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

Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> and construct the complete set of Casimir invariants that determine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> coadjoint orbits, using the notion of vorticity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> . This allows us to introduce spin charges for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> as the generators of area-preserving diffeomorphisms forming its isotropy subalgebra. To elucidate the parallelism between our analysis and the Poincaré case, we clarify several features of the Poincaré embedding in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> and reveal the presence of condensate fields associated with the symmetry breaking from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> to Poincaré. 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–Lubański pseudo-vector. Finally, we obtain the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtext mathvariant="sans-serif">gbms</mml:mtext> </mml:mrow> </mml:math> moment map, which we use to construct the gravitational spin charges and gravitational Casimirs from their dual algebra counterparts.