The Schwarzschild metric is given by

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

(i) Show that geodesics in the Schwarzschild spacetime obey the equation

$$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2} E^{2}, \quad \text { where } V(r)=\frac{1}{2}\left(1-\frac{2 M}{r}\right)\left(\frac{L^{2}}{r^{2}}-Q\right)$$

where $E, L, Q$ are constants and the dot denotes differentiation with respect to a suitably chosen affine parameter $\lambda$.

(ii) Consider the following three observers located in one and the same plane in the Schwarzschild spacetime which also passes through the centre of the black hole:

• Observer $\mathcal{O}_{1}$ is on board a spacecraft (to be modeled as a pointlike object moving on a geodesic) on a circular orbit of radius $r>3 M$ around the central mass $M$.

• Observer $\mathcal{O}_{2}$ starts at the same position as $\mathcal{O}_{1}$ but, instead of orbiting, stays fixed at the initial coordinate position by using rocket propulsion to counteract the gravitational pull.

• Observer $\mathcal{O}_{3}$ is also located at a fixed position but at large distance $r \rightarrow \infty$ from the central mass and is assumed to be able to see $\mathcal{O}_{1}$ whenever the two are at the same azimuthal angle $\phi$.

Show that the proper time intervals $\Delta \tau_{1}, \Delta \tau_{2}, \Delta \tau_{3}$, that are measured by the three observers during the completion of one full orbit of observer $\mathcal{O}_{1}$, are given by

$$\Delta \tau_{i}=2 \pi \sqrt{\frac{r^{2}\left(r-\alpha_{i} M\right)}{M}}, \quad i=1,2,3$$

where $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$ are numerical constants that you should determine.

(iii) Briefly interpret the result by arranging the $\Delta \tau_{i}$ in ascending order.