The Bondi—Metzner—Sachs group in the nuclear topology

The Bondi-Metzner-Sachs group is topologized as a nuclear Lie group, and it is shown that irreducible representations arise from either (i) transitive $SL$(2,$C$) actions on supermomentum space, or (ii) cylinder measures in supermomentum space with respect to which the $SL$(2, $C$) action is strictly ergodic. The irreducibles arising from transitive actions are shown to be induced, and most of the theorems from a previous analysis (in which the group was given a Hilbert topology) are generalized so as to apply here. All non-discrete closed subgroups of $SL$(2, $C$) are found, and this analysis is used to construct all induced representations whose little groups are not both discrete and infinite. In the previous analysis, there were exactly two connected little groups, $SU$(2) and $\Gamma $($\Gamma $ double covers $SO$(2)). In the present analysis, exactly one additional connected little group $\Delta $ (which double covers $E$(2)) arises for faithful representations (that is, those for which the mass squared is defined); the associated mass squared value is zero. Exactly one further connected little group arises; it is associated with unfaithful representations (for which the mass squared is undefined) and is $SL$(2, $R$). The expected wide variety of new little groups only arises if the requirement of connectivity is dropped, their being several new non-connected little groups. Whenever the little group is non-compact the terms in the invariant supermomenta responsible for faithful representations have precisely defined directions. This suggests that the new faithful representations describe scattering states of gravitating systems. The new faithful representations are all decomposed with respect to restriction to the Poincare subgroup.