The symmetrical energy–momentum tensor derived by parametrization

Abstract

With reference to a paper by Goedecke in this journal attention is drawn to the fact that already in his original paper on the subject Rosenfeld proved the equality of the results of the two general procedures of symmetrizing energy–momentum tensors, i.e., the procedure of Belinfante (1939), utilizing the angular–momentum tensor, and the procedure of Rosenfeld (1940), taking the Lorentz metric limit of the manifestly symmetrical energy–momentum tensor of Riemannian space. Since Rosenfeld’s presentation of his procedure may give the misleading impression that it has something to do with curved spaces, general relativity, or gravitational theory, we show in the present paper how his scheme can be recast in a form, where one merely takes resort to an infinitesimal transformation of the ordinary Lorentz coordinates to arbitrary curvilinear coordinates, describing the same original Lorentz space of zero curvature. This transformation, of course, means a parametrization of the variational principle, and the analysis can thus be performed by means of a generalization of the theory of parameter-invariant variational principles. An expression for the symmetrized energy–momentum tensor is given, which is equivalent to that given by Rosenfeld, and in which the transformation functions are seen to vanish identically. The procedure is thus seen to be not so much a limiting process as a transformation to curvilinear coordinates, construction of a symmetrical energy–momentum tensor, and a transformation back again.