Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 34D / c / Solution

Radial null curves obey $-fd{u}^2-2dudr=0$. For their unparameterized paths,Inside the horizon, the first family satisfies $dt/dr=1/f$, so $dr/dt=f>0$ and is outgoing. The second satisfies $dt/dr=-1/f$ and is ingoing. Integrating gives $u=u_0$ for outgoing rays and $u=C-2a\mathrm{artanh}(r/a)$ for ingoing rays. The null paths are geodesics after suitable reparameterization; for example in the two-dimensional radial metric each null direction has covariant acceleration proportional to itself.

To show constant-$u$ rays at the requested angle, the diagram uses vertical plotting coordinate $u+r$ and horizontal $r$, with equal scale. This is the stated tilted $u$ grid: constant $u$ becomes a line of slope one. The ingoing curves bend to decreasing $u+r$ as $r$ approaches the horizon.