Self-Interaction Correction to Black Hole Radiance

Frank Wilczek; Per Kraus

REVIEW

Self-Interaction Correction to Black Hole Radiance

Per Kraus, Frank Wilczek

General Relativity and Quantum Cosmology · Submitted 2 Aug 1994 · 12 references

Pith's verdict: A real, concrete correction to Hawking's spectrum — the derivation is sound in structure, but the one algebraic step that fixes the ω² term is asserted rather than shown.

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arXiv IDgr-qc/9408003 (version 1)

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CitationFrank Wilczek, Per Kraus. "Self-Interaction Correction to Black Hole Radiance." arXiv:gr-qc/9408003 (1994). https://pith.science/paper/YFNYJHBT

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Editor's note

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Worth your time, with one caveat about where to look. Kraus and Wilczek take the s-wave spherical-shell truncation of BCMN, solve the Hamiltonian constraints to get an effective action for the shell, quantize by WKB, and extract a definite non-thermal correction to the Hawking flux. Eq. (5.14) gives the flux with exp(8π(M−ω)ω) − 1 in the denominator, i.e. the exponential is set by the mass after emission. That is a computed statement where previously there was only the observation that the temperature depends on which mass you use. The introduction's claim that no concrete correction had been derived in twenty years looks right to me, and the machinery is the interesting part: the characteristics are no longer geodesics, the action along a null ray is no longer zero, and both facts fall out of the constraint solution rather than being put in by hand. The imaginary-time saddle point at t = 4iπ(M−ω) is a nice piece of physics, and the pair-production analogy in §5 is more than decoration. The footnote on p. 3 is honest about the UV problem: the neglected virtual transitions are formally infinite, and only the exponential factor is claimed to be robust. That is the right thing to concede and the right thing to claim. On the stress-test note: it half lands. The ω² coefficient is not floating free — at the saddle x = −1/2, Eq. (5.11) gives M′ = M − ω exactly within the model, and the modulus of the integrand is e^{4πωM′}. So the two routes to the saddle do agree at the order that matters, provided (5.11) and (4.16) are right. That proviso is the soft spot. The step from (4.15) to (4.16) is dismissed as 'some tedious algebra', and the finite parts dropped between (4.11) and (4.16) are exactly the kind of terms that could shift the O(ω) piece. Nobody can check that from the text. Same for the claim that (3.18) differentiates back to (3.13) — asserted, not displayed. These are fixable in a revision, not fatal to the program. Also worth flagging: the answer coincides with the obvious guess T_after, which the authors say themselves in Comment 1. That does not make it wrong, but it does mean the result confirms a guess rather than surprising anyone. Send it out. The referee should be someone who will redo the (4.15)→(4.16) algebra and check the WKB validity at complex t.

Referee report

Significance

If the result holds, this is the first concrete correction to Hawking's radiance formula derived beyond the fixed-background approximation, and it is a sharp one: the deviation from Planckian form is a specific ω² term in the exponent, with no fitted constants and no free parameters anywhere in the chain from the constraint solution to the saddle point. That parameter-free character is a genuine strength, as is the explicit statement that the correction is calculable rather than merely estimable. The imaginary-time reading of the saddle point (t = 4iπ(M−ω)) is a useful and checkable interpretive addition, and the authors are explicit that the machinery is designed to be extended to two-particle geometries and hence to correlations in the radiation. The paper is also candid about the truncation, conceding in the footnote on p. 3 that the neglected virtual transitions are formally infinite absent a UV completion.

Major objections (3)

[§4, Eqs. (4.11)–(4.16)] The passage from the integral expression (4.11) to the closed-form phase (4.16) is summarized as occurring 'after some tedious algebra,' with finite terms discarded on the grounds that they do not 'contribute to the late-time radiation.' The saddle point of the Bogoliubov integral (5.12) lies at imaginary time t = 4iπ(M−ω), so terms that are finite and negligible on the real axis need not be negligible there. Please state explicitly which terms are dropped between (4.11) and (4.16) and verify that they are analytic in the relevant domain and do not contribute at the order retained. As written, the phase that fixes the entire result is not reproducible from the text.
[§5, Eqs. (5.10)–(5.14) and §6.1] The ω² term in the exponent 8π(M−ω)ω is precisely what distinguishes T_after from T_before; at leading order in ω/M the two are indistinguishable, so the interpretive claim of §6.1 rests entirely on this next-to-leading term. The two routes to the saddle point — direct evaluation using M′(t) from (5.11), and the Hamilton–Jacobi relation M₊ = M − ω — are shown to agree only to leading order, while the ω² coefficient depends on the O(ω) shift of the saddle. Note in particular that M′(t) in (5.11) is itself ω-dependent and is evaluated at x = −1/2, where x/(1+x) = −1, giving M′ ≈ M − ω. It is not shown that this shift, together with the terms dropped in (4.16), leaves the ω² coefficient unchanged. Please give a systematic expansion in ω/M carried through the order that fixes the ω² term, or else state the accuracy to which the exponent is claimed.
[§5, choice r = 2(M+ω_k)] The entire ω-dependence of the final result enters through the identification of the evaluation point r = 2(M+ω_k), and the justification given ('the solution u_k(r)e^{−iω_k t} cannot be extended past that point') is asserted rather than derived. Please justify this choice more carefully and demonstrate the sensitivity of (5.13) to it, for instance by showing what happens for r = 2M + ε with ε ≪ ω_k. If the coefficient of ω² in the exponent depends on this choice, the claim that the effective temperature is exactly T_after is correspondingly weakened.

