(i) The worldline $x^{a}(\lambda)$ of a massive particle moving in a spacetime with metric $g_{a b}$ obeys the geodesic equation

$$\frac{d^{2} x^{a}}{d \tau^{2}}+\Gamma^a_{bc} \frac{d x^{b}}{d \tau} \frac{d x^{c}}{d \tau}=0$$

where $\tau$ is the particle's proper time and $\Gamma^a_{bc}$ are the Christoffel symbols; these are the equations of motion for the Lagrangian

$$L_{1}=-m \sqrt{-g_{a b} \dot{x}^{a} \dot{x}^{b}}$$

where $m$ is the particle's mass, and $\dot{x}^{a}=d x^{a} / d \lambda$. Why is the choice of worldline parameter $\lambda$ irrelevant? Among all possible worldlines passing through points $A$ and $B$, why is $x^{a}(\lambda)$ the one that extremizes the proper time elapsed between $A$ and $B$ ?

Explain how the equations of motion for a massive particle may be obtained from the alternative Lagrangian

$$L_{2}=\frac{1}{2} g_{a b} \dot{x}^{a} \dot{x}^{b} .$$

What can you conclude from the fact that $L_{2}$ has no explicit dependence on $\lambda$ ? How are the equations of motion for a massless particle obtained from $L_{2}$ ?

(ii) A photon moves in the Schwarzschild metric

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

Given that the motion is confined to the plane $\theta=\pi / 2$, obtain the radial equation

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

where $E$ and $h$ are constants, the physical meaning of which should be stated.

Setting $u=1 / r$, obtain the equation

$$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$$

Using the approximate solution

$$u=\frac{1}{b} \sin \phi+\frac{M}{2 b^{2}}(3+\cos 2 \phi)+\ldots$$

obtain the standard formula for the deflection of light passing far from a body of mass $M$ with impact parameter $b$. Reinstate factors of $G$ and $c$ to give your result in physical units.