Hawking Radiation as Tunneling

Frank Wilczek; Maulik K. Parikh

REVIEW

Hawking Radiation as Tunneling

Maulik K. Parikh, Frank Wilczek

High Energy Physics - Theory · Submitted 1 Jul 1999 · 9 references

Pith's verdict: A clean, standard derivation whose leading result is right and whose novel ω² term rests on a borrowed shell-geodesic input rather than a fresh computation.

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arXiv IDhep-th/9907001 (version 3)

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CitationFrank Wilczek, Maulik K. Parikh. "Hawking Radiation as Tunneling." arXiv:hep-th/9907001 (1999). https://pith.science/paper/YXYCLKWD

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Editor's note

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Short version: the derivation is clean and the leading Boltzmann factor is right, but the part everyone remembers — the ω² correction and the exp(ΔS) form — is the part that leans on a cited input rather than a fresh computation. That is fine, but it should be read that way. What is actually new is not the tunneling picture, which is old heuristic, and not the leading exponential, which the paper says outright agrees with Kraus–Keski-Vakkuri and Massar–Parentani. New is the concrete implementation: Painlevé–Gullstrand coordinates so the horizon is not singular, the contour deformation in Eqs. 5–8 that picks up the pole at r = 2M, and the resulting Im S = 4πω(M − ω/2). The charged extension in Eq. 16 is a real second result, and the extremality bound falling out of the square root is a nice touch. On the stress-test concern: I do not think it lands as stated. The M → M − ω replacement is Eq. 4, and the paper is explicit that it is using the Kraus–Wilczek Hamiltonian-gravity result for self-gravitating shells, a published calculation, not an invention. What the paper does not do is re-derive it, and it does not track back-reaction beyond that shell approximation. That is a limitation of the framework, not a hidden gap. Where I would push: the vacuum choice. Section II asserts that the Painlevé-time vacuum differs strictly from the standard Unruh vacuum but that the difference shows up only in transients. That is load-bearing for the sign and the direction of the radiation, and it is asserted, not shown. Same for the claim that the two channels only concern the pre-factor — plausible, unproved. The greybody factor in Eq. 11 is inserted by hand. And the exact-result-must-be-correct paragraph is entropy counting, which the paper presents as a consistency check, correctly. The non-thermality reading is the part I would flag to a reader: exp(ΔS) is what a microcanonical treatment gives, and whether the ω² term is information-carrying is not settled here. The paper hedges, which is the right call. Who should read it: anyone doing tunneling or back-reaction calculations, and anyone who wants the cleanest short derivation of the Boltzmann factor. It is a standard reference for a reason. Send it out.

Referee report

Significance

If correct, this is the most direct realization of the heuristic tunneling picture of Hawking radiation: it requires neither Euclideanization nor an explicit collapse phase, and it is parameter-free, with no adjustable constants introduced anywhere in the derivation. The leading result reproduces the standard Hawking temperature, the exponentiated entropy form exp(ΔS_BH) is compact and extends naturally to charged holes, and the limiting cases are physically sensible — the ω → M limit is checked against a state-counting argument, and the square root in Eq. (16) makes extremality unreachable. The comparison with electric discharge in §III is a genuinely useful pedagogical contribution that clarifies why the gravitational process requires tunneling at all. These are real strengths and should be weighed in the assessment.

Major objections (4)

