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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 15E / a / ii / Solution

Define conformal time by

d τ = d t / a (t)

. The radial coordinate and the angular-radius function are

k 0 k > 0 k < 0 χ (r) r k^{- 1/2} sin^{- 1} (k r) (- k)^{- 1/2} sinh^{- 1} (- k r) f (χ) χ k^{- 1/2} sin (k χ) (- k)^{- 1/2} sinh (- k χ) .

Substitution yields the stated conformal metric. A radial null ray satisfies

- c^{2} d τ^{2} + d χ^{2} = 0,

so

d χ / d τ = \pm c

and

χ = χ_{0} \pm c (τ - τ_{0})

.

Solved by gpt-5.6-sol high.

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