Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 38A (General Relativity)

Spherical symmetry puts any geodesic in a plane through the origin, chosen as $\theta =\pi /2$. The cyclic coordinates giveThe null condition givesFor $u=1/r$, $(du/d\varphi {)}^2={\dot{r}}^2/L^2$, hence

Differentiation gives $u^{\prime \prime }+u=\frac{3}{2}{r}_s{u}^2$. With $u=b^{-1}(v_0+\epsilon v_1+\cdots )$,The symmetry selects $v_0=\sin \varphi$, the straight line at closest distance $b$. Since
$\frac{3}{2}{\sin }^2\varphi =\frac{3}{4}(1-\cos 2\varphi )$,The two asymptotic zeros shift from $0,\pi$ to $-\epsilon ,\pi +\epsilon$, so the deflection is $2\epsilon =2r_s/b=4GM/(b{c}^2)$, the Schwarzschild light deflection. It is tested by stellar deflection near the Sun and gravitational lensing.