Abstract We propose a solution to a classic problem in gravitational physics consisting of deïŹning the spin associated with asymptotically-ïŹat spacetimes. We advocate that the correct asymptotic symmetry algebra to approach this problem is the generalizedâBMS 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 deïŹning 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 diïŹeomorphisms forming its isotropy subalgebra. To elucidate the parallelism between our analysis and the PoincarĂ© case, we clarify several features of the PoincarĂ© embedding in gbms and reveal the presence of condensate ïŹelds associated with the symmetry breaking from gbms to PoincarĂ©. 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 PauliâLubaĆski 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.
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