The Schwarzschild line element is given by

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

where $F=1-r_{s} / r$ and $r_{s}$ is the Schwarzschild radius. Obtain the equation of geodesic motion of photons moving in the equatorial plane, $\theta=\pi / 2$, in the form

$$\left(\frac{d r}{d \tau}\right)^{2}=E^{2}-\frac{h^{2} F}{r^{2}}$$

where $\tau$ is proper time, and $E$ and $h$ are constants whose physical significance should be indicated briefly.

Defining $u=1 / r$ show that light rays are determined by

$$\left(\frac{d u}{d \phi}\right)^{2}=\left(\frac{1}{b}\right)^{2}-u^{2}+r_{s} u^{3}$$

where $b=h / E$ and $r_{s}$ may be taken to be small. Show that, to zeroth order in $r_{s}$, a light ray is a straight line passing at distance $b$ from the origin. Show that, to first order in $r_{s}$, the light ray is deflected through an angle $2 r_{s} / b$. Comment briefly on some observational evidence for the result.