Past exam of the general relativity course of the University of Oxford / 2022 / 2 / f

The only nonconstant entry of the metric tensor is $g_{\varphi \varphi }=R^2{\sin }^2\theta$, whose partial derivative is ${\partial }_{\theta }{g}_{\varphi \varphi }=2R^2\sin \theta \cos \theta$. The nonzero Christoffel symbols areAlong a fixed latitude with $0<\theta <\pi$, parallel transport givesSet $X=V^{\theta }$ and $Y=\sin \theta V^{\varphi }$. Since $\theta$ is fixed, this linear ordinary differential equation becomesIt is a rotation at constant rate $-\cos \theta$. Its general solution, also valid at the equator where the rate is zero, isThe initial data give $A=1$ and $B=0$. At $\theta =\pi /3$, a complete circuit has $\varphi \cos \theta =\pi$, soThe physical squared length uses the metric tensor, rather than the Euclidean sum of coordinate components:It is constant and equals $R^2$ for these data. More generally, metric compatibility gives $dg(V,V)/d\varphi =({\nabla }_{{\partial }_{\varphi }}g)(V,V)+2g({\nabla }_{{\partial }_{\varphi }}V,V)=0$. The final change of direction is holonomy, exemplified by parallel transport around a latitude of the unit sphere; parallel transport preserves length without requiring the direction to return unchanged.