The tensorial representation of the distributional stress-energy quadrupole and its dynamics

TL;DR

This paper develops a tensorial framework for the distributional stress-energy tensor up to quadrupole order, deriving its dynamical equations and clarifying the tensorial nature of multipole components.

Contribution

It introduces a tensorial representation of stress-energy multipoles, including derivation of their dynamical equations and the relation to tensor field moments.

Findings

Components depend on a choice of vector along the worldline

All multipoles can be expressed using covariant derivatives

Components are unique for each multipole

Abstract

We investigate stress-energy tensors constructed from the covariant derivatives of delta functions on a worldline. Since covariant derivatives are used all the components transform as tensors. We derive the dynamical equations for the components, up to quadrupole order. The components do, however, depend in a non-tensorial way, on a choice of a vector along the worldline. We also derive a number of important results about general multipoles, including that their components are unique, and all multipoles can be written using covariant derivatives. We show how the components of a multipole are related to standard moments of a tensor field, by parallelly transporting that tensor field.