Nonlinear electrodynamics as a symmetric hyperbolic system

TL;DR

This paper establishes conditions under which nonlinear electrodynamics theories, derived from a Lagrangian, are well-posed by analyzing the intersection of effective metric cones, with implications for models like Born-Infeld and Euler-Heisenberg.

Contribution

It provides a criterion for symmetric hyperbolicity in nonlinear electrodynamics based on the intersection of effective metric cones, ensuring well-posed initial value problems.

Findings

The theory is symmetric hyperbolic if and only if the effective metric cones intersect.

The intersection condition guarantees the existence of positive definite symmetrizers for the system.

Application to models like Born-Infeld and Euler-Heisenberg demonstrates the criterion's relevance.

Abstract

Nonlinear theories generalizing Maxwell's electromagnetism and arising from a Lagrangian formalism have dispersion relations in which propagation planes factor into null planes corresponding to two effective metrics which depend on the point-wise values of the electromagnetic field. These effective Lorentzian metrics share the null (generically two) directions of the electromagnetic field. We show that, the theory is symmetric hyperbolic if and only if the cones these metrics give rise to have a non-empty intersection. Namely that there exist families of symmetrizers in the sense of Geroch which are positive definite for all covectors in the interior of the cones intersection. Thus, for these theories, the initial value problem is well-posed. We illustrate the power of this approach with several nonlinear models of physical interest such as Born-Infeld, Gauss-Bonnet and Euler-Heisenberg.