Minor objections (8)

[Abstract vs. §4] The abstract states that 'the particles no longer move along geodesics,' but §4 shows that the characteristics are 'simply the null geodesics of the metric' (4.6), whose mass parameter is M₊ = M − p_t. The two statements are reconcilable only if 'geodesics' in the abstract means geodesics of the fixed background; please make that explicit so the abstract is not read as contradicting §4.
[Eq. (3.18)] In the third line, the second logarithm has 2M/ˆR under the square root but 2M₊/ˆR in the prefactor denominator; presumably both should be M₊. Similarly, the first line appears to be missing the RR′ factor that multiplies the corresponding logarithm in (3.14).
[Eq. (5.5)] The normalization 1/(2π u_k(r)) of the Bogoliubov coefficients at fixed r deserves a sentence of explanation; as written, it is not obvious that the r-dependence cancels in the combination used in (5.8).
[§5, discussion below Eq. (5.12)] The statement that for α 'the saddle point is reached when e^{k/2M′−t/4M′} → ∞' describes a limiting region rather than a stationary point of the integrand. Please clarify how the α integral is dominated and why the ratio |α/β| is insensitive to the treatment of that end of the integration contour.
[Eq. (5.14)] The flux formula is quoted without a domain of validity. For ω ≳ M the exponent changes sign and the expression is no longer a particle flux; please state the regime ω ≪ M in which the result is claimed.
[§6.1] The phrase 'to lowest order' is misleading here, since at lowest order in ω/M the temperatures T_before and T_after coincide; the identification of T_after requires the O(ω²) term. Please rephrase.
[Footnote, p. 3] The authors acknowledge that the neglected virtual transitions are formally infinite without a UV completion, and assert that such transitions 'will not alter the exponential factors we compute.' A sentence explaining why the anticipated additive ω²/Λ² corrections do not feed into the exponent would strengthen an otherwise unsupported assertion.
[Eqs. (3.1), (4.14), (4.7)] Typographical issues: 'sin θ²' in (3.1) should read sin²θ; the closing symbol in (4.14) reads 'M ”' where M′ is intended; and the outgoing branch of (4.7) contains log(√r − √2M₊), which is singular at the horizon, so a note that this is the Kruskal-extended branch would help the reader.

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The paper

The paper claims

Kraus and Wilczek argue that once the emitted radiation's own gravity is included, Hawking radiation stops being exactly thermal: the exponential suppression of the flux is governed by the black hole's mass after emission, not before, and a

The reading

Hawking's original derivation treats the collapsing geometry as a fixed background, and in that approximation the outgoing radiation is exactly thermal. Kraus and Wilczek ask what happens when the radiation is allowed to gravitate, so that the hole's mass changes as it emits. They truncate the problem to a single spherical shell (the s-wave) interacting with a black hole of fixed interior mass, solve the Hamiltonian constraints of spherically symmetric gravity to obtain an effective action for the shell alone, and quantize it in the WKB approximation. The resulting particle trajectories are no longer geodesics, the action along a massless ray is no longer zero, and the ratio of Bogoliubov coefficients becomes $|e^{4(4M-4M-4M)}|$ — in their notation $|e^{4(4M-4M-4M)}|$ with the shift set by the emitted energy. The payoff is a specific, calculable departure from thermality in the emission rate, which matters because exact thermality is precisely the premise behind the information-loss puzzles.

