 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/37d/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Let

φ = ϕ - F (R)

, so

d ϕ = d φ + F^{'} (R) d R .

The new coefficient of

d R^{2}

- \frac{a ^{2} s i n ^{2} θ}{Δ} + Δ sin^{2} θ [F^{'} (R)]^{2} .

It vanishes if

F^{'} (R) = \frac{a}{R ^{2} + a ^{2}} .

One choice that tends to zero as

R \to \infty

and also has a regular

a \to 0

limit is

F (R) = - arctan \frac{a}{R} .

d s^{2} = - d u^{2} - 2 d u d R + 2 a sin^{2} θ d R d φ + (R^{2} + a^{2} cos^{2} θ) d θ^{2} + (R^{2} + a^{2}) sin^{2} θ d φ^{2} .

There is no

d R^{2}

term, so

g_{RR} = 0

and

\partial_{R}

is null for every

a \geq 0

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