Approximate twistors and positive mass

TL;DR

This paper introduces an invariant based on twistor equations that quantifies the deviation of initial data from Minkowski space and provides a proof of the positivity of mass in vacuum Einstein equations.

Contribution

It extends previous work using Killing spinors to analyze initial data by employing twistor equations, establishing a new invariant that detects mass positivity.

Findings

Invariant vanishes if and only if mass is zero

Provides a new proof of mass positivity

Connects twistor solutions to geometric properties of spacetime

Abstract

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 [T. B\"ackdahl, J.A. Valiente Kroon, Phys. Rev. Lett. 104, 231102 (2010)] 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, in the present article 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 of the mass vanishes. Hence, we get a proof of the positivity of mass.