 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/37b/b/solution

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

For the given metric, direct substitution of the stated connection coefficients into the curvature formula leaves

R_{uu} = - \frac{1}{2} (H_{, xx} + H_{, yy}) .

g^{uu} = 0

, and every other Ricci component vanishes, so

R = g^{α β} R_{α β} = 0.

Thus the full, nonlinear vacuum Einstein equation is precisely the transverse Laplace equation from a plane-fronted gravitational wave:

H_{, xx} + H_{, yy} = 0.

The dependence on

u

is unrestricted.

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