Improved outer boundary conditions for Einstein's field equations

TL;DR

This paper develops and refines boundary conditions for Einstein's equations that effectively absorb gravitational waves, reducing reflections and improving accuracy for various spacetime configurations.

Contribution

It generalizes and enhances boundary conditions for Einstein's equations, including local and non-local forms that account for curvature and backscatter effects.

Findings

Local boundary condition C_2 reduces spurious reflections.

Non-local boundary condition D_2 is exact with first order corrections.

Methods improve gravitational wave absorption in simulations.

Abstract

In a recent article, we constructed a hierarchy B_L of outer boundary conditions for Einstein's field equations with the property that, for a spherical outer boundary, it is perfectly absorbing for linearized gravitational radiation up to a given angular momentum number L. In this article, we generalize B_2 so that it can be applied to fairly general foliations of spacetime by space-like hypersurfaces and general outer boundary shapes and further, we improve B_2 in two steps: (i) we give a local boundary condition C_2 which is perfectly absorbing including first order contributions in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of the spacetime and R is a typical radius of the outer boundary) and which significantly reduces spurious reflections due to backscatter, and (ii) we give a non-local boundary condition D_2 which is exact when first order corrections in 2M/R for both curvature and backscatter are considered, for quadrupolar radiation.