Implementation of higher-order absorbing boundary conditions for the Einstein equations

TL;DR

This paper develops and tests higher-order absorbing boundary conditions for Einstein equations, improving the simulation of gravitational waves by reducing spurious reflections at the boundary.

Contribution

It reformulates boundary conditions as ODE systems for auxiliary variables, enabling constraint-preserving, well-posed boundary conditions for Einstein equations in a first-order formulation.

Findings

Higher-order boundary conditions eliminate spurious reflections in linearized gravitational wave simulations.

The method effectively matches the angular momentum number l, reducing boundary reflections.

Lower-order conditions like freezing-Psi_0 are less effective in absorbing outgoing waves.

Abstract

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.