Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4

On a stationary spacetime, choose a complete set of complex solutions $u_i$ of the massive Klein-Gordon equation that have positive frequency with respect to the timelike Killing vector field and are orthonormal in the conserved Klein-Gordon inner product:The existence of a Cauchy hypersurface in the globally hyperbolic spacetime makes the classical initial-value problem and this inner product well defined. Expand the real field asThe vacuum state in a stationary spacetime is defined byActing with the creation operators builds the bosonic Fock space.

The early and late stationary regions define distinct positive-frequency mode bases and hence an in-vacuum and out-vacuum. Write their Bogoliubov transformation asThe corresponding operators satisfyUsing ${\hat{a}}_j^{\mathrm{in}}|0_{\mathrm{in}}⟩=0$ and the canonical commutator gives the particle number from Bogoliubov coefficients:Time dependence in the sandwich region can make ${\beta }_{ij}\ne 0$, so the in-vacuum contains out-particles.

A gravitational-collapse spacetime is stationary and approximately empty in the remote past and settles to a stationary black-hole exterior in the future, with a dynamical region between them. The same comparison of in- and out-positive-frequency modes therefore applies. Tracing an outgoing late-time mode backward through the collapse produces an exponentially blueshifted mixture of early positive and negative frequencies. Its nonzero Bogoliubov beta coefficient yields the thermal occupation numbers of Hawking radiation, while partner modes pass through the event horizon.

The horizon generator must be null at $r=r_{+}$. Substitution of $\xi ={\partial }_v+{\Omega }_H{\partial }_{\chi }$ in the Kerr black hole metric givesThe surface gravity and Hawking temperature are

For $a=0$, $T_H=(8\pi M{)}^{-1}$, so the Schwarzschild black hole has negative heat capacity. A small energy gain from a reservoir lowers its temperature below the reservoir temperature and causes further absorption; a small energy loss raises its temperature and causes further emission. The equilibrium is therefore unstable in the canonical ensemble.

DefineThe Bekenstein-Hawking entropy and Hawking temperature becomeAt fixed $J$,The fixed-$J$ first law gives $(\partial S_{BH}/\partial M{)}_J=1/T_H$, soTherefore the Kerr black-hole heat capacity at fixed angular momentum isAt $a=0$, this correctly reduces to $C_J=-8\pi M^2$.

Local thermal stability against energy exchange with a fixed-temperature reservoir requires $C_J>0$. Since $s\ge 0$, this meansEquivalently,The lower endpoint is the divergent-heat-capacity transition. The extremal endpoint $|a|=M$ has $T_H=0$ and $C_J=0$ and is obtained only as a limiting case.