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In classical continuum mechanics the balance or unbalance equation of pseudomomentum reflects the material invariance of the system under study. It relates the time derivative of pseudo-momentum and the flux of the Eshelby stress. It is legitimate to inquire whether this structure is conserved in a relativistic four-dimensional background. We examine here the relativistic definition of pseudo/matertal momentum using simultaneously variational and direct approaches (the latter using the canonical projection of space-time onto the material manifold). It appears that the truly material entities, just as those in a proper frame, should be the basic ones, being independent of the relativity framework used.