Cross-section continuity of definitions of angular momentum

We introduce a notion of "cross-section continuity' as a criterion for the viability of definitions of angular momentum, J, at null infinity: If a sequence of cross-sections, C-n, of null infinity converges uniformly to a cross-section C, then the angular momentum, J(n), on C-n should converge to the angular momentum, J, on C. The Dray-Streubel (DS) definition of angular momentum automatically satisfies this criterion by virtue of the existence of a well defined flux associated with this definition. However, we show that the one-parameter modification of the DS definition proposed by Compere and Nichols-which encompasses numerous other alternative definitions-does not satisfy cross-section continuity. On the other hand, we prove that the Chen-Wang-Yau definition does satisfy the cross-section continuity criterion.