Hawking Radiation

For sixty years a black hole was the blackest object imaginable: a one-way membrane that swallows light and returns nothing. The classical laws of black hole mechanics, worked out in 1973, hinted otherwise — the horizon area never decreases, exactly as entropy never decreases, and the surface gravity is constant over the horizon, exactly as temperature is uniform in equilibrium. Yet everyone, including the people who wrote those laws, read the analogy as a formal accident. A body with a genuine temperature must radiate, and a classical black hole cannot. Stephen Hawking dissolved the paradox in 1974 by doing something no one had bothered to do: quantizing a field on the spacetime of a collapsing star and asking what a detector at infinity actually registers at late times. The answer was a steady, featureless flux of particles with a perfectly Planckian spectrum at temperature $T_H = \hbar\kappa/(2\pi k_B c)$ . The surface gravity $\kappa$ , a purely geometric quantity, turned out to be a literal thermodynamic temperature, and the horizon area a literal entropy, $S = k_B c^3 A/(4G\hbar)$ . The mechanism is not "energy leaking out." It is that particle is not an observer-independent notion in a dynamical spacetime. The vacuum an observer defines in the distant past — no positive-frequency excitations of the early-time modes — is not the vacuum a late-time observer defines, because outgoing waves that reach infinity late traced back through an exponentially blueshifting near-horizon region. The linear map between the two mode bases, the Bogoliubov transformation, mixes positive and negative frequencies, and that mixing is exactly thermal. A black hole therefore evaporates, slowly for a star, explosively at the end, and the apparently information-free spectrum sets up the deepest known tension between gravity and quantum mechanics: the information paradox.

Key takeaways

A black hole is not perfectly black: quantizing a field on the spacetime of a collapsing star gives a steady thermal flux at $T_H = \hbar\kappa/(2\pi k_B c)$ , with $\kappa$ the surface gravity of the horizon.
For a Schwarzschild hole $T_H = \hbar c^3/(8\pi G M k_B) \approx 6.2\times10^{-8}\ \mathrm{K}\times(M_\odot/M)$ — small holes are hot, large holes cold (negative heat capacity).
The radiation exists because "vacuum" is observer-dependent in a dynamical spacetime: the in-vacuum contains, in the out-basis, a thermal crowd of quanta, counted by the Bogoliubov coefficient $|\beta_{\omega\omega'}|^2$ .
Fixing the temperature fixes the entropy through the first law: the Bekenstein–Hawking entropy is $S = k_B c^3 A/(4G\hbar)$ , one quarter of the horizon area in Planck units.
Energy conservation forces evaporation: $\mathrm{d}M/\mathrm{d}t \propto -M^{-2}$ and a lifetime $\tau \propto M^3$ (about $2\times10^{67}$ yr for the Sun), ending in an explosive Planck-scale burst.
The exactly thermal, no-hair spectrum threatens unitarity — the black hole information paradox — and drives holography, the Page curve, and the AdS/CFT program.
Key idea: particle number is not observer-invariant in a dynamical spacetime; the exponential blueshift of outgoing modes near the forming horizon makes the collapse-vacuum a thermal state of out-quanta, a Bogoliubov mixing of positive and negative frequencies that yields exactly the Planck spectrum at $T_H = \hbar\kappa/(2\pi k_B c)$ .

The core insight

Particle number is not a coordinate- or observer-invariant notion in a dynamical spacetime. The vacuum of the collapsing-star spacetime — defined by modes purely positive-frequency in the far past — contains, re-expanded in the late-time out-mode basis, a thermal population of outgoing quanta. The exponential blueshift of outgoing modes as they trace back to the near-horizon region produces, via a Bogoliubov mixing of positive and negative frequencies, exactly the Planck spectrum at temperature $T_H = \hbar\kappa/(2\pi k_B c)$ .

Where this lives today

No one has caught a black hole radiating — the astrophysical ones are far too cold — but three research fronts keep Hawking's result at the centre of physics. First, primordial black hole searches: a hole of mass $\sim10^{11}\ \mathrm{kg}$ would be exploding now in a burst of GeV–TeV gamma rays, and observatories such as Fermi-LAT, HAWC, H.E.S.S. and LHAASO scan for these bursts, turning non-detections into some of the tightest limits on any primordial-hole contribution to dark matter and on the small-scale primordial power spectrum. Second, analogue gravity: because the effect is kinematic — a horizon for waves plus mode mixing — it can be staged in the laboratory. Jeff Steinhauer's group created a sonic horizon in a flowing Bose–Einstein condensate and, across a decade of experiments, measured correlated phonon pairs whose spectrum matches a thermal Hawking emission at the analogue temperature, while water-tank and nonlinear-optics setups probe the same physics classically. Third, the information paradox has become quantitative: the 2019–2020 "island" and replica-wormhole computations reproduce the Page curve — entanglement entropy that rises and then falls, as unitarity demands — directly from a semiclassical gravity path integral, the strongest evidence yet that evaporation is unitary after all. Hawking radiation remains the one hard number every candidate theory of quantum gravity is checked against.

