Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 2 / a / ii / Solution

After the shell, spherical symmetry and stationarity put the horizon at

r = 2 (M + E)

. Before the shell, an outgoing radial null geodesic in the mass-

M

region has

r

linear in an affine parameter. The normalizations

r (0) = 2 M

and

r (1) = 2 (M + E)

therefore give

r (λ) = 2 M + 2 E λ (λ < 1) .

The degenerate intrinsic metric of the horizon has no

d λ^{2}

term, so

d s_{H^{+}}^{2} = {4 (M + E λ)^{2} d Ω^{2}, 4 (M + E)^{2} d Ω^{2}, λ < 1, λ > 1.

The two expressions agree at the shell,

λ = 1

.

Solved by gpt-5.6-sol high.

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