When a spacetime takes Bondi radiating metric and is vacuum and asymptotically flat at spatial infinity which ensures the positive mass theorem, we prove that the standard ADM energy--momentum is the past limit of the Bondi energy--momentum. We also derive a formula relating the ADM energy--momentum of any asymptotically flat spacelike hypersurface to the Bondi energy--momentum of any null hypersurface. The formula indicates that the Bondi mass is always less than the ADM total energy if the system has news. The assumed asymptotic flatness precludes gravitational radiation. We therefore study further the relation between the ADM total energy and the Bondi mass when gravitational radiation emits. We find that, in this case, the ADM total energy is no longer the past limit of the Bondi mass. They differ by certain quantity relating to the news of the system.
"On the relation between ADM and Bondi energy--momenta." Adv. Theor. Math. Phys. 10 (2) 261 - 282, April 2006.