Dynamical Origin of the Entropy of a Black Hole

Igor Novikov; Valery Frolov

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Dynamical Origin of the Entropy of a Black Hole

Valery Frolov, Igor Novikov

General Relativity and Quantum Cosmology · Submitted 2 Sep 1993 · 23 references

Pith's verdict: The definition in Eqs. (2)-(3) is the real contribution and it holds up; the 1/4 coefficient is a consistency check with an undetermined O(1) in it, not a derivation, and the paper mostly says so.

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arXiv IDgr-qc/9309001 (version 2)

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CitationIgor Novikov, Valery Frolov. "Dynamical Origin of the Entropy of a Black Hole." arXiv:gr-qc/9309001 (1993). https://pith.science/paper/CXHFRNAA

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Worth your time, with one caveat the abstract papers over. The real contribution is the definition, not the number. Eqs. (2)-(3) — trace the initial state over modes that cross the horizon, take the von Neumann entropy of what's left — is a clean, invariant, statistical-mechanical statement, and it is the same object Srednicki and Bombelli-Koul-Lee-Sorkin were circling at the same moment. The note added after submission says so and draws the right distinction: the horizon is null, the flat-space sphere is timelike, so the analogy is not an identity. That is honest, and it is the kind of note more people should write. The calculation is laid out step by step and the area law falls out robustly: S_H0 = 1/(360λ) with λ = α l_Pl²/r_g² gives S ∝ A_H/l_Pl². The mechanism — thermally excited trapped modes piling up in a thin shell near the horizon — is the part that survives, and it survives independent of the details. Where it is soft is the 1/4. γ = 1/(360πα), so matching A/4 means α = 1/(360π). The paper then offers an independent estimate, α ≈ 1.88×10⁻³ ξ from mass-measurement noise, giving γ ≈ 0.47/ξ. For ξ = 1/2 you get 0.94; for ξ = 1 you get 0.47. So the 'good agreement' is a factor-of-two statement dressed as a 6% one, and the stress-test is right about that. I would push back on calling it tuned, though: the estimate is genuinely independent and lands close for the minimal uncertainty product. It is a consistency check with an undetermined O(1) in it, not a derivation. The authors know — 'one cannot expect to determine the entropy with accuracy higher than ... thermodynamical fluctuations' is in the text. Two smaller things. The claim that the result is independent of the number of fields works by having α grow with N while the per-field entropy also grows with N; that cancellation is asserted, not shown. And note [22] admits the T-mode bookkeeping and the non-additivity of S_H and S_rad are unresolved — the classifier says no limitations are stated, which is wrong; the paper is candid in the body. Who should read it: anyone working on entanglement entropy and horizons. It is a foundational reference and it has been undercited relative to Srednicki. Send it out.

Referee report

Significance

The mechanism is physically well motivated and the calculation is unusually clean for this subject: the mode-counting constant 1/360 in Eq. (37) is a definite number that I have checked (∫₀^∞ z² s(z) dz = 4π⁴/45, giving S_H0 = 1/(360λ) exactly), and the correction ΔS is computed explicitly rather than dropped. The definition in Eqs. (1)–(3) is genuinely invariant under independent Bogoliubov transformations of the visible and invisible sectors, which is a real improvement over the 'Cheshire-Cat' reading of black-hole entropy criticized in the introduction, and the identification of the entropy as a surface effect concentrated in a thin near-horizon shell is a substantive, falsifiable-in-principle statement. The comparison with Srednicki and Bombelli-Koul-Lee-Sorkin in the Note added is appropriately careful about the null-versus-timelike distinction. The weakness is quantitative rather than structural: the area law follows robustly, but the coefficient is obtained by matching one undetermined parameter rather than predicted, so the abstract's 'good agreement' is a consistency check whose strength depends on a choice the paper does not justify.

