Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 57 / 1

A stationary black hole behaves mechanically like an equilibrium thermodynamic system. Its horizon area A, asymptotic energy E, angular momentum J and electric charge Q are linked by the laws of black-hole mechanics. Let ${\chi }^a=t^a+{\Omega }_H{\varphi }^a$ generate its Killing horizon, with the time translation normalized at infinity; its surface gravity is defined by ${\chi }^b{\nabla }_b{\chi }^a=\kappa {\chi }^a$ on that horizon. We use $c=1$ and keep G explicit. The horizon electrostatic potential ${\Phi }_H$ is measured relative to infinity.

The Zeroth law of black-hole mechanics states that kappa is constant on each connected stationary horizon, under the Einstein equations and suitable energy/regularity assumptions. It parallels the zeroth law of thermodynamics, which makes temperature uniform in equilibrium. Kappa need not be constant in a dynamical geometry. For an asymptotically flat static hole, local observers see gravitationally redshifted temperatures; the equilibrium quantity comparable to kappa is the temperature referenced to infinity.

The First law of black-hole mechanics relates nearby stationary solutions:The same leading relation describes a small physical process which perturbs a horizon and subsequently settles to equilibrium, with the appropriate hypotheses on fluxes and final state. The work terms correspond to rotational and electromagnetic work. The structural analogy with the first law of thermodynamics, $dE=TdS+\text{work}$, identifies area as an entropy-like variable and surface gravity as a temperature-like variable. Classically this fixes their product in the first law, rather than their separate normalizations.

The second law of black-hole mechanics, or black-hole area theorem, states that future event-horizon area does not decrease classically. Its essential geometric ingredient is the Raychaudhuri equation for affinely parametrized null generators. In four dimensions a hypersurface-orthogonal congruence obeysThe null energy condition and Einstein equations make the last term nonpositive. If the expansion became negative, the inequality would focus the generators to a conjugate point in finite affine parameter. Under future completeness and predictability assumptions, that would conflict with their remaining on an achronal event horizon. Thus $\vartheta \ge 0$, and the area element, satisfying $d\log dA/d\lambda =\vartheta$, cannot decrease. For a merger the final area is at least the sum of the initial component areas. These assumptions matter: the theorem is a classical statement, not a prohibition of quantum evaporation.

The third law of black-hole mechanics is an unattainability statement: a nonextremal hole cannot be driven to zero surface gravity in a finite physical process satisfying the usual bounded-stress and regularity conditions. This parallels the unattainability form of the third law of thermodynamics. Extremal stationary solutions exist, so the law concerns how an initially nonextremal hole approaches them. The stronger entropy form of Nernst's theorem does not transfer unchanged: extremal holes can have finite area and entropy, depending on their conserved charges.

Quantum field theory turns the mechanical analogy into a physical thermodynamic identification. Hawking radiation is thermal at infinity to leading semiclassical order, withThese give $T_H\delta S_{\mathrm{BH}}=\kappa \delta A/(8\pi G)$ exactly, so the mechanical first law becomes the thermodynamic first law with Bekenstein-Hawking entropy. Scattering through the exterior adds greybody factors to the emitted spectrum but leaves this temperature identification intact. Equilibrium with an external bath and isolated evaporation are different physical states of the field.

During black-hole evaporation, the area can decrease because the semiclassical stress tensor need not satisfy the classical null energy condition. The proposed generalized second law applies instead to $S_{\mathrm{gen}}=S_{\mathrm{BH}}+S_{\mathrm{outside}}$, asserting nondecrease of the combined entropy. A Schwarzschild hole illustrates an unusual thermodynamic property: $T_H\propto E^{-1}$, so its heat capacity is negative. It becomes hotter as it loses energy and is unstable in a simple canonical heat bath, while entropy and energy laws still describe an isolated system. The quantum temperature, area entropy and generalized second law supply the correspondence; the mechanical laws alone do not establish a thermal emission spectrum or a microscopic entropy count.