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In a collapse spacetime, choose the quantum field's incoming state to have no incoming particles at past null infinity and to be regular in freely falling coordinates near the forming event horizon. The late outgoing modes then have the Hawking temperature, while there is no incoming thermal bath. This collapse-state boundary condition differs from an equilibrium state on an eternal black hole. The Hawking exponential ray map and the thermal ratio of Hawking Bogoliubov coefficients determine the leading late-time particle spectrum; greybody factors describe scattering outside the event horizon.

Use $G=\hslash =c_{\mathrm{light}}=k_B=1$ and normalize the stationary time at infinity. For a regular stationary, asymptotically flat electrovacuum black hole, the Kerr-Newman metric hasInitially assume $M>0$ and $d>0$, so the outer Killing horizon is nondegenerate. Its surface gravity, angular velocity, electric potential and area areThese are horizon quantities for the generator $\chi ={\partial }_t+{\Omega }_H{\partial }_{\varphi }$.

The laws of black-hole mechanics provide the first reason to regard $\kappa$ and area as thermodynamic variables. The Zeroth law of black-hole mechanics makes $\kappa$ constant on an equilibrium horizon. The first law for the Kerr-Newman family readsFor example, write the horizon relation as $M=(r_{+}^2+a^2+Q^2)/(2r_{+})$ and use $J=Ma$; differentiating eliminates $d{r}_{+}$ and $da$ to give this law. The last two terms are rotational and electric work. The classical second law of black-hole mechanics says that area cannot decrease under the null energy condition and appropriate global horizon assumptions. The third law of black-hole mechanics concerns unattainability of a zero-surface-gravity regular horizon by a finite physical process.

This analogy alone does not determine the temperature scale: if entropy were $\eta A_H$, the first law would only give $T=\kappa /(8\pi \eta )$. The decisive input is quantum field theory in the collapsing or stationary background. In a collapse vacuum that is regular for freely falling observers, the late outgoing retarded time $u$ and a regular affine null coordinate $U$ satisfy the Hawking exponential ray mapAn outgoing mode $e^{-i\omega u}$ therefore behaves as $(-U/C{)}^{i\omega /\kappa }$. Continuing this power through the horizon changes its amplitude by the factor $e^{-\pi \omega /\kappa }$. Its positive- and negative-frequency decomposition consequently has the thermal ratio of Hawking Bogoliubov coefficientsCombining that ratio with bosonic normalization $|\alpha {|}^2-|\beta {|}^2=1$ gives a Planck occupation $1/(e^{2\pi \omega /\kappa }-1)$. The fermionic normalization instead gives the corresponding Fermi factor. Thus Hawking radiation fixesFor rotating charged modes, the horizon energy is $\tilde{\omega }=\omega -m_{\varphi }{\Omega }_H-q{\Phi }_H$, so the emission spectrum contains the same temperature with angular-momentum and charge chemical potentials. Exterior scattering supplies greybody factors; it changes the received flux, not the horizon temperature. Superradiant bosonic modes require the usual signed absorption factor, and a globally regular thermal bath need not exist throughout an asymptotically flat rotating exterior. The collapse-state emission argument is the relevant one.

A complementary check is the Euclidean black-hole regularity condition. The local corotating nondegenerate horizon geometry is Rindler-like:Smoothness at $\rho =0$ requires $\tau$ to have period $2\pi /\kappa$. Imaginary-time periodicity is precisely inverse temperature, agreeing with the radiation calculation. Matching the resulting temperature to the first law fixes the Bekenstein-Hawking entropy to $S=A_H/4$. The generalized second law then uses $S_{\mathrm{outside}}+A_H/4$: the classical area theorem alone does not apply to the negative-energy quantum flux responsible for evaporation.