[§II, Eq. (4)] The step from Eq. (3) to Eq. (4) — replacing M by M − ω in the geodesic — is the sole source of the ω² term and therefore of the paper's headline non-thermality claim, yet it is asserted rather than derived. The cited Kraus-Wilczek result [6] covers the complementary case (hole mass held fixed, total ADM mass varied, giving M → M + ω); the 'fix the total mass and allow the hole mass to fluctuate' argument that yields M − ω is the authors' own and is not shown. Please either derive Eq. (4) from the Hamiltonian constraint for a self-gravitating shell in the Painlevé geometry, or state explicitly the transformation that takes the [6] result to Eq. (4) and the approximation under which it holds. Without this, the ω² correction reads as a plausible ansatz rather than a derived result.
[§II, second paragraph] The statement that the Painlevé vacuum 'differs strictly from the standard Unruh vacuum' but that 'the difference, however, shows up only in transients, and does not affect the late-time radiation' is load-bearing and undemonstrated. The positive-frequency condition with respect to t fixes the sign of Im S, the direction of the radiation, and the identification of the outgoing mode, so a reader cannot check the claim from the text as written. Please supply the argument — for instance a comparison of the late-time mode functions of the two vacua, or a citation that establishes it — or soften the claim to match what is shown.
[§II, Eq. (10) versus Eq. (11) and the abstract] The abstract states that 'the exact spectrum is not precisely thermal,' but the operational spectral flux quoted in Eq. (11) is the standard Planck form, obtained precisely by dropping the ω² term. As written, the non-thermality resides only in the exponentiated entropy form of Eq. (10), and it is not clear whether it is an observable departure or a rewriting of the same physics. Please state what 'not precisely thermal' means operationally — whether Eq. (11) should itself carry the correction, and over what range of ω/M the departure is measurable — so that the abstract's claim is tied to a specific formula.
[§II, discussion following Eq. (8)] The contour argument fixes the sign, but the accompanying trajectory narrative is hard to reconcile with the geodesic. Along the outgoing branch ṙ = +1 − √(2(M−ω′)/r) is positive at r = 2M, so r increases, whereas the stated endpoints rin = 2M and rout = 2(M−ω) with rin > rout describe a decrease in r. The resolution via the contracting horizon is plausible but is compressed into a single sentence. Since the sign of Im S is the whole result, please make the relation between the contour ordering and the physical trajectory explicit.

Minor objections (6)

[§III, Eq. (18)] The second term qR/(r² − R²) is dimensionally inconsistent with the first, q(Q − q)/r; the image-charge contribution should presumably read q²R/(r² − R²). Please check the expression.
[§I, first paragraph] 'quantitively' should read 'quantitatively.'
[§II, text after Eq. (15)] The contour deformation is written w′ → w′ − iε while the integration variable is ω′; please use ω′ consistently throughout.
[End of §II] The exp(S_BH) counting argument is introduced as a check 'on physical grounds'; it would help to label it explicitly as a consistency check rather than an independent derivation, since it employs the result it is meant to support.
[§II, after Eq. (16)] The phrase 'the third law of black hole thermodynamics is here manifestly enforced' is slightly loose: the square root shows that the rate becomes imaginary past extremality, i.e. that emission cannot proceed through it, which is not quite the third law as usually stated (extremality is unreachable). Consider rephrasing.
[Eq. (11)] The placement of the dω/2π factor and the definition of |T(ω)|² could be stated more explicitly for readers comparing against standard greybody conventions.

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The paper

The paper claims

Hawking radiation can be derived as a tunneling process, and enforcing energy conservation during the emission makes the resulting spectrum slightly non-thermal, with a correction that grows as the hole shrinks.

The reading

Parikh and Wilczek want to derive black hole radiance from the picture physicists most often use informally: a particle escapes the horizon by tunneling through a barrier that classical physics forbids it to cross. Their complaint about earlier derivations is that they hold the geometry fixed, so the emitted quantum's energy comes from nowhere; here the hole's mass drops by exactly the energy carried away, so total energy is conserved at every step. Working in coordinates that stay smooth across the horizon, they compute the imaginary part of the tunneling action and find it equals one quarter of the change in the hole's Bekenstein-Hawking entropy, so the emission rate goes as $e^{4S}$. Expanding that result, the leading term is the familiar Boltzmann factor at the Hawking temperature $1/(8M)$, and the next term is a definite, calculable departure from exact thermality caused by the hole lightening as it radiates. The same structure carries over to a charged hole, where the rate's reality condition forbids radiating past extremality, making the third law of black hole thermodynamics automatic rather than imposed.

What carries the argument

The load-bearing device is the imaginary part of the WKB action for a classically forbidden trajectory, $S = ( {r_{out}}{r_{in}} p_r r)$, rewritten via Hamilton's equation into an integral over the energy $'$ of the emitted shell. Because the radial null geodesic speed $= 1 - {2(M-')/r}$ vanishes at the horizon, the integrand has a pole there, and the contour prescription $' - i$ picks up a residue $-r$; the limits then force $r_{in} = 2M$ and $r_{out} = 2(M-)$, so the particle emerges just outside the contracted horizon. The choice of Painlevé-type coordinates — stationary but not static, and non-singular at $r = 2M$ — is what makes a positive-frequency condition definable across the horizon and gives the ingoing/outgoing sign its meaning.

