Twistors and the BMS Group

TL;DR

This paper explores the behavior of angular momentum at null infinity in gravitational fields, revealing anomalies in boosts and translations linked to the shear quadrupole moment, with implications for supertranslation invariance.

Contribution

It demonstrates that angular momentum components exhibit anomalous translations due to shear quadrupole moments, a result consistent across weak-field limits and rooted in supertranslation invariance.

Findings

Boosts are supplemented by anomalous translations.

Anomalies are proportional to shear quadrupole moments.

Effect persists in the weak-field limit.

Abstract

We compute the motions of null infinity to which the components of the angular momentum of the gravitational field, as defined by Penrose, are conjugate. We find that the boosts are supplemented by anomalous translations. If $c_{ab}$ is the skew bivector determining the component of the angular momentum in question, the anomaly is proportional to $$c_{ab}t^b\phi ^{ad}\, ,$$ where $t^a$ is a unit vector in the direction of the Bondi--Sachs momentum and $\phi _{ab}$ is the quadrupole moment of the shear of the cut at which the angular momentum is evaluated. This effect persists in the weak--field limit. This surprising result is a general consequence of the requirement that angular momentum be supertranslation--invariant in a quiescent regime, and not some essentially twistorial peculiarity.