What carries the argument

The load-bearing object is the reduced Hamiltonian effective action $S = 4dt[p_c 4r + p_t]$ with $p_c$ given by Eq. (3.24), $p_c = 4(2M r) - 4(2M_+ r) + r (4)$, obtained by solving the momentum constraints of spherically symmetric gravity across the shell and fixing the gauge $L=1$, $R=r$. $M_+$ plays the role of the Hamiltonian and is related to the shell momentum by $p = (M_+ - M)/(1 - 4(2M_+/r))$. Because the action is nonlocal, quantization proceeds through the WKB/Hamilton-Jacobi route: the characteristics of the resulting first-order partial differential equation replace geodesics, and the classical action evaluated along them supplies the phase of the mode. The Bogoliubov ratio then follows from a saddle-point evaluation of the time integral, with the saddle displaced into the complex $t$ plane by $t = 4i4M(M-4M)$.

Extended reading notes

Core claim

The paper's central claim is that self-gravitation corrects black hole radiance in a definite, computable way. Working in the Hamiltonian (BCMN) formulation of spherically symmetric gravity, the authors reduce the coupled gravity-shell system to a single degree of freedom, find the canonical momentum conjugate to the shell radius, and identify the exterior mass $M_+$ as the Hamiltonian. Because the effective action is nonlocal, they quantize semiclassically: the phase of the wavefunction obeys a Hamilton-Jacobi equation solved by the method of characteristics, and those characteristics depend on the particle's energy in a way that geodesics do not. For an initial phase $S(0,r)=kr$ near the horizon they carry the calculation through to the late-time form of the mode, then extract the Bogoliubov coefficients by saddle point. The result is $|e^{4(4M-4M-4M)}|$, giving an outgoing flux $F_+ + 8M_+ - 4M_+$ of the form $d4M_+/(2) (4)/(e^{8(4M-4M)}-1)$ — a Planck-like spectrum whose exponential is set by $M-4M$, the mass the hole has after emission.

Load-bearing premise

The whole result rests on truncating to one s-wave spherical shell with the interior mass held fixed and quantizing the resulting nonlocal effective action only semiclassically, and the authors themselves note in a footnote that the neglected virtual transitions to higher partial waves are formally infinite without a ultraviolet completion.

Editorial extensions

If this is right

The exponential suppression of the emission rate is $e^{-8(4M-4M)4M}$ rather than the standard $e^{-8(4M)4M}$, so the effective temperature is that of the hole after it has radiated, resolving the ambiguity of which mass to use in the thermal formula.
Massless particles no longer propagate along null geodesics, and the action along a ray is nonzero even for a massless particle, so the geometrical-optics picture of horizon-skimming modes is modified by the particle's own gravitational field.
The radiation is not exactly Planckian, which removes one of the premises on which the information-loss paradox is usually built.
The emission can be reinterpreted as negative-energy quanta crossing the horizon in an imaginary time interval $4(4M-4M)$, sharpening the analogy with pair production in an electric field treated by instanton methods.
The same machinery is stated to extend to Reissner-Nordström and dilaton holes, to charged or massive scalar fields, and to two-shell geometries that would allow the question of correlations in the radiation to be posed.

Reading between the lines

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

Because the correction sits in the exponent and is proportional to the emitted energy, it cannot be absorbed into a greybody factor or a rescaled overall temperature; the spectrum drifts as the hole shrinks, so a late-time spectrum is not a single Planck distribution at any one temperature.
The imaginary-time interval $4(4M-4M)$ is shorter than the $4(4M)$ that would correspond to the initial mass, which suggests reading the correction as a consequence of the horizon area shrinking during emission rather than as an independent dynamical effect.
The authors' own footnote concedes that virtual transitions to higher partial waves are formally infinite without a ultraviolet completion; if those additive corrections exponentiate rather than stay additive, the numerical coefficient in the exponent would change and the result would be a truncation artifact rather than a physical prediction.
For astrophysical holes the fractional shift in the exponent is of order $4M/M$, far below observability, so the claim is best read as a consistency statement about the structure of semiclassical gravity rather than as a predicted spectral line.

Original abstract

We consider the modification of the formulas for black hole radiation, due to the self-gravitation of the radiation. This is done by truncating the coupled particle-hole system to a small set of modes, that are plausibly the most significant ones, and quantizing the reduced system. In this way we find that the particles no longer move along geodesics, nor is the action along the rays zero for a massless particle. The radiation is no longer thermal, but is corrected in a definite way that we calculate. Our methods can be extended in a straightforward manner to discuss correlations in the radiation, or between incoming particles and the radiation.

