Classical Electrodynamics of Extended Bodies.

We study the classical electrodynamics of extended bodies. Currently in the literature, no self-consistent dynamical theory of such bodies exists. Causally correct equations can be produced only in the point-charge limit; this has the unfortunate result of infinite self-energies, requiring some renormalization procedure, and perturbative methods to account for radiation. We review the history that has led to the understanding of this fact. We then investigate possible self-consistent, non-point-charge, classical electrodynamic theories from an energy-momentum perspective. Limiting to a purely electromagnetic system, where the only dynamical degrees of freedom are in the electromagnetic field, a strict constraint is found that leads to a specific stress-energy addition, yielding self-consistent dynamical equations. Due to stable objects being extended, direct current-current interactions appear if two objects overlap; these could be interpreted as non-electromagnetic short-range forces. Also, in curved space-time, solutions appear to be able to only admit stable objects with quantized charge. The Lagrangian associated with the stress-energy addition is also found. Interestingly, to conserve charge as the metric is varied, the electromagnetic current density must be held constant, rather than the electromagnetic potential. The resulting variational field equations (the Proca equation), while possibly satisfactory inside an object, are obviously unsatisfactory outside (they imply zero field). But, rather than requiring the action be invariant against arbitrary field variations, if variations are due only to an infinitesimal active coordinate transformation (diffeomorphism invariance), Einstein's equations and conservation of energy-momentum become the fundamental field equations, and the theory is fully self-consistent, without any apparent pathologies.