Maxwell's equations in accelerated reference frames and their application in computational electromagnetism

In many engineering applications the interaction between the electromagnetic field and moving bodies is of great interest. E.g., motional induced eddy currents have to be taken into account correctly for the modelling and simulation of high-speed solenoid actuators. In connection with computational electromagnetism, it seems natural to use a Lagrangian (also called material) description. The unknowns are defined on the mesh, which moves and deforms together with the considered objects. What is the correct form of Maxwell's and the constitutive equations under such circumstances? Since the bodies might undergo accelerated motion, this question cannot in general be answered by the application of Lorentz transforms. Consequently, Maxwell's equations do not necessarily have their usual form in accelerated frames of reference. This was demonstrated in a classical paper by Schiff [1], where it is shown that a significant difference occurs even at "low" velocities, which are small compared to the velocity of light. In contrast, it is convenient to perform the analysis of rotating induction machines from the rotor's point of view. Despite the acceleration, starting from the usual form of Maxwell's equations yields the correct results. How could that be possible? There are only few publications that address the subject from a general point of view and not only for a restricted class of examples, e.g. [2,3]. The aim of the present paper is to tackle the problem once more by using the language of differential forms (DFs). DFs are especially well suited for such considerations, since they allow to seamlessly migrate from the (3+1)-to the four-dimensional formulation of electrodynamics, which are both briefly reviewed. Moreover, DFs allow separating the topological from the metric part of the theory. Using a noninertial frame induces a metric that is different from the standard Lorentz metric. This metric enters the formulation only through the coordinate expression for the four-dimensional Hodge operator. A localization transform can be introduced, to revert to a (3+1)-dimensional description. This is connected to the concept of a co-moving observer. The result is a relativistically correct Lagrangian form of Maxwell's and the constitutive equations. For "small" accelerations, i.e. if the extension of the system is neglectable compared to the radii of curvature, a concise set of transforms for all the relevant field quantities can be derived. These transforms are well suited for the implementation into numerical field computation codes.