Strongly hyperbolic systems in General Relativity

TL;DR

This paper explores the concept of strong hyperbolicity in general relativity, providing covariant definitions, analyzing hypersurface deformations, and examining implications for systems with constraints and second-order derivatives.

Contribution

It introduces covariant definitions of strong hyperbolicity and demonstrates its invariance under hypersurface deformations, addressing limitations of symmetric hyperbolic systems in GR.

Findings

Strong hyperbolicity is preserved under hypersurface deformations.

Systems with constraints inherit hyperbolicity properties from the evolution system.

Second-order in space derivatives systems like ADM and BSSN are analyzed for hyperbolicity.

Abstract

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that is a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any near by one. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what is the maximal propagation speed in any given direction. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives, they include certain versions of the ADM and the BSSN families of equations. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.