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Belinfante-Rosenfeld tensor and the inertia principle

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.