On the weak‐field approximation of nonlinear vacuum electrodynamics of the Plebański class

Plebański's class of nonlinear vacuum electrodynamics is considered, which is for several reasons of interest at the present time. In particular, the question is answered under which circumstances Maxwell's original field equations are recovered approximately and which ‘post‐Maxwellian’ effects could arise. To this end, a weak field approximation method is developed, allowing to calculate ‘post‐Maxwellian’ corrections up to Nth order. In some respect, this is analogue of determining ‘post‐Newtonian’ corrections from relativistic mechanics by a low velocity approximation. As a result, we got a series of linear field equations that can be solved order by order. In this context, the solutions of the lower orders occur as source terms inside the higher order field equations and represent a ‘post‐Maxwellian’ self‐interaction of the electromagnetic field, which increases order by order. It becomes apparent that one has to distinguish between problems with and without external source terms because without sources also high frequency solutions can be approximately described by Maxwell's original equations. The higher order approximations, which describe ‘post‐Maxwellian’ effects, can give rise to experimental tests of Plebańksi's class. Finally, two boundary value problems are discussed to have examples at hand.