Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations

TL;DR

This paper develops and analyzes boundary conditions for the Einstein equations in a generalized harmonic formulation, ensuring stability, constraint preservation, and control of incoming gravitational radiation, supported by theoretical and numerical evidence.

Contribution

It introduces boundary conditions that maintain stability and constraint preservation for the Einstein equations, including a criterion to detect weak instabilities.

Findings

The system is boundary-stable.

No weak instabilities with polynomial growth are found.

Numerical tests support well-posedness of the initial-boundary value problem.

Abstract

This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with subluminal normal component) are analyzed using the Fourier-Laplace technique. We show that the system is boundary-stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well-posed even if nonzero initial and source data are included.