Hawking Radiation

The result and how Hawking got there

By 1973 it was already strange that black holes seemed to obey their own four-law thermodynamics. Bardeen, Carter and Hawking had derived a "zeroth law" — the surface gravity κ of a stationary horizon is constant — and a "first law" — dM = (κ/8π) dA + Ω dJ + Φ dQ — that mimicked dE = T dS + work terms. The analogy looked formal: classically nothing comes out, so what would T even mean? Jacob Bekenstein, then a Princeton graduate student, insisted in 1972 that the analogy must be physical — a black hole's entropy had to be proportional to its horizon area, otherwise an outside observer could let entropy disappear by dropping a teacup over the edge and violate the second law of thermodynamics.

Hawking thought Bekenstein was wrong. To prove it, in 1973-1974 he worked through the propagation of a quantum scalar field on the spacetime of a star collapsing to form a black hole. The calculation tracks Bogoliubov coefficients relating "in" modes at past null infinity to "out" modes at future null infinity. To his surprise, the result was non-zero: an observer far from the hole sees a thermal flux of particles at temperature

T_H = ℏ κ / (2π k_B c)   (general)
    = ℏ c³ / (8π G M k_B)  (Schwarzschild)

which is exactly what was needed to make the black-hole "thermodynamics" a literal thermodynamics with S = k_B A / (4 ℓ_P²). The two papers — "Black hole explosions?" (Nature, 1974) and "Particle creation by black holes" (Commun. Math. Phys., 1975) — turned a formal analogy into a physical prediction and reframed black holes as proper thermodynamic objects. Bekenstein, it turned out, had been right; Hawking had supplied the coefficient.

Putting numbers on the formula

Plug ℏ = 1.055 × 10⁻³⁴ J·s, c = 3 × 10⁸ m/s, G = 6.674 × 10⁻¹¹ N·m²/kg², k_B = 1.381 × 10⁻²³ J/K, M☉ = 1.989 × 10³⁰ kg:

T_H = ℏc³ / (8π G M k_B)
    ≈ 6.17 × 10⁻⁸ K × (M☉ / M)

So a one-solar-mass black hole has a Hawking temperature of about 62 nanokelvin. The cosmic microwave background is 2.7 K — over 4 × 10⁷ times hotter. Every known astrophysical black hole is therefore net absorbing CMB photons and gaining mass faster than it loses it via Hawking emission. Only when the universe cools below T_H of a particular hole does that hole begin to net evaporate — for the Sun, that requires another factor of 10⁷ expansion, roughly 10⁹⁰ years from now.

Object	Mass	T_H	Photon λ_peak	τ (years)
Supermassive (M87*)	6.5 × 10⁹ M☉	9 × 10⁻¹⁸ K	3 × 10¹⁴ m	5 × 10⁹⁵
Supermassive (Sgr A*)	4 × 10⁶ M☉	1.5 × 10⁻¹⁴ K	2 × 10¹¹ m	10⁸⁷
Stellar-mass	10 M☉	6.2 × 10⁻⁹ K	4.7 × 10⁵ m	2 × 10⁷⁰
Smallest known stellar	3 M☉	2.1 × 10⁻⁸ K	1.4 × 10⁵ m	5 × 10⁶⁸
Lunar mass	7.3 × 10²² kg	1.7 K	1.7 mm	2 × 10⁴²
Asteroid (1 km, ρ ~ ρ_ast)	10¹⁵ kg	1.2 × 10⁸ K	2.4 × 10⁻¹¹ m	8 × 10²⁰
Evaporating today	≈ 5 × 10¹¹ kg	≈ 2 × 10¹¹ K	1.4 × 10⁻¹⁴ m	13.8 × 10⁹
Planck mass	2.2 × 10⁻⁸ kg	1.4 × 10³² K	2 × 10⁻³⁵ m	τ → 0

The middle of this table is dramatic. A lunar-mass hole is just a little colder than the room you're reading this in. A 10¹² kg hole — about the mass of a small mountain compressed inside its own 10⁻¹⁵ m horizon — would emit at gigakelvin temperatures and complete its evaporation in a closing-second flare of hard gamma rays. Whether any such "primordial" holes exist is one of the open questions of cosmology.

