Spatial-symmetry violating electromagnetic fields corrected by nonlinear Lagrangian

We investigate a general nonlinear electromagnetic Lagrangian belonging to a subclass of the Plebański class. Depending on the form of nonlinear Lagrangian model, in an electrostatic problem, not only the electric field but also the electric flux density can vary from that given by linear classical electromagnetism. The variance is a correction which stems from the nonlinearity of Lagrangian. The nonlinear correction is a divergence-free field and possesses a mathematical vector potential. By considering a two charge system with a non-perturbative approach, we derive a necessary condition on nonlinear Lagrangian such that the nonlinear correction of the electric flux density becomes zero in the entire space. Several frequently considered nonlinear Lagrangian models do not satisfy this condition. As another important result, we show that the vector potential of the nonlinear correction violates mirror symmetry. (paper)