Electromagnetic Shocks in Strong Magnetic Fields

The nonlinear properties of electromagnetic waves traveling through a magnetized vacuum is of particular interest in the study of neutron stars. Heisenberg and Euler [1] and Weisskopf [2] first derived nonlinear corrections to the Maxwell equations of the electromagnetic field. Lutzky and Toll [3] and Zheleznyakov and Fabrikant [4] applied the weak-field expansion to show that shock waves can develop in the electromagnetic field. Bialynicka-Birula [5] used the full expression for the nonlinear correction to the Lagrangian to study the generation of harmonics and other nonlinear phenomena in the propagation of EM radiation. In this paper, we use the method of characteristics to study the evolution of waves governed by an arbitrary Lagrangian and then apply these techniques to the Heisenberg-Euler-Weisskopf-Schwinger Lagrangian [1,2,6]. We find a concise expression for the opacity to shocking for disturbances travelling through an arbitrarily strong mangetic field. After deriving a form for the energy-momentum tensor for an arbitrary electromagnetic Lagrangian, we use relativitistic fluid mechanics to derive the shock jump conditions and the long-term evolution of electromagnetic disturbances propagating in magnetic fields.