Instantaneous Action-at-a-Distance Formulation of Classical Electrodynamics

Abstract

The possibility of formally translating the interaction of charges from charge ↔ field ↔ charge to charge ↔ charge, where the orbits satisfy Newtonian (second order in t), yet covariant, equations of motion, is exploited for the Wheeler‐Feynman interaction. A method for computing the forces on the charges correct to second order in the coupling constant e² is presented, and ten constants of the motion correct to e² are found. The integration is effected via the Noether theorem with the inhomogeneous Lorentz group as symmetry transformations. An important result is that a well‐known correction to the Coulomb interaction which accounts for the uniform motion of charges is revealed to be, to first order in e², a frame‐invariant expression. The consequent corrected Coulomb dynamics admits first‐order integrals identical to those of the Wheeler‐Feynman dynamics.