The physical picture

Follow one outgoing light ray backward from a distant detector. If it arrives at late retarded time $u$ , it must have left the star just before the horizon formed, having lingered exponentially long in the strong-field region and been blueshifted there without bound. The later the arrival, the closer to the forming horizon the ray began, and the relation between late time at infinity and advanced time near the horizon is logarithmic: $u = -\tfrac{1}{\kappa}\ln\big(\kappa(v_0-v)\big)$ , where $v_0$ labels the very last ray that escapes. That single logarithm is the entire origin of the thermal spectrum — a log-frequency map Fourier-transforms a pure tone into a Planck factor. The first figure puts the two faces of the mechanism side by side: the near-horizon crowding of crests, and the correlated pair it leaves behind.

Figure

The two faces of the mechanism. Left, the collapse produces correlated pairs across the horizon: the outgoing partner escapes as a Hawking quantum while the ingoing partner falls in, so the outside sees a mixed thermal state at T_H = ħκ/(2πk_B c). Right, evenly spaced outgoing crests at infinity trace back through u = −(1/κ) ln(κ(v₀ − v)) to crests crowded exponentially against the last escaping ray v₀ — the logarithm turns a pure tone into a Planck spectrum

There is a second way to say the same thing that avoids ray-tracing entirely. "Vacuum" means no positive-frequency excitations, but positive frequency is defined relative to a time coordinate, and the collapse changes which time coordinate is natural. Written in late-time modes, the early-time vacuum is a two-mode squeezed state: correlated pairs, one member of each escaping to infinity, its partner trapped behind the horizon. Discard the trapped partner — an observer outside cannot access it — and the reduced state of the outgoing radiation is exactly a thermal density matrix. The temperature is set entirely by $\kappa$ . The second figure is the number a physicist actually reaches for: how that temperature runs across thirty decades of black-hole mass, and where it crosses the one temperature every real black hole is bathed in, the cosmic microwave background.

Figure

Hawking temperature versus mass, T_H = 6.17×10⁻⁸ K × (M⊙/M), on log axes. A primordial hole of 5×10¹¹ kg glows at ~2×10¹¹ K (tens of MeV, gamma rays); a solar-mass hole sits at 6.2×10⁻⁸ K; Sagittarius A* at 4×10⁶ M⊙ is 1.5×10⁻¹⁴ K. Only holes lighter than ~4.5×10²² kg are hotter than the 2.7 K CMB and so net emitters today

The same mathematics governs a uniformly accelerated observer in flat spacetime: the Unruh effect, a thermal bath at $T_U = \hbar a/(2\pi k_B c)$ . A static observer hovering outside a black hole must accelerate to avoid falling, and the surface gravity is precisely that proper acceleration redshifted out to infinity. Hawking's $\kappa/2\pi$ and Unruh's $a/2\pi$ are two projections of one fact about quantum fields and horizons.

The core equations

The master formula assigns a temperature to any Killing horizon from its surface gravity alone. It is the statement that geometry — $\kappa$ — is thermodynamics — $T$ .

T_H = \dfrac{\hbar\,\kappa}{2\pi k_B c} \qquad (1)

where: $T_H$ — temperature of the radiation measured at infinity; $\kappa$ — surface gravity of the Killing horizon (dimensions of acceleration in the convention used here); $\hbar$ — reduced Planck constant; $k_B$ — Boltzmann constant; $c$ — speed of light. Reading: every factor except $\kappa$ is a fundamental constant, so the horizon's geometry fixes the temperature outright. Specializing to the simplest hole gives the number everyone quotes, and exposes the hallmark inverse-mass scaling.

T_H = \dfrac{\hbar c^3}{8\pi G M k_B} \approx 6.17\times10^{-8}\ \mathrm{K}\times\dfrac{M_\odot}{M} \qquad (2)

where: $M$ — mass of the Schwarzschild hole; $M_\odot$ — solar mass. Reading: because $T_H \propto 1/M$ , a hole cools as it grows and heats as it shrinks — the negative heat capacity that drives runaway evaporation. The spectrum itself is not merely "warm" but exactly Planckian, mode by mode, up to a frequency-dependent filter.

\langle N_\omega \rangle = \dfrac{\Gamma_\omega}{e^{2\pi\omega/\kappa} - 1} \qquad (3)

where: $\langle N_\omega\rangle$ — mean number of quanta emitted in the mode of frequency $\omega$ (units $G=c=\hbar=1$ ); $\Gamma_\omega$ — greybody factor, the probability that the same mode incident from infinity is absorbed by the hole; $\omega$ — frequency at infinity. Reading: the bracket is the exact Bose–Einstein occupation at $T_H=\kappa/2\pi$ ; $\Gamma_\omega$ is the back-scattering off the spacetime's potential barrier, so the hole is a grey body, not a perfect one. Fixing the temperature forces the entropy through the first law, and the answer is geometrical: one quarter of the area in Planck units.

