Multipole structure of current vectors in curved spacetime

TL;DR

This paper introduces a method for constructing divergence-free current vectors from multipole moments in curved spacetime, with applications to modeling charge and particle flux in classical extended bodies.

Contribution

It provides an exact construction method for conserved currents from multipole moments and explores their properties and limitations in curved spacetime.

Findings

Can generate all smooth, bounded currents with a given total charge.

Shows limitations on constant moments, affecting quasirigid motion models.

Derives conditions for currents to exist in different spacetimes with identical moments.

Abstract

A method is presented which allows the exact construction of conserved (i.e. divergence-free) current vectors from appropriate sets of multipole moments. Physically, such objects may be taken to represent the flux of particles or electric charge inside some classical extended body. Several applications are discussed. In particular, it is shown how to easily write down the class of all smooth and spatially-bounded currents with a given total charge. This implicitly provides restrictions on the moments arising from the smoothness of physically reasonable vector fields. We also show that requiring all of the moments to be constant in an appropriate sense is often impossible; likely limiting the applicability of the Ehlers-Rudolph-Dixon notion of quasirigid motion. A simple condition is also derived that allows currents to exist in two different spacetimes with identical sets of multipole moments (in a natural sense).