Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 1 / b

Every metric coefficient is independent of $w$, sois a Killing vector field.

Forthe only potentially nonzero components of ${\mathcal{L}}_{X}g$ areandThus the necessary and sufficient conditions areThe second-order linear ODE has a two-dimensional solution space, giving a two-parameter family

Interchanging the two transverse directions changes the sign of the quadratic profile. Henceis another two-parameter family of Killing fields.

Fix $u=u_0$. Initial data $p(u_0),p^{\prime }(u_0)$ and $q(u_0),q^{\prime }(u_0)$ may be prescribed independently for the two oscillator equations. In particular, values of $p(u_0)$ and $q(u_0)$ generate arbitrary translations in $x$ and $y$, while their derivatives merely add controllable $w$ components. The independent Killing field ${\partial }_w$ supplies any remaining translation in $w$. These Killing fields span the tangent space of each constant-$u$ wavefront, so their flows act transitively. Therefore an isometry maps any point $r$ on such a surface to any other point $s$, realizing a homogeneous wavefront of a plane gravitational wave.