Multipole analysis on stationary massive vector and symmetric tensor fields

TL;DR

This paper derives multipole expansions for stationary massive vector and symmetric tensor fields using symmetric trace-free formalism, revealing Yukawa-like dependence and novel multipole moments not present in massless fields.

Contribution

It provides the first comprehensive multipole expansion expressions for massive vector and tensor fields, including new types of multipole moments and their dependence on source distribution.

Findings

Multipole expansions exhibit Yukawa-like decay with distance.

Stationary massive vector fields have two types of magnetic multipole moments.

Additional multipole moments appear in massive symmetric tensor fields.

Abstract

The multipole expansions for massive vector and symmetric tensor fields in the region outside spatially compact stationary sources are obtained by using the symmetric and trace-free formalism in terms of the irreducible Cartesian tensors, and the closed-form expressions for the source multipole moments are provided. The expansions show a Yukawa-like dependence on the massive parameters of the fields, and the integrals of the stationary source multipole moments are all modulated by a common radial factor related to the source distribution. For stationary massive vector field, there are two types of ``magnetic'' multipole moments, among which one is the generalization of that of the magnetostatic field, and another, being an additional set of multipole moments of the stationary massive vector field, can not be transformed away. As to the stationary massive symmetric tensor field, its multipole expansion is presented when the trace of its spatial part is specified, where besides the counterparts of the mass and spin multipole moments of massless symmetric tensor field, three additional sets of multipole moments also appear. The multipole expansions of the tensor field under two typical cases are discussed, where it is shown that if the spatial part of the tensor field is trace-free, the monopole and dipole moments in the corresponding expansion will vanish.