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

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.