The Relativistic Action at a Distance Two Body Problem

Abstract

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 $\mathbf{x}_n(t)$ and $\mathbf{p}_n(t)$, for $n= 1,2,3,\ldots N$; 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 $V(\mathbf{x}_1(t), \mathbf{x}_2(t))$; for Galiliean invariance, V may be a scalar function of the difference, i.e., $V(\mathbf{x}_1(t)-\mathbf{x}_2(t))$. It is usually understood that $\mathbf{x}_1$ and $\mathbf{x}_2$ 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.