On certain quasi-local spin-angular momentum expressions for small spheres

TL;DR

This paper compares quasi-local spin-angular momentum expressions for small spheres in various spacetimes, revealing their leading order behaviors and how they relate to energy-momentum and Bel-Robinson tensors.

Contribution

It demonstrates the equivalence of different quasi-local spin-angular momentum expressions at leading order for small spheres and clarifies their dependence on spacetime properties.

Findings

In non-vacuum, leading order is r^4, involving energy-momentum tensor contraction.

In vacuum, leading order is r^6, involving Bel-Robinson tensor contraction.

Expressions coincide in leading order, differing mainly by sign in non-vacuum cases.

Abstract

The Ludvigsen-Vickers and two recently suggested quasi-local spin-angular momentum expressions, based on holomorphic and anti-holomorphic spinor fields, are calculated for small spheres of radius $r$ about a point $o$. It is shown that, apart from the sign in the case of anti-holomorphic spinors in non-vacuum, the leading terms of all these expressions coincide. In non-vacuum spacetimes this common leading term is of order $r^4$, and it is the product of the contraction of the energy-momentum tensor and an average of the approximate boost-rotation Killing vector that vanishes at $o$ and of the 3-volume of the ball of radius $r$. In vacuum spacetimes the leading term is of order $r^6$, and the factor of proportionality is the contraction of the Bel-Robinson tensor and an other average of the same approximate boost-rotation Killing vector.