Major objections (4)

[Eq. (39) and the paragraph immediately following; abstract:s3] The sentence 'The result (39) reproduces the standard expression for the black-hole's entropy A_H/4l_Pl² for the value of the parameter α = (360π)^{-1}' is a matching, not a derivation: γ = 1/(360πα) is fixed to unity by choosing α, and α is the one quantity in the chain that the mode sum does not supply. The independent estimate in Eq. (43), α ≈ 1.88×10⁻³ ξ, gives γ ≈ 0.47/ξ, so the quoted γ ≈ 0.94 obtains only for ξ = 1/2, the minimal uncertainty-product value, and γ ≈ 0.47 for the equally standard ξ = 1. The paper gives no argument for saturating the bound ξ ≥ 1/2, so the headline agreement is a factor-of-two statement presented as a 6% one. Concrete fix: report γ explicitly as a function of ξ over, say, ξ ∈ [1/2, 2], and restate the abstract's 'good agreement' as agreement up to the O(1) uncertainty in the horizon-fluctuation parameter. This is load-bearing for abstract:s3 and should not be left to the reader to reconstruct from Eqs. (39) and (43).
[Eq. (24) and the final paragraph on the number of fields] The relation λ = α l_Pl²/r_g² is asserted with a citation to York and to two conference essays, and the entire coefficient in Eq. (39) scales as 1/α, so this single un-derived proportionality controls the quantitative claim. A short derivation, or an explicit statement of exactly what is assumed about the relation between the mass dispersion δM and the horizon smearing scale, would make the argument self-contained. The same assumption carries the closing claim that the entropy is independent of the number N of fields: that cancellation requires α ∝ N (via n̄ ∝ N), so if α does not scale with N the total entropy becomes N-dependent and the claim fails. Please state the N-scaling of α as an explicit assumption and show the cancellation in one line, rather than leaving it as an observation.
[Eqs. (36)–(38) and the sentence 'It is important to stress that S_H0 does not depend on r_0'] The correction ΔS ≈ (9x_0 - 23)×10⁻³ depends linearly on x_0 = r_0/r_g, so the stated justification that ΔS 'is much smaller than unity' fails for a large cavity: at x_0 = 10², ΔS ≈ 0.9, and at x_0 = 10⁴ it is of order 90. The physical conclusion survives, because ΔS remains negligible compared with S_H0 = 1/(360λ), which is enormous; but the argument as written compares ΔS to the wrong scale and the accompanying r_0-independence sentence is contradicted by the authors' own formula. Concrete fix: compare ΔS with S_H0 rather than with unity, and state the residual r_0 dependence as a relative correction of order ΔS/S_H0.
[Footnote [22] and Eqs. (26)–(35)] The authors acknowledge in footnote [22] that 'it is not quite clear whether one can also change the states of T-modes without any changing of the black hole parameters and whether these T-modes might give an additional contribution to the entropy.' This is not a peripheral caveat: T-modes supply the 1/(4πλ) term that becomes S_H0 in Eq. (37), so the footnote appears to question the status of the very degrees of freedom that produce the area law. The manuscript should say plainly whether the footnote concerns only the wormhole argument of the concluding section or whether it qualifies the identification of T-modes as invisible degrees of freedom used in the main calculation. If the latter, the counting in Eqs. (26)–(35) needs to be re-justified; if the former, the footnote should be reworded so it does not read as undercutting the central result.

Minor objections (7)

