POSITIVITY AND DEFINITIONS OF MASS

This paper deals with problems and properties of definitions of mass in general relativity. We prove that it is sufficient for the spinor field determining the Nester-Witten 2-form to satisfy only one complex scalar equation in order that the integral over a convex 2-surface is positive when the dominant energy condition holds. As corollaries we show that the quasilocal momentum of Ludvigsen and Vickers is future-pointing, and we obtain a proof of the positivity of the Bondi mass which uses a different spinor propagation equation to previous proofs. As another corollary, we show that there are infinitely many ways of defining a quasilocal 4-momentum which is future-pointing, allows definition of a corresponding positive mass, gives the Bondi and ADM momenta at infinity, is zero in flat spacetime and is correct in linearized theory and in the spherically symmetric case.