Past exam of the general relativity course of the University of Oxford / 2021 / 1 / c / Solution

Write $\mu =GM/c^2$ and $r_0=R_E+h$. The Christoffel symbols follow fromThe relevant time and radial entries, using $x^0=ct$, areFor the angular geodesic equations, ${\Gamma }_{r\theta }^{\theta }={\Gamma }_{r\varphi }^{\varphi }=1/r$, ${\Gamma }_{\varphi \varphi }^{\theta }=-\sin \theta \cos \theta$, and ${\Gamma }_{\theta \varphi }^{\varphi }=\cot \theta$. Thus, with primes on coordinates here denoting proper time derivatives,Initial conditions ${\theta }^{\prime }={\varphi }^{\prime }=0$ are preserved by these ordinary differential equations. The motion is therefore radial, with constant angular coordinates. Choose the polar axis so the apple is away from the angular coordinate singularities.

The exact remaining radial geodesic equation and four-velocity normalization areEliminating $dt/d\tau$ gives the particularly simple exact result $r^{\prime \prime }=-GM/r^2$. The supported observer's proper time $T$ differs from the apple's proper time by relative corrections of order $GM/(R_E{c}^2)$ and $v^2/c^2$. Consequently, to leading order in the weak-field approximation and $h/R_E$, $d^{2}r/d{T}^2\simeq -GM/R_E^2=-g$, with $r(0)=r_0$ and zero initial radial velocity. HenceThe proper height measured along a static radial ruler is ${\int }_{R_E}^{r}dr/\sqrt{f}\simeq r-R_E$ at this accuracy. The use of $T$ means the ground clock's elapsed time, with propagation delays corrected; raw visual arrival times include the changing light travel time.

If variation of $g$ over the fall is retained but relativistic corrections are neglected, a useful parametric solution isIts small-$\eta$ expansion gives the same constant-$g$ result. For an exact check on the choice of clock, the initial coordinate acceleration measured using $T$ is $-GMf(r_0)/[r_0^{2}f(R_E)]$, which reduces to the displayed approximation.