Algebraic stability analysis of constraint propagation

TL;DR

This paper introduces an algebraic method to analyze the stability of constraint propagation in computational gravity, focusing on how gauge choices affect the intrinsic constraint violations in Einstein's equations.

Contribution

It presents a new algebraic technique to study the stability of constraint propagation and examines the impact of gauge parameters on constraint violations in the Weyl system.

Findings

Constraint violations can originate from boundary conditions and intrinsic instabilities.

Gauge parameters influence the stability of the constraint surface.

The Weyl system's stability depends on the choice of time foliation.

Abstract

The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations.