Whom actually do multipole moments belong to?

Using an integral definition given in Hernández-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.