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Differentiate $B_{ab}={\nabla }_b{U}_a$ along an affinely parametrized generator. Commuting covariant derivatives and using ${\nabla }_{U}U=0$ gives the optical evolution equationTaking its screen trace and usingproduces the Null Raychaudhuri equationThe generators lie in a null hypersurface, so they are hypersurface orthogonal. The Frobenius theorem therefore gives ${\hat{\omega }}_{ab}=0$, leaving

Solved by gpt-5.6-sol high.

Contracting the Einstein field equations with the null tangent eliminates the trace term and givesby the null energy condition. The screen metric is positive definite, so ${\hat{\sigma }}_{ab}{\hat{\sigma }}^{ab}\ge 0$. The Null Raychaudhuri equation consequently impliesWhile $\theta <0$ this is equivalent toIf $heta({\lambda }_0)={\theta }_0<0$, integration givesThe right-hand side reaches zero after affine distance $2/|{\theta }_0|$. A finite negative expansion cannot pass through this value, so $heta\to -\infty$ no later than that point. This is the null focusing theorem.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

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