I review the construction of Hertz potentials in vector calculus starting from Maxwell's equations. From here, I lay the minimal foundations of differential geometry to construct Hertz potentials for a general (spatially compact) Lorentzian manifold with or without boundary. In this general framework, I discuss "scalar" Hertz potentials as they apply to the vector calculus situation, and I consider their possible generalization, showing which procedures used by previous authors fail to generalize and which succeed, if any. I give specific examples, including the standard at coordinate systems and an example of a non-flat metric, specifically a spherically symmetric black hole. Additionally, I generalize the introduction of gauge terms, and I present techniques for introducing gauge terms of arbitrary order. Finally, I give a treatment of one application of Hertz potentials, namely calculating electromagnetic Casimir interactions for a couple of systems.

We extend previous work (2020Class. Quantum Grav. 37 075001)to the case of Maxwell’s equations with a source. Our work shows how to construct a vector potential for the Maxwell field on the Kerr–Newman background in a radiation gauge. The vector potential has a ‘reconstructed’ term obtained from a Hertz potential solving Teukolsky’s equation with a source, and a ‘correction’ term which is obtainable by a simple integration along outgoing principal null rays. The singularity structure of our vector potential is discussed in the case of a point particle source