Internal spin angular momentum of an asymptotically flat spacetime

Abstract

In this paper we investigate the manner in which the internal spin angular momentum of a spinor field is encoded in the gravitational field at asymptotic infinity. The inclusion of internal spin requires us to reanalyze our notion of asymptotic flatness. In particular, the Poincar\'e symmetry at asymptotic infinity must be replaced by a spin-enlarged Poincar\'e symmetry. Likewise, the generators of the asymptotic symmetry group must be supplemented to account for the internal spin. In the Hamiltonian framework of first-order Einstein-Cartan gravity, the extra generator comes from the boundary term of the Gauss constraint in the asymptotically flat context. With the additional term, we establish the relations among the Noether charges of a Dirac field, the Komar integral, and the asymptotic Arnowitt-Deser-Misner-like geometric integral. We show that by imposing mild restraints on the generating functionals of gauge transformations at asymptotic infinity, the phase space is rendered explicitly finite. We construct the energy-momentum and the new total ($\mathrm{\text{spin}}+\mathrm{\text{orbital}}$) angular momentum boundary integrals that satisfy the appropriate algebra to be the generators of the spin-enlarged Poincar\'e symmetry. This demonstrates that the internal spin is encoded in the tetrad at asymptotic infinity. In addition, we find that a new conserved and (spin-enlarged) Poincar\'e invariant charge emerges that is associated with the global structure of a gauge transformation.