  Penrose's twistorial approach to the definition of angular momentum at null
infinity is developed so that angular momenta at different cuts can be
meaningfully compared. This is done by showing that the twistor spaces
associated with different cuts of scri can be identified as manifolds (but not
as vector spaces). The result is a well-defined, Bondi-Metzner-Sachs-invariant
notion of angular momentum in a radiating space-time; the difficulties and
ambiguities previously encountered are attached to attempts to express this in
special-relativistic terms, and in particular to attempts to identify a single
Minkowski space of origins. Unlike the special-relativistic case, the angular
momentum cannot be represented by a purely j=1 quantity M_{ab}, but has
higher-j contributions as well. Applying standard kinematic prescriptions,
these higher-j contributions are shown to correspond precisely to the shear.
Thus it appears that shear and angular momentum should be regarded as different
aspects of a single unified concept.
