Poisson algebra of quasilocal angular momentum and its asymptotic limit

We study the previously proposed quasilocal angular momentum of gravitational fields in the absence of isometries. The quasilocal angular momentum $L(\xi )$ has the following attractive properties; ({\it i}) it follows from the Einstein’s constraint equations, ({\it ii}) it satisfies the Poisson algebra $\{L(\xi ),L(\eta ){\}}_{\mathrm{P}\mathrm{.}\mathrm{B}\mathrm{.}}=(1\mathrm{/}16\pi )L([\xi ,\eta {]}_{\mathrm{L}})$, ({\it iii}) its Poisson algebra reduces to the standard $SO(3)$ algebra of angular momentum at null infinity, and ({\it iv}) it reproduces the standard value for the Kerr spacetime at null infinity. It will be argued that our definition is a quasilocal and canonical generalization of A. Rizzi’s geometric definition at null infinity. We also propose a new definition of an {\it invariant} quasilocal angular momentum $L^2$ such that $\{L^2,L(\xi ){\}}_{\mathrm{P}\mathrm{.}\mathrm{B}\mathrm{.}}=0$, which becomes $(ma{)}^2$ at the null infinity of the Kerr spacetime. Therefore, it may be regarded as a quasilocal generalization of the Casimir invariant of ordinary angular momentum in the flat spacetime.