Write
$$
f(r)=1-\frac{2M}{r}.
$$
For a radial null ray, the <Schwarzschild spacetime> line element gives
$$
0=-f\,dt^2+f^{-1}dr^2,
\qquad
\frac{dt}{dr}=\pm f^{-1}.
$$
Outside the horizon an ingoing ray has $dr/dt<0$, so it takes the minus sign. Since the <Schwarzschild tortoise coordinate> obeys $dr_*/dr=f^{-1}$,
$$
\frac{d}{dr}(t+r_*)=0.
$$
It follows that
$$
\boxed{v=t+r+2M\log\left|\frac{r}{2M}-1\right|}
$$
is constant along every ingoing radial light ray.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 37E / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/37e/a/solution