Substitution gives the Kerr-Newman horizon temperatureIt reduces to $1/(8\pi M)$ for a Schwarzschild black hole. The nonextremal limit towards $d=0$ gives zero temperature. At exact extremality the Euclidean horizon is degenerate, so the elementary conical-period argument does not itself fix a period; zero temperature here is the limiting semiclassical result. The reasoning assumes ordinary Einstein–Maxwell dynamics, a regular horizon, the stated normalization at infinity and a regime where quantum fields on a slowly evolving classical geometry are a useful approximation. The mechanical laws, radiation spectrum and Euclidean regularity agree under these assumptions, which is substantially stronger evidence than the classical analogy alone.

Use a free real scalar field obeying a linear Klein-Gordon equation, for example $(\square -{\mu }^2-\xi R)\varphi =0$. This linearity is the assumption that permits mode evolution by a Bogoliubov transformation. Work with a complete set of normalized modes, or normalizable wave packets when the spectrum is continuous. The stationary asymptotic regions supply time translations relative to which positive-frequency solutions and their complex conjugates can be distinguished.

For solutions define the conserved Klein-Gordon inner productHere $\Sigma$ is a Cauchy surface with future-directed normal. Its conservation follows by applying the wave equation to the divergence of its current, provided boundary flux is included or vanishes. Let $u_j$ be a positive-frequency basis in the past and $v_s$ one in the future, each propagated as a solution through the intermediate geometry. Normalize them by $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and $(u_i,u_j^{\ast })=0$, with the corresponding identities for the future basis. The two field expansions areThe canonical commutation relations give $[a_i,a_j^{†}]={\delta }_{ij}$ and $[b_s,b_t^{†}]={\delta }_{st}$. The in-vacuum satisfies $a_j|0_{-}⟩=0$, while the out-vacuum satisfies $b_s|0_{+}⟩=0$. They are generally different vacua.

Fix the coefficient convention by expanding future modes in past modes:The minus sign is due to the negative norm of the conjugate modes. Taking the Klein-Gordon inner product with the field, which is antilinear in its first entry, givesBoth frequency sectors are needed: if $\beta \ne 0$, a future annihilation operator contains a past creation operator. This is the mechanism of particle number from Bogoliubov coefficients, not a classical nonzero mean field.

Computing $[b_s,b_t^{†}]$ and $[b_s,b_t]$ explicitly gives the canonical identities for a bosonic Bogoliubov transformation:Completeness also gives the inverse $a_j={\sum }_s({\alpha }_{sj}{b}_s+{\beta }_{sj}^{\ast }{b}_s^{†})$. Thus the evolved past vacuum, viewed in the future Fock space, obeysFor a finite-mode treatment with invertible $\alpha$, solve this condition by a multimode squeezed vacuum:The canonical identities make $Z$ symmetric. Commuting $b_s$ through the exponential gives $b_s|\Psi ⟩={\sum }_t{Z}_{st}{b}_t^{†}|\Psi ⟩$, which directly verifies the vacuum condition. Its normalization is $|C|=\det (I-Z{Z}^{†}{)}^{1/4}$. A general initial state is obtained by applying its initial creation-operator polynomial and expressing those operators in the future basis. In infinitely many modes, a common unitary Fock representation requires the creation-mixing coefficients to be Hilbert-Schmidt; finite wave-packet counts avoid treating delta-normalized modes as individual normalizable states. These are the bosonic mode mixing implementability conditions, rather than an assumption of arbitrary finite total creation over an infinite time interval.

For the requested number operator of one future mode, set $N_s=b_s^{†}{b}_s$. Expectations below are in the evolved past vacuum $|\Psi ⟩$, equivalently in $|0_{-}⟩$ using the Heisenberg-mode operators above. Acting on the past vacuum gives $b_s|0_{-}⟩=-{\sum }_j{\beta }_{sj}^{\ast }{a}_j^{†}|0_{-}⟩$, whose squared norm isThe mean particle number vanishes exactly when that mode has no negative-frequency mixing.

