Hyperbolicity and Constrained Evolution in Linearized Gravity

TL;DR

This paper analyzes the hyperbolicity and effectiveness of constrained evolution schemes in linearized gravity, providing insights into maintaining constraints during numerical simulations of Einstein's equations.

Contribution

It offers a detailed analysis of constrained evolution methods applied to linearized gravitational waves, highlighting their properties and potential relevance to full nonlinear Einstein equations.

Findings

Hyperbolicity of unconstrained Einstein equations studied

Explicit construction of constrained evolution effects

Insights relevant to nonlinear Einstein equations

Abstract

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). Since computational solution of differential equations introduces almost inevitable errors, it is clearly "more correct" to introduce a scheme which actively maintains the constraints by solution ({\it constrained} evolution). This has shown promise in computational settings, but the analysis of the resulting mixed elliptic hyperbolic method has not been completely carried out. We present such an analysis for one method of constrained evolution, applied to a simple vacuum system, linearized gravitational waves. We begin with a study of the hyperbolicity of the unconstrained Einstein equations. (Because the study of hyperbolicity deals only with the highest derivative order in the equations, linearization loses no essential details.) We then give explicit analytical construction of the effect of initial data setting and constrained evolution for linearized gravitational waves. While this is clearly a toy model with regard to constrained evolution, certain interesting features are found which have relevance to the full nonlinear Einstein equations.