The relativistic action at a distance two body problem

Models with action at a distance potentials, such as the Coulomb potential, have been very useful in nonrelativistic mechanics. They provide a simpler framework than the perhaps more fundamental field mediated models for interaction, and are also straightforwardly amenable to rigorous mathematical analysis. In this Newtonian-Galilean view, all events directly interacting dynamically occur simultaneously; the dynamical phase space of N particles contains the points x_n(t) and p_n(t), for n=1,2,3,...N; these points move through the phase space as a function of the parameter t, following some prescribed equations of motion. Two particles may be thought of as interacting through a potential function V(x₁(t), x₂(t)); for Galiliean invariance, V may be a scalar function of the difference, i.e., V(x₁(t) − x₂(t)). It is usually understood that x₁ and x₂ are taken to be at equal time, corresponding to a correlation between the two particles consistent with the Newtonian-Galilean picture. With the advent of special relativity, it became a challenge to formulate dynamical problems on the same level as that of the nonrelativistic theory.