Why evaporation runs away

A blackbody of temperature T and area A radiates at the Stefan-Boltzmann rate σ T⁴ A. Plug in T = T_H ∝ 1/M and A ∝ M² (since R_S = 2GM/c²) and you get

dM/dt = -L / c² ∝ -1 / M²

Separating variables, M² dM = -k dt, integrating from M_0 to 0 gives

τ_evap = (5120 π G² / ℏ c⁴) M³
       ≈ 2.1 × 10⁶⁷ yr × (M / M☉)³

Three features of this expression are worth pausing on.

• Negative heat capacity. dT_H/dM < 0 — losing energy raises the temperature, which raises the luminosity, which loses more energy faster. This is the same instability that makes a star contract when energy is removed (Lane-Emden 1869). For black holes it means evaporation accelerates without limit, ending in a brief flare rather than a long fade.
• Cube-law sensitivity to mass. A factor-of-two change in mass changes the lifetime by a factor of eight. The mass scale where τ matches the age of the universe is therefore extraordinarily narrow: a tenfold range from 10¹¹ to 10¹² kg.
• Final-flare phenomenology. The last second of a primordial-BH evaporation releases on the order of 10²² J — comparable to a 5-megaton bomb, in a 1-second pulse of TeV gamma rays. Fermi-LAT, HESS, HAWC and Milagro have searched for these signatures and set upper limits on the local-galaxy density of evaporating PBHs.

Primordial black holes — the only practical detection target

Stars never collapse to less than about 3 M☉ — the Tolman-Oppenheimer-Volkoff limit. Anything lighter has to have been produced some other way. Zel'dovich and Novikov in 1967, and independently Hawking in 1971, pointed out that the very early universe contained density fluctuations large enough that small over-densities could have collapsed directly into black holes of nearly arbitrary mass — primordial black holes (PBHs). The PBH mass formed at time t after the Big Bang is roughly the mass within the Hubble horizon at that time:

M_PBH ~ c³ t / G  ≈  10¹⁵ g × (t / 10⁻²³ s)

So PBHs formed at the QCD epoch (t ~ 10⁻⁵ s, M ~ M☉) would still be around. PBHs formed much earlier, at t ~ 10⁻²³ s, would weigh 10¹² kg — and have evaporation lifetimes matching the present age of the universe. Constraints from Fermi-LAT's non-detection of gamma-ray bursts with the right spectrum exclude PBHs of this mass from being more than a small fraction of dark matter. Larger PBHs of asteroid mass (10¹⁷-10²² g) escape that constraint and remain a viable dark-matter candidate; LIGO has reopened interest in stellar-mass PBHs after the GW150914-class detections.

Bekenstein-Hawking entropy and area

The entropy that goes with T_H is

S_BH = k_B A / (4 ℓ_P²)        ℓ_P = √(ℏG/c³) ≈ 1.616 × 10⁻³⁵ m

For a solar-mass Schwarzschild hole, A ≈ 1.1 × 10⁸ m² and S ≈ 1.0 × 10⁷⁷ k_B. The entropy of the star that collapsed to make that hole is of order 10⁵⁸ k_B — twenty orders of magnitude smaller. Forming a black hole therefore creates entropy more efficiently than any other process in the universe; on the largest scales, supermassive black holes are believed to dominate the entropy budget of the visible universe (Egan & Lineweaver 2010).

That S scales with area, not volume, is the central clue physicists take to "holography" — that the fundamental quantum description of a volume of spacetime lives on its boundary. The AdS/CFT correspondence (Maldacena 1997) makes this exact in a particular setting and is one of two main testing grounds (the other being de-Sitter cosmology) for what a quantum theory of gravity might look like.

The information paradox in one paragraph

Quantum mechanics evolves pure states into pure states. A star collapsing to a black hole is a pure state. Hawking's 1974 calculation says it ends up as thermal radiation — which is, by definition, a maximally mixed state. So Hawking's calculation implies that black holes destroy information. Either the calculation is wrong, or unitarity is wrong, or both — and physicists have spent fifty years sorting through the options.

