Regular hyperbolicity, dominant energy condition and causality for Lagrangian theory of maps

TL;DR

This paper explores the relationship between energy conditions and hyperbolicity in Lagrangian field theories, analyzes hyperbolicity breakdown in the Skyrme model, and summarizes regular hyperbolicity for relativity.

Contribution

It establishes a general theorem on dominant energy conditions for Lagrangian maps and analyzes hyperbolicity issues, including well-posedness in the Skyrme model.

Findings

Proved a general theorem on dominant energy conditions for Lagrangian theories.

Analyzed hyperbolicity breakdown in the Skyrme model.

Provided a summary of regular hyperbolicity framework for relativity.

Abstract

The goal of the present paper is three-fold. First is to clarify the connection between the dominant energy condition and hyperbolicity properties of Lagrangian field theories. Second is to provide further analysis on the breakdown of hyperbolicity for the Skyrme model, sharpening the results of Crutchfield and Bell and comparing against a result of Gibbons, and provide a local well-posedness result for the dynamical problem in the Skyrme model. Third is to provide a short summary of the framework of regular hyperbolicity of Christodoulou for the relativity community. In the process, a general theorem about dominant energy conditions for Lagrangian theories of maps is proved, as well as several results concerning hyperbolicity of those maps.