Carter separable electromagnetic fields

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The purely electromagnetic analogue in flat space of Kerr’s metric in general relativity is only rarely considered. Here we carry out in flat space a programme similar to Carter’s investigation of metrics in general relativity in which the motion of a charged particle is separable. We concentrate on the separability of the motion (be it classical, relativistic or quantum) of a charged particle in electromagnetic fields that lie in planes through an axis of symmetry. In cylindrical polar coordinates (t,R,φ,z) the four-vector potential takes the form is the unit toroidal vector. The forms of the functions Φ(R,z) and A(R,z) are sought that allow separable motion. This occurs for relativistic motion only when AR,Φ and A2−Φ2 are all of the separable form ζ(λ)−η(μ)]/(λ−μ), where ζ and η are arbitrary functions, and λ and μ are spheroidal coordinates or degenerations thereof. The special forms of A and Φ that allow this are deduced. They include the Kerr metric analogue, with E+iB=−∇{q[(r−ia)·(r−ia)]−1/2}. Rather more general electromagnetic fields allow separation when the motion is non-relativistic. The investigation is extended to fields that lie in parallel planes. Connections to Larmor’s theorem are remarked upon.