Cross-Section Continuity of Definitions of Angular Momentum

TL;DR

This paper introduces a criterion called cross-section continuity for evaluating angular momentum definitions at null infinity, showing that some definitions satisfy it while others do not, thereby guiding the selection of physically consistent angular momentum measures.

Contribution

It establishes the cross-section continuity criterion for angular momentum at null infinity and compares several existing definitions against this standard.

Findings

DS definition satisfies cross-section continuity due to flux existence.

CN modification does not satisfy the continuity criterion.

CWY definition satisfies the cross-section continuity criterion.

Abstract

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, ${\mathcal C}_n$, of null infinity converges uniformly to a cross-section ${\mathcal C}$, then the angular momentum, $J_n$, on ${\mathcal C}_n$ should converge to the angular momentum, $J$, on ${\mathcal C}$. 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 (CN) -- which encompasses numerous other alternative definitions -- does not satisfy cross-section continuity. On the other hand, we prove that the Chen-Wang-Yau (CWY) definition does satisfy the cross-section continuity criterion.