The second moment needs the anomalous contractionThe canonical commutation relations give $N_s^2=b_s^{†2}{b}_s^2+N_s$. Moreover,Its vacuum and two-particle pieces are orthogonal. Applying the commutation relations twice showsThe two pairings each contribute $n_s^2$ to the squared norm. Hence the full answer isThus $\mathrm{Var}(N_s)=n_s(n_s+1)+|c_s{|}^2$. These are the out-mode number fluctuations in the in-vacuum. Omitting $c_s$ would assume a property not given in the general question. A thermal single mode has $c_s=0$, giving $⟨N_s^2⟩=2n_s^2+n_s$. A single-mode squeezed vacuum state instead has $|c_s{|}^2=n_s(n_s+1)$, giving $⟨N_s^2⟩=3n_s^2+2n_s$.

If the counted observable is the total number over a finite set of future modes, its moments follow by retaining cross-mode correlations as well. Define $C_{st}=⟨b_s^{†}{b}_t⟩$ and $M_{st}=⟨b_s{b}_t⟩$. The same vacuum pairings give $⟨N_s{N}_t⟩=C_{ss}{C}_{tt}+|C_{st}{|}^2+|M_{st}{|}^2+{\delta }_{st}{C}_{ss}$. Thus $⟨N_{\mathrm{tot}}⟩=\mathrm{tr}C$ and $⟨N_{\mathrm{tot}}^2⟩=(\mathrm{tr}C{)}^2+\mathrm{tr}(C^2)+\mathrm{tr}(M^{†}M)+\mathrm{tr}C$, with $C_{st}={\sum }_j{\beta }_{sj}{\beta }_{tj}^{\ast }$ and $M_{st}=-{\sum }_j{\alpha }_{sj}^{\ast }{\beta }_{tj}^{\ast }$. Independent thermal counts cannot be assumed for a general correlated squeezed state.

For Hawking radiation, choose the incoming vacuum at past null infinity of a collapse spacetime and propagate late outgoing modes backwards. Near a nonextremal forming event horizon, the logarithmic tortoise-coordinate divergence leads to the Hawking exponential ray mapwhere $u$ is late retarded time, $U$ is an early regular affine null coordinate and $\kappa >0$ is the final surface gravity. A late positive-frequency mode $e^{-i\omega u}$ therefore becomes proportional to $(U_H-U{)}^{i\omega /\kappa }$ on rays escaping just before horizon formation. This is not positive-frequency with respect to $U$: it has both frequency signs and hence nonzero $\beta$.

The thermal factor can be seen directly. With $x=U_H-U>0$, $a=\omega /\kappa$ and a regulator $\epsilon >0$, the relevant Fourier integrals areAs $\epsilon \downarrow 0$, the arguments of the two complex factors approach $\pm \pi /2$. Their squared-modulus ratio for the suppressed and enhanced frequency branches is $e^{-2\pi a}$. Thus the thermal ratio of Hawking Bogoliubov coefficients, combined with the canonical normalization, yields the bosonic occupationin units $\hslash =c=k_B=1$. Scattering through the exterior potential multiplies the asymptotic occupation by the appropriate greybody factor.

A complete future mode description includes modes crossing the horizon as well as modes reaching future null infinity. The global incoming vacuum develops correlations between these sectors; restricting to the exterior gives approximately thermal outgoing occupation rather than a claim that the complete pure state became a thermal mixed state. This is the particle-production picture of a collapsing black hole, not an incoming thermal bath on an eternal geometry. Including backreaction turns the emitted positive energy into black-hole mass loss. The renormalized quantum stress tensor is not subject to the pointwise classical positivity used in the black-hole area theorem, so this does not contradict its classical proof.

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 $G=c=\hslash =k_B=1$.

