Multipole structure of current vectors in curved spacetime

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).