Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 37B / b / Solution

For the given metric, direct substitution of the stated connection coefficients into the curvature formula leavesThe inverse metric has $g^{uu}=0$, and every other Ricci component vanishes, soThus the full, nonlinear vacuum Einstein equation is precisely the transverse Laplace equation from a plane-fronted gravitational wave:The dependence on $u$ is unrestricted.