Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / b / Solution

For an affine parameter $\lambda$, the geodesic equation and the connection coefficients from part (a) givewhere dots denote derivatives with respect to $\lambda$.

On a line of constant $\varphi$, one has $\dot{\varphi }=0$. The equations reduce to $\ddot{r}=0$, so $r$ itself can be chosen as an affine parameter. Every such radial line is therefore a geodesic.

For a circle $r=r_0>0$, the first equation instead requiresBoth hyperbolic factors are positive when $r_0>0$, so $\dot{\varphi }=0$. This gives only a constant point, not a parametrized circle. Thus no circle of constant positive $r$ is a geodesic, in agreement with the coordinate geodesics of the hyperbolic polar metric.