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Use $\hslash =c=k_B=1$ and normalize the time-translation Killing vector field at infinity. The simple outer zero and positivity on its exterior imply $b=V^{\prime }(r_0)>0$. Near the horizon, write $x=r-r_0$; then $V=bx+O(x^2)$. The Wick rotation $t=-i{t}_E$ gives the radial Euclidean metricDefine $\rho =2\sqrt{x/b}$, so $x=b{\rho }^2/4$. Its leading near-horizon form isThus $b{t}_E/2$ is a polar angle. Smoothness at the origin requires period $2\pi$, rather than a cone or a multiple cover with a branched origin. The Euclidean black-hole regularity condition isA thermal quantum state has imaginary-time period $\beta =1/T$. Since $V\to 1$, this time coordinate is normalized to the clocks at infinity, and the Hawking temperature isA static clock at radius $r>r_0$ measures the redshifted local temperature $T/\sqrt{V(r)}$; that local quantity is not the temperature requested with the asymptotic normalization.

The static coordinates are singular on the horizon, so calculate its surface gravity in a regular ingoing chart. With $v=t+\int dr/V$, the metric and the same normalized Killing vector field areThe inverse radial block has $g^{vr}=1$, $g^{rr}=V$, $g^{vv}=0$. Direct calculation givesSince $k$ has constant coordinate components, its acceleration isOn the future horizon, $V=0$ and $k$ is null, so the defining equation gives $\kappa =b/2$. In the outgoing chart regular on the past branch, the radial cross-term is $-2dudr$ and the same calculation gives $\kappa =-b/2$ for $k={\partial }_u$. Hence the branch-independent physical statement isThis proves the Euclidean temperature and signed horizon surface gravity relation directly. Rescaling the Killing vector field would rescale its surface gravity; the condition at infinity fixes that otherwise arbitrary normalization.

Let $r_h=\sqrt{2m}$ and $f(r)=1-r_h^2/r^2$. The geometry is the five-dimensional Schwarzschild-Tangherlini metric. Introduce a tortoise coordinate by $d{r}_{\ast }/dr=f^{-1}$. Decomposing the integrand givesAn irrelevant additive constant has been fixed. Set $v=t+r_{\ast }$ and $u=t-r_{\ast }$. Substituting $dt=dv-dr/f$ or $dt=du+dr/f$ respectively gives the Ingoing Eddington-Finkelstein coordinates and Outgoing Eddington-Finkelstein coordinates:The radial-time block in each has determinant $-1$, so the apparent singularity at $f=0$ has disappeared. The ingoing chart crosses the future event horizon, and the outgoing chart crosses the past horizon. Radial null geodesics in the ingoing chart obey either $v=\text{constant}$ or $dr/dv=f/2$, displaying how outward light is trapped when $r<r_h$.

For the maximal extension, the exterior radial metric is $-fdudv$. Since the surface gravity is $\kappa =f^{\prime }(r_h)/2=1/r_h$, choosein the right exterior. Their product and the extended metric areThe coefficient at the horizon is finite and nonzero, $-4r_h^2{e}^{-2}$. The product relation therefore extends smoothly through either horizon. For $r>r_h$, $UV<0$ gives two exterior regions. For $0<r<r_h$, $0<UV<1$ gives the future black-hole interior with $U,V>0$ and the past white-hole interior with $U,V<0$. The Kruskal extension of the five-dimensional Schwarzschild-Tangherlini metric has a bifurcation three-sphere at $U=V=0$.

The singularity $r=0$ corresponds to $UV=1$. It is a curvature singularity: for example, its Kretschmann scalar is $72r_h^4/r^8$. Since $g^{ab}{\partial }_{a}r{\partial }_{b}r=f<0$ in the interior, constant-$r$ surfaces are spacelike there, and the singularities are spacelike boundaries. Define $P=\arctan U$, $Q=\arctan V$, then compactified coordinates $T=(Q+P)/2$, $X=(Q-P)/2$. Radial null curves have $T\pm X=\text{constant}$. Moreover, $\tan P\tan Q=1$ gives $T=\pm \pi /4$ at the two singularities. This constructs the Penrose diagram, not just its horizon labels:

Each point away from the singular boundaries represents a three-sphere, not a two-sphere. The diagram has two asymptotically flat exteriors with separate past and future null infinities, a future black hole and a past white hole. The diagram of a collapse spacetime would omit the second exterior and the white-hole region; the maximal extension of the metric given here contains both.

For the Komar integral, normalize the stationary Killing vector field as $k={\partial }_t$, with unit norm at infinity. The Killing equation makes ${\nabla }^a{k}^b$ antisymmetric, and its divergence is proportional to $R^b{}_c{k}^c$. In a vacuum exterior this vanishes, so Stokes theorem makes the flux independent of the enclosing sphere. In the weak-field limit its radial component measures the derivative of the gravitational potential; its large-sphere flux is therefore proportional to the asymptotic gravitational mass. Normalization of $k$ is needed because rescaling it would rescale the integral.

