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

Write

f (r) = 1 - \frac{2 M}{r} .

For a radial null ray, the Schwarzschild spacetime line element gives

0 = - f d t^{2} + f^{- 1} d r^{2}, \frac{d t}{d r} = \pm f^{- 1} .

Outside the horizon an ingoing ray has

d r / d t < 0

, so it takes the minus sign. Since the Schwarzschild tortoise coordinate obeys

d r_{*} / d r = f^{- 1}

,

\frac{d}{d r} (t + r_{*}) = 0.

It follows that

v = t + r + 2 M lo g \frac{r}{2 M} - 1

is constant along every ingoing radial light ray.

Solved by gpt-5.6-sol high.

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