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The BTZ black hole is a quotient of three-dimensional Anti-de Sitter spacetime. In lapse-and-shift coordinates its outer and inner horizon radii are $r_{+}$ and $r_{-}$, and its outer-horizon surface gravity is

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.

For a Euclidean radial-time metric $C(r)(f(r)d{t}_E^2+d{r}^2/f(r))$, assume $f$ has a simple horizon zero and $C$ is smooth, finite and positive there. A proper radial coordinate $\rho =2\sqrt{C(r_H)(r-r_H)/f^{\prime }(r_H)}$ puts the leading metric in the form $d{\rho }^2+f^{\prime }(r_H{)}^2{\rho }^{2}d{t}_E^2/4$. Smoothness fixes the displayed period independently of $C(r_H)$. When the Lorentzian time is normalized at infinity, the period is the inverse Hawking temperature in natural units. A conformal factor singular at the horizon need not preserve this argument.

// Target: general-relativity.bigb

For a normalized static metric $d{s}^2=-Vd{t}^2+d{r}^2/V+r^{2}d{\Omega }^2$ with a simple outer zero at $r_0$, set $b=V^{\prime }(r_0)>0$. The Wick rotation and $r-r_0=b{\rho }^2/4$ give $d{\rho }^2+(b\rho /2{)}^{2}d{t}_E^2$ to leading order, so smoothness fixes period $4\pi /b$. In an ingoing null chart, ${\nabla }_{k}k=(b/2)k$ on the future horizon; the outgoing chart gives the opposite sign on the past branch. The Hawking temperature is $b/(4\pi )=|\kappa |/(2\pi )$ and uses the time generator normalized at infinity.

An extremal black hole lies at the parameter bound where its inner and outer horizons coincide. Its surface gravity and hence its semiclassical Hawking temperature vanish.

Differentiate $M=(r_{+}^2+a^2+Q^2)/(2r_{+})$, $J=Ma$ and $A=4\pi (r_{+}^2+a^2)$. Eliminating $d{r}_{+}$ and $da$ gives the displayed relation, with ${\Omega }_H=a/(r_{+}^2+a^2)$ and ${\Phi }_H=Q{r}_{+}/(r_{+}^2+a^2)$. Matching Hawking temperature to this First law of black-hole mechanics fixes the Bekenstein-Hawking entropy as $A/4$.

The laws of black-hole mechanics relate surface gravity, horizon area, mass, angular momentum, and charge in close analogy with the laws of thermodynamics. Quantum theory identifies the analogy physically through Hawking temperature and Bekenstein-Hawking entropy.

The laws organize the equilibrium and process constraints of thermodynamics: the zeroth law of thermodynamics makes thermal equilibrium transitive, the first law of thermodynamics expresses energy balance, the Second law of thermodynamics restricts entropy changes, and the third law of thermodynamics describes the zero-temperature limit. Their analogy with horizon mechanics becomes quantitative through Hawking temperature and Bekenstein-Hawking entropy.

In units $G=\hslash =c=k_B=1$, a Schwarzschild black hole has Hawking temperature $T=1/(8\pi M)$, so its heat capacity is $dM/dT=-8\pi M^2$. It heats up as it loses mass, preventing stable canonical equilibrium with an unlimited thermal reservoir.

In natural units, periodic Euclidean time of period beta is the inverse temperature of a thermal quantum state. The horizon regularity calculation therefore gives the Hawking temperatureThis agrees with the radiation calculation based on the near-horizon redshift. Restoring constants with $c=1$ multiplies the expression ${\kappa }_{+}/(2\pi )$ by $\hslash /k_B$. In the neutral case $T_H=1/(2\pi \sqrt{2M})$; the temperature is positive for every nonextremal outer horizon.

The extremal limit is different. At $|Q|=M$, $V\sim 4(r-r_h{)}^2/M$, so the proper Euclidean radial distance to the horizon diverges logarithmically. This extremal Euclidean throat is an infinite end, with no finite polar origin on which to impose the cone-removal condition. Local smoothness therefore does not select a finite period by the preceding argument. The vanishing surface gravity and the nonextremal limit give $\boxed{T_H=0\text{ at extremality}}$, rather than a universally nonzero temperature for all horizon solutions. A geometry with $|Q|>M$ has no black-hole horizon and no Hawking temperature inferred from this horizon argument.

Use geometrized units $G=c=1$ and positive mass $M$. The Schwarzschild metric initially describes an exterior region, not the whole black hole:Here $r$ is the areal radius and $t$ is Schwarzschild time, normalized at infinity. We must distinguish a removable coordinate singularity from a genuine curvature singularity. The Kretschmann scalar isIt is finite at $r=2M$ and diverges at $r=0$. Thus $r=2M$ can be crossed in a better chart, whereas $r=0$ cannot be made a regular point by changing the coordinate chart.

