  In a recent letter we show that for an isolated system with a non symmetric
energy momentum tensor the usual forms of the center of mass motion theorem are
not valid. This was illustrated with a particular configuration of a magnet and
a point charge for which it was shown that what is usually regarded as the
center of mass of the system does not remain stationary even if the system is
isolated. In a subsequent work we demonstrated that the violation of the center
of mass motion theorem for isolated systems with spin is a direct consequence
of the conservation of total angular momentum. We also show that there exists a
generalized center of mass and spin which moves with constant velocity. In this
letter we show that this center of mass and spin corresponds to the center of
mass defined by the Belinfante-Rosenfeld tensor. We also show that, if the spin
density instead of being of microscopic origin appears by a scaling process,
the macroscopic Belinfante-Rosenfeld tensor emerges from the average of the
microscopic energy-momentum tensor as the true macroscopic energy momentum
tensor. This implies that in general spin has to be considered as a source of
the gravitational field in general relativity.
