Instantaneous Action-at-a-Distance in Classical Relativistic Mechanics

The possibility of describing orbits in classical relativistic mechanics in instantaneous action-at-a-distance fashion by second-order differential equations (as in Newton's gravitational theory) is investigated with particular emphasis on the two-body problem of classical relativistic electrodynamics. Differential conditions are stated to guarantee world-line invariance and form-invariance of the equations of motion under Lorentz transformation for such a description of an N-particle system in three dimensions. A pair of integrodifferential equations for the equations of motion are derived to provide an explicit means of passing from a description via direct interaction along light cones to an instantaneous action-at-a-distance description for a two-body problem. These integrodifferential equations are applicable to the two-body problem of classical electrodynamics with either retarded interactions and radiation damping or with half-advanced plus half-retarded interactions.