Extended reading notes

Core claim

The central claim is that the exponential part of the semi-classical emission rate for a Schwarzschild black hole is $e^{-2S}$ with $S = 4(M - /2)$, so that $e^{-2S} = e^{S_{B-H}}$: the logarithm of the emission probability is precisely the change in the hole's Bekenstein-Hawking entropy. The derivation treats the outgoing particle as a self-gravitating shell, so the geometry it moves through has mass $M - '$ rather than $M$. The action integral is evaluated without solving the wave equation, by multiplying and dividing by Hamilton's equation $= dH/dp_r$, converting the momentum integral into an energy integral, and deforming the contour so that positive-energy solutions decay in time. The result reduces to the standard Boltzmann factor at inverse temperature $8M$ when the quadratic term is dropped, and the surviving $-8^2$ term is the paper's substantive addition: the spectrum is not exactly thermal because the hole's temperature rises as it loses mass. The authors argue the exact form must be right on statistical grounds, since a shell carrying the entire mass of the hole is one state among $e^{S_{B-H}}$, giving probability $e^{-S_{B-H}}$. The charged generalization replaces the mass shift appropriately and yields a rate whose reality enforces the third law.

Load-bearing premise

The argument needs the positive-frequency condition defined with respect to the Painlevé time $t$ to coincide, at late times, with the standard outgoing radiation condition, and the paper asserts this rather than demonstrating it.

Editorial extensions

If this is right

The emission rate for an uncharged black hole is $e^{-8(M - /2)}$, which reduces to a thermal Boltzmann factor at the Hawking temperature to first order in the emitted energy and predicts a specific negative correction at second order.
For a Reissner-Nordström hole the rate is $e^{S_{B-H}}$ exactly, and its reality condition cuts off radiation at extremality, so the third law needs no separate postulate.
Because the spectrum is not exactly thermal, the radiation can in principle carry correlations, which the authors note opens a route to the information question.
The derivation needs no Euclidean continuation and no explicit collapse phase; only local field theory plus global energy conservation enters.
The comparison with electric discharge identifies what makes the gravitational case different: the horizon contracts, enlarging the external volume over which the negative field energy is integrated, and that is the kinematic source of the radiation's kinetic energy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The entropy-change form of the rate looks like a unitarity-friendly statement: if each emission has probability $e^{S}$ where $S$ is the entropy drop, the full decay sequence has a naturally multiplicative structure that a strictly thermal spectrum lacks.
The correction term scales as $^2/M^2$, so it is negligible for solar-mass holes but order-unity for Planck-mass remnants, which suggests the late stage of evaporation is where any observational or theoretical distinction from Hawking's fixed-background result would have to be sought.
The reliance on a particular time coordinate to define positive frequency invites the question of whether the result is coordinate-invariant; a natural check is whether the same $S$ emerges from a genuinely dynamical collapse metric rather than a sequence of static slices.
The electric-discharge analogy suggests a laboratory-side question: whether an analogous barrier-from-image-charges calculation for a small charged conductor reproduces an exponential rate of the same entropy-change form.

Original abstract

We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry. The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature. Because the derivation respects conservation laws, the exact spectrum is not precisely thermal. We compare and contrast the problem of spontaneous emission of charged particles from a charged conductor.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.81 · mentions code or data availability 0.01.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry. The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature. Because the derivation respects conservation laws, the exact spectrum is not precisely thermal. We compare and contrast the problem of spontaneous emission of charged particles from a charged conductor.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a short and direct derivation of Hawking radiation as a tunneling process, based on particles in a dynamical geometry.	contribution 0.99	0.96	0.57	asserted 1.00
s1	The imaginary part of the action for the classically forbidden process is related to the Boltzmann factor for emission at the Hawking temperature.	result 0.83	0.79	0.29	asserted 1.00
s2	Because the derivation respects conservation laws, the exact spectrum is not precisely thermal.	result 0.87	0.86	0.28	asserted 1.00
s3	We compare and contrast the problem of spontaneous emission of charged particles from a charged conductor.	contribution 0.75	0.65	0.03	unclear 0.57

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

Works cited (9)

[6]

P. Kraus and F. Wilczek, Nucl. Phys. B433, 403 (1995)

View on Pith 1995
[1]

