Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 2 / b / Solution

In the weak-field, slow-source approximation, the leading luminosity is the quadrupole formulain units $G=c=1$, whereOnly the $\ell =2$ symmetric trace-free coefficient contributes. Indeed,and contraction with traceless $a_{kl}$ removes the first term. ThereforeThe power crossing the large sphere at time $t$ is consequentlyThe monopole is conserved total mass, the dipole is center-of-mass motion, and every $\ell \ne 2$ term is orthogonal to the trace-free quadrupole at this leading order.