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

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, 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, 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.