S. W. Hawking, Commun. Math. Phys. 43, 199 (1975)

View on Pith 1975
[2]

G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977)

View on Pith 1977
[3]

P. Painlev´ e, C. R. Acad. Sci. (Paris), 173, 677 (1921)

View on Pith 1921
[4]

P. Kraus and F. Wilczek, Mod. Phys. Lett. A 9 , 3713 (1994)

View on Pith 1994
[5]

W. G. Unruh, Phys. Rev. D 14, 870 (1976)

View on Pith 1976
[7]

P. Kraus and E. Keski-Vakkuri, Nucl. Phys. B491, 249 (1997)

View on Pith 1997
[8]

S. Massar and R. Parentani, Nucl. Phys. B575 333 (2000)

View on Pith 2000
[9]

P. Kraus and F. Wilczek, Nucl. Phys. B437, 231 (1995). 10

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284 later papers cite it

Evidence: substantial. 282 of the citing papers share no author with it, and 75 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 575 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

15 later papers by other authors are listed below: 15 used its method or tool.

Uses its method or tool 15 of 60 papers with this label

Followed the standard procedure of this paper to calculate the emission rate of particles due to tunneling across the LC.

Hawking-like radiation of charged particles via tunneling across the lightcylinder of a rotating magnetosphere (2021) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's tunneling argument of black hole radiation as a tunneling process.

Davies Critical Point and Tunneling (2010) · 1.00 · paraphrased: the licence does not allow quoting
This is done by studying the tunneling process close to the horizon using the Parikh-Wilczek method [ref] , which takes two processes contributing to the spontaneous emission of massless particles into account.

Thermodynamics of analogue black holes in a non-Hermitian tight-binding model (2025) · Introduction · 1.00
We apply the null geodesics method which is well-known as the Parikh-Wilczek method [ref] .

Cosmological LTB Black Hole in a Quintom Universe (2023) · The Parikh-Wilczek Method · 1.00
In Chapter [ref], we study the Hawking emission of charged particles from the scalar-hairy Reissner-Nordström black hole using the tunnelling method, developed by Parikh and Wilczek [ref] .

On the effect of scalar fields on Hawking radiation and quasinormal modes of black holes (2021) · Outline of the thesis · 1.00
To derive the probability rate of charged particles tunneling through the black hole and/or cosmological horizon we follow the approach of Parikh and Wilczek [ref] .

Extremal Black Hole Decay in de Sitter Space (2023) · Black Hole Decay as a Tunneling Process · 1.00
Used this paper's tunneling method.

Constructing a counterexample to the black hole complementarity (2008) · 1.00 · paraphrased: the licence does not allow quoting
Followed this paper, employing the tunnelling probability through a classically forbidden region.

Verifying the upper bound on the speed of scrambling with the analogue Hawking radiation of trapped ions (2020) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's method, which differs from the Hamilton-Jacobi method adopted in another work.

Massive charged particle's tunneling from spherical charged black hole (2010) · 0.99 · paraphrased: the licence does not allow quoting
Chose the Painlevé coordinates following this paper.

Tunneling of massive particles from noncommutative inspired Schwarzschild black hole (2010) · 0.99 · paraphrased: the licence does not allow quoting
Following this paper, under the s-wave approximation, made the replacement M to M minus omega in an equation.

Noncommutative Black Hole Thermodynamics (2008) · 0.99 · paraphrased: the licence does not allow quoting
Applied the Hamilton-Jacobi method and this paper's approach to show thermal radiation for the apparent horizon of the FRW universe.

Hawking Radiation of Apparent Horizon in a FRW Universe (2008) · 0.97 · paraphrased: the licence does not allow quoting
The imaginary part of the action for that shell is given by [ref] , [equation] Using the Hamilton's equation of motion $ṙ=dH/dp_r|_r$ the last equality of the above equation is written as, [equation] where, instead of momentum, energy is used as the variable of integration.

Quantum Tunneling in Black Holes (2011) · Radial null geodesic method · 0.97
Used the tunneling method proposed in this paper on a quantum black hole derived in the Quantum Einstein Gravity framework.

Evaporation of (quantum) black holes and energy conservation (2013) · 0.96 · paraphrased: the licence does not allow quoting
Wrote the imaginary part of the action of an s-wave incoming particle crossing the horizon as in this paper.

