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

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