Approximate twistors and positive mass

In this paper, the problem of comparing initial data to a reference solution for the vacuum Einstein field equations is considered. This is not done in a coordinate sense, but through quantification of the deviation from a specific symmetry. In a recent paper (Backdahl and Valiente Kroon 2010 Phys. Rev. Lett. 104 231102), this problem was studied with the Kerr solution as a reference solution. This analysis was based on valence 2 Killing spinors. In order to better understand this construction, we analyse the analogous construction for valence 1 spinors solving the twistor equation. This yields an invariant that measures how much the initial data deviates from Minkowski data. Furthermore, we prove that this invariant vanishes if and only if the mass vanishes. Hence, we get a proof of the positivity of mass.