S_{BH} = \dfrac{k_B c^3}{4 G\hbar}\,A = \dfrac{A}{4 \ell_P^2}\,k_B \qquad (4)

where: $S_{BH}$ — Bekenstein–Hawking entropy; $A$ — horizon area; $\ell_P = \sqrt{G\hbar/c^3}$ — Planck length. Reading: entropy scales as area, not volume — the germ of holography — and for a solar-mass hole $S/k_B \approx 1.0\times10^{77}$ , dwarfing the $\sim10^{58}$ of the star that formed it. The four laws of black-hole mechanics now read as the four laws of thermodynamics; the first law is an exact differential identity among horizon quantities.

\mathrm{d}(M c^2) = T_H\,\mathrm{d}S_{BH} + \Omega_H\,\mathrm{d}J + \Phi_H\,\mathrm{d}Q \qquad (5)

where: $\Omega_H$ — angular velocity of the horizon; $\Phi_H$ — its electrostatic potential; $J, Q$ — angular momentum and charge. Reading: the $\Omega_H\,\mathrm{d}J$ and $\Phi_H\,\mathrm{d}Q$ terms are the rotational and electrical work, exactly as in ordinary thermodynamics. Treating the hole as a grey body of temperature $T_H$ and area $A$ gives the mass-loss rate and hence a finite lifetime.

\dfrac{\mathrm{d}M}{\mathrm{d}t} = -\dfrac{\alpha\,\hbar c^4}{G^2 M^2}, \qquad \tau \sim \dfrac{5120\,\pi\,G^2 M^3}{\hbar c^4} \qquad (6)

where: $\alpha$ — dimensionless constant collecting greybody factors and the number of emitted species, $\mathcal{O}(10^{-3}\!-\!10^{-4})$ ; $\tau$ — evaporation lifetime (photon-only estimate). Reading: $A T_H^4 \propto M^2\cdot M^{-4} = M^{-2}$ , so $\dot M \propto -M^{-2}$ and $\tau \propto M^3$ — about $2\times10^{67}$ years for the Sun. Finally, the local twin of the whole effect: acceleration alone, with no horizon and no gravity, already produces a thermal bath.

T_U = \dfrac{\hbar\, a}{2\pi k_B c} \qquad (7)

where: $a$ — proper acceleration of a uniformly accelerated (Rindler) observer in flat space; $T_U$ — temperature that observer perceives. Reading: replace $a$ by the redshifted surface gravity $\kappa$ and Unruh becomes Hawking — the two are the same statement about quantum fields seen through a horizon.

Derivation AHawking temperature from Euclidean cone smoothness

The fastest fully rigorous route to $T_H$ bypasses mode-counting altogether and rests on a single geometric demand: that the black-hole geometry, continued to imaginary time, be smooth. The idea is that a quantum field in equilibrium at temperature $T$ has correlation functions periodic in imaginary time with period $\hbar/(k_B T)$ — the KMS condition — so if the Euclidean geometry itself forces a periodicity, that period is an inverse temperature. We expand the metric near the horizon, Wick-rotate, recognize a plane in polar coordinates, and read off the period that removes the conical singularity.

Step 1. Write a static horizon metric as $\mathrm{d}s^2 = -f(r)\,\mathrm{d}t^2 + f(r)^{-1}\mathrm{d}r^2 + \cdots$ with $f(r_+)=0$ , and expand $f(r)\approx f'(r_+)(r-r_+)$ just outside the horizon. The surface gravity in this convention is $\kappa = \tfrac{1}{2}c^2 f'(r_+)$ . Why: only the leading behaviour of $f$ near its zero matters for the near-horizon geometry, and it is exactly that slope that defines $\kappa$ .

Step 2. Introduce the proper radial distance $\rho = \int \mathrm{d}r/\sqrt{f}$ , which near the horizon gives $\rho \propto \sqrt{r-r_+}$ and hence $f \approx \kappa^2\rho^2$ . The metric collapses to the Rindler form $\mathrm{d}s^2 \approx -\kappa^2\rho^2\,\mathrm{d}t^2 + \mathrm{d}\rho^2$ . Why: proper distance is the coordinate in which the near-horizon geometry is manifestly that of an accelerated frame, stripping away the coordinate singularity of $r$ .

Step 3. Wick-rotate $t = -i\tau$ . The line element becomes Euclidean, $\mathrm{d}s_E^2 = \rho^2\,\mathrm{d}(\kappa\tau)^2 + \mathrm{d}\rho^2$ , which is precisely the flat plane in polar coordinates with radius $\rho$ and angle $\theta = \kappa\tau$ . Why: imaginary time turns the Lorentzian near-horizon wedge into an ordinary two-dimensional plane, where smoothness has an elementary meaning.

