 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-4/37a/a/solution

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

Let

f = 1 - r^{2} / b^{2}

. Then

U_{μ} = (1, - 1/ f, 0, 0)

and

g^{μν} U_{μ} U_{ν} = - 1/ f + f / f^{2} = 0

. Since

d t = d u + d r / f

, the metric becomes

d s^{2} = - fd u^{2} - 2, d u, d r + r^{2} d Ω^{2},

which is regular at

r = b

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