On the Energy-Momentum Tensor [1940b]

Abstract

The work [1] of Lorentz, Hilbert, De Donder, F. Klein and Weyl has established the close relation which connects the energy-momentum tensor of an arbitrary system of physical entities, such as material particles or electromagnetic fields, to the gravitational field. In principle, this relation leads automatically to a well-determined, symmetric form of this tensor, as soon as the Lagrangian of the system under consideration has been expressed in a way which is invariant under arbitrary transformations of the space-time coordinates. But it may appear at first sight that this general procedure for deriving the energy-momentum tensor will run into practical difficulties if it is applied, since it seems to require in each case special considerations and calculations, relating to the gravitational potentials, i.e. on variables not directly related to the problem, and whose influence in general is negligible empirically. It is for this reason that procedures which are less direct, but which can be applied immediately to a Lagrangian invariant only under the Lorentz group, have often been preferred. On the other hand, these procedures also require special considerations in each case, especially to ensure that a symmetric form of the tensor is found [2], so that no practical advantage compensates for the unsatisfactory neglect of the profound relation which exists between the energy-momentum tensor and the general invariance of the Lagrangian.