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, , of null infinity converges uniformly to a cross-section , then the angular momentum, J _n , on should converge to the angular momentum, J, on . 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.