For the metric

$$\mathrm{d} s^{2}=\frac{1}{r^{2}}\left(-\mathrm{d} t^{2}+\mathrm{d} r^{2}\right), \quad r \geqslant 0,$$

obtain the geodesic equations of motion. For a massive particle show that

$$\left(\frac{\mathrm{d} r}{\mathrm{~d} t}\right)^{2}=1-\frac{1}{k^{2} r^{2}}$$

for some constant $k$. Show that the particle moves on trajectories

$$r^{2}-t^{2}=\frac{1}{k^{2}}, \quad k r=\sec \tau, \quad k t=\tan \tau$$

where $\tau$ is the proper time, if the origins of $t, \tau$ are chosen appropriately.