Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 1 / a

Under parallel transport, the gyroscope spin obeys $u^b{\nabla }_b{s}^a=0$, and its freely falling circular orbit is a geodesic, $u^b{\nabla }_b{u}^a=0$. Metric compatibility therefore givesThus $s^a{u}_a$ is constant along the orbit.

Initially $s^a$ is radial, whereas the circular-orbit four-velocity has only $t$ and $\varphi$ components. The diagonal Schwarzschild metric therefore gives $s^a{u}_a=0$ at $t=0$. By part i,throughout the orbit.