Black hole initial data by numerical integration of the parabolic-hyperbolic form of the constraints

TL;DR

This paper presents a high-order numerical method for constructing initial data of black holes, including highly boosted and spinning ones, by integrating the parabolic-hyperbolic form of Einstein's constraints without boundary conditions in the strong field.

Contribution

The authors develop and verify a fourth-order accurate numerical scheme for solving the parabolic-hyperbolic constraints, enabling initial data construction for a wide range of black hole parameters.

Findings

Successfully constructed initial data for highly boosted black holes.

Validated the numerical method with convergence tests and analytical solutions.

Able to generate initial data with significant gravitational wave content.

Abstract

The parabolic-hyperbolic form of the constraints is integrated numerically. The applied numerical stencil is $4^{th}$ order accurate (in the spatial directions) while 'time'-integration is made by using the method of lines with a $4^{th}$ order accurate Runge-Kutta scheme. The proper implementation of the applied numerical method is verified by convergence tests and monitoring the relative and absolute errors determined by comparing numerical and analytically known solutions of the constraints involving boosted and spinning vacuum single black hole configurations. The main part of our investigations is, however, centered on construction of initial data for distorted black holes which, in certain cases, have non-negligible gravitational wave content. Remarkably the applied new method is unprecedented in that it allows to construct initial data for highly boosted and spinning black holes, essentially for the full physical allowed ranges of these parameters. In addition, the use of the evolutionary form of the constraints is free from applying any sort of boundary conditions in the strong field regime.