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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.

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.