Decay of the cosmological constant by Hawking radiation as quantum tunneling (2008) · 0.80 · paraphrased: the licence does not allow quoting

How the literature uses it

228 arXiv papers cite this in 473 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 282 sentences without a confident label

Contests it 1 paper

In Ref. [ref] the authors take into account only the contribution of ([ref]); even if they considered also the back-reaction, while here we are ignoring this problem (i.e. we are considering the limit in which particle energy is much less than the Schwarzschild mass), the back-reaction considerations should not justify the omission of the temporal contribution ([ref]).

The time factor in the semi-classical approach to the Hawking radiation · Appendix 2: Photons in Painlev$é$ coordinates · contest 0.86

Builds on it 2 papers

[ref] but generalizing it to a quantum tunneling particle with mass and charge.

Massive charged particle's tunneling from spherical charged black hole · Conclusion · 0.99
We expect that, following [ref] , similar results should hold for the Reissner-Nordström black hole.

Horizon Tunneling Revisited: The Case of Higher Dimensional Black Holes · Conclusions · 0.89

Uses its method or tool 52 papers

We use the null-geodesic method of Parikh and Wilczek [ref] to calculate the action of a tunneling particle.

Hawking temperature of Kerr-Newman-AdS black hole from tunneling · Introduction · 1.00
We use the Parikh-Wilczek's method [ref] which is different from the Hamilton-Jacobi method adopted in ref.

Massive charged particle's tunneling from spherical charged black hole · Hawking radiation via particle's tunneling · 0.99
Following the standard procedure [ref] , we can calculate the emission rate of particles due to the tunneling across the LC by using the above metric.

Hawking-like radiation of charged particles via tunneling across the lightcylinder of a rotating magnetosphere · Tunneling across the LC · 1.00

Compares with it 13 papers

In addition, this result can now be compared directly with the tunneling method of [ref] .

Extremal Tunneling and Anti-de Sitter Instantons · The tunneling perspective · 1.00
Our result coincides with the probability of black hole decay into a single shell found within the tunneling approach to Hawking radiation [ref] and is consistent with interpretation of the Bekenstein–Hawking entropy as the number of black hole microstates [ref] .

Semiclassical S-matrix for black holes · Conclusions and outlook · 0.96
For the classical Schwarzschild spacetime [equation] and so by equation ([ref]) the value of $K_0(M)$ is [equation] Substituting these in ([ref]) and then integrating over $ω'$ we have [equation] Now according to the WKB-approximation method the tunneling probability is given by, [equation] So the modified tunneling probability due to back reaction effects is, [equation] The exponential factor of the tunneling probability was previously obtained by Parikh and Wilczek [ref] .

Quantum Tunneling and Back Reaction · Introduction · 0.83

Cites it as support 3 papers

This is also consistent with the accepted boundary condition of zero energy-momentum tensor at the horizon [ref] .

Combining The Tunneling And Anomaly Phenomena In Deriving the Gravitational Anomaly · Introduction · 0.98
In addition, and perhaps most importantly, there are a number of simple derivations of the Hawking effect that do not require event horizons, indeed depend on local horizons for their derivation [ref] .

Black holes as local horizons · Brief history · 0.99
Unitarity is not violated since physically it is possible to envisage the tunneling as the shrinking of the horizon forwarded in time rather than the particle travelling backward in time [ref] .

Quantum Tunneling in Black Holes · Outline of the thesis · 0.96

Cites it as background 78 papers

As far as the relatively recent research on the tunneling picture of Hawking radiation is concerned, two different approaches are currently very popular, one is the Parikh-Wilczek's semi-classical tunneling method [ref] (based upon the early developments [ref] ), the other is the Hamilton-Jacobi tunneling method [ref] which is developed from the method of complex-path integral proposed in [ref] .

Hawking radiation from rotating AdS black holes in conformal gravity · Introduction · 0.87
It was subsequently suggested by Parikh and Wilczek that these results could also be interpreted, and more easily computed, in terms of the amplitude of a single particle tunneling through the horizon [ref] .

Extremal Tunneling and Anti-de Sitter Instantons · Introduction · 0.85
In 1999, Parikh and Wilczek presented [ref] a concise and direct derivation of the Hawking radiation as a tunneling process.

Massive charged particle's tunneling from spherical charged black hole · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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