Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 37E / c / Solution

For a point mass, $\Phi =-GM/r$, and part (b) givesThe linearized Ricci tensor and scalar obeyFor a static perturbation with $h_{0i}=0$, this becomesAway from the source, ${\nabla }^2{h}_{00}=0$. For $h_{ij}=f(r)x_i{x}_j$, one has $h_{ii}=r^{2}f$, so the two supplied identities giveThe vacuum trace equation $R=0$ therefore reduces towhose general solution for $r>0$ isThe scalar equation alone leaves $C$ arbitrary. Imposing the remaining vacuum components $R_{ij}=0$ and matching the mass fixes $C=2GM$, as in the linearized point-mass metric in radial Cartesian form.