Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 2 / b / Solution

Let $k^a=(d/d\lambda {)}^a$ be an affine horizon generator, normalized so that the horizon Killing field is ${\xi }^a=\kappa \lambda k^a$. To first order in the small total injected energy, expansion and shear-squared terms are second order, and the Null Raychaudhuri equation on the horizon becomesThe flux of Killing energy through all the shells isSubstitute the linearized focusing equation and integrate by parts. Stationarity before and after the process makes the endpoint term $\lambda \theta$ vanish, while $dA/d\lambda =\theta A$ to first order. ThereforeUsing the Hawking temperature $T_H=\kappa /(2\pi )$ and Bekenstein-Hawking entropy $S_{BH}=A/4$ gives the Physical-process first law of black-hole mechanicsThe derivation is linear in the stress tensor, so separated shells simply contribute additively.

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