We investigate the Plebański class of electrodynamical theories, i.e., theories of nonlinear vacuum electrodynamics that derive from a Lorentz-invariant Lagrangian (or Hamiltonian). In any such theory the light rays are the lightlike geodesics of two optical metrics that depend on the electromagnetic background field. A set of necessary and sufficient conditions is found whose fulfillment secures that the optical metrics are causal in the sense that the light rays are lightlike or timelike with respect to the underlying space-time metric. Thereupon we derive conditions on the Lagrangian, or the Hamiltonian, of the theory such that the causality conditions are satisfied for all allowed background fields. (The allowed values of the field strength tensor are those for which the excitation tensor is finite and real.) The general results are illustrated with several examples.

We discuss the theoretical foundations for testing nonlinear vacuum electrodynamics with Michelson interferometry. Apart from some nondegeneracy conditions to be imposed, our discussion applies to all nonlinear electrodynamical theories of the Plebański class, i.e., to all Lagrangians that depend only on the two Lorentz-invariant scalars quadratic in the field strength. The main idea of the experiment proposed here is to use the fact that, according to nonlinear electrodynamics, the phase velocity of light should depend on the strength and on the direction of an electromagnetic background field. There are two possible experimental setups for testing this prediction with Michelson interferometry. The first possibility is to apply a strong electromagnetic field to the beam in one arm of the interferometer and to compare the situation where the field is switched on with the situation where it is switched off. The second possibility is to place the whole interferometer in a strong electromagnetic field and to rotate it. If an electromagnetic field is placed in one arm, the interferometer could have the size of a gravitational wave detector, i.e., an arm length of several hundred meters. If the whole interferometer is placed in an electromagnetic field, one would have to do the experiment with a tabletop interferometer. As an alternative to a traditional Michelson interferometer, one could use a pair of optical resonators that are not bigger than a few centimeters. Then the whole apparatus would be placed in the background field and one would either compare the situation where the field is switched on with the situation where it is switched off or one would rotate the apparatus with the field kept switched on. We derive the theoretical foundations for these types of experiments, in the context of an unspecified nonlinear electrodynamics of the Plebański class, and we discuss their feasibility. A null result of the experiment would place bounds on the parameters of the theory. We specify the general results to some particular theories of the Plebański class; in particular, we give numerical estimates for Born, Born-Infeld and Heisenberg-Euler theories.

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We analyze the behavior of shock waves in nonlinear theories of electrodynamics. For this, by use of generalized Hadamard step functions of increasing order, the electromagnetic potential is developed in a series expansion near the shock wave front. This brings about a corresponding expansion of the respective electromagnetic field equations which allows for deriving relations that determine the jump coefficients in the expansion series of the potential. We compute the components of a suitable gauge-normalized version of the jump coefficients given for a prescribed tetrad compatible with the shock front foliation. The solution of the first-order jump relations shows that, in contrast to linear Maxwell's electrodynamics, in general the propagation of shock waves in nonlinear theories is governed by optical metrics and polarization conditions describing the propagation of two differently polarized waves (leading to a possible appearance of birefringence). In detail, shock waves are analyzed in the Born and Born-Infeld theories verifying that the Born-Infeld model exhibits no birefringence and the Born model does. The obtained results are compared to those ones found in literature. New results for the polarization of the two different waves are derived for Born-type electrodynamics.