Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 4

Choose complete orthonormal sets of complex Klein-Gordon solutions $\{u_i^{\mathrm{in}}\}$ and $\{u_i^{\mathrm{out}}\}$ that have positive frequency with respect to the asymptotic timelike Killing fields in the remote past and future. Their normalization uses the conserved Klein-Gordon inner product:The real quantum field has either mode expansion of a free field

Completeness relates the mode bases by the Bogoliubov transformationand hencePreservation of the canonical commutation relations requiresThe past and future notions of positive frequency therefore define the generally different in-vacuum and out-vacuum. Nonzero $\beta$ means that the in-vacuum is a squeezed state containing out-particles.

The in-vacuum satisfies $a_j^{\mathrm{in}}|0_{\mathrm{in}}⟩=0$. For the out-mode number operator $N_i^{\mathrm{out}}=a_i^{\mathrm{out}†}{a}_i^{\mathrm{out}}$, substitute the operator transformation and use the canonical commutation relations. Only the $a_j^{\mathrm{in}}{a}_k^{\mathrm{in}†}$ contraction survives, giving the particle number from Bogoliubov coefficientsFor a continuous mode label, the sum becomes the corresponding integral with Dirac-delta normalization.

In a collapsing spacetime, positive-frequency in-modes are defined at past null infinity and outgoing modes at future null infinity. Tracing a late outgoing wave packet backward toward the forming event horizon produces an exponential blueshift. Its rapidly varying phase has both positive- and negative-frequency parts relative to the past time coordinate, so the Bogoliubov coefficient $\beta$ is nonzero. The resulting bosonic occupation numbers have the Bose-Einstein distribution atwhere $\kappa$ is the surface gravity. The outgoing quanta form Hawking radiation, while their correlated partners cross the horizon.

For a metric $d{s}^2=-f(r)d{v}^2+2dvdr+r^{2}d{\Omega }_2^2$ with a simple Killing horizon, the surface gravity of $k={\partial }_v$ is $\kappa =f^{\prime }(r_{+})/2$. HeresoThe Hawking temperature of the Schwarzschild--anti-de Sitter black hole is therefore

LetThe one-form dual to $k={\partial }_v$ is $k^{♭}=-fdv+dr$. For the $r_{+}\to 0$ anti-de Sitter background, ${\bar{k}}^{♭}=-(1+r^2/L^2)dv+dr$, soWith the stated orientation and metric volume form,The Background-subtracted Komar energy is consequently

The horizon area is $A=4\pi r_{+}^2$, whileThus the First law of black-hole mechanicsholds with the Bekenstein-Hawking entropy $S_{BH}=A/4$.

For equilibrium with an infinite fixed-temperature reservoir, local stability requires positive heat capacity. Since $dM/d{r}_{+}>0$, its sign is the sign ofTherefore the Canonical stability of a Schwarzschild--anti-de Sitter black hole occurs precisely forThe equality is the minimum-temperature, marginal case; smaller black holes have negative heat capacity and unstable thermal equilibrium.