 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/38c/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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} .

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