The Internal Spin Angular Momentum of an Asymptotically Flat Spacetime

TL;DR

This paper explores how internal spin of a spinor field is represented in the gravitational field at infinity in asymptotically flat spacetimes, leading to a spin-enlarged Poincaré symmetry and new conserved charges.

Contribution

It introduces a modified asymptotic symmetry group incorporating internal spin, and constructs boundary integrals for total angular momentum including spin contributions.

Findings

Internal spin is encoded in the asymptotic tetrad.

A new conserved charge related to gauge transformations is identified.

The phase space becomes finite with appropriate boundary conditions.

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 re-analyze our notion of asymptotic flatness. In particular, the Poincar\'{e} symmetry at asymptotic infinity must 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 ADM-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 (spin+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.