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

TL;DR

This paper constructs symmetric hyperbolic systems for a broad class of tensor fields in any dimension, emphasizing superenergy tensors and their role in ensuring physical characteristics and energy estimates.

Contribution

It introduces a unified method to formulate symmetric hyperbolic systems for tensor fields using superenergy tensors, applicable to various physical systems and higher order equations.

Findings

Hyperbolic systems depend on 2r-1 arbitrary timelike vectors.

Superenergy tensors provide positive matrices for hyperbolization.

Characteristics are always physical, related to null directions of superenergy tensors.

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.