Step 4. A plane in polar coordinates is smooth at the origin $\rho=0$ only if the angle has period exactly $2\pi$ ; any other period leaves a conical defect. So $\theta \cong \theta + 2\pi$ forces $\kappa\tau \cong \kappa\tau + 2\pi$ , i.e. Euclidean time is periodic with period $\beta = 2\pi/\kappa$ . Why: the horizon $\rho=0$ is the tip of the cone, and physical regularity of the geometry there is what pins the period — no field theory input yet.

Step 5. Impose the KMS condition: equilibrium at temperature $T$ means periodicity $\hbar/(k_B T)$ in imaginary time. Matching $\hbar/(k_B T_H)\cdot c^{-1}\cdot\ldots$ — restoring units, $\beta = 2\pi/\kappa$ — gives

\dfrac{1}{k_B T_H} = \dfrac{2\pi}{\kappa}\dfrac{\hbar}{c} \;\Rightarrow\; T_H = \dfrac{\hbar\kappa}{2\pi k_B c}.

Why: geometry supplies the period; thermal quantum field theory interprets that period as an inverse temperature, and the two must agree for a smooth, regular state.

Go deeper · Derivation B · 1 minSurface gravity, temperature, and entropy of the Schwarzschild hole

Derivation BSurface gravity, temperature, and entropy of the Schwarzschild hole

Here we make the abstract $\kappa$ concrete for the one black hole every reader knows, and follow the chain all the way to the entropy so that the factor of one quarter appears with nothing put in by hand. The plan is: read $\kappa$ off the metric function $f(r) = 1 - 2GM/rc^2$ , substitute into the master formula to get $T_H(M)$ , attach the numerical value, and then integrate the first law $\mathrm{d}(Mc^2) = T_H\,\mathrm{d}S$ to recover the Bekenstein–Hawking area law with its exact coefficient.

Step 1. The horizon sits at $f(r_s)=0$ , i.e. $r_s = 2GM/c^2$ . With $f'(r) = 2GM/(r^2 c^2)$ , the surface gravity is $\kappa = \tfrac{1}{2}c^2 f'(r_s) = \tfrac{1}{2}c^2\cdot\dfrac{2GM}{r_s^2 c^2} = \dfrac{c^4}{4GM}$ . Why: this is the single geometric input; everything thermodynamic below is bookkeeping on top of this one number, which carries units of acceleration.

Step 2. Insert into equation (1):

T_H = \dfrac{\hbar\kappa}{2\pi k_B c} = \dfrac{\hbar}{2\pi k_B c}\cdot\dfrac{c^4}{4GM} = \dfrac{\hbar c^3}{8\pi G M k_B}.

Why: the geometry-to-thermodynamics dictionary is now explicit, and the $1/M$ dependence — hotter as it shrinks — is manifest.

Step 3. Put in numbers. With $\hbar, c, G, k_B$ and $M = M_\odot = 1.989\times10^{30}\ \mathrm{kg}$ one finds $T_H \approx 6.17\times10^{-8}\ \mathrm{K}$ , about 60 nK — four thousand times colder than intergalactic space and far below the 2.7 K microwave background. Why: it shows at a glance why no astrophysical hole radiates detectably; they all absorb the CMB faster than they emit.

Step 4. Integrate the first law. Using $\mathrm{d}(Mc^2) = T_H\,\mathrm{d}S$ ,

S = \int \dfrac{c^2\,\mathrm{d}M}{T_H} = \int \dfrac{8\pi G k_B M}{\hbar c}\,\mathrm{d}M = \dfrac{4\pi G k_B M^2}{\hbar c}.

Why: the temperature we just derived is exactly what converts the mechanical first law into a thermodynamic one, and the integral is elementary because $T_H\propto 1/M$ .

Step 5. Rewrite in terms of area. With $A = 4\pi r_s^2 = 16\pi G^2 M^2/c^4$ , substitute $M^2 = A c^4/(16\pi G^2)$ to get

S_{BH} = \dfrac{4\pi G k_B}{\hbar c}\cdot\dfrac{A c^4}{16\pi G^2} = \dfrac{k_B c^3}{4 G\hbar}\,A.

The coefficient is exactly $1/4$ , and for a solar-mass hole $S/k_B \approx 1.0\times10^{77}$ . Why: nothing was tuned — the quarter emerges from the same $\kappa$ that set the temperature, which is why the entropy–area law is regarded as a genuine prediction rather than a definition.

