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

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.