First integrate the Schwarzschild tortoise coordinate:The radial metric becomes $f(-d{t}^2+d{r}_{\ast }^2)$. Set $v=t+r_{\ast }$; the ingoing Eddington-Finkelstein coordinates giveThe radial block has determinant minus one and smooth coefficients at $r=2M$, so it extends through the future Schwarzschild event horizon. Its ingoing radial null geodesics have constant $v$, while the outgoing radial null geodesics obey $dr/dv=f/2$. In the future interior both future-directed radial null families decrease $r$; $r=0$ is therefore an unavoidable future boundary for causal motion there. The outgoing chart $u=t-r_{\ast }$ similarly gives $d{s}^2=-fd{u}^2-2dudr+r^{2}d{\Omega }^2$, extending through the past horizon. A single one of these null charts does not display both horizons or the entire extension.

To regularize both horizons together, introduce Kruskal–Szekeres coordinates in the original exterior:ThenFor $r>0$ the right side defining $UV$ decreases monotonically from one to minus infinity. In fact its derivative is $-r{e}^{r/(2M)}/(4M^2)$. Thus it determines a unique smooth $r=r(UV)$ wherever $UV<1$, including $UV=0$. At $r=2M$ the radial metric coefficient is finite and nonzero, proving actual regularity of both horizons and their intersection, rather than merely relabelling the old coordinate divergence. Allowing all real $U,V$ with $UV<1$ produces the simply connected maximal analytic extension, the Kruskal spacetime.

There are four regions of the Kruskal spacetime. The original exterior is $U<0,V>0$; a second asymptotically flat exterior is $U>0,V<0$. The future black hole has $U>0,V>0$ and $0<r<2M$; the past white hole has $U<0,V<0$ and the same range of areal radius. The horizon branches are $U=0$ and $V=0$. Their intersection $U=V=0$ is a regular bifurcation surface, a two-sphere of radius $2M$, not the centre of the spacetime. Each point of the radial diagram represents a symmetry two-sphere.

Put $T_K=(V+U)/2$ and $X_K=(V-U)/2$. Radial null geodesics have slopes $d{T}_K=\pm d{X}_K$, and the curvature singularities haveThe upper branch is the future spacelike Schwarzschild singularity; the lower branch is the white hole past singularity. The Killing vector field extends as ${\partial }_t=(V{\partial }_V-U{\partial }_U)/(4M)$. It is future directed in the original exterior and past directed in the other exterior for the chosen global time orientation; inside, this stationary vector is spacelike. In the future interior, $r$ is a time coordinate decreasing toward the singularity. For example, a radial infaller of unit specific Killing energy obeys $dr/d\tau =-\sqrt{2M/r}$ and reaches $r=0$ from the horizon in finite proper time $4M/3$. The curvature divergence and this geodesic endpoint show why the extension stops there.

For the global causal picture, compactify the null coordinates using $p=\arctan U$, $q=\arctan V$ and let $\mathcal{T}=2(p+q)/\pi$, $\mathcal{X}=2(q-p)/\pi$. The conformal compactification rescales the metric to remove the coordinate factors from the radial metric while preserving its null directions. Since $UV=1$ gives $p+q=\pm \pi /2$ on the two singular branches, they become the horizontal lines $\mathcal{T}=\pm 1$. The resulting Penrose diagram displays the two exteriors, each with its own past null infinity, future null infinity and spatial infinity, plus the future and past interiors. The singularities are spacelike boundaries; the horizons are interior null surfaces. Future event horizons separate events that can escape to an exterior's future null infinity from events in the black-hole interior.

The time-symmetric spatial slice has a bridge between the exteriors, but it is not a traversable passage. Future-directed causal curves have nondecreasing $U$ and $V$. Moving from the right exterior to the left would require $V$ to decrease from positive to negative; the reverse passage would require $U$ to decrease. Both are impossible. This proves the non-traversability of the Schwarzschild bridge. The complete eternal vacuum extension has two exteriors, a black hole and a white hole; it has no causal route between its exteriors. A black hole made by stellar collapse is a different global spacetime: its matter-filled past replaces the white hole region, and it normally has only one exterior. The vacuum extension constructed here should not be mistaken for the collapse history of a single star.

