Spin and Angular Momentum in General Relativity

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.