Hamiltonian two-body system in special relativity

TL;DR

This paper develops a covariant Hamiltonian framework for a two-body system in special relativity, extending previous results and analyzing special cases like extreme mass ratios and circular orbits.

Contribution

It generalizes earlier restricted models to a comprehensive covariant formalism for relativistic two-body systems with detailed analysis of motion and energy conservation.

Findings

Relative motion resembles nonrelativistic one-body motion in a stationary potential

Center of mass is a center of energy, not a point

In the extreme mass ratio limit, the heavy body approaches the center of mass

Abstract

We consider an isolated system made of two pointlike bodies interacting at a distance in the nonradiative approximation. Our framework is the covariant and a priori Hamiltonian formalism of "predictive relativistic mechanics", founded on the equal-time condition. The center of mass is rather a center of energy. Individual energies are separately conserved and the meaning of their positivity is discussed in terms of world-lines. Several results derived decades ago under restrictive assumptions are extended to the general case. Relative motion has a structure similar to that of a nonrelativistic one-body motion in a stationnary external potential, but its evolution parameter is generally not a linear function of the center-of-mass time, unless the relative motion is circular (in this latter case the motion is periodic in the center-of-mass time). Finally the case of an extreme mass ratio is investigated. When this ratio tends to zero the heavy body coincides with the center of mass provided that a certain first integral, related to the binding energy is not too large.