A static black hole in a five-dimensional spacetime is described by the metric

$$d s^{2}=-\left(1-\frac{\mu}{r^{2}}\right) d t^{2}+\left(1-\frac{\mu}{r^{2}}\right)^{-1} d r^{2}+r^{2}\left[d \psi^{2}+\sin ^{2} \psi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$$

where $\mu>0$ is a constant.

A geodesic lies in the plane $\theta=\psi=\pi / 2$ and has affine parameter $\lambda$. Show that

$$E=\left(1-\frac{\mu}{r^{2}}\right) \frac{d t}{d \lambda} \quad \text { and } \quad L=r^{2} \frac{d \phi}{d \lambda}$$

are both constants of motion. Write down a third constant of motion.

Show that timelike and null geodesics satisfy the equation

$$\frac{1}{2}\left(\frac{d r}{d \lambda}\right)^{2}+V(r)=\frac{1}{2} E^{2}$$

for some potential $V(r)$ which you should determine.

Circular geodesics satisfy the equation $V^{\prime}(r)=0$. Calculate the values of $r$ for which circular null geodesics exist and for which circular timelike geodesics exist. Which are stable and which are unstable? Briefly describe how this compares to circular geodesics in the four-dimensional Schwarzschild geometry.