Go deeper · Derivation C · 2 minThe thermal spectrum from Bogoliubov mixing of in and out modes

Derivation CThe thermal spectrum from Bogoliubov mixing of in and out modes

This is the heart of Hawking's 1974 calculation, and it explains why the spectrum is thermal rather than merely warm. The strategy is to propagate a late-time outgoing wavepacket backward through the collapsing body to past null infinity, discover that the relation between out-time and in-time is logarithmic, and show that a logarithmic phase relation forces the Bogoliubov coefficients into the single analyticity relation $|\alpha| = e^{\pi\omega/\kappa}|\beta|$ from which the Planck factor follows with no further assumptions. We work in the geometric-optics limit, tracing null rays from future infinity $\mathcal{I}^+$ back to past infinity $\mathcal{I}^-$ .

Step 1. Trace an outgoing null ray that reaches $\mathcal{I}^+$ at retarded time $u$ backward through the star to an ingoing ray at advanced time $v$ . Rays that emerge later skimmed closer to the forming horizon, giving the logarithmic relation

u = -\dfrac{1}{\kappa}\ln\big(\kappa(v_0 - v)\big) + \text{const}, \qquad v < v_0,

with $v_0$ the advanced time of the last ray that escapes. Why: the exponential redshift of the near-horizon region is the whole physical content; the logarithm is its inverse and everything thermal descends from it.

Step 2. A positive-frequency out-mode is $\phi^{\rm out}_\omega \sim e^{-i\omega u}$ . Substituting the ray relation converts it, as a function of $v$ , into a branch-cut power law:

\phi^{\rm out}_\omega \sim \exp\!\Big[\tfrac{i\omega}{\kappa}\ln\big(\kappa(v_0-v)\big)\Big] \propto (v_0 - v)^{\,i\omega/\kappa},

supported only for $v<v_0$ . Why: the out-mode, which is a clean oscillation at infinity, is a wildly oscillating power law near $v_0$ — the seed of infinitely many in-frequencies and hence of particle creation.

Step 3. Project onto in-modes $e^{\pm i\omega' v}$ to read off the Bogoliubov coefficients:

\alpha_{\omega\omega'},\ \beta_{\omega\omega'} \propto \int_{-\infty}^{v_0}\!\mathrm{d}v\; e^{\pm i\omega' v}\,(v_0-v)^{\,i\omega/\kappa}.

Both are gamma-function (Mellin) integrals; $\alpha$ pairs the out-mode with positive in-frequency, $\beta$ with negative. Why: a non-zero $\beta$ — mixing of positive and negative frequencies — is exactly the statement that the in-vacuum contains out-particles.

Step 4. The two integrals differ only by the sign of $\omega'$ , i.e. by rotating the branch cut of $(v_0-v)^{i\omega/\kappa}$ through the lower-half plane, which multiplies the result by $e^{\pi\omega/\kappa}$ . Hence the single relation

|\alpha_{\omega\omega'}| = e^{\pi\omega/\kappa}\,|\beta_{\omega\omega'}|.

Why: this one line, pure analyticity, encodes all of the thermality — nothing else about the collapse survives into the final spectrum.

Step 5. Combine with the Bogoliubov normalization $\int(|\alpha|^2 - |\beta|^2)\,\mathrm{d}\omega' = 1$ . Writing $|\alpha|^2 = e^{2\pi\omega/\kappa}|\beta|^2$ gives $|\beta|^2(e^{2\pi\omega/\kappa}-1) = 1$ per mode, so

\langle N_\omega\rangle = \int |\beta_{\omega\omega'}|^2\,\mathrm{d}\omega' = \dfrac{1}{e^{2\pi\omega/\kappa} - 1}.

This is the Bose–Einstein distribution at $T_H = \kappa/2\pi$ ; restoring $\Gamma_\omega$ for back-scattering gives equation (3). Why: the exponent $2\pi\omega/\kappa$ is precisely $\hbar\omega/(k_B T_H)$ , so the geometric factor $e^{\pi\omega/\kappa}$ has become a genuine thermal Boltzmann weight.

Limiting cases

Case	Result	Meaning
Schwarzschild (mass only)	$T_H = \dfrac{\hbar c^3}{8\pi G M k_B}$	$\approx 6.2\times10^{-8}\ \mathrm{K}\times(M_\odot/M)$ — negligible for any astrophysical hole, below the CMB, so they currently absorb more than they emit.
CMB crossover	$T_H = T_{\rm CMB} \Rightarrow M \approx 4.5\times10^{22}\ \mathrm{kg}$	Only holes lighter than about half a lunar mass are net emitters today; heavier ones are warmed by the microwave background and grow.
Kerr–Newman (mass, charge, spin)	$T_H = \dfrac{\hbar c}{2\pi k_B}\,\dfrac{r_+ - r_-}{2(r_+^2 + a^2)}$	With $a = J/Mc$ and $r_\pm = M \pm \sqrt{M^2 - a^2 - Q^2}$ (geometrized). Temperature falls as the hole spins up or charges toward extremality.
Extremal limit	$M^2 \to a^2 + Q^2 \Rightarrow \kappa \to 0,\ T_H \to 0$	Zero surface gravity, zero temperature — the third law: extremality is unreachable in finitely many steps.
Late-stage evaporation	$M \to M_{\rm Pl} = \sqrt{\hbar c/G} \approx 2.2\times10^{-8}\ \mathrm{kg}$	$T_H$ diverges and the semiclassical treatment fails near the Planck mass; the endpoint needs quantum gravity.

