= Solution
The <laws of black-hole mechanics> establish the classical analogy with <thermodynamics>. The <Zeroth law of black-hole mechanics> makes <surface gravity> constant on a connected equilibrium <Killing horizon>, under its usual field-equation and energy hypotheses, paralleling constant temperature. For neighboring stationary Einstein–Maxwell solutions, the <First law of black-hole mechanics> is
$$
\delta M=\frac{\kappa}{8\pi G}\delta A+\Omega_H\delta J+\Phi_H\delta Q.
$$
This parallels $\delta E=T\delta S$ plus work terms; $M$ denotes energy in $c=1$ units. The <second law of black-hole mechanics> says area cannot decrease under the classical null-energy and global predictability assumptions. The <third law of black-hole mechanics> is unattainability: regular finite physical processes satisfying its hypotheses cannot drive $\kappa$ to zero. It parallels unattainability of absolute zero, not a universal assertion of zero extremal entropy.
Classically a hole absorbs without emitting, so the analogy alone does not identify a measured temperature. Quantum <Hawking radiation> supplies $T_H=\hbar\kappa/(2\pi)$, with $c=k_B=1$ and future-horizon normalization at infinity. Comparing the area terms in the first laws gives \b[the <Bekenstein-Hawking entropy>]
$$
\delta S_{\rm BH}=\frac{\delta A}{4G\hbar},\qquad\boxed{S_{\rm BH}=\frac{A}{4G\hbar}},
$$
up to a conventionally fixed additive constant. In ordinary units it is $k_Bc^3A/(4G\hbar)$. Entropy scales with area, not interior volume.
For the massless scalar in <quantum field theory in curved spacetime>, <global hyperbolicity> ensures well-posed Cauchy evolution. The conserved <Klein-Gordon inner product> is
$$
(f,g)=i\int_\Sigma d\Sigma^\mu(f^*\nabla_\mu g-g\nabla_\mu f^*).
$$
Its current has zero divergence by the wave equation, making it slice-independent with appropriate boundary behavior. Positive frequency relative to the asymptotic past and future Minkowski times defines complete bases $u_j,v_i$, with $(u_i,u_j)=\delta_{ij}$, $(u_i^*,u_j^*)=-\delta_{ij}$ and vanishing mixed products. Expand the field as
$$
\Phi=\sum_j(a_j^{\rm in}u_j+a_j^{{\rm in}\dagger}u_j^*)=\sum_i(a_i^{\rm out}v_i+a_i^{{\rm out}\dagger}v_i^*).
$$
Continuous labels replace sums by integrals; wave packets avoid artificial normalization infinities. <Particle creation by a nonstationary spacetime> occurs when evolution mixes frequency signs through a <Bogoliubov transformation>:
$$
v_i=\sum_j(\alpha_{ij}u_j+\beta_{ij}u_j^*),\qquad a_i^{\rm out}=\sum_j(\alpha_{ij}^*a_j^{\rm in}-\beta_{ij}^*a_j^{{\rm in}\dagger}).
$$
The inner product gives $\alpha\alpha^\dagger-\beta\beta^\dagger=I$ and $\alpha\beta^{\mathsf T}=\beta\alpha^{\mathsf T}$. For the initial vacuum annihilated by all $a_j^{\rm in}$, \b[$\boxed{\langle N_i^{\rm out}\rangle_{\rm in}=\sum_j|\beta_{ij}|^2}$.] Thus nonzero negative-frequency mixing makes the in-vacuum non-vacuum for final observers. The <vacuum state in a stationary spacetime> depends on its positive-frequency splitting; this is not inconsistent with deterministic mode evolution. A unitary implementation on one fixed infinite-mode <Fock space> further needs the Hilbert–Schmidt condition on $\beta$.
For collapse forming a <Schwarzschild black hole>, a late outgoing ray traced back to past null infinity obeys the <Hawking exponential ray map>
$$
U_H-U=C e^{-\kappa u},\qquad\kappa=1/(4GM).
$$
Pulling $e^{-i\omega u}$ back gives $(U_H-U)^{i\omega/\kappa}$ for $U<U_H$. Its past Fourier transform has both frequency signs. For $y=\omega/\kappa$ and damping $\epsilon>0$, the <Gamma function> evaluates the Fourier integrals as
$$
I_\pm=\int_0^\infty x^{iy}e^{-\epsilon x\pm i\omega' x}dx=\frac{\Gamma(1+iy)}{(\epsilon\mp i\omega')^{1+iy}}.
$$
As $\epsilon\downarrow0$, the two denominator arguments approach $\mp\pi/2$, so $|I_+|/|I_-|=e^{-\pi y}$. Thus the <thermal ratio of Hawking Bogoliubov coefficients> is $|\beta|^2/|\alpha|^2=e^{-2\pi\omega/\kappa}$. Together with the bosonic normalization difference, this yields $1/(e^{2\pi\omega/\kappa}-1)$. Potential scattering adds a <greybody factor>:
$$
\frac{dN}{dt\,d\omega}=\frac1{2\pi}\sum_{\ell,m}\frac{\Gamma_\ell(\omega)}{e^{2\pi\omega/\kappa}-1},\qquad\boxed{T_H=\frac{\hbar}{8\pi GM}}.
$$
The collapse state has outgoing flux at future infinity without an incoming thermal bath and is regular for infall. It is not the eternal-hole equilibrium state. Horizon-entering modes complete the future basis; tracing over their correlated partners gives the approximately thermal exterior state. The collapsing geometry is not literally Minkowskian everywhere in the far future: asymptotically flat null-infinity modes are the relevant application of the earlier in/out construction.
Emission reduces the isolated hole's mass. Schwarzschild temperature is proportional to $M^{-1}$, giving the <negative heat capacity of a Schwarzschild black hole>: losing energy makes it hotter. Dimensional estimates give luminosity proportional to $M^{-2}$ and evaporation time proportional to $M^3$, with species and greybody factors determining coefficients. Area may shrink because quantum stress violates the energy assumptions of <Hawking's area theorem>. The <generalized second law> instead concerns $S_{\rm gen}=A/(4G\hbar)+S_{\rm outside}$, incorporating radiation entropy and preventing ordinary thermodynamic violations by disposal of entropy into a hole.
Pair correlations also distinguish thermal reduced states from the complete pure state. The <black hole information paradox> asks whether complete evaporation preserves quantum information: an exactly thermal final exterior with no remaining partners would appear inconsistent with unitary pure-state evolution. The semiclassical calculation controls late-time flux, not the Planck-scale endpoint, and does not itself settle this question. \b[Quantum emission gives the temperature–entropy identification physical meaning; generalized entropy replaces the classical area alone when radiation back-reacts.]
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