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The Physical-process first law for a rotating black hole states that a small flux of matter through an initially and finally stationary horizon obeysin units $G=1$. Let $k^a$ be an affinely parametrized horizon tangent with affine parameter $\lambda =0$ on a bifurcation surface. Constancy of the surface gravity from the Zeroth law of black-hole mechanics and Gaussian null coordinates give the horizon generatorTo first order about a stationary horizon, the squared expansion and shear in the Null Raychaudhuri equation are second order. The Einstein field equations reduce it toThe final stationary condition $\theta (\infty )=0$ givesSince $d(\delta A)/d\lambda =\int \theta dA$, reversing the order of integration yieldsThe stress-energy current from a Killing vector gives the horizon Killing-energy fluxwhich proves the stated law.

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