Common trapHawking radiation is a virtual pair at the horizon, one falling in

This cartoon is the single most over-quoted line in the subject, and it is misleading on three counts. First, it is not the derivation and yields no correct number — not the temperature, not the spectrum, not the greybody filtering. The real result is a global statement about the mismatch between the in-vacuum on $\mathcal{I}^-$ and the out-vacuum on $\mathcal{I}^+$ , and it depends on the entire collapse geometry. Second, the emission does not come from "at the horizon": the dominant flux originates a few Schwarzschild radii outside, where the gravitational potential barrier (the greybody factor) shapes the spectrum; the near-horizon region merely supplies the exponential redshift. Third, $T_H$ is the temperature measured at infinity. The locally measured, Tolman-redshifted temperature $T_{\rm loc} = T_H/\sqrt{-g_{tt}}$ diverges as one approaches the horizon, so "the temperature of the horizon" is not a single number. Keep the pair picture only as a mnemonic for the sign of the energy flux, never as physics.

Worked examples

1 · A solar-mass black hole: temperature, entropy, and lifetime (fully worked)

Take a Schwarzschild hole of one solar mass, $M = 1.989\times10^{30}\ \mathrm{kg}$ . Find its Hawking temperature, its entropy in units of $k_B$ , and its evaporation lifetime, and decide whether it is emitting or absorbing today.

Step 1. Temperature. Using equation (2), $T_H = 6.17\times10^{-8}\ \mathrm{K}\times(M_\odot/M) = \mathbf{6.17\times10^{-8}\ K}$ for $M=M_\odot$ . Equivalently $\kappa = c^4/4GM = 1.5\times10^{13}\ \mathrm{m/s^2}$ , and $\hbar\kappa/(2\pi k_B c)$ returns the same 62 nK.

Step 2. Entropy. The horizon radius is $r_s = 2GM/c^2 = 2.95\ \mathrm{km}$ , so $A = 4\pi r_s^2 = 1.10\times10^{8}\ \mathrm{m^2}$ . With $\ell_P^2 = G\hbar/c^3 = 2.61\times10^{-70}\ \mathrm{m^2}$ , the entropy is $S/k_B = A/(4\ell_P^2) = \mathbf{1.05\times10^{77}}$ . For comparison, the thermodynamic entropy of the progenitor star is only $\sim10^{58}\,k_B$ : collapse multiplies the entropy by nineteen orders of magnitude, which is why the generalized second law is not in danger.

Step 3. Lifetime. From equation (6), $\tau \approx 5120\pi\,G^2 M^3/(\hbar c^4)$ . Plugging in gives $\tau \approx 6.6\times10^{74}\ \mathrm{s} = \mathbf{2.1\times10^{67}\ years}$ — some fifty-seven orders of magnitude longer than the present age of the Universe.

Step 4. Emit or absorb? At 62 nK the hole is far colder than the 2.7 K CMB it is bathed in, so it absorbs far more than it radiates and slowly grows. It will only begin to truly evaporate in the unimaginably distant future, after the CMB has redshifted below $T_H$ . A solar-mass hole is a 62 nK, $10^{77}\,k_B$ , $10^{67}$ -year object that is currently gaining mass, not losing it.

2 · How heavy is a black hole evaporating right now?

Primordial black holes could have formed with any mass in the early Universe. Which initial mass has an evaporation lifetime equal to the age of the Universe, $t_0 = 13.8\ \mathrm{Gyr} = 4.35\times10^{17}\ \mathrm{s}$ , and how hot is it as it dies?

Step 1. Invert the lifetime formula. Setting $\tau = t_0$ in $\tau = 5120\pi\,G^2 M^3/(\hbar c^4)$ gives $M = \big(t_0\,\hbar c^4 / 5120\pi G^2\big)^{1/3}$ . Numerically $M^3 \approx 5.2\times10^{33}\ \mathrm{kg^3}$ , so $M \approx \mathbf{1.7\times10^{11}\ kg}$ — a mountain's mass packed inside a proton-sized horizon ( $r_s \approx 2.5\times10^{-16}\ \mathrm{m}$ ).

Step 2. Its temperature. From equation (2), $T_H = 6.17\times10^{-8}\ \mathrm{K}\times(1.989\times10^{30}/1.7\times10^{11}) \approx \mathbf{7\times10^{11}\ K}$ , or $k_B T_H \approx 60\ \mathrm{MeV}$ . This is a genuinely hot object radiating in gamma rays and relativistic particles, nothing like the frozen giants of example 1.

