Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 36D / b / Solution

For a nonradial null geodesic,It tends to $-\infty$ as $r\to 0^{+}$, rises to one maximum, and tends to $-L^2{H}^2$ as $r\to \infty$. A circular null geodesic requires $V^{\prime }(r)=0$, soSince $V^{\prime \prime }(3m)<0$, this photon sphere is unstable. Its energy satisfies $E^2=L^2(1/(27m^2)-H^2)$.