First-order horizon-area response to matter flux ($\delta A=8\pi \int \lambda \delta T_{kk}d\lambda d{A}_0$)

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.