(i) Show that the geodesic equation follows from a variational principle with Lagrangian

$$L=g_{a b} \dot{x}^{a} \dot{x}^{b}$$

where the path of the particle is $x^{a}(\lambda)$, and $\lambda$ is an affine parameter along that path.

(ii) The Schwarzschild metric is given by

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

Consider a photon which moves within the equatorial plane $\theta=\frac{\pi}{2}$. Using the above Lagrangian, or otherwise, show that

$$\left(1-\frac{2 M}{r}\right)\left(\frac{d t}{d \lambda}\right)=E, \text { and } r^{2}\left(\frac{d \phi}{d \lambda}\right)=h$$

for constants $E$ and $h$. Deduce that

$$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

Assume further that the photon approaches from infinity. Show that the impact parameter $b$ is given by

$$b=\frac{h}{E}$$

By considering the equation $(*)$, or otherwise

(a) show that, if $b^{2}>27 M^{2}$, the photon is deflected but not captured by the black hole;

(b) show that, if $b^{2}<27 M^{2}$, the photon is captured;

(c) describe, with justification, the qualitative form of the photon's orbit in the case $b^{2}=27 M^{2}$.