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

Let ${\Phi }_s$ be the local flow generated by the vector field $X$. For a covariant tensor, its Lie derivative isFor contravariant indices one equivalently uses the differential of the inverse flow, and the construction extends to every tensor product by the Leibniz rule. It measures the infinitesimal change of $T$ under transport by the flow of $X$.

At a point where $X\ne 0$, the flow-box theorem supplies a transverse hypersurface with coordinates $x^i$. Flow each point on it for parameter $t$ and keep the $x^i$ constant along the flow. In the resulting coordinates,Because the coordinate basis is transported by this flow, the Lie derivative of any tensor is obtained by differentiating its coordinate components:

For a function, pullback by the flow givesIn flow-box coordinates $X={\partial }_t$, write $Y=Y^{\mu }{\partial }_{\mu }$. ThenBoth sides are vector fields defined invariantly, so equality in these coordinates provesin every coordinate system.

For one-forms, the defining pullback or the identitygivesApplying the Leibniz rule to both covariant slots of a type-$(0,2)$ tensor yields

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.