Momentum, angular momentum and their quasi-local null surface extensions

Let M be an asymptotically flat space-time which satisfies the dominant energy condition and let N be a null hypersurface of M which meets I+ in a space-like cross section S given by r= infinity , where r is an affine parameter along the generators of N. It is shown that there exists a quasi-local momentum Pa(r), defined by means of an integral over the r=constant cross sections of N, which tends to the Bondi momentum as r to infinity , and which satisfies the radial 'mass-gain condition' Pa(r2)ka>or=Pa(r1)ka when r2>or=r1, where ka is a constant future pointing null vector. This condition is then used to show that, in certain circumstances, the Bondi momentum is always future pointing or, in other words, that the Bondi mass is always positive. A quasi-local angular momentum is also defined, which tends to Bramson's (1975) angular momentum as r to infinity .