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 imply
$$
g_{rr}=g_{ri}=0,
\qquad
g_{r\lambda}=1,
\qquad
g_{\lambda\lambda}|_{r=0}=g_{\lambda i}|_{r=0}=0.
$$
Because $\lambda$ is affine on the generators, $\partial_rg_{\lambda\lambda}|_{r=0}=0$. Smoothness then factors the remaining components as $g_{\lambda\lambda}=r^2F$ and $g_{\lambda i}=rh_i$, giving
$$
\boxed{ds^2=2\,dr\,d\lambda+r^2F\,d\lambda^2+2rh_i\,d\lambda\,dy^i+h_{ij}\,dy^i\,dy^j}.
$$

Solution

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

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Gaussian null coordinates adapt one coordinate to the generators of a <null hypersurface>, one affine coordinate to transverse null geodesics, and the remaining coordinates to a spacelike cross-section. If the generator parameter is affine, the metric can locally be written
$$
ds^2=2\,dr\,d\lambda+r^2F\,d\lambda^2+2rh_i\,d\lambda\,dy^i+h_{ij}\,dy^i\,dy^j.
$$

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