Specify the surface orientation to fix the sign and factor. On a static exterior slice take future unit normal $u^a=f^{-1/2}({\partial }_t{)}^a$, outward radial normal $n^a=f^{1/2}({\partial }_r{)}^a$ andThen $d{\Sigma }^{tr}=-dA$ and $d{\Sigma }^{rt}=dA$, with both antisymmetric components included. Since $k_t=-f$,The three-sphere area is $2{\pi }^2{r}^3$, and $f^{\prime }=2r_h^2/r^3=4m/r^3$. ThusReversing the binormal orientation gives $+8{\pi }^{2}m$; a convention with half the antisymmetric binormal gives half this magnitude. The question does not fix those conventions, so the choice above is explicit.

For comparison with physical mass, the Komar mass normalization in higher dimensions isIn five dimensions this givesThe metric parameter $m$ has dimensions of length squared and is not itself the physical mass. An independent horizon check is $I_K=-2\kappa A_H$ with $A_H=2{\pi }^2{r}_h^3$, which reproduces the same integral. The dimension-adjusted normalization also agrees with the ADM mass; the four-dimensional Komar prefactor should not be reused unchanged.

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.

Take the first centre at the origin. If some listed centres coincide, first combine their poles; let $M$ be the total residue at the origin, so $M=M_1$ when the centres are distinct. The contributions from centres away from the origin are real analytic there. Their constant term iswhere $F$ is analytic in $r$ and in local angular charts. This follows by substituting $\mathbf{x}=r\mathbf{n}(\theta ,\varphi )$ in the ordinary Taylor expansions of the regular contributions. The constant $h$ is independent of angle, which makes the hinted coordinate differential integrable.

Put $A=M+hr$, $Q=rH=A+r^{2}F$, and defineinitially for $r>0$. Substitution into the metric givesThe apparently singular radial coefficient is actuallysince $Q-A=r^{2}F$. All coefficients therefore extend analytically to negative and positive $r$ near zero, with $Q(0)=M>0$. At the horizon $g_{vv}=0$, $g_{vr}=1$, $g_{rr}=4MF(0,\theta ,\varphi )$ and the angular metric is $M^{2}d{\Omega }_2^2$. The $v,r$ determinant is $-1$ both before and at the extension, so the metric remains nondegenerate and Lorentzian. Angular coordinate singularities are handled by ordinary sphere charts. This explicitly constructs the analytic extension of a Majumdar–Papapetrou horizon.

Inverting the radial block gives $g^{rr}=r^2/Q^2$, which vanishes at $r=0$. Thus the hypersurface is null. Its normal agrees there with the metric dual of $\xi ={\partial }_v$: ${\xi }_{a}d{x}^a=dr$ on the horizon. The metric is independent of $v$, so $\xi$ is a Killing vector field and the null hypersurface is a Killing horizon.

The surface gravity obeys ${\nabla }_a({\xi }^2)=-2\kappa {\xi }_a$ on a Killing horizon. Here ${\xi }^2=-r^2/Q^2$ has a double zero and its entire differential vanishes at $r=0$, while ${\xi }_a$ is the nonzero normal. ConsequentlyThe horizon is degenerate. Its cross-sectional area is $4\pi M^2$, independent of the centres away from the origin. The essential cancellation would fail with an arbitrary constant in place of the displayed regular part $h$.

Use geometrized units $G=c=1$. The Physical-process first law of black-hole mechanics concerns a small perturbation of a rotating black hole in an initially stationary spacetime by infalling matter, followed by relaxation to another stationary spacetime. For an uncharged black hole, or neutral accretion with charge held fixed, its first-order statement isHere $\kappa >0$ is the background surface gravity, ${\Omega }_H$ is the background horizon angular velocity, and $\delta A$ is the change in event horizon area. The energy $\delta E$ and angular momentum $\delta J$ are the fluxes carried into the event horizon; when these account for the changes of the black hole, they equal $\delta M$ and its change of angular momentum. The result is a linearized law, not an exact identity for a finite violent accretion event. The perturbation must be small enough that focusing does not produce new caustics or invalidate the expansion about the existing generators.

