Relativistic angular momentum for asymptotically flat Einstein-Maxwell manifolds

Abstract

Treating general relativity as a Yang-Mills theory based on Lorentz invariance of the second kind leads to the derivation of six spin currents, which are linear combinations of the Ricci rotation coefficients, together with a Gauss theorem for the integrated charges. For asymptotically flat Einstein-Maxwell manifolds these charges are calculated by performing two-surface integrals at future null infinity. The gauge dependence of these charges, which arises because the underlying group is non-Abelian, is removed by making a canonical alinement of the asymptotic frames of reference. The Lorentz generators then arise as a field in asymptotic spin space and defined on the set of all outgoing null cones, which is the carrier space of the B.M.S. group. In the absence of outgoing gravitational radiation the Lorentz generators transform under translations as a moment of momentum. However, the points about which moments are taken belong not to the original Riemannian manifold but rather to a four-parameter family of cones isomorphic to Minkowski space-time. From the angular momentum structure there arises a natural twistor structure. Points on the twistor and the centre of mass of the system are defined as world lines in the manifold of cones. The remainder of the paper is devoted to formulating physical laws in cone space. These include a Poynting’s theorem for the radiation of spin.