Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 3 / c

Only the transverse coordinates occur in the perturbation, so its tensor components are unchanged by replacing $(t,z)$ with $(u,v)$. The plane gravitational wave in linearized gravity isEvery component depends on $u$ alone. The Minkowski wave operator in null coordinates therefore givesThus the field satisfies the vacuum Linearized Einstein equations.