Whom actually do multipole moments belong to?

Using an integral definition given in Hernandez-Pastora et al. (Class Quantum Gravity 33:225009, 2016) 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.