With electric charge, spherical electrovacuum instead gives the Reissner-Nordstrom metric, in an electromagnetic normalization absorbing the charge conversion constants:For $0<|Q|<M$, there are two simple horizons. Outside $r_{+}$ the metric is static; between $r_{-}$ and $r_{+}$, $f<0$ and $r$ is timelike. Inside $r_{-}$, $f>0$ again, so $r$ is spacelike and the $r=0$ singularity is timelike. The outer horizon is the event horizon; the inner horizon is a Cauchy horizon. The same logarithmic tortoise-coordinate and exponential-null-coordinate method crosses each simple horizon locally, but one exponential scale does not regularize both simultaneously: their surface gravities have different magnitudes,Continuing the exact metric through successive inner horizons produces repeated blocks with further exteriors, and evolution beyond a Cauchy horizon is not uniquely fixed by data on the original Cauchy hypersurface. The timelike singularity can be avoided by some trajectories in this ideal extension, unlike the unavoidable spacelike singularity of the future Schwarzschild spacetime interior. However, the enormous inner-horizon blueshift makes this exact continuation unstable: perturbations can cause mass inflation. This is an important qualification on the global structure of a charged spherical black hole.

At $|Q|=M$ the two horizons coalesce, $f=(1-M/r{)}^2$ and surface gravity is zero. The horizon is a degenerate Killing horizon, the tortoise-coordinate divergence becomes a pole and the ordinary nondegenerate Kruskal spacetime construction must be replaced. The exterior develops an infinitely long spatial throat and the formal Hawking temperature vanishes. For $|Q|>M$, no real zero of $f$ remains: the electrovacuum solution has a naked singularity, not a black hole. Consequently arbitrarily allowing charge changes the causal structure, not just the horizon radius.

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.

Use geometrized units with $\hslash =k_B=1$ initially, and write $f(r)=1-2M/r$, $h(r)=1-Q^2/(Mr)$. The given nonextremality assumption makes $h_H=h(2M)=1-Q^2/(2M^2)>0$. Thus the static Killing vector field has a simple zero of its norm atThe exterior $r>2M$ is static. For radial null geodesics, $dt/dr=\pm 1/f$, independently of $h$. With $v=t+\int dr/f$, the metric iswhich is regular at $r_H$ since $h_H>0$. The outgoing null equation is $dr/dv=f/2$. Inside the future horizon these rays move to smaller radius, while outside they can escape to infinity. Hence the zero is the event horizon. The inner zero of $h$ is not another regular Killing horizon: the Killing norm there diverges rather than vanishes. The strict charge bound matters when making the regular extension.

For the Euclidean black-hole regularity condition, continue $t=i{t}_E$, consistent with the sign of the given quantum time-translation operator. Set $x=r-2M$. To leading order $f=x/(2M)$ and $h=h_H$, so the radial–time part of the Euclidean metric isIntroduce the radial proper length coordinate $\rho$ by $x=h_H{\rho }^2/(8M)$. ThenThe first two terms are a polar plane with angle $t_E/(4M)$. Smoothness at its origin requiresA different period would make a conical singularity. The time coordinate is normalized to proper time at infinity, so no further asymptotic redshift factor is needed. The cancellation of $h_H$ proves the charge independence of the Euclidean horizon period under a regular radial conformal factor; this argument would fail at $h_H=0$.

To justify the thermal interpretation, analytic continuation of the time-evolution operator $U(t)=e^{iHt}$ gives $U(i{t}_E)=e^{-H{t}_E}$. Its matrix element is an imaginary-time propagation kernel. Identifying the initial and final configurations and integrating over them takes its trace, giving the Euclidean path integral $Z=\mathrm{Tr}{e}^{-\beta H}$. In an energy eigenbasis each state has weight $e^{-\beta E}$, exactly the Boltzmann factor of a canonical ensemble. Thus the geometrically required period is the inverse Hawking temperature. With dimensions restored, an imaginary time period $\Delta t_E$ gives $\beta =\Delta t_E/\hslash =1/(k_{B}T)$. ConsequentlyHere $\mathcal{M}$ is mass in ordinary units; for a neutral hole $M=G\mathcal{M}/c^2$ and the physical imaginary-time period is $8\pi G\mathcal{M}/c^3$. Bosonic thermal fields are periodic and fermionic fields antiperiodic over that same Euclidean period. The relation to the canonical density operator, not just dimensional reasoning, identifies the temperature.

For an evaporation estimate take a neutral, isolated Schwarzschild black hole and approximate its radiated power with the Stefan–Boltzmann law. Its area is $A=16\pi G^2{\mathcal{M}}^2/c^4$, and the photon Stefan-Boltzmann constant is ${\sigma }_{SB}={\pi }^2{k}_B^4/(60{\hslash }^3{c}^2)$. Substituting the temperature givesThe black-hole evaporation speeds up as the mass decreases. Integrating, rather than dividing the initial rest energy by the initial power, givesThe corresponding initial Hawking temperature is about $6\times 10^{-8}$K. This is the Stefan-Boltzmann estimate of Schwarzschild evaporation time at the level of accuracy requested. Greybody factors and other emitted species alter the numerical coefficient, but not the leading cubic scaling of a large isolated hole. The estimate assumes negligible accretion or ambient radiation absorption; a stellar-mass hole in the present cosmic radiation bath is not currently losing mass at that isolated-hole rate. Finally the semiclassical formula does not describe the Planck-scale endpoint: extrapolating it to zero mass is an estimate of the macroscopic lifetime, not a proof of what the final quantum object is.