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We consider the modification of the formulas for black hole radiation, due to the self-gravitation of the radiation. This is done by truncating the coupled particle-hole system to a small set of modes, that are plausibly the most significant ones, and quantizing the reduced system. In this way we find that the particles no longer move along geodesics, nor is the action along the rays zero for a massless particle. The radiation is no longer thermal, but is corrected in a definite way that we calculate. Our methods can be extended in a straightforward manner to discuss correlations in the radiation, or between incoming particles and the radiation.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the modification of the formulas for black hole radiation, due to the self-gravitation of the radiation.	contribution 0.72	0.38	0.07	asserted 0.51
s1	This is done by truncating the coupled particle-hole system to a small set of modes, that are plausibly the most significant ones, and quantizing the reduced system.	method 1.00	0.92	0.05	asserted 0.93
s2	In this way we find that the particles no longer move along geodesics, nor is the action along the rays zero for a massless particle.	result 0.99	0.91	0.23	asserted 1.00
s3	The radiation is no longer thermal, but is corrected in a definite way that we calculate.	result 1.00	0.94	0.68	asserted 1.00
s4	Our methods can be extended in a straightforward manner to discuss correlations in the radiation, or between incoming particles and the radiation.	future work 1.00	0.37	0.04	asserted 0.62

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References

Works cited (12)

[1]

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

View on Pith 1975
[2]

J. Preskill, P. Schwarz, A. Shapere, S. Trivedi, and F. Wi lczek. Mod. Phys. Lett. A6 (1991), 2353

View on Pith 1991
[3]

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974)

View on Pith 1974
[4]

B. K. Berger, D. M. Chitre, V. E. Moncrief, and Y. Nutku. Ph ys. Rev. D5 (1972), 2467

View on Pith 1972
[5]

W. Fischler, D. Morgan, and J. Polchinski. Phys. Rev. D42 (1990), 4042

View on Pith 1990
[6]

R. Arnowitt, S. Deser, and C. Misner, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962)

View on Pith 1962
[7]

T. Regge, and C. Teitelboim, Ann. Phys. 88 (1974) 286

View on Pith 1974
[8]

P. Kraus, and F. Wilczek. “Some Applications of a Simple Stationary Line Element for the Schwarzschild Geometry,” preprint PUPT 1474, IASSNS 94/96, June 1994

View on Pith 1994
[9]

See e.g. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cam- bridge University Press, Cambridge, 1982)

View on Pith 1982
[10]

I. K. Aﬄeck, O. Alvarez, and N. S. Manton. Nucl. Phys. B197 (1982), 509

View on Pith 1982
[11]

G. ’t Hooft, Nucl. Phys b335 138 (1990)

View on Pith 1990
[12]

K. Schoutens, H. Verlinde, and E. Verlinde. ”Black Hole Evaporation and Quantum Gravity,” Princeton preprint, PUPT-1430, April 1993. 20

View on Pith 1993

Forward citations

Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Both tunnelling methods give the standard Hawking temperature for non-extremal Rindler, Kerr-Newman, and Taub-NUT-AdS spacetimes, and break down for extremal Reissner-Nordstrom.
Hawking Radiation as Tunneling for Extremal and Rotating Black Holes
hep-th 2005-03 conditional novelty 5.0 of 10

Deforming the tunneling contour in the invariant proper distance, rather than in the radial coordinate, reproduces the standard Hawking temperature and forces extremal black holes to zero temperature.
Radiation from a collapsing object is manifestly unitary
gr-qc 2015-03 reject novelty 4.0 of 10

Keeping off-diagonal correlations in the radiation density matrix leaves its trace squared at unity, which the authors read as unitarity of gravitational collapse.
Noncommutative Black Hole Thermodynamics
hep-th 2008-01 conditional novelty 4.0 of 10

A model-independent tunneling derivation of T = K/2π is applied to a noncommutative black hole with back reaction, yielding a closed-form temperature that remains finite at small horizon radius.
Quantum Tunneling and Back Reaction
hep-th 2007-12 conditional novelty 4.0 of 10

Inserting a one-loop corrected surface gravity into the tunneling integral yields a prefactor in the emission probability and reproduces the known logarithmic entropy correction and temperature shift.
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hep-th 2011-08 conditional novelty 3.0 of 10

The information paradox survives every proposed resolution except the fuzzball construction, which removes the horizon the paradox depends on.
Alternative Derivations of Hawking Radiation
gr-qc 2025-05 unverdicted novelty 2.0 of 10

A review chapter that reproduces and compares the tunneling, anomaly, and Green's function derivations of Hawking radiation, recovering the Hawking temperature and noting open justifications.

How the literature uses it

49 arXiv papers cite this in 131 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 · 77 sentences without a confident label

Contests it 1 paper

For both the neutral and charged case this does not exactly reproduce the result of the original articles [ref] due to a technical error, which was in fact corrected in [ref] for the neutral case.

