On gravitational radiation and the energy flux of matter
  A suitable derivative of Einstein's equations in the framework of the teleparallel equivalent of general relativity (TEGR) yields a continuity equation for the gravitational energy-momentum. In particular, the time derivative of the total gravitational energy is given by the sum of the total fluxes of gravitational and matter fields energy. We carry out a detailed analysis of the continuity equation in the context of Bondi and Vaidya's metrics. In the former space-time the flux of the gravitational energy is given by the well known expression in terms of the square of the news function. It is known that the energy definition in the realm of the TEGR yields the ADM (Arnowitt-Deser-Misner) energy for appropriate boundary conditions. Here we show that the same definition also describes the Bondi energy. The analysis of the continuity equation in Vaidya's space-time shows that the variation of the total gravitational energy is given by the energy flux of matter only. 
J. W. Maluf and F. F. Faria