= Solution
Work initially in units $G=c=\hbar=k_B=1$ and consider gravitational collapse settling to a nonextremal <Schwarzschild black hole>. Classically, outgoing rays sufficiently close to the forming <event horizon> suffer an arbitrarily large redshift. Quantum mechanically, the meaning of <positive frequency> at <past null infinity> differs from that at <future null infinity>; a <Bogoliubov transformation> between these two mode descriptions can therefore mix <creation operators> and <annihilation operators>. Hawking's argument computes that mixing and obtains a thermal occupation, even when the incoming state has no particles.
To see why the frequency mixing is universal, use final <Schwarzschild time> $t$, the <Schwarzschild tortoise coordinate> $r_*$, and retarded time $u=t-r_*$. Near the final <event horizon>, $r_*\sim(2\kappa)^{-1}\log(r-2M)$, with <Schwarzschild surface gravity> $\kappa=1/(4M)$. On a fixed advanced-time section, $u=v-2r_*$, so $r-2M$ is proportional to $e^{-\kappa u}$. Tracing a ray back through the regular collapsing region maps this small separation smoothly to its separation from the last escaping incoming ray. If $U$ is an affine incoming <null coordinate> and $U_H$ labels that limiting ray, the <Hawking exponential ray map> is
$$
U_H-U=Ae^{-\kappa u}(1+o(1)),\qquad A>0.
$$
The logarithm at a simple horizon root fixes the exponent. Smooth propagation through the earlier collapse changes $A$ and phases but not the leading exponential redshift.
For a massless bosonic test <scalar field>, an outgoing mode of frequency $\omega>0$ at <future null infinity> behaves as $e^{-i\omega u}$. Trace it backward in the high-frequency <geometric optics> approximation. Its relevant incoming dependence is, up to normalization and a phase,
$$
p_\omega(U)\sim\Theta(U_H-U)\left(\frac{U_H-U}{A}\right)^{ia},\qquad a=\frac\omega\kappa.
$$
Here $\Theta$ is the <Heaviside step function>. Incoming rays with $U>U_H$ do not escape. This truncated logarithmic phase is not purely <positive frequency> with respect to $U$. Decomposing it into incoming <positive-frequency solutions> $e^{-i\omega' U}$ and negative modes $e^{+i\omega' U}$ gives the <Bogoliubov transformation>. With $x=U_H-U$, the two relevant <Fourier transform> integrals are
$$
I_-(\omega')=\int_0^\infty x^{ia}e^{-(\epsilon+i\omega')x}\,dx,
\qquad
I_+(\omega')=\int_0^\infty x^{ia}e^{-(\epsilon-i\omega')x}\,dx,
\qquad\epsilon>0.
$$
The former is the positive-frequency coefficient and the latter is the negative-frequency coefficient; the <Klein-Gordon inner product> introduces the same frequency normalization into their modulus ratio. By the defining integral of the <gamma function> and analytic continuation in its Laplace parameter,
$$
I_\mp=\Gamma(1+ia)(\epsilon\pm i\omega')^{-1-ia}.
$$
For a complex number $z$ in the right half-plane, $|z^{-1-ia}|=|z|^{-1}e^{a\arg z}$. As $\epsilon\downarrow0$, the two arguments tend to $+\pi/2$ and $-\pi/2$. This gives the <thermal ratio of Hawking Bogoliubov coefficients>:
$$
\boxed{\frac{|\beta_{\omega\omega'}|^2}{|\alpha_{\omega\omega'}|^2}=e^{-2\pi\omega/\kappa}.}
$$
The regulator fixes the branches and is essential to the sign of the thermal exponent.
Expand the outgoing annihilator as $b=\int d\omega'(\alpha_{\omega\omega'}^*a_{\omega'}-\beta_{\omega\omega'}^*a_{\omega'}^\dagger)$, including a complete set of incoming channels. The <canonical identities for a bosonic Bogoliubov transformation> express $[b,b^\dagger]=1$. For normalized outgoing <wave packets>, they give $\int(|\alpha|^2-|\beta|^2)\,d\omega'=1$. In the incoming <Fock vacuum>, the outgoing mean of the <particle number operator> is $N_\omega=\int|\beta|^2\,d\omega'$. Combining the two identities with the thermal ratio, in the late-time narrow-frequency packet limit, yields
$$
N_\omega=\frac1{e^{2\pi\omega/\kappa}-1},\qquad
\boxed{T_H=\frac\kappa{2\pi}=\frac1{8\pi M}.}
$$
<Wave packets> avoid interpreting the divergent normalization of infinitely long continuum modes as a finite particle count. The temperature is the <Hawking temperature>, and the occupation is the zero-chemical-potential <Bose-Einstein distribution>. Restoring constants, the <Schwarzschild black hole> temperature is $T_H=\hbar c^3/(8\pi G M k_B)$ for physical mass $M$.
The preceding thermal occupation describes the horizon-originating channel before exterior scattering. The exterior angular and curvature potential partly reflects it. For a <scalar field>, the number flux at <future null infinity> is consequently
$$
\frac{dN}{dt\,d\omega}=\frac1{2\pi}\sum_{\ell=0}^\infty(2\ell+1)\frac{\Gamma_\ell(\omega)}{e^{\omega/T_H}-1},
$$
where the <greybody factor> $\Gamma_\ell$ is the transmission probability of a partial wave. Multiplication by $\omega$ gives its energy flux. Thus the asymptotic spectrum has the thermal denominator but is not an exact blackbody spectrum. For fermionic fields the <canonical anticommutation relations> replace the minus sign in the occupation denominator by a plus sign, at the same <Hawking temperature>.
The relevant state is the <collapse vacuum for Hawking radiation>: no incoming thermal bath at <past null infinity>, and regular short-distance behavior in freely falling coordinates at the forming <event horizon>. It predicts an outgoing flux, unlike equilibrium with a thermal bath on an eternal <black hole>. The derivation uses <quantum field theory in curved spacetime> on a prescribed classical geometry; it does not require outgoing particles to follow forbidden classical paths out of the interior. Positive energy carried to infinity is accompanied, in the semiclassical description, by negative Killing-energy flux into the <black hole>. Including that flux in slow backreaction decreases its <mass>, giving <black-hole evaporation>.
The approximation applies to the late radiation of a large nonextremal <black hole> while its <surface gravity> changes slowly on a time scale $\kappa^{-1}$. Backward propagation involves very high locally measured frequencies, so the calculation assumes the usual regular short-distance quantum-field state. It determines the leading radiation and temperature, not the Planck-scale endpoint of <black-hole evaporation>. \b[The exponential horizon redshift forces positive/negative frequency mixing with a thermal ratio, giving outgoing Hawking radiation at $T_H=\kappa/(2\pi)$.]
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