Multipolar Expansions for Closed and Open Systems of Relativistic Particles

TL;DR

This paper develops a relativistic multipole framework for systems of particles, extending classical concepts like inertia to special relativity, and explores applications to both closed and open subsystems, with potential relevance to astrophysics.

Contribution

It introduces a relativistic multipole technique using Dixon's formalism within the rest-frame instant form, addressing both closed and open systems of particles and fields, and clarifies the concept of collective motion in these systems.

Findings

Multipole formalism is adapted to relativistic systems using Wigner hyper-planes.

Unique world-line for collective motion exists for closed systems but not for open subsystems.

The approach may aid in relativistic modeling of binary star systems in gravity.

Abstract

Dixon's multipoles for a system of N relativistic positive-energy scalar particles are evaluated in the rest-frame instant form of dynamics. The Wigner hyper-planes (intrinsic rest frame of the isolated system) turn out to be the natural framework for describing multipole kinematics. Classical concepts like the {\it barycentric tensor of inertia} turn out to be extensible to special relativity only by means of the quadrupole moments of the isolated system. Two new applications of the multipole technique are worked out for systems of interacting particles and fields. In the rest-frame of the isolated system of either free or interacting positive energy particles it is possible to define a unique world-line which embodies the properties of the most relevant centroids introduced in the literature as candidates for the collective motion of the system. This is no longer true, however, in the case of open subsystems of the isolated system. While effective mass, 3-momentum and angular momentum in the rest frame can be calculated from the definition of the {\it subsystem energy-momentum tensor}, the definitions of effective center of motion and effective intrinsic spin of the subsystem are not unique. Actually, each of the previously considered centroids corresponds to a different world-line in the case of open systems. The pole-dipole description of open subsystems is compared to their description as effective extended objects. Hopefully, the technique developed here could be instrumental for the relativistic treatment of binary star systems in metric gravity.