Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 3

Write $g_{\mu \nu }={\eta }_{\mu \nu }+ϵh_{\mu \nu }$. The linearized inverse metric is $g^{\mu \nu }={\eta }^{\mu \nu }-ϵh^{\mu \nu }$, and all quadratic Christoffel products are $O({ϵ}^2)$. The wave-coordinate condition linearizes toUsing the supplied Ricci formula then gives $G_{\mu \nu }^{(1)}=-\frac{1}{2}\square {\bar{h}}_{\mu \nu }$. Consequently the Linearized Einstein equations in Lorenz gauge in linearized gravity areThis Lorenz condition is the first-order form of the harmonic coordinate equations.

To first order in the angular velocity, $T^{00}=\rho$ and $T^{i0}=\rho v^i$ with $v=(-\Omega y,\Omega x,0)$. The conservation equation ${\partial }_{\mu }{T}^{\mu 0}=0$ and time independence giveThus $\boxed{x{\partial }_y\rho -y{\partial }_x\rho =0}$: the density is invariant under rotations about the z-axis.

Stationarity changes $\square$ to the spatial Laplace operator. The Green function of the Poisson equation therefore givesHere the bar denotes the trace-reversed metric perturbation; this is the variable denoted by $h$ in the displayed field equation of the question.

For $r\gg R$, the multipole expansion isThe center of mass condition removes the mass dipole. Axisymmetry makes the mixed second moments vanish and gives $\int \rho x^{\prime 2}=\int \rho y^{\prime 2}$. Undoing trace reversal therefore yieldswhereThus $a=J_z/m$ is the specific angular momentum, and the cross term is the dipole part of the slowly rotating weak-field metric.