= First-order horizon-area response to matter flux
{title2=$\delta A=8\pi\int\lambda\delta T_{kk}\,d\lambda\,dA_0$}
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 $\delta A=8\pi\int\lambda\delta T_{kk}\,d\lambda\,dA_0$. Combining this with <affine horizon-generator scaling> converts area response to horizon Killing-energy flux.
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