[Eq. (31)] The expression for N^T_jl is introduced with 'It is possible to show that' and is central to the T-mode contribution; a line or two of derivation, or a pointer to where the intermediate steps appear, would help the reader verify the change of counting surface from Σ^λ_v to Σ^λ_t.
[Eq. (30)] The definition of A_νl has an unbalanced bracket: 'Aνl = −i exp[−i(ν ln(µl/2) −φν],'. Please correct.
[Eq. (10)] The two Killing-vector relations carry opposite signs for the UP and SIDE modes; a sentence fixing the frequency convention (and hence the sign in the thermal factor exp(−2πν)) would prevent confusion when comparing with Eq. (15).
[Eq. (19) and footnote [14]] The step-function approximation Γ = 1 or 0 for the wave-packet transmission is justified by geometric optics but the size of the error is not estimated; a brief remark on why the neglected modes do not shift the 1/360 constant would strengthen the result.
[Eq. (22) and footnote [15]] The replacement of the l-sum by an integral drops the Euler-Maclaurin end terms; the argument that large l dominates is plausible but should be quantified, since the omitted terms are exactly the ones the footnote identifies as important for the luminosity calculation.
[Reference [19]] The numerical value n̄ = 6.644×10⁻⁴ M⁻¹ is taken from an unpublished PhD thesis. The DeWitt analytic value quoted alongside agrees to about 2%, which mitigates the concern, but a published source for n̄ would be preferable.
[Throughout] There are numerous typographical errors that should be corrected: 'Bekenstien' in refs. [2] and [4], 'raduis' (p. 5), 'priciple byy' (p. 6), 'exited' for 'excited' (p. 6), 'decribed' (p. 3), 'disscused' in ref. [11], and the broken spacing in the author affiliations ('Edm onton', 'P rofsoyuznaya', 'De nmark').

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The paper

The paper claims

The paper argues that a black hole's entropy is the entanglement entropy of quantum field modes trapped inside the horizon, and that counting thermally excited near-horizon modes reproduces the area law once a single horizon-jitter constant

The reading

The paper asks where a black hole's entropy actually lives, and answers that it lives in the quantum field modes that sit inside the horizon and can never be seen by a distant observer. The authors define the hole's entropy as the entropy of the state obtained by tracing out everything visible outside the horizon, then count how many thermally excited 'invisible' modes cross a surface near the horizon when the field is in the Hartle-Hawking (thermal equilibrium) state. Summing those contributions for a conformal massless scalar field gives an entropy proportional to the horizon area, $S_H = S_{H0}+s$mall corrections with $S_{H0} = A_H/(4l_{Pl}^2)$ up to a coefficient $1/(360pi alpha)$, where $alpha$ sets the scale of quantum jitter of the horizon. Matching the standard coefficient requires $alpha = (360pi)^{-1}$, and an independent heuristic estimate of that jitter gives a coefficient of about $0.94$ instead of $1$, which the authors call good agreement. The appeal of the argument is that it turns the area law from a bookkeeping convention into a counting problem about real dynamical degrees of freedom, and it makes the entropy vanish in the classical limit.

What carries the argument

The mechanism is a mode-counting argument in the near-horizon region. Modes are packaged into wave packets labelled by $J = jnlm$, and two new sets $f^B_J$ and $f^W_J$ are formed as normalized superpositions of SIDE and UP modes with weight $w_J = exp(-pi nu_J)$; these combinations are positive-frequency with respect to the affine parameter on the past horizon, so the Hartle-Hawking state takes the simple thermal form $rho_{init,J} = rho_{0J} : exp[-alpha^{*B}_J alpha^B_J - alpha^{*W}_J alpha^W_J] :$. Tracing over the visible member of each pair leaves a thermal density matrix for the invisible one, giving per-mode entropy $s(z) = z/(e^z-1) - ln[1-e^{-z}]$ with $z = 2pi nu_J$. The escaping and trapped contributions are then integrated over angular momentum and frequency in a homogeneous-gravitational-field (Rindler-like) approximation near the horizon. The area law emerges because the dominant contribution comes from a thin shell of modes hugging the horizon, and the divergent count of those modes is cut off by quantum spreading of the horizon over a dimensionless distance $lambda = alpha l_{Pl}^2/r_g^2$, which is what converts the mode sum into a finite multiple of the horizon area.

Extended reading notes

Core claim

The central claim is that a black hole's entropy is an ordinary statistical-mechanical entropy of field modes hidden behind the horizon, defined invariantly as $S_H = -Tr_{inv}(rho_H ln rho