The Schwarzschild metric is given by

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

where $M$ is the mass in gravitational units. By using the radial component of the geodesic equations, or otherwise, show for a particle moving on a geodesic in the equatorial plane $\theta=\pi / 2$ with $r$ constant that

$$\left(\frac{d \phi}{d t}\right)^{2}=\frac{M}{r^{3}}$$

Show that such an orbit is stable for $r>6 M$.

An astronaut circles the Earth freely for a long time on a circular orbit of radius $R$, while the astronaut's twin remains motionless on Earth, which is assumed to be spherical, with radius $R_{0}$, and non-rotating. Show that, on returning to Earth, the astronaut will be younger than the twin only if $2 R<3 R_{0}$.