Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 35E / Solution

The metric is nonsingular on $r>0$; $r=0$ is a coordinate boundary at which the displayed coefficients diverge. The nonzero Christoffel symbols areWriting dots for derivatives with respect to proper time, the geodesic equations areThe first gives $\dot{t}=k{r}^2$ for a constant $k$, chosen positive for a future-directed particle. Normalization of a timelike four-velocity gives $(-{\dot{t}}^2+{\dot{r}}^2)/r^2=-1$, so ${\dot{r}}^2=k^2{r}^4-r^2$. Dividing by ${\dot{t}}^2$ givesIntegrating on each side of the turning point yields $\sqrt{r^2-k^{-2}}=|t-t_0|$, hence $r^2-(t-t_0{)}^2=k^{-2}$. Choose $t_0=0$ and let $\tau =0$ at the minimum radius. ThenDifferentiating verifies $\dot{t}={\sec }^2\tau /k=k{r}^2$, the normalization and the radial Geodesic equation. proper time therefore remains finite as the coordinate radius and time tend to infinity at the endpoints of this chart.