Extremal Tunneling and Anti-de Sitter Instantons · Evaluating the Bogoliubov coefficients · contest 0.92

Builds on it 2 papers

We extended the treatment of [ref] to more than one shell [ref] proving on the way the universality of the reduced conjugate momentum within the family of Painlevé gauges [ref] .

Reduced Hamiltonian for intersecting shells and Hawking radiation · Concluding remarks · 1.00
The perfect-fluid shell model we develop is a generalization of the dust shell model used by Kraus and Wilczek in their attempt to calculate self-interaction corrections to standard Hawking radiation [ref] - [ref] .

Self-gravitating Interferometry and Intrinsic Decoherence · Action Principle · 0.99

Uses its method or tool 11 papers

We shall keep the formalism for more that one shell as close as possible to the original formalism of [ref] which can be summarized in words that we shall adopt a Painlevé gauge.

Reduced Hamiltonian for intersecting shells and Hawking radiation · Introduction · 0.85
The method we use to obtain a four-dimensional backreacted decay probability for charged black holes is the tunneling approach to Hawking radiation developed by [ref] .

Extremal Black Hole Decay in de Sitter Space · Introduction · 0.99
Ensuring that the positive energy solutions decay in time (i.e. into the lower half of $ω'$ plane and $r_in>r_out$) we have after $r$ integration [A similar treatment can be done by taking the contour in the upper half plane but then one has to replace $M→M+ω$ [ref] .], [equation]

Quantum Tunneling and Back Reaction · Introduction · 0.86

Compares with it 8 papers

It can also be shown [ref] that the expression ([ref]) for the reduced canonical shell momentum $P_c$ is equivalent to the form analogous to the expression given by Kraus and Wilczek [ref] , [equation] despite the different minus sign placement.

Bootstrapping Time Dilation Decoherence · Reduced Phase Space Quantization · 0.91
To show this, it suffices to calculate [equation] From the flux of outgoing particles with frequencies between $ω$ and $ω+ dω$ [ref] : [equation] We finally arrive at: [equation] Comparing with the Planck distribution [equation] we find that [equation] which is the same expression as the expression for the temperature obtained using the surface gravity in [ref] .

Quantum Gravitational Corrections to Particle Creation by Black Holes · Quantum Gravitational corrections to Hawking radiation · 0.82
Along the way we will show how the Kraus-Wilczek computation [ref] reduces to the one performed by Parikh and Wilczek [ref] .

Extremal Tunneling and Anti-de Sitter Instantons · Inclusion of gravitational backreaction · 0.80

Cites it as background 15 papers

In paper [ref] Kraus and Wilczek introduced a semiclassical treatment of Hawking radiation by considering the mechanics of a thin self-gravitating shell of matter in a spherical gravitational field.

On the semiclassical treatment of Hawking radiation · Introduction · 0.95
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
The semiclassical treatment of the Hawking radiation was introduced by Kraus and Wilczek in [ref] after which several developments followed.

Late-time expansion in the semiclassical theory of the Hawking radiation · Introduction · 1.00

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.

[1] [1]

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

View on Pith 1975

[2] [2]

J. Preskill, P. Schwarz, A. Shapere, S. Trivedi, and F. Wi lczek. Mod. Phys. Lett. A6 (1991), 2353

View on Pith 1991

[3] [3]

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974)

View on Pith 1974

[4] [4]

B. K. Berger, D. M. Chitre, V. E. Moncrief, and Y. Nutku. Ph ys. Rev. D5 (1972), 2467

View on Pith 1972

[5] [5]

W. Fischler, D. Morgan, and J. Polchinski. Phys. Rev. D42 (1990), 4042

View on Pith 1990

[6] [6]

R. Arnowitt, S. Deser, and C. Misner, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962)

View on Pith 1962

[7] [7]

T. Regge, and C. Teitelboim, Ann. Phys. 88 (1974) 286

View on Pith 1974

[8] [8]

P. Kraus, and F. Wilczek. “Some Applications of a Simple Stationary Line Element for the Schwarzschild Geometry,” preprint PUPT 1474, IASSNS 94/96, June 1994

View on Pith 1994

[9] [9]

See e.g. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cam- bridge University Press, Cambridge, 1982)

View on Pith 1982

[10] [10]

I. K. Aﬄeck, O. Alvarez, and N. S. Manton. Nucl. Phys. B197 (1982), 509

View on Pith 1982

[11] [11]

G. ’t Hooft, Nucl. Phys b335 138 (1990)

View on Pith 1990

[12] [12]

K. Schoutens, H. Verlinde, and E. Verlinde. ”Black Hole Evaporation and Quantum Gravity,” Princeton preprint, PUPT-1430, April 1993. 20

View on Pith 1993