Vacuum momenta of small spheres

It has been shown for some definitions of quasilocal mass and momentum that for small spheres in vacuum there is a dominating fifth-order contribution determined by the Bel-Robinson tenser. While for Hawking's quasilocal mass this contribution is always strictly positive except in Minkowski spacetime, we show that in type-N spacetimes, but in no other spacetimes, the momenta defined by Dougan-Mason and by Ludvigsen-Vickers are to leading order null vectors parallel to the unique principal null direction of the Weyl tenser. This means that the corresponding rest masses are zero to fifth order and that the momenta for such plane gravitational waves point in the propagation direction of the waves.