On the energy tensor of spinning massive matter in classical field theory and general relativity

Abstract

We are looking for the energy tensor which gives the correct local distribution of the energy of spinning massive matter fields (“localization of energy”). This is particularly important for the formulation of a consistent gravitational theory. We start with a Lagrangian formalism in flat spacetime. In (20, 21) we derive the set of all possible energy-momentum and spin tensors compatible with the conservation laws. They depend on two unknown fields, U ijk and Z ijk , which have to be specified for the solution of the localization problem. In particular, the most general symmetric energy-momentum tensor is made explicit in (26), the symmetrization procedure of the Belinfante–Rosenfeld type being a special case of our simpler and more general prescription. For energy and spin localization, which are related by angular momentum conservation, our special relativistic formalism naturally suggests one of the two alternatives: U = canonical spin and Z = 0, or U = 0 and Z = 0. Applied to gravitational theory, the first alternative leads to Einstein's general relativity, but to some unsatisfactory features for spinning massive fields. The second alternative includes spin on a dynamical basis and leads to U 4 theory (general relativity plus torsion), i.e. to a space-time described by a Riemann–Cartan geometry. Spin fits more naturally into the U 4 scheme and the unpleasant features of Einstein's theory disappear. We eventually recognize the canonical energy-momentum tensor as the “true” one for spinning massive fields.