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

For a slowly moving particle in a static weak field, the spatial part of the geodesic equation reduces toTo first order in the perturbation,and henceNewtonian acceleration is $-{\delta }^{ij}{\partial }_j\Phi$, so $h_{00}=-2\Phi$. Substitution into part (a), followed by the Poisson equation ${\nabla }^2\Phi =4\pi G\rho$, givesThus the Newtonian limit of general relativity fixes