The electromagnetic Debye potentials reviewed: including sources and new solutions

We review the Debye potentials formalism for an electromagnetic test field on a fixed vacuum type D background and extend it in two different directions. (a) We show how to introduce sources (electric currents and charge distributions) in the framework of the Debye potentials. For the derivation all the three distinct possible Debye potentials are necessary but, fortunately, the fields outside the sources can still be described by a single Debye potential. (b) We present a new class of solutions (within the framework of the Debye potentials) which have been overlooked for a long time. The Maxwell equations represent in fact 3rd order partial differential equations for the Debye potentials. The known equations for the Debye potentials represent only a sufficient condition for the existence of source-free Maxwell field. This is exactly where the new solutions are “hidden”. Moreover, we found a completely new exact solution which describes physical process of charging of the black hole by charges infalling onto it along north and south axis at the speed of light. Finally, we seek for possible connections between the Debye potentials and the recently found Lunin potential. We also provide physical interpretation of some explicit solutions obtained by the Lunin potential.