On the definition of the spin charge in asymptotically-flat spacetimes

We propose a solution to a classic problem in gravitational physics consisting of defining the spin associated with asymptotically-flat spacetimes. We advocate that the correct asymptotic symmetry algebra to approach this problem is the generalized-BMS algebra gbms instead of the BMS algebra used hitherto in the literature for which a notion of spin is generically unavailable. We approach the problem of defining the spin charges from the perspective of coadjoint orbits of gbms and construct the complete set of Casimir invariants that determine gbms coadjoint orbits, using the notion of vorticity for gbms. This allows us to introduce spin charges for gbms as the generators of area-preserving diffeomorphisms forming its isotropy subalgebra. To elucidate the parallelism between our analysis and the Poincar\'e case, we clarify several features of the Poincar\'e embedding in gbms and reveal the presence of condensate fields associated with the symmetry breaking from gbms to Poincar\'e. We also introduce the notion of a rest frame available only for this extended algebra. This allows us to construct, from the spin generator, the gravitational analog of the Pauli--Luba\'nski pseudo-vector. Finally, we obtain the gbms moment map, which we use to construct the gravitational spin charges and gravitational Casimirs from their dual algebra counterparts.