The exterior metric has $f(r)=1-2M/r$. Introduce the tortoise coordinate and outgoing and ingoing null coordinates,The surface gravity is $\kappa =1/(4M)$. The logarithm in $r_{\ast }$ 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 consequentlyHere 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 isThis is not purely positive frequency in $v$. Its decomposition into incoming positive-frequency modes $f_{{\omega }^{\prime }}\sim e^{-i{\omega }^{\prime }v}/\sqrt{4\pi {\omega }^{\prime }}$ and their complex conjugates is a Bogoliubov transformation: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, areThe 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 $\sqrt{{\omega }^{\prime }/\omega }{I}_{+}/(2\pi )$ and $\beta$ is $\sqrt{{\omega }^{\prime }/\omega }{I}_{-}/(2\pi )$. Taking $ϵ\downarrow 0$ on the principal branches, the arguments of $ϵ\pm i{\omega }^{\prime }$ 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 coefficientsIn particular, using $|\Gamma (1+ia){|}^2=\pi a/\sinh (\pi a)$,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 occupationThese are the Bose-Einstein distribution and the Hawking temperature. Strictly, monochromatic modes have delta-function normalization, and $\int d{\omega }^{\prime }|{\beta }_{\omega {\omega }^{\prime }}{|}^2$ contains the infinite-duration divergence $\int d{\omega }^{\prime }/{\omega }^{\prime }$. 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 potentialOnly 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 isThe 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 $M_{\mathrm{phys}}$,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.

Work initially in units $G=c=\hslash =k_B=1$ 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 $r_{\ast }$, and retarded time $u=t-r_{\ast }$. Near the final event horizon, $r_{\ast }\sim (2\kappa {)}^{-1}\log (r-2M)$, with Schwarzschild surface gravity $\kappa =1/(4M)$. On a fixed advanced-time section, $u=v-2r_{\ast }$, 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 isThe 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,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 $e^{-i{\omega }^{\prime }U}$ and negative modes $e^{+i{\omega }^{\prime }U}$ gives the Bogoliubov transformation. With $x=U_H-U$, the two relevant Fourier transform integrals areThe 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,For a complex number $z$ in the right half-plane, $|z^{-1-ia}|=|z{|}^{-1}{e}^{a\arg z}$. As $ϵ\downarrow 0$, the two arguments tend to $+\pi /2$ and $-\pi /2$. This gives the thermal ratio of Hawking Bogoliubov coefficients:The regulator fixes the branches and is essential to the sign of the thermal exponent.

Expand the outgoing annihilator as $b=\int d{\omega }^{\prime }({\alpha }_{\omega {\omega }^{\prime }}^{\ast }{a}_{{\omega }^{\prime }}-{\beta }_{\omega {\omega }^{\prime }}^{\ast }{a}_{{\omega }^{\prime }}^{†})$, including a complete set of incoming channels. The canonical identities for a bosonic Bogoliubov transformation express $[b,b^{†}]=1$. For normalized outgoing wave packets, they give $\int (|\alpha {|}^2-|\beta {|}^2)d{\omega }^{\prime }=1$. In the incoming Fock vacuum, the outgoing mean of the particle number operator is $N_{\omega }=\int |\beta {|}^{2}d{\omega }^{\prime }$. Combining the two identities with the thermal ratio, in the late-time narrow-frequency packet limit, yieldsWave 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 $T_H=\hslash c^3/(8\pi GM{k}_B)$ 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 consequentlywhere 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. The exponential horizon redshift forces positive/negative frequency mixing with a thermal ratio, giving outgoing Hawking radiation at $T_H=\kappa /(2\pi )$.

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 isThis 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 $T_H=\hslash \kappa /(2\pi )$, with $c=k_B=1$ and future-horizon normalization at infinity. Comparing the area terms in the first laws gives the Bekenstein-Hawking entropyup to a conventionally fixed additive constant. In ordinary units it is $k_B{c}^{3}A/(4G\hslash )$. 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 isIts 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 $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and vanishing mixed products. Expand the field asContinuous 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:The inner product gives $\alpha {\alpha }^{†}-\beta {\beta }^{†}=I$ and $\alpha {\beta }^{\mathsf{T}}=\beta {\alpha }^{\mathsf{T}}$. For the initial vacuum annihilated by all $a_j^{\mathrm{in}}$, $\boxed{⟨N_i^{\mathrm{out}}{⟩}_{\mathrm{in}}=\sum _j|{\beta }_{ij}{|}^2}$. 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 mapPulling $e^{-i\omega u}$ 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 $ϵ>0$, the Gamma function evaluates the Fourier integrals asAs $ϵ\downarrow 0$, 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: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 $S_{\mathrm{gen}}=A/(4G\hslash )+S_{\mathrm{outside}}$, 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. Quantum emission gives the temperature–entropy identification physical meaning; generalized entropy replaces the classical area alone when radiation back-reacts.