Let $k^a$ be the stationary Killing vector field normalized at infinity, $m^a$ the axial Killing vector field with $2\pi$-periodic orbits, andOn the background Killing horizon, ${\chi }^a$ is its future-directed null generator and satisfies ${\chi }^b{\nabla }_b{\chi }^a=\kappa {\chi }^a$. Choose a parameter $v$ with $\chi ={\partial }_v$. The null expansion in this parameter is $\theta ={\partial }_v\log dA$. The generator is not affinely parametrized; if $\ell ={\partial }_{\lambda }$ is affine, then $\chi =\kappa \lambda \ell$ after an appropriate choice of affine origin, with $\lambda \propto e^{\kappa v}$. Rescaling the Null Raychaudhuri equation givesThe null twist vanishes because the generators are normal to the event horizon. By the Einstein field equations, $R_{ab}{\chi }^a{\chi }^b=8\pi T_{ab}{\chi }^a{\chi }^b$. Background null expansion and null shear vanish on a Killing horizon of the background stationary spacetime. Thus $\theta$ and $\hat{\sigma }$ are first order, their squares are second order, and the linearized equation isThe final stationary spacetime supplies the boundary condition $\theta \to 0$ in the future. Explicitly,This future boundary condition reflects the global definition of the event horizon. For localized infall, $\theta$ also tends to zero in the remote past. Integrating the linearized equation along the generators, and then over a background cross-section with area element $d{A}_0$, givesUsing the background area element introduces only second-order corrections.

With orientations chosen so that future infalling positive energy has positive flux, the conserved Killing energy and axial angular momentum giveConsequently $\delta E-{\Omega }_H\delta J={\int }_{\mathcal{H}}\delta T_{ab}{\chi }^a{\chi }^{b}dvd{A}_0$, proving the displayed Physical-process first law of black-hole mechanics. Restoring $G$ replaces $8\pi$ by $8\pi G$. The null energy condition makes the integral nonnegative and therefore implies area increase in this regime; that condition is needed for the sign conclusion, not for the linearized flux identity itself. Charged infall would require the additional electrostatic work term, and an extremal background needs separate treatment because the proof uses $\kappa >0$.

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.

In advanced coordinates, $K={\partial }_v$ is a Killing vector field and $K^2=-f$. The normal covector to $r=2m$ is $dr$; raising it gives $g^{ar}{\partial }_a={\partial }_v+f{\partial }_r$, which equals $K$ on the horizon. Thus $r=2m$ is a null hypersurface normal to $K$, and hence a Killing horizon.

The surface gravity is defined by ${\nabla }_{K}K=\kappa K$ on this horizon, equivalently ${\nabla }_a(K^2)=-2\kappa K_a$. Here $K_a=(-f,1,0,0)$, so the radial component gives $\kappa =f^{\prime }(2m)/2$. Because $F(2m)=0$,It is positive for all finite real $\alpha$ and $m>0$; the stationary Killing vector field retains its asymptotic normalization.

Classically, the laws of thermodynamics and black-hole thermodynamics have parallel structures, but an ordinary black hole absorbs and does not emit: its temperature would appear to be zero, despite nonzero surface gravity. Hawking radiation changes this by giving an actual temperature measurable from a thermal particle spectrum. In units $G=\hslash =c=k_B=1$,Scattering through the exterior modifies the observed flux by greybody factors; for rotation and charge the occupation involves $\omega -m_{\varphi }{\Omega }_H-q{\Phi }_H$. These transmission and chemical-potential factors do not change the underlying Hawking temperature.

The Zeroth law of black-hole mechanics makes $\kappa$ constant on a stationary horizon, so it gives a uniform equilibrium temperature, matching the zeroth law of thermodynamics. The full First law of black-hole mechanics isComparing it with $dE=TdS+\Omega dJ+\Phi dQ$ and using the actual Hawking temperature fixesWith constants restored, $S_{BH}=k_B{c}^{3}A/(4G\hslash )$. If $\kappa$ is defined geometrically with dimensions of inverse length, $T_H=\hslash c\kappa /(2\pi k_B)$; if surface gravity is expressed as an acceleration, the equivalent formula is $\hslash \kappa /(2\pi c{k}_B)$. The first-law comparison fixes entropy up to an additive constant. This is the Bekenstein-Hawking entropy, so the mechanical analogy becomes a quantitative thermodynamic identification, including its normalization.

The classical second law of black-hole mechanics then corresponds to increasing black-hole entropy. Quantum evaporation can reduce $A$, because the classical energy-condition hypotheses fail, while outgoing radiation carries entropy. The appropriate thermodynamic statement is the generalized second law,Finally, the third law of black-hole mechanics, that zero surface gravity cannot be achieved by a finite physical process under its regularity assumptions, becomes the unattainability of zero temperature. It does not assert the Nernst entropy formulation: an extremal horizon can have nonzero area. Hawking radiation supplies the physical temperature, the first law fixes entropy, and the generalized second law includes both the black hole and its exterior matter.

Use the exterior-normalized Killing vector field $K={\partial }_t={\partial }_v$ from the previous part. In Ingoing BTZ coordinates, its only nonzero acceleration component on $r=r_{+}$ isThe defining equation for surface gravity, ${\nabla }_{K}K=\kappa K$, therefore gives the positive future-branch valueThis value uses exactly the prescribed Killing vector field normalization. A constant rescaling of the Killing vector field rescales its surface gravity.

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.