(a) The Schwarzschild metric is

$$d s^{2}=-\left(1-r_{s} / r\right) d t^{2}+\left(1-r_{s} / r\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

(in units for which the speed of light $c=1$ ). Show that a timelike geodesic in the equatorial plane obeys

$$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2} E^{2}$$

where

$$2 V(r)=\left(1-\frac{r_{s}}{r}\right)\left(1+\frac{h^{2}}{r^{2}}\right)$$

and $E$ and $h$ are constants.

(b) For a circular orbit of radius $r$, show that

$$h^{2}=\frac{r^{2} r_{s}}{2 r-3 r_{s}} .$$

Given that the orbit is stable, show that $r>3 r_{s}$.

(c) Alice lives on a small planet that is in a stable circular orbit of radius $r$ around a (non-rotating) black hole of radius $r_{s}$. Bob lives on a spacecraft in deep space far from the black hole and at rest relative to it. Bob is ageing $k$ times faster than Alice. Find an expression for $k^{2}$ in terms of $r$ and $r_{s}$ and show that $k<\sqrt{2}$.