Stress energy momentum in terms of geodesic accelerations and variational tensors including torsion

Abstract

General relativity and its extensions including torsion identify stress energy momentum as being proportional to the Einstein tensor, thus ensuring both symmetry and conservation. Here we visualize stress energy and momentum by identifying the associated relative fractional accelerations of geodesics encoded in the Einstein tensor. This also provides an intuitive explanation for the vanishing divergence of the Einstein tensor. In order to obtain this same energy and momentum for other actions such as that of Dirac theory including torsion, we then review the various stress energy momentum tensors resulting from the variation of different quantities derived from parallel transport, and detail their interrelationships. This provides an opportunity to revisit some classic material from a geometric point of view, including Einstein–Cartan theory, the Sciama–Kibble formalism, and the Belinfante–Rosenfeld relation, whose derivation in the mostly pluses signature would seem to not be otherwise readily available.