Satisfaction of deDonder and Trautman conditions by radiative solutions of the Einstein field equations

We show that when the Einstein field equations for the gravitational field are modified by imposing the deDonder coordinate conditions these equations can be ‘solved’ in terms of source functions using the retarded Green's function for the d'Alembertian in flat space. The ‘solution’, which becomes an actual solution in the fast-motion approximation, is shown to satisfy the deDonder conditions if and only if the stress-energy tensor of the sources of the gravitational field is covariantly conserved. It is also shown to satisfy the Trautman outgoing radiation condition.