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 $T_H=\kappa /(2\pi )$.

In $c=k_B=\hslash =1$ units, the Hawking temperature is $T_H=|\kappa |/(2\pi )$ for the stated time normalization. The inertial central observer sees the de Sitter horizon temperature $T_0=1/(2\pi R)$. The Tolman temperature law states $T\sqrt{-k^2}=\text{constant}$ in static thermal equilibrium. HenceWith part b, $T^2=(a^2+R^{-2})/(4{\pi }^2)$ and $T/(a/(2\pi ))=R/r\to 1$ at the horizon. This tends to the Unruh effect temperature for the increasingly accelerated observer.

Let $s$ be proper distance inward from the horizon. Then $R-r\simeq s^2/(2R)$, $F\simeq s^2/R^2$, givingThe radial factor has Rindler coordinates: $T_{\mathrm{M}}=s\sinh (t/R)$ and $X_{\mathrm{M}}=s\cosh (t/R)$ make it Minkowskian. Fixed $s$ has acceleration $1/s$ and temperature $1/(2\pi s)$, agreeing with the leading behavior. At the center $a=0$ but the finite de Sitter temperature remains; $T=a/(2\pi )$ is only the near-horizon limit here.

The two identities. Set $F_{\beta \gamma }={\nabla }_{\beta }{K}_{\gamma }$. The Killing equation says $F_{\beta \gamma }=-F_{\gamma \beta }$. The Levi-Civita connection has symmetric lower connection indices, so they cancel in the antisymmetric difference:Use normalized antisymmetrization, soPut $a^{\sigma }=K^{\tau }{\nabla }_{\tau }{K}^{\sigma }$. Contracting with $F^{\beta \gamma }$, the second term is $a^{\gamma }{F}_{\gamma \alpha }$ and the third is the same after using antisymmetry. ThusRearranging proves the Killing derivative contraction identity

Restriction to a Killing horizon. On a Killing horizon, the generator $K$ is normal to the horizon as well as tangent to its null generators. Consequently $K_{[\alpha }{\nabla }_{\beta }{K}_{\gamma ]}=0$ there. One can see this without assuming hypersurface orthogonality away from the horizon: if the horizon is locally $\Phi =0$, write its dual one-form as $K=fd\Phi +\Phi q$ near it; then $K\wedge dK=0$ on $\Phi =0$. By the definition of surface gravity, $a^{\sigma }=\kappa K^{\sigma }$ on the horizon, and lowering that relation gives $K^{\sigma }{\nabla }_{\sigma }{K}_{\alpha }=\kappa K_{\alpha }$. The contraction identity becomesAway from points where $K$ vanishes, cancel a nonzero component of $K_{\alpha }$; at a regular bifurcation surface extend by continuity. ThereforeThis is surface gravity from the Killing derivative; it concerns the horizon, not every point in the exterior.

Static spherical metric. Choose the Killing vector $K={\partial }_t$. Its dual one-form is $-Adt$, and the first identity immediately giveswith all other components zero. HenceUsing the horizon identity and the simultaneous simple zeros,Here the positive value is chosen for a regular outer Killing horizon with $A,B>0$ on the exterior side. This is surface gravity of a static spherical horizon. A different normalization $K\mapsto cK$ rescales surface gravity by $c$; when an asymptotically flat normalization is wanted, the time coordinate is chosen so $A\to 1$ at infinity.

Euclidean regularity and temperature. Let $\delta =r-r_{+}$, $a=A^{\prime }(r_{+})$ and $b=B^{\prime }(r_{+})$. On the exterior side, define the proper radial coordinate $R=2\sqrt{\delta /b}$. After Wick rotation, the near-horizon metric isThe radial-time plane is a polar plane with angular coordinate $\vartheta =\kappa \tau$. Its circumference-to-radius ratio is $\kappa \Delta \tau$; the conical singularity disappears precisely whenThis is the Euclidean black-hole regularity condition. If a signed surface gravity convention is used, the period uses $|\kappa |$.

In a thermal quantum field theory, imaginary-time periodicity is the inverse-temperature condition expressed by the KMS condition. Therefore smoothness identifies the Hawking temperaturein units $\hslash =k_B=1$, or $T_H=\hslash \kappa /(2\pi k_B)$ when $\kappa$ is measured as an inverse time. It provides the thermal interpretation of surface gravity in black-hole thermodynamics. The regular Euclidean construction describes an equilibrium thermal state; the collapse calculation of Hawking radiation in the last solution supplies the outgoing spectrum. The simple-zero hypothesis excludes an extremal black hole, for which this polar-plane argument changes.

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 $|0⟩$.