Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 2 / a

Solution

From

γ = c / r + O (r^{- 2})

\partial_{r} γ = - \frac{c}{r ^{2}} + O (r^{- 3}) .

\partial_{r} β = \frac{c ^{2}}{2 r ^{3}} + O (r^{- 4}),

and hence

β = β_{0} (u, θ) - \frac{c ^{2}}{4 r ^{2}} + O (r^{- 3}) .

Asymptotic flatness in the standard Bondi frame sets the integration function

β_{0} = 0

\frac{\partial _{θ} c + 2 c c o t θ}{r ^{2}} + O (r^{- 3}) .

Therefore

\partial_{r} (r^{4} \partial_{r} U) = 2 (\partial_{θ} c + 2 c cot θ) + O (r^{- 1}) .

D = \partial_{θ} c + 2 c cot θ

and integrating twice,

U = U_{0} (u, θ) - \frac{D}{r ^{2}} - \frac{U _{3} ( u , θ )}{3 r ^{3}} + O (r^{- 4}) .

Standard asymptotic flatness sets

U_{0} = 0

;

U_{3}

is free integration data at the next order.

Solved by gpt-5.6-sol high.

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