Step 3. Reconcile with the standard $\sim5\times10^{11}\ \mathrm{kg}$ . The formula above counts only photons. A 60 MeV hole radiates every species lighter than its temperature — photons, neutrinos, electrons, and more — so the true $\alpha$ is several times larger, the lifetime shorter at fixed mass, and the initial mass that survives exactly one Hubble time rises to $\approx 5\times10^{11}\ \mathrm{kg}$ . A hole finishing its life today started near $10^{11}$ – $10^{12}\ \mathrm{kg}$ and dies at tens of MeV — which is why gamma-ray observatories hunt for its final burst.

Physics of the portrait

Hawking radiation is where relativity, thermodynamics, and quantum theory were forced into a single sentence, and it was one person who wrote it.

Origins · 1 minStephen Hawking and how this was discovered

Stephen Hawking

1942–2018 · quantum fields on collapsing spacetimes

Stephen Hawking began the 1970s as a classical relativist. With Roger Penrose he had proved the singularity theorems, and with James Bardeen and Brandon Carter he had written down the four laws of black-hole mechanics — while insisting, against Jacob Bekenstein's 1972 suggestion, that a black hole's "entropy" and "temperature" were only analogies, because a body with a real temperature must radiate and a black hole plainly could not. In 1973–74 he set out to demolish Bekenstein's idea properly by computing quantum particle production during gravitational collapse, fully expecting a null result. Instead the calculation returned an exactly thermal flux at $T = \kappa/2\pi$ , converting his own objection into a proof: the entropy is real, and it is one quarter of the horizon area. He announced it in a 1974 talk titled, provocatively, "Black hole explosions?" and published the full derivation in 1975. Two years later he drew out its most disturbing consequence — if the radiation is exactly thermal and depends only on mass, charge, and spin, then the information about what collapsed is lost, and quantum mechanics is not unitary. That "information paradox" occupied him, and much of theoretical physics, for the next forty years, seeding holography, the AdS/CFT correspondence, and the recent island computations of the Page curve. Working almost entirely in his head as motor-neurone disease advanced, Hawking left the field with the one number nearly everyone agrees a full theory of quantum gravity must reproduce: $S = A/4$ .

History

1972

— Bekenstein proposes that a black hole carries entropy proportional to its horizon area

1973

— Bardeen, Carter and Hawking publish the four laws of black hole mechanics

1974

— Hawking announces that black holes radiate thermally

1975

— Hawking's full derivation of the Planckian spectrum appears in Commun. Math. Phys.

1976

— Hawking formulates the black hole information paradox

Going further

Texts. Hawking, "Particle creation by black holes", Commun. Math. Phys. 43, 199 (1975) · Wald, General Relativity (1984), Ch. 14, and Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (1994) · Birrell & Davies, Quantum Fields in Curved Space (1982), Ch. 8 · Carroll, Spacetime and Geometry (2004), Ch. 9 · Misner, Thorne & Wheeler, Gravitation (1973), §33 · Bardeen, Carter & Hawking, "The four laws of black hole mechanics", Commun. Math. Phys. 31, 161 (1973) · Bekenstein, "Black holes and entropy", Phys. Rev. D 7, 2333 (1973). The key results collected here are: (1) $T_H = \hbar\kappa/(2\pi k_B c)$ with Schwarzschild $\kappa = c^4/4GM$ giving $T_H = \hbar c^3/(8\pi GM k_B) \approx 6.17\times10^{-8}\ \mathrm{K}\,(M_\odot/M)$ — solar-mass 62 nK; (2) $S_{BH} = k_B c^3 A/(4G\hbar)$ from the first law with $A = 16\pi G^2 M^2/c^4$ , giving $S/k_B \approx 1.05\times10^{77}$ for $M_\odot$ ; (3) the Bogoliubov relation $|\alpha| = e^{\pi\omega/\kappa}|\beta|$ leading to $\langle N\rangle = (e^{2\pi\omega/\kappa}-1)^{-1}$ ; (4) the lifetime $\tau = 5120\pi G^2 M^3/(\hbar c^4) \approx 2.1\times10^{67}\ \mathrm{yr}$ for $M_\odot$ , and a primordial mass $\sim10^{11}$ – $10^{12}\ \mathrm{kg}$ finishing at $t_0$ ; (5) the Unruh temperature $T_U = \hbar a/(2\pi k_B c)$ as the local analogue. The constant $\alpha$ in the mass-loss rate is left symbolic since it depends on emitted species and greybody factors.

Prerequisites. Killing Horizons and Surface Gravity · Schwarzschild Black Hole Structure · Event Horizon and Apparent Horizon · Four Laws of Black Hole Mechanics. Leads to. Black Hole Information Paradox · Bekenstein–Hawking Entropy and Holography · Superradiance and the Black Hole Bomb · Unruh Effect.

