have a cone geometry instead. In order to remedy this and approximate asymptotically Euclidean data sets, we then propose and test an iterative numerical scheme. *

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

“…With no control over the asymptotics it is therefore possible that the method generates initial data sets that lack a physical interpretation. Exactly this issue has been explored recently in [10,12,19]. It was confirmed that generic solutions of these equations are not asymptotically flat (this notion is defined in Section 4 below).…”

“…In this subsection we introduce data sets (without imposing the constraints yet) of Kerr-Schild form. Such data sets were the basis of our previous work in [10,12] and we shall continue to use them in particular in Section 6.1. In this paper now we introduce such data sets as follows.…”

“…As in [10,12] we restrict now to the case Σ = R 3 \B where B is an arbitrary fixed ball in R 3 in all of what follows. Moreover, we assume that the level sets of ρ are diffeomorphic to 2-spheres.…”

“…In [12] we proposed an iterative approach to, at least partly, address the asymptotic flatness problem. In this paper here we provide strong evidence (by a combination of analytic and numerical techniques, see below) that a small change of how the free data for Rácz's parabolic-hyperbolic formulation are specified is sufficient to resolve this issue and to generate vacuum initial data sets which are generically asymptotically flat.…”

“…This will allow us to use the same numerical pseudo-spectral methods developed in [6,8,9,11] based on the ð-and the spinweight formalism. As in [10,12], our focus is on the asymptotics exclusively. In particular, we are not concerned with the properties of the solutions in the strong field regime, e.g.…”