Cross-Section Continuity of Definitions of Angular Momentum

TL;DR

This work defines cross-section continuity as a stringent viability criterion for angular momentum definitions at null infinity in general relativity. It analyzes three proposed notions—the Dray-Streubel (DS), Compere-Nichols (CN), and Chen-Wang-Yau (CWY) angular momenta—showing that DS satisfies continuity due to a well-defined flux, CN fails except when α=1 (in which case it reduces to DS), and CWY satisfies the criterion by coupling a local notion of pure rotation to data on each cross-section. The key contributions include a detailed continuity proof for CWY via flux relations and elliptic estimates, and a concrete no-go result for CN’s one-parameter modification. The results bolster CWY as a robust, cross-section-continuous angular momentum notion at null infinity, clarifying the boundaries of viable definitions in the presence of supertranslations and memory effects.

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.

Theorems & Definitions (2)

• Lemma 4.1
• proof