Classical Electrodynamics of Extended Bodies of Charge

We investigate the classical dynamics of charged bodies. Calculating their dynamics is difficult, primarily due to electromagnetic self-interaction (which produces radiation). This remains an unsolved problem: in the literature, no self-consistent dynamical theory of extended charged bodies exists, which can satisfy causality and include radiative (self-interaction) effects. Deterministic, causally correct equations of motion can be produced only in the point charge limit; however, this has the unfortunate effect of infinite self-energies, requiring some renormalization procedure. This paper reviews the history of the development of Electrodynamics. We then investigate limitations on possible self-consistent electrodynamic theories, which do not assume the point charge limit. With fairly broad assumptions (compatibility with General Relativity, Maxwell's equations, only quadratic terms in the Lagrangian, and requiring that initial conditions be in terms of the current and field), we find a very restrictive limitation, leading to only one possible, self-consistent theory. To develop Lagrangians for extended charge distributions, we show that in order to conserve charge, the electromagnetic current vector density must be held constant when varying the metric, rather than the electromagnetic potential 1-form. In the self-consistent theory of this paper, non-electromagnetic, short-range forces are immediately realized due to contact interactions. We show that no charged, static, spherically symmetric solutions exist. However, when gravity is primarily responsible for binding the charge together, we find the behavior of the charge density near the center of a static, spinning charge distribution would be constrained in such a way, that if some rotation model and angular momentum were set, the charge would be set; i.e. only a single charge would be allowed (it would be quantized).