Angular momentum, spinors and twistors

Abstract

Twistors appear to provide a satisfactory treatment of angular momentum for gravitationally radiating systems. The approach is manifestly Bondi-Metzner-Sachs (BMS) invariant, and there are no supertranslation ambiguities. The resulting definitions of center of mass and spin are appealing: unphysical contributions from bad cuts are canceled off from the center of mass, and the spin appears formally as a displacement of the center of mass into the complex. For transitions between asymptotically Minkowskian regimes (non-radiative regimes with purely electric Bondi shear), when there is no supertranslation offset (equivalently, radation memory) between the regimes the results are in agreement with those deducible from other approaches. However, when there is an offset, the results are different. The twistor-derived change-of-origin formula is closely parallel to the specialrelativistic one, with an algebraic cross-product between the energy-momentum and a directiondependent translation derived from the supertranslation. (No supermomenta appear.) There is also a "longitudinal" contribution to the emitted angular momentum (one sensitive to the total energymomentum and not just the emitted energy-momentum), and terms which are both linear and cubic in the gravitational radiation (whereas BMS-based definitions give purely quadratic contributions). The first-order terms mean that the supertranslation offsets can contribute to the exchange of angular momentum with gravitational radiation, even in weak-field limits. This is illustrated with a simple almost-special-relativistic model.