Symmetric hyperbolic systems for a large class of fields in arbitrary dimension

Abstract

Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r-1 arbitrary timelike vectors. The importance of the so-called "superenergy" tensors, which provide the necessary symmetric positive matrices, is emphasized and made explicit. Thereby, a unified treatment of many physical systems is achieved, as well as of the sometimes called "higher order" systems. The characteristics of these symmetric hyperbolic systems are always physical, and directly related to the null directions of the superenergy tensor, which are in particular principal null directions of the tensor field solutions. Generic energy estimates and inequalities are presented too.

24 pages, no figures

References in corpus (14)

• New Formalism for Numerical Relativity
• Super-energy tensors
• Theorems on existence and global dynamics for the Einstein equations
• Null cone preserving maps, causal tensors and algebraic Rainich theory
• General Electric-Magnetic decomposition of fields, positivity and Rainich-like conditions
• General study and basic properties of causal symmetries
• On the Causal Propagation of Fields
• Remarks on Superenergy Tensors
• Symmetric hyperbolic systems for Bianchi equations
• Positivity and conservation of superenergy tensors
• Causal symmetries
• The universal 'energy' operator
• A weighted de Rham operator acting on arbitrary tensor fields and their local potentials
• Einstein Bianchi equations with sources

Cited by in corpus (5)

• Green-hyperbolic operators on globally hyperbolic spacetimes
• Characterization of (asymptotically) Kerr-de Sitter-like spacetimes at null infinity
• Kerr Initial data
• The universal 'energy' operator
• Non-linear electrodynamics non-minimally coupled to gravity:symmetric-hyperbolicity and causal structure