= Solution
The <Physical-process first law for a rotating black hole>, in four-dimensional <general relativity> with $G=1$, is
$$
\boxed{\frac{\kappa}{8\pi}\delta A=\delta E-\Omega_H\delta J.}
$$
It applies to a small neutral matter influx into an initially stationary, nonextremal <black hole> that settles to another stationary state. The background <surface gravity> and angular velocity are $\kappa$ and $\Omega_H$, and $\delta E,\delta J$ are the <Killing energy> and <angular momentum> delivered through the <event horizon>. Other conserved charges are held fixed. The perturbation must be small enough that its horizon generators do not develop caustics, and all equalities below are to first order.
Let $t^a$ be the stationary <Killing vector field> normalized at infinity, and $\psi^a$ the axial <Killing vector field> with $2\pi$-periodic orbits. The horizon generator is $\xi^a=t^a+\Omega_H\psi^a$. Choose an affine tangent $k^a$ and parameter $\lambda$ with zero at the past stationary limiting section, so <affine horizon-generator scaling> gives, on the background horizon
$$
\xi^a=\kappa\lambda k^a.
$$
The background <null expansion> and <null shear> vanish, and <null twist> vanishes for the hypersurface-orthogonal horizon generators. Linearizing the <Null Raychaudhuri equation> and using the <Einstein field equations> gives
$$
\frac{d\,\delta\theta}{d\lambda}=-8\pi\delta T_{ab}k^ak^b.
$$
The quadratic <null expansion> and <null shear> terms are second order. With background cross-sectional measure $dA_0$, the first-order area change is $\delta A=\int_{\mathcal H}\delta\theta\,d\lambda\,dA_0$. Multiplying the evolution equation by $\lambda$ and integrating by parts gives the <first-order horizon-area response to matter flux>
$$
\delta A=8\pi\int_{\mathcal H}\lambda\,\delta T_{ab}k^ak^b\,d\lambda\,dA_0.
$$
The boundary term $[\lambda\delta\theta]$ vanishes: $\lambda=0$ at the past limiting section, and the final stationary boundary condition gives zero expansion in the settled future. Equivalently, assume a sufficiently localized influx with the required late-time decay.
The energy and angular-momentum fluxes have signs
$$
\delta E=\int_{\mathcal H}\delta T_{ab}t^ak^b\,d\lambda\,dA_0,\qquad\delta J=-\int_{\mathcal H}\delta T_{ab}\psi^ak^b\,d\lambda\,dA_0.
$$
These correspond to the particle conventions $E=-p\cdot t$ and $J=p\cdot\psi$. Their combination is
$$
\delta E-\Omega_H\delta J=\int_{\mathcal H}\delta T_{ab}\xi^ak^b\,d\lambda\,dA_0=\kappa\int_{\mathcal H}\lambda\delta T_{ab}k^ak^b\,d\lambda\,dA_0=\frac\kappa{8\pi}\delta A.
$$
This proves the requested <Physical-process first law of black-hole mechanics>. For charged infall the corresponding law contains an additional horizon-potential term $\Phi_H\delta Q$; the energy-minus-angular-momentum version assumes that work term is absent.
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