Poisson algebra of quasilocal angular momentum and its asymptotic limit

Abstract

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 $\{L(\xi), L(\eta) \}_{\rm P.B.} =({1/16\pi)}\, L( [\xi, \eta]_{\rm L} )$, ({\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 $\{ L^2, L(\xi) \}_{\rm P.B.} = 0$, 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.