Constraint Preserving Boundary Conditions for Hyperbolic Formulations of Einstein's Equations

TL;DR

This paper develops boundary conditions that preserve constraints in hyperbolic formulations of Einstein's equations, ensuring stable numerical solutions in bounded domains for linearized gravity models.

Contribution

It introduces a technique to find well-posed, constraint-preserving boundary conditions for symmetric hyperbolic systems related to Einstein's equations.

Findings

Derived well-posed boundary conditions for hyperbolic Einstein formulations.

Ensured the preservation of constraints in bounded numerical domains.

Established equivalence with linearized ADM system.

Abstract

Einstein's system of equations in the ADM decomposition involves two subsystems of equations: evolution equations and constraint equations. For numerical relativity, one typically solves the constraint equations only on the initial time slice, and then uses the evolution equations to advance the solution in time. Our interest is in the case when the spatial domain is bounded and appropriate boundary conditions are imposed. A key difficulty, which we address in this thesis, is what boundary conditions to place at the artificial boundary that lead to long time stable numerical solutions. We develop an effective technique for finding well-posed constraint preserving boundary conditions for constrained first order symmetric hyperbolic systems. By using this technique, we study the preservation of constraints by some first order symmetric hyperbolic formulations of Einstein's equations derived from the ADM decomposition linearized around Minkowski spacetime with arbitrary lapse and shift perturbations, and the closely related question of their equivalence with the linearized ADM system. Our main result is the finding of well-posed maximal nonnegative constraint preserving boundary conditions for each of the first order symmetric hyperbolic formulations under investigation, for which the unique solution of the corresponding initial boundary value problem provides a solution to the linearized ADM system on polyhedral domains.