 Solution

In Lorenz gauge, ${\partial }_{\mu }{\bar{h}}^{\mu \nu }=0$. A leading outgoing far-zone field depends on retarded time $u=t-r$, so ${\partial }_j=-{\hat{x}}_j{\partial }_u+O(r^{-1})$. Taking $\nu =i$ and using ${\bar{h}}^{0i}=-{\bar{h}}_{0i}$ givesDropping a nonradiative integration constant and using part i,