Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 4

The second law of black-hole mechanics states that the area of spatial cross-sections of a future event horizon cannot decrease toward the future, provided the null energy condition and the standard global assumptions hold.

Let $k^a$ be an affinely parametrized horizon generator. It is hypersurface-orthogonal, so its null twist vanishes. The Null Raychaudhuri equation in $D$ dimensions and the Einstein equation giveIf $\theta ({\lambda }_0)<0$, integration implies that $\theta$ diverges to $-\infty$ within affine distance at most $(D-2)/|\theta ({\lambda }_0)|$. The resulting focal point would make the generator leave the achronal boundary that defines the event horizon, contradicting its assumed future completeness. Hence $\theta \ge 0$ everywhere. Since null expansion obeysevery horizon area element, and therefore every complete cross-section area, is nondecreasing.

For this metric, $\sqrt{-g}=a^2$, while $g^{\eta \eta }=-a^{-2}$, $g^{zz}=a^{-2}$ and $g^{ij}={\delta }^{ij}$. The massless Klein-Gordon equation is consequentlyInsert the spatial Fourier transform given in the question. Each wavenumber mode then satisfiesThus every field mode is a simple harmonic oscillator with a time-dependent angular frequency.

In the two constant regions defineThe normalized positive-frequency modes areup to the common spatial Fourier normalization. They satisfy the unit Wronskian conditionwhich is the mode form of the Klein-Gordon inner product. Because the mode equation contains no delta function at $\eta =0$, both $\Phi$ and ${\Phi }^{\prime }$ are continuous there.

Continue the incoming positive-frequency mode through $\eta =0$ asContinuity of the mode and its first derivative gives the sudden frequency quench coefficientsThe associated Bogoliubov transformation says that the incoming vacuum has expected outgoing occupation numberIt vanishes when $a_{+}=a_{-}$, as required when there is no change of geometry.