 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/15e/a/i/solution

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

At fixed time and angles, proper radial distance is

D = a (t) \int_{0}^{Δ r} \frac{d r}{1 - k r ^{2}} .

Thus

D = ⎩ ⎨ ⎧ a Δ r, \frac{a}{k} sin^{- 1} (k Δ r), \frac{a}{- k} sinh^{- 1} (- k Δ r), k = 0, k > 0, k < 0.

These are respectively flat Euclidean, positively curved spherical, and negatively curved hyperbolic spatial geometries.

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