Proposal	Year	Idea
Information lost	Hawking 1976	Quantum gravity violates unitarity at horizons. Now widely rejected after AdS/CFT.
Remnant scenario	various, 1980s	Evaporation halts at Planck scale, leaving a "remnant" carrying the lost information. Difficult to embed in known physics.
BH complementarity	Susskind-Thorlacius-Uglum 1993	Infalling and external observers see different but mutually consistent realities. No global pure-state conflict.
Soft hair	Hawking-Perry-Strominger 2016	Asymptotic symmetries imprint "soft" zero-energy charges on horizons that store missing information.
Firewalls (AMPS)	Almheiri-Marolf-Polchinski-Sully 2012	Forces a contradiction: smoothness of horizon, monogamy of entanglement, and unitarity can't all hold. Some must give.
ER = EPR	Maldacena-Susskind 2013	Entanglement between Hawking pairs is geometrically a microscopic Einstein-Rosen bridge — quantum entanglement = wormhole.
Page curve / islands	Penington 2019; AHMST 2019-2020	Including replica-wormhole / entanglement-island contributions reproduces Page's predicted unitary entropy curve. Currently the leading consensus picture.

The "Page curve" result is now widely viewed as a partial resolution: the radiation's entanglement entropy rises and then falls, just as it must for the joint state to be pure. The mechanism — extremal surfaces inside the hole that carry the radiation's information after the "Page time" t_P ~ τ/2 — is computed within semiclassical gravity using the replica-trick path integral, and is the closest thing yet to a derivation of unitarity from first-principles gravity.

Worked example — when does a stellar BH start to net evaporate?

Take a 10 M☉ stellar-mass black hole. Its Hawking temperature is T_H ≈ 6.2 × 10⁻⁹ K. The CMB is currently T_CMB(t_0) ≈ 2.725 K. Net evaporation requires T_H > T_CMB, i.e. the universe to cool by a factor of

f = T_CMB / T_H = 2.725 / 6.2e-9 ≈ 4.4 × 10⁸

The CMB temperature scales as 1/a, where a is the cosmological scale factor. In ΛCDM, future expansion is exponential at the de-Sitter rate H_∞ ≈ √(Λ/3) ≈ 1.85 × 10⁻¹⁸ s⁻¹, so a(t) ∝ exp(H_∞ t). Set exp(H_∞ Δt) = 4.4 × 10⁸:

Δt = ln(4.4e8) / H_∞ ≈ 20.0 / (1.85 × 10⁻¹⁸ s⁻¹)
   ≈ 1.08 × 10¹⁹ s
   ≈ 3.4 × 10¹¹ yr

So our 10 M☉ hole begins to net evaporate about 340 billion years from now — once the CMB has cooled below 6 nK. From that point on the hole shrinks; the remaining evaporation takes the cube-law time, ~10⁷⁰ years, dominated by the cool early phase. The entire "evaporation era" is therefore preceded by an "absorption era" of comparable order in the exponent but utterly different in physics. Almost all of a stellar hole's lifetime is spent passively accreting cold backgrounds; only the last 10⁻²⁰ of its life is the dramatic flare.

The four laws of black-hole mechanics

Law	Thermodynamics	Black-hole mechanics
Zeroth	T constant in equilibrium	Surface gravity κ constant over a stationary horizon
First	dE = T dS + work	dM = (κ/8π) dA + Ω dJ + Φ dQ
Second	dS_total ≥ 0	dA ≥ 0 (Hawking's area theorem, classically) — replaced by generalised dS_total = d(S_matter + A/4) ≥ 0
Third	T → 0 unattainable in finite time	κ = 0 (extremal Kerr/Reissner-Nordström) unattainable in finite operations

What Hawking's 1974 derivation does is collapse the analogy: the "T" in column three really is the temperature in column two, with k_B T = ℏκ/(2π c). The four laws stop being a useful coincidence and become four predictions of a unified quantum theory of gravity, however we eventually formulate it.

Analog Hawking experiments — sonic horizons

If you can't watch a real black hole evaporate, you can watch an analog. Unruh pointed out in 1981 that a transonic flow — a fluid that goes from subsonic to supersonic — has a sonic horizon for sound: phonons inside the supersonic region cannot propagate against the flow, just as photons inside a Schwarzschild horizon cannot escape. The corresponding "Hawking temperature" for the sound field is k_B T = ℏ |∇v|/(2π) at the sonic horizon — a few nanokelvin for typical BEC experiments.

Jeff Steinhauer's group at Technion realised this experimentally in a sodium BEC (Nature 2016, Nature 2019). They report a thermal phonon spectrum at the predicted Hawking temperature and — more remarkably — correlations between in-going and out-going phonons consistent with quantum entanglement of Hawking pairs. The result is not a direct test of gravitational Hawking radiation, but it confirms the kinematic mechanism (mode mixing across a horizon producing a thermal spectrum) and is the closest thing we have to an experimental signature.

