Numerical scalar curvature deformation and a gluing construction

Abstract

In this work a new numerical technique to prepare Cauchy data for the initial value problem (IVP) formulation of Einstein's field equations (EFE) is presented. Our method is directly inspired by the exterior asymptotic gluing (EAG) result of Corvino (2000). The argument assumes a moment in time symmetry and allows for a composite, initial data set to be assembled from (a finite subdomain of) a known asymptotically Euclidean initial data set which is glued (in a controlled manner) over a compact spatial region to an exterior Schwarzschildean representative. We demonstrate how (Corvino, 2000) may be directly adapted to a numerical scheme and under the assumption of axisymmetry construct composite Hamiltonian constraint satisfying initial data featuring internal binary black holes (BBH) glued to exterior Schwarzschild initial data in isotropic form. The generality of the method is shown in a comparison of properties of EAG composite initial data sets featuring internal BBHs as modelled by Brill-Lindquist and Misner data.

The underlying geometric analysis character of gluing methods requires work within suitably weighted function spaces, which, together with a technical impediment preventing (Corvino, 2000) from being fully constructive, is the principal difficulty in devising a numerical technique. Thus the single previous attempt by Giulini and Holzegel (2005) (recently implemented by Doulis and Rinne (2016)) sought to avoid this by embedding the result within the well known Lichnerowicz-York conformal framework which required ad-hoc assumptions on solution form and a formal perturbative argument to show that EAG may proceed. In (Giulini and Holzegel, 2005) it was further claimed that judicious engineering of EAG can serve to reduce the presence of spurious gravitational radiation - unfortunately, in line with the general conclusion of (Doulis and Rinne, 2016) our numerical investigation does not appear to indicate that this is the case.

Concretising the sought initial data to be specified with respect to a spatial manifold with underlying topology R×S² our method exploits a variety of pseudo-spectral (PS) techniques. A combination of the eth-formalism and spin-weighted spherical harmonics together with a novel complex-analytic based numerical approach is utilised. This is enabled by our Python 3 based numerical toolkit allowing for unified just-in-time compiled, distributed calculations with seamless extension to arbitrary precision for problems involving generic, geometric partial differential equations (PDE) as specified by tensorial expressions. Additional features include a layer of abstraction that allows for automatic reduction of indicial (i.e., tensorial) expressions together with grid remapping based on chart specification - hence straight-forward implementation of IVP formulations of the EFE such as ADM-York or ADM-York-NOR is possible. Code-base verification is performed by evolving the polarised Gowdy T³ space-time with the above formulations utilising high order, explicit time-integrators in the method of lines approach as combined with PS techniques.

As the initial data we prepare has a precise (Schwarzschild) exterior this may be of interest to global evolution schemes that incorporate information from spatial-infinity. Furthermore, our approach may shed light on how more general gluing techniques could potentially be adapted for numerical work. The code-base we have developed may also be of interest in application to other problems involving geometric PDEs.