The positivity of the Bondi mass

Abstract

Two component spinor techniques, similar to those used by Witten, are used to express the Bondi momentum of an asymptotically flat space-time in the form of an integral over an asymptotically null, space-like hypersurface 2. It is then shown that the Bondi mass is positive, given the existence of a Green function for the 'Witten equation' on 2. In a recent paper Witten (1981) has given a simple proof of the positivity of the ADM mass of an asymptotically flat space-time by means of a simple argument using Dirac spinors. Witten's proof of the positive energy theorem is not, however, quite correct as it stands as it uses an invalid three-dimensional truncation of the four-dimensional Gauss divergence theorem. By giving a new four-dimensional covariant expression for the ADM four-momentum, Nester (1981) has been able to avoid this difficulty and give a simple proof of the positivity of the ADM mass. We show here how similar techniques may be used in the case of the Bondi mass. Let M be an asymptotically flat space-time with future null infinity $+. The Bondi-Sachs (Bondi et a1 1962, Sachs 1962) four-momentum Pa(S) of M is a four-vector function, defined on the space of all space-like cross sections (cuts) of 4+, which lies in the Minkowski space of BMS translations T. If we let T = YO 9 where Y is the space of two-spinors, then, on using the Penrose abstract index notation (Penrose 1968), we may write