Canonical mode mixing and its inverse. Treat annihilation operators as column vectors and keep complex conjugation, transpose and Hermitian adjoint distinct. Computing the canonical commutation relations for $a^{\prime }=\alpha a+\beta a^{†}$ givesConsequently the required conditions areFor complete invertible mode mixing, putThe canonical identities for a bosonic Bogoliubov transformation are $\mathcal{T}J{\mathcal{T}}^{†}=J$, giving ${\mathcal{T}}^{-1}=J{\mathcal{T}}^{†}J$. Its upper blocks yieldEquivalently, the column identities are ${\alpha }^{†}\alpha -{\beta }^T\overline{\beta }=I$ and ${\alpha }^{†}\beta ={\beta }^T\overline{\alpha }$. In finite dimension invertibility follows from the canonical matrix identity. In infinitely many modes, the row identities alone need not give a complete inverse: $a_i^{\prime }=a_{i+1}$ preserves them but discards one mode. Here completeness of the two field-mode expansions supplies the needed invertible transformation.

Obtaining coefficients from the modes. Use the Klein-Gordon inner product, antilinear in its first argument,The normalized positive-frequency solutions satisfy $(p_i,p_j)={\delta }_{ij}$, $({\overline{p}}_i,{\overline{p}}_j)=-{\delta }_{ij}$ and $(p_i,{\overline{p}}_j)=0$. Current conservation makes this product independent of the Cauchy hypersurface when boundary flux vanishes. Taking the product of the field with a mode gives $a_i=(p_i,\hat{\Phi })$ and henceWith this operator convention, the corresponding mode expansion isThe minus sign comes from the negative norm of conjugate modes. Using a different convention for mode coefficients can move this sign and the complex conjugates; the operator formula fixes them unambiguously here.

Particle count. In the in-vacuum, only the contraction $⟨0|a_j{a}_k^{†}|0⟩={\delta }_{jk}$ survives. ThereforeThis is particle number from Bogoliubov coefficients: the old vacuum contains that expected number of particles in the new mode $i$. Nonzero negative-frequency mixing is the source of particle production. For continuum modes one uses normalized wave packets and replaces the sum by the corresponding integral.

The squeezed-state relation. First use finitely many modes, or an implementable infinite-mode limit. SeekThe canonical commutation relations giveAll further nested commutators vanish since $\hat{F}$ contains only creation operators. ThusThe old annihilation conditions hold exactly whenThe inverse transformation's canonical identity $A{B}^T=B{A}^T$ shows $A^{-1}B=(A^{-1}B{)}^T$, so the required symmetry is automatic. Also $A{A}^{†}-B{B}^{†}=I$ givesHence the singular values of $M$ are less than one. The Autonne-Takagi factorization $M=U\mathrm{diag}(s_j)U^T$ changes to independent canonical oscillators. For each factor, the squared norm of $\exp (s_j{b}_j^{†2}/2)|0⟩$ is the even-occupation seriesMultiplying these norms gives the finite-mode normalizationThe phase of $C$ is arbitrary. The resulting multimode squeezed vacuum is annihilated by all old annihilation operators, so uniqueness of the normalized Fock vacuum identifies it with $|0⟩$.

The PDF's squeezed-vacuum claim needs qualification for infinitely many modes. Bosonic mode mixing implementability requires $\beta$ to be a Hilbert-Schmidt operator, ${\sum }_{ij}|{\beta }_{ij}{|}^2<\infty$, for a common ordinary bosonic Fock space. For a counterexample, take $\alpha =\cosh sI$ and $\beta =\sinh sI$ on countably many modes with fixed $s>0$. The algebraic canonical conditions hold, but $M=\tanh sI$ and the finite-$N$ normalization is $C_N=(\cosh s{)}^{-N/2}\to 0$. Every mode has a fixed positive probability of nonzero occupation in the required product state; the probability that all but finitely many modes are empty is zero. Ordinary Fock space vectors instead have total occupation finite with probability one, even if their expected occupation is infinite. Thus there is no nonzero common-Fock-space vector of the prescribed form. The squeezed expression is a normalizable vacuum relation with a finite-mode regulator or the implementability condition, not solely from the commutator identities.

