Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 2 / c

Choose a spacelike two-dimensional cross-section $S$ of the null hypersurface $\mathcal{N}$ and coordinates $y^i$ on $S$. Extend $y^i$ along the null generators, and choose their parameter $\lambda$ to be affine, so $U={\partial }_{\lambda }$ on $\mathcal{N}$. At every point choose the other null normal $L$ with $g(U,L)=1$, shoot out the affinely parametrized null geodesic with tangent $L$, and call its affine parameter $r$. Transport $(\lambda ,y^i)$ along those transverse geodesics.

This construction gives Gaussian null coordinates. The hypersurface is $r=0$, and the coordinate conditions implyBecause $\lambda$ is affine on the generators, ${\partial }_r{g}_{\lambda \lambda }{|}_{r=0}=0$. Smoothness then factors the remaining components as $g_{\lambda \lambda }=r^{2}F$ and $g_{\lambda i}=r{h}_i$, giving

On $r=0$, the screen metric is $h_{ij}$ and $U={\partial }_{\lambda }$. Therefore the null expansion isThe derivative formula for the determinant givesand henceThus $\theta$ is the fractional rate of change of an infinitesimal transverse area carried along the generators: positive expansion enlarges the beam and negative expansion focuses it.