Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 3 / a

For a small amount of matter falling into an initially and finally stationary rotating black hole, the Physical-process first law for a rotating black hole isLet $q^a$ be an affinely parametrized horizon generator, with affine parameter $\lambda =0$ at the background bifurcation surface. The horizon Killing vector field iswhere constancy of $\kappa$ is the Zeroth law of black-hole mechanics. The flux of the conserved Killing current through the horizon is

On the stationary background, expansion and shear vanish. To first order in the perturbation, the quadratic optical terms in the Null Raychaudhuri equation may be dropped, and the Einstein equation givesImpose the teleological final condition $\theta (\infty )=0$. Integration followed by reversal of integration order givesComparison with the Killing-energy flux proves the stated law.