Physical-process first law of black-hole mechanics

The physical-process first law derives the area change caused by a small flux of matter through an initially and finally stationary horizon. Linearized Null Raychaudhuri equation evolution converts the Killing-energy flux into $\Delta M=\kappa \Delta A/(8\pi )$.

For a small perturbation of a stationary horizon, the linearized Null Raychaudhuri equation gives $\delta {\theta }^{\prime }=-8\pi \delta T_{kk}$. With vanishing boundary term $[\lambda \delta \theta ]$, integration by parts gives $\delta A=8\pi \int \lambda \delta T_{kk}d\lambda d{A}_0$. Combining this with affine horizon-generator scaling converts area response to horizon Killing-energy flux.

For horizon generator $\xi =k+{\Omega }_{H}m$, a small matter flux into an initially and finally stationary rotating black hole satisfies