Non-linear electrodynamics non-minimally coupled to gravity:symmetric-hyperbolicity and causal structure

It is shown here that symmetric hyperbolicity, which guarantees well-posedness, leads to a set of two inequalities for matrices whose elements are determined by a given theory. As a part of the calculation, carried out in a mostly-covariant formalism, the general form for the symmetrizer, valid for a general Lagrangian theory, was obtained. When applied to nonlinear electromagnetism linearly coupled to curvature, the inequalities lead to strong constraints on the relevant quantities, which were illustrated with applications to particular cases. The examples show that non-linearity leads to constraints on the field intensities, and non-minimal coupling imposes restrictions on quantities associated to curvature.