Cosmological consequences

• Constraint on dark matter. If dark matter were made of 10¹² kg PBHs, every one would be evaporating now, producing TeV gamma rays. Fermi-LAT, HESS, HAWC and Milagro non-detections rule out f_PBH > 10⁻⁸ at that mass.
• Heat-death timescales. In the standard "five ages of the universe" (Adams & Laughlin 1997), the black-hole era runs from 10⁴⁰ to 10¹⁰⁰ years, dominated by stellar BHs evaporating one at a time. After that, only photons, neutrinos, and possibly long-lived elementary particles remain.
• Soltan-type bookkeeping. Because S_BH = A/4 and BHs dominate the universe's entropy budget, Hawking radiation eventually returns that entropy to the radiation field — the highest-entropy state achievable in cosmology is empty de-Sitter plus thermal Hawking radiation at the de-Sitter horizon temperature ~10⁻²⁹ K.
• Quantum gravity laboratory. Every claimed approach to QG — string theory, loop quantum gravity, asymptotic safety, causal sets, holographic dualities — must reproduce S = A/4 in the appropriate limit. The entropy-area law is the most-used numerical check for any candidate theory.

Variants and extensions

• Kerr-Hawking temperature. A spinning Kerr hole has T_H = ℏκ/(2π k_B c) with κ depending on both mass and spin; the temperature drops to zero in the extremal limit a = M, where the surface gravity vanishes. Near-extremal holes are exceptionally cold and long-lived — the basis of Strominger-Vafa's 1996 microstate counting that reproduced S = A/4 for near-extremal supersymmetric black holes from string theory.
• Greybody factors. The emission is not exactly Planckian; the spectrum is modulated by frequency-dependent absorption coefficients (greybody factors) arising from the curved spacetime potential barrier around the hole. For Schwarzschild, photons see a barrier that suppresses long wavelengths.
• Unruh effect. A uniformly accelerating observer in flat space sees a thermal bath at T_U = ℏ a / (2π c k_B). Mathematically identical to Hawking: replace surface gravity by proper acceleration. The shared structure points to a deep connection between horizons (event, apparent, Rindler) and thermal physics.
• Generalised second law. Bekenstein 1973: the total entropy of matter outside a black hole plus its A/4 cannot decrease. Wall (2011) and Bousso et al. proved this rigorously in semiclassical gravity, anchoring the consistency of black-hole thermodynamics.
• Page curve. Don Page (1993) argued that if Hawking radiation is a unitary process, its entanglement entropy must rise, peak at the "Page time" (about half the evaporation), then fall back to zero. The 2019-2020 entanglement-island calculations reproduced exactly this curve, considered the cleanest theoretical evidence that information escapes the hole.

Common pitfalls

• "One particle escapes, the other falls in." Useful only as a slogan. The actual derivation tracks quantum-field modes through a forming-collapse spacetime; particles are not produced inside the horizon and then crawl out. The slogan is closer to the truth in the Parikh-Wilczek tunnelling derivation, but even there the "tunnelling" is a contour integral, not a particle's worldline.
• Confusing Hawking with Unruh. Hawking radiation is real radiation from a black hole; Unruh radiation is what an accelerated detector in flat space registers from the Minkowski vacuum. The mathematics is identical; the physics is observer-dependent in one case and not in the other.
• Treating T_H as a coronal temperature. T_H is a thermodynamic temperature of the horizon, not the temperature of the accretion disk a few r_g out. Accretion disks of stellar BHs sit at 10⁷ K; their Hawking temperatures are 10⁻⁸ K. Twenty-three orders of magnitude separate the two.
• Ignoring the CMB ceiling. Every known black hole is colder than the CMB, so every known black hole is presently net absorbing photons. There are no astrophysical Hawking detections to look for. Direct detection requires PBH-scale objects or analog systems.
• Reading τ ∝ M³ off as instant for small holes. The cube law is steep but not infinite. A 10⁹ kg micro-BH still has τ ~ 10⁻⁷ s of life — long enough to release ~10²⁰ J as a TeV burst. The "instant" limit is asymptotic, not literal.