= Solution
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> is
$$
\boxed{\Delta M-\Omega_H\Delta J
=\frac{\kappa}{8\pi}\Delta A}.
$$
Let $q^a$ be an affinely parametrized horizon generator, with affine parameter $\lambda=0$ at the background <bifurcation surface>. The horizon <Killing vector field> is
$$
\xi^a=k^a+\Omega_Hm^a=\kappa\lambda q^a,
$$
where constancy of $\kappa$ is the <Zeroth law of black-hole mechanics>. The flux of the conserved Killing current through the horizon is
$$
\Delta M-\Omega_H\Delta J
=\int_{\mathcal H}T_{ab}\xi^aq^b\,d\lambda\,dA
=\kappa\int_{\mathcal H}\lambda T_{ab}q^aq^b\,d\lambda\,dA.
$$
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 gives
$$
\frac{d\theta}{d\lambda}=-8\pi T_{ab}q^aq^b.
$$
Impose the teleological final condition $\theta(\infty)=0$. Integration followed by reversal of integration order gives
$$
\Delta A=\int_{\mathcal H}\theta\,d\lambda\,dA
=8\pi\int_{\mathcal H}\lambda T_{ab}q^aq^b\,d\lambda\,dA.
$$
Comparison with the Killing-energy flux proves the stated law.
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