Detecting ill posed boundary conditions in General Relativity

TL;DR

This paper analyzes the well-posedness of boundary conditions in linearized Einstein equations using Laplace-Fourier techniques, identifying conditions that prevent ill posed modes in numerical relativity.

Contribution

It introduces a systematic analysis of boundary conditions in hyperbolic formulations of Einstein equations, highlighting how parameter choices affect well posedness.

Findings

Constraint-preserving boundary conditions are well posed.

Ill posed modes can arise if parameters are not carefully chosen.

Certain boundary conditions lead to constraint violating modes.

Abstract

A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace-Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: i) Boundary conditions that have been shown to preserve the constraints, ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero and iii) boundary conditions that result from considering the projection of Einstein's equations along the normal to the boundary surface. While we show that in case i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases.