= Solution
<Hawking radiation> is a prediction of <quantum field theory in curved spacetime>, not of classical collapse alone. Consider a free massless <Klein-Gordon field> in the <collapse vacuum for Hawking radiation>: initially no incoming particles from <past null infinity>, with the short-distance state regular for freely falling observers through the forming <event horizon>. At late times use the approximately fixed exterior <Schwarzschild metric>; neglect backreaction during the observation interval. First set $G=c=\hbar=k_B=1$.
The exterior metric has $f(r)=1-2M/r$. Introduce the <tortoise coordinate> and outgoing and ingoing null coordinates,
$$
r_*=r+2M\log\left(\frac r{2M}-1\right),\qquad u=t-r_*,\qquad v=t+r_*.
$$
The <surface gravity> is $\kappa=1/(4M)$. The logarithm in $r_*$ makes a late escaping ray spend a long exterior coordinate time close to the <event horizon>. One of the regular outgoing <Kruskal–Szekeres coordinates> is proportional to $-e^{-\kappa u}$. Smooth propagation backwards through the collapsing body relates this regular coordinate to the incoming coordinate on <past null infinity>. If $v_H$ labels the last ray able to escape, the late-time <Hawking exponential ray map> is consequently
$$
\boxed{v_H-v=C e^{-\kappa u},\qquad
u(v)=-\frac1\kappa\log\left(\frac{v_H-v}{C}\right),\quad C>0.}
$$
Here the symbol $u(v)$ is the outgoing retarded time evaluated on the ray; the first equation is the leading asymptotic relation, with subleading corrections that vanish at late times. The constant $C$ depends on collapse history, whereas the exponential rate is determined by the final <surface gravity>. The relation follows from a nondegenerate <Killing horizon>; a linear ray map would not have the same effect.
An outgoing positive-frequency mode at <future null infinity> behaves as $p_\omega\sim e^{-i\omega u}/\sqrt{4\pi\omega}$, where $\omega>0$ is measured with respect to the asymptotic stationary time. In the near-horizon <geometric optics> approximation its backward-propagated form on <past null infinity> is
$$
p_\omega(v)\sim\frac{1}{\sqrt{4\pi\omega}}\,
\Theta(v_H-v)\left(\frac{v_H-v}{C}\right)^{i\omega/\kappa}.
$$
This is not purely positive frequency in $v$. Its decomposition into incoming positive-frequency modes $f_{\omega'}\sim e^{-i\omega'v}/\sqrt{4\pi\omega'}$ and their complex conjugates is a <Bogoliubov transformation>:
$$
p_\omega=\int_0^\infty\left(\alpha_{\omega\omega'}f_{\omega'}+\beta_{\omega\omega'}f_{\omega'}^*\right)d\omega'.
$$
The coefficients follow from the <Klein-Gordon inner product>, or equivalently from the <Fourier transform> with the indicated flux normalization. Writing $y=v_H-v$ and $a=\omega/\kappa$, the two Fourier integrals, after a convergence regulator, are
$$
I_\pm=\int_0^\infty y^{ia}e^{-(\epsilon\pm i\omega')y}\,dy
=\Gamma(1+ia)(\epsilon\pm i\omega')^{-1-ia},\qquad\epsilon>0.
$$
The equality is the defining integral of the <Gamma function>, continued to these arguments from the right half-plane. Up to phases of unit modulus, $\alpha$ is $\sqrt{\omega'/\omega}\,I_+/(2\pi)$ and $\beta$ is $\sqrt{\omega'/\omega}\,I_-/(2\pi)$. Taking $\epsilon\downarrow0$ on the principal branches, the arguments of $\epsilon\pm i\omega'$ tend to $\pm\pi/2$. Thus $|I_+|\propto e^{\pi a/2}$ and $|I_-|\propto e^{-\pi a/2}$, giving the <thermal ratio of Hawking Bogoliubov coefficients>
$$
\boxed{\frac{|\beta_{\omega\omega'}|^2}{|\alpha_{\omega\omega'}|^2}=e^{-2\pi\omega/\kappa}.}
$$
In particular, using $|\Gamma(1+ia)|^2=\pi a/\sinh(\pi a)$,
$$
|\beta_{\omega\omega'}|^2=\frac{1}{2\pi\kappa\omega'}\frac{1}{e^{2\pi\omega/\kappa}-1}.
$$
The collapse-dependent phases and $C$ disappear from the ratio. The <canonical identities for a bosonic Bogoliubov transformation> state that the positive-frequency norm minus the negative-frequency norm is one for a normalized outgoing mode. Combining that identity with the thermal ratio yields the late-time mean occupation
$$
\boxed{\langle N_\omega\rangle=\frac{1}{e^{2\pi\omega/\kappa}-1},\qquad
T_H=\frac{\kappa}{2\pi}=\frac{1}{8\pi M}.}
$$
These are the <Bose-Einstein distribution> and the <Hawking temperature>. Strictly, monochromatic modes have delta-function normalization, and $\int d\omega'\,|\beta_{\omega\omega'}|^2$ contains the infinite-duration divergence $\int d\omega'/\omega'$. Normalized <wave packets> with a finite frequency band and a finite retarded-time window remove this artifact. For sufficiently late packets the occupation is the frequency average of the displayed thermal factor, tending to it for a narrow band. Fermionic fields instead give the corresponding denominator $e^{2\pi\omega/\kappa}+1$ because their canonical normalization uses anticommutators.
Four-dimensional propagation adds an important qualification to the word thermal. Separation of the massless <Klein-Gordon equation> into <spherical harmonics> gives an exterior radial scattering potential
$$
V_\ell(r)=f(r)\left(\frac{\ell(\ell+1)}{r^2}+\frac{2M}{r^3}\right).
$$
Only a fraction $\Gamma_\ell(\omega)$ of the outgoing mode is transmitted to <future null infinity>; this is the <greybody factor>. The scalar particle flux there is
$$
\boxed{\frac{dN}{du\,d\omega}=\frac{1}{2\pi}\sum_{\ell=0}^\infty
(2\ell+1)\frac{\Gamma_\ell(\omega)}{e^{8\pi M\omega}-1}.}
$$
The factor $2\ell+1$ counts the angular modes, and multiplication by $\omega$ gives the spectral <energy> flux. An observer at infinity therefore receives an outgoing thermal population filtered by curvature scattering, rather than an exact unfiltered blackbody spectrum. In physical units, for mass $M_{\mathrm{phys}}$,
$$
\boxed{T_H=\frac{\hbar c^3}{8\pi G k_BM_{\mathrm{phys}}}.}
$$
There is no incoming thermal bath in the <collapse vacuum for Hawking radiation>. The outgoing flux has positive <Killing energy>; its counterpart inside the <event horizon> can carry negative <Killing energy>. Including the expectation value of the <stress-energy tensor> in the gravitational evolution allows the mass to decrease. The fixed-background derivation describes the slowly evolving semiclassical regime and does not determine the endpoint of evaporation. Quantum expectation values need not satisfy the classical <null energy condition>, so this loss of area is not a contradiction of the classical <second law of black-hole mechanics>. The essential late-time universality is the combination of a regular initial quantum state and the exponential redshift at the newly formed <event horizon>.
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