= Solution
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 $\chi^a=t^a+\Omega_H\phi^a$ 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:
$$
\boxed{\delta E=\frac{\kappa}{8\pi G}\delta A+\Omega_H\delta J+\Phi_H\delta Q.}
$$
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>, $dE=T\,dS+\text{work}$, 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 obeys
$$
\frac{d\vartheta}{d\lambda}=-\frac12\vartheta^2-\sigma_{ab}\sigma^{ab}-R_{ab}k^ak^b.
$$
The <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 $\vartheta\geq0$, 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, with
$$
\boxed{T_H=\frac{\hbar\kappa}{2\pi k_B},\qquad S_{\rm BH}=\frac{k_B A}{4G\hbar}.}
$$
These give $T_H\delta S_{\rm BH}=\kappa\delta A/(8\pi G)$ 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 $S_{\rm gen}=S_{\rm BH}+S_{\rm outside}$, 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.
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