Black-hole radiation. In a collapse spacetime, choose early positive-frequency solutions with respect to affine incoming time $v$ on past null infinity, and late outgoing modes proportional to $e^{-i\omega u}$ on future null infinity. The early state is their in-vacuum. The Hawking exponential ray map hasfor late outgoing rays, with $\kappa >0$. Tracing a late mode backwards therefore gives a profile proportional toIt is not a pure positive-frequency incoming wave, so the Bogoliubov transformation has nonzero $\beta$.

To see the thermal factor, put $x=v_H-v>0$ and $a=\omega /\kappa$. The positive- and negative-frequency Fourier pieces have, apart from common normalization and phases, the regulated integralsAs $\epsilon \downarrow 0$, the positive-frequency piece has squared-modulus factor $e^{+\pi a}$ and the negative-frequency piece has $e^{-\pi a}$. Consequently the thermal ratio of Hawking Bogoliubov coefficients isFor a normalized narrow-frequency outgoing wave packet, combine this ratio with ${\sum }_j(|{\alpha }_{ij}{|}^2-|{\beta }_{ij}{|}^2)=1$. The occupation is the Bose-Einstein distributionThe Hawking temperature is thus the same one found by Euclidean regularity. For Schwarzschild, $\kappa =1/(4M)$ gives $T_H=1/(8\pi M)$ in natural units. Propagation through the exterior potential multiplies the asymptotic occupation by the greybody factor ${\Gamma }_{\ell }(\omega )$.

The outgoing radiation has partner-mode correlations across the horizon: the total state can be pure while the reduced outgoing state is thermal. A complete mode basis must include modes entering the horizon as well as those reaching infinity. The particle statement is made for normalized packets and finite observation intervals; idealized infinite-duration emission need not define the two vacua in one global Fock space. This explains both the squeezed-state structure and the physically measurable Hawking radiation.

Work first in units $G=c=\hslash =k_B=1$. Black-hole thermodynamics starts with the laws of black-hole mechanics. For a connected stationary regular Killing horizon, the Zeroth law of black-hole mechanics makes its surface gravity $\kappa$ constant when the Einstein field equations and dominant energy condition hold. This parallels uniform equilibrium temperature. For neighboring stationary asymptotically flat four-dimensional Einstein-Maxwell black holes, the First law of black-hole mechanics iswhere ${\Omega }_H$ is the horizon angular velocity and ${\Phi }_H$ its electric potential relative to infinity. The last terms are rotational and electromagnetic work, analogous to the work terms in $\delta E=T\delta S+\text{work}$. The second law of black-hole mechanics, expressed by Hawking's area theorem, makes the total future horizon area nondecreasing for classical matter obeying the null energy condition and global assumptions such as strong asymptotic predictability. The third law of black-hole mechanics is an unattainability statement: subject to the usual regularity assumptions, the weak energy condition, and a bounded stress-energy tensor, no finite physical process reduces the surface gravity of a regular horizon to zero. It does not assert that extremal black holes have zero area or zero entropy.

Classical geometry alone fixes an analogy, not a nonzero physical temperature. Quantum field theory supplies Hawking temperature $T_H=\kappa /(2\pi )$. Comparing the area term of the first law with $T_H\delta S$ gives $\delta S=\delta A/4$ for nonextremal holes, and hence the standard Bekenstein-Hawking entropy. Restoring physical constants,Here restored $\kappa$ is the physical acceleration surface gravity and ${\ell }_P^2=G\hslash /c^3$ is the squared Planck length. The first-law comparison determines the area coefficient, leaving an additive entropy constant unspecified; the displayed formula is the usual convention. Its area scaling, rather than ordinary volume scaling, signals that the available thermodynamic degrees of freedom of a gravitating system are constrained by the horizon geometry.