Whom actually do multipole moments belong to?

TL;DR

This paper establishes a link between relativistic multipole moments and the source material in static axisymmetric spacetimes, providing a method to compute the quadrupole moment from the source's energy-momentum tensor.

Contribution

It introduces a generalized relativistic Gauss theorem and demonstrates how to relate multipole moments to the source's interior properties in Weyl exterior fields.

Findings

Derived a formula linking RMM to source energy-momentum tensor

Established a method to compute the relativistic quadrupole moment

Proved the connection between volume integrals and multipole moments in static spacetimes

Abstract

Using an integral definition to calculate the relativistic multipole moments (RMM), and the ensuing generalized relativistic Gauss theorem, we prove that the evaluation of that volume integral in Erez-Rosen coordinates, leads to a specific link between the RMM and the source of the exterior space--time, provided we have a global static axisymmetric metric in that coordinate system for any Weyl exterior field. This result allows to establish a relationship between the RMM and certain volume integral expressions involving the material content of the source from its energy-momentum tensor as well as the interior metric. In particular the relativistic quadrupole moment for the Erez-Rosen space-time is obtained.