Ask the tutor: Ask me to walk through why the exponential redshift near the horizon turns a pure tone into a Planck spectrum, or to estimate the temperature, entropy, and lifetime of any black hole you name.

Is the "virtual pair at the horizon" picture actually wrong?

It is a mnemonic, not a mechanism. The picture gets one thing right — the sign of the energy, with the escaping quantum carrying positive energy and its partner effectively negative energy as seen from outside — but it produces no correct number and misplaces the physics. Hawking's derivation never invokes pair creation at a point; it compares the field's vacuum defined in the far past with the vacuum defined in the far future and finds them inequivalent, a global statement over the whole collapse. The emission, moreover, comes mainly from a region a few Schwarzschild radii outside the horizon, where the potential barrier shapes the spectrum. Use the pair cartoon to remember that energy leaves, never to compute how much.

Where is the radiation actually produced?

Not at the horizon, and not at a single place. The rigorous statement is that outgoing modes reaching infinity at late times are exponentially blueshifted when traced back near the forming horizon, and it is that blueshift, integrated over the whole geometry, that populates the out-modes thermally. The flux that a distant detector receives is dominated by wavelengths comparable to the hole's size, filtered by the greybody factor as the waves climb out through the curved region a few radii outside the horizon. So the "surface" of a black hole as a radiator is fuzzy and extended, of order the Schwarzschild radius, not a thin shell hugging $r_s$ .

Why does a black hole have negative heat capacity, and what does that mean?

Because $T_H \propto 1/M$ : remove energy and the temperature rises. A hotter hole radiates faster, loses mass faster, and heats further — a runaway with no stable end short of complete evaporation. The immediate consequence is that a Schwarzschild hole cannot sit in stable equilibrium with an infinite heat bath: if the surroundings are hotter it absorbs and cools away from equilibrium; if colder it radiates and heats away from equilibrium. Only a finite box (or a negative cosmological constant, as in AdS) can stabilize it. This negative heat capacity is also why the evaporation ends explosively rather than fading out.

How is Hawking radiation related to the Unruh effect?

They are the same effect seen from two vantage points. Unruh showed that a uniformly accelerated observer in flat, empty spacetime perceives a thermal bath at $T_U = \hbar a/(2\pi k_B c)$ , purely because acceleration defines a horizon (the Rindler horizon) and a different notion of positive frequency. Near a black hole, a static observer must accelerate to hover, and the surface gravity $\kappa$ is exactly that proper acceleration redshifted out to infinity. Replace $a$ by $\kappa$ and Unruh's formula becomes Hawking's. The equivalence is not a coincidence but a theorem about quantum fields restricted to a wedge bounded by a horizon (the Bisognano–Wichmann / KMS result).

Does Hawking radiation really destroy information?

That is the open question, not a settled fact. The semiclassical calculation gives a spectrum that is exactly thermal and depends only on $M$ , $J$ , $Q$ , so it appears to carry away none of the detail of what fell in; a pure initial state would then evolve to a mixed thermal state, violating the unitarity of quantum mechanics. Most physicists now believe unitarity survives and the information leaks out in subtle correlations that the leading-order calculation misses. The recent island and replica-wormhole computations recover the Page curve — entropy that rises then falls back to zero — from gravity itself, strong evidence that evaporation is unitary. Exactly how a distant observer would decode the information remains unresolved.

Has Hawking radiation ever been observed?

Not from a real black hole, and realistically it never will be by direct detection: a solar-mass hole radiates at 62 nK, swamped a hundred-million-fold by the microwave background. What has been observed is the same physics in analogue systems. Jeff Steinhauer's Bose–Einstein-condensate experiments create a sonic horizon where the flow outruns the speed of sound, and measure correlated phonon pairs whose spectrum is thermal at the predicted analogue temperature — a laboratory realization of the Bogoliubov mechanism. These support the kinematics of Hawking's argument (horizon plus mode mixing) without settling the gravitational and information-theoretic questions, which require a real evaporating horizon.

Why is the Hawking temperature quoted "at infinity" rather than near the hole?

Because temperature in a gravitational field is redshift-dependent. The single invariant number $T_H = \hbar\kappa/(2\pi k_B c)$ is what a detector infinitely far away measures. A detector held static at radius $r$ measures the Tolman-shifted value $T_{\rm loc} = T_H/\sqrt{-g_{tt}} = T_H/\sqrt{1 - 2GM/rc^2}$ , which grows without bound as $r \to r_s$ . So there is no single "temperature of the horizon"; the horizon is locally infinitely hot to a hovering observer (who must also accelerate infinitely to stay there), while the radiation that actually reaches infinity, redshifted all the way out, arrives at the modest $T_H$ . Quoting the infinity value is what makes the number well-defined and observer-independent.