Outer boundary conditions for Einstein's field equations in harmonic coordinates

TL;DR

This paper develops and analyzes a class of boundary conditions for Einstein's vacuum equations in harmonic coordinates, ensuring well-posedness and reducing gravitational wave reflections, with numerical tests on black hole perturbations.

Contribution

It introduces a broad, constraint-preserving boundary condition class for Einstein's equations, including recent proposals, and proves their well-posedness in a simplified setting.

Findings

Boundary conditions reduce spurious gravitational wave reflections.

The initial-boundary value problem is well posed in the frozen coefficient approximation.

Numerical implementation shows effectiveness in black hole perturbation tests.

Abstract

We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-Psi0 boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole.