= Solution
The <Physical-process first law for a rotating black hole> states that a small flux of matter through an initially and finally stationary horizon obeys
$$
\boxed{\Delta M-\Omega_H\Delta J=\frac{\kappa}{8\pi}\Delta A}
$$
in 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 generator
$$
\xi^a=t^a+\Omega_Hm^a=\kappa\lambda k^a.
$$
To 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 to
$$
\frac{d\theta}{d\lambda}=-8\pi T_{ab}k^ak^b.
$$
The final stationary condition $\theta(\infty)=0$ gives
$$
\theta(\lambda)=8\pi\int_\lambda^\infty
T_{ab}k^ak^b\,d\lambda'.
$$
Since $d(\delta A)/d\lambda=\int\theta\,dA$, reversing the order of integration yields
$$
\Delta A=8\pi\int_H\lambda T_{ab}k^ak^b\,d\lambda,dA.
$$
The <stress-energy current from a Killing vector> gives the horizon Killing-energy flux
$$
\Delta M-\Omega_H\Delta J
=\int_HT_{ab}\xi^ak^b\,d\lambda,dA
=\kappa\int_H\lambda T_{ab}k^ak^b\,d\lambda,dA,
$$
which proves the stated law.
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