Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 50 / 4 / a / Solution

Vary the geodesic Lagrangian before imposing the normalization of proper time. With $u^{\mu }=d{x}^{\mu }/d\tau$,The Euler-Lagrange equation isequivalently the affinely parametrized geodesic equation. For the static weak metric, its leading spatial connection coefficient isAll terms with two spatial velocities are suppressed by $v^2$. Therefore $d^2{x}^i/d{\tau }^2\simeq -{\partial }_i\Phi (dt/d\tau {)}^2$.

Because $t$ is cyclic, the same Euler-Lagrange equation gives $(1+2\Phi )dt/d\tau =\text{constant}$. On converting to coordinate time, the additional term $(d^{2}t/d{\tau }^2)(d{x}^i/dt)$ is of order $v^2\nabla \Phi$ and can be dropped. Henceto leading order, the Newtonian limit of general relativity. Restoring SI units gives $\Phi ={\Phi }_{\mathrm{SI}}/c^2$, so the acceleration is $-\nabla (c^2\Phi )$. Substituting $L=1$ from proper-time normalization before varying would incorrectly erase the dynamics.