Spin and Angular Momentum in General Relativity

TLDR

In general relativity, coordinate transformations generate four point‑to‑point conservation laws typically identified with energy and linear momentum. The authors introduce a local orthonormal vierbein field and employ superpotential techniques to construct a contravariant stress‑energy tensor containing symmetric Dirac and Maxwell terms plus asymmetric gravitational contributions, a method applicable to any metric‑based covariant theory. They find that conservation laws require spin angular momentum terms that must be accompanied by orbital angular momentum, and these expressions constitute the orbital angular momentum.

Abstract

In the general theory of relativity, the group of coordinate transformations gives rise to four point-to-point conservation laws, which are usually identified with energy and linear momentum. In the presence of a semiclassical Dirac field, it is convenient to introduce at each point of space-time an arbitrary set of four orthonormal vectors (quadrupeds, "beine") and to consider the group of "bein" transformations, which then play the role of local, nonholonomic lorentz transformations. A search for the corresponding conservation laws leads to terms that have the form of a spin angular momentum and which, in order to be conserved, must be supplemented by terms representing the orbital angular momentum. The technique of the so-called superpotentials has enabled us to introduce, in addition to the canonical stress-energy, a "contravariant" stress-energy which contains the usual symmetric Dirac and Maxwell terms and also asymmetric, purely gravitational terms. It is this set of expressions which enters into the orbital angular momentum. The techniques presented here are applicable to more general covariant theories, provided the gravitational field is represented by a metric tensor.