Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 3 / a

Assume a spatially compact source of size $d\ll r=|\mathbf{x}|$, internal speeds much smaller than one, and wavelength much larger than $d$. In the radiation zone,soDefine the mass quadrupole momentTwice using ${\partial }_{\mu }{T}^{\mu \nu }=0$, discarding boundary terms, givesHence

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,