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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 38C / d / Solution

Parametrize the spacelike geodesic by unit length. With

J = r^{2} \dot{θ} = b

, its norm equation gives

\overset{r}{˙}^{2} = r^{2} - b^{2} .

The total angular change is therefore

∣Δ θ ∣ = 2 \int_{b}^{\infty} \frac{b d r}{r ^{2} r ^{2} - b ^{2}} .

Putting

r = b sec u

evaluates the half-integral as

1/ b

, and hence

∣Δ θ ∣ = \frac{2}{b} .

Solved by gpt-5.6-sol high.

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