First law of black-hole mechanics

For a nonrotating uncharged stationary black hole, the first law is $\delta M=\kappa \delta A/(8\pi )=T_H\delta S_{BH}$.

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 $\Delta M=\kappa \Delta A/(8\pi )$.