Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 3 / a

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.