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

Abstract

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.