Centers of Mass and Rotational Kinematics for the Relativistic N-Body Problem in the Rest-Frame Instant Form

TL;DR

This paper develops a relativistic kinematic framework for the N-body problem using the rest-frame instant form, defining centers of mass, relative variables, and rotational kinematics consistent with special relativity.

Contribution

It introduces a relativistic N-body kinematic formalism with new definitions of centers of mass and rotational variables within the rest-frame instant form.

Findings

Defined external and internal centers of mass in relativistic context

Established a canonical basis of relative variables for N-body systems

Extended non-relativistic rotational kinematics concepts to relativistic framework

Abstract

In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem. The Wigner hyperplanes define the intrinsic rest frame and realize the separation of the center-of-mass. Three notions of {\it external} relativistic center of mass can be defined only in terms of the {\it external} Poincar\'e group realization. Inside the Wigner hyperplane, an {\it internal} unfaithful realization of the Poincar\'e group is defined. The three concepts of {\it internal} center of mass weakly {\it coincide} and are eliminated by the rest-frame conditions. An adapted canonical basis of relative variables is found. The invariant mass is the Hamiltonian for the relative motions. In this framework we can introduce the same {\it dynamical body frames}, {\it orientation-shape} variables, {\it spin frame} and {\it canonical spin bases} for the rotational kinematics developed for the non-relativistic N-body problem.