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

Assume the test body moves slowly, $d{x}^i/dt=O(ϵ)$, and that the field is quasistatic, so time derivatives are of higher order. The spatial geodesic equation, parametrized by coordinate time, then reduces at leading order toDefining the Newtonian potential bygivesthe Newtonian equation of motion. Terms involving $h_{0i}$, spatial velocity, or time derivatives enter beyond the stated order.

The nonrelativistic fluid assumptions giveAssume again a stationary field and asymptotic flatness. The harmonic-gauge Linearized Einstein equations becomewhile the other trace-reversed components are smaller. Trace reversal then gives $h_{00}={\bar{h}}_{00}/2$ to leading order, soUsing $h_{00}=-2\Phi$ yieldsin units $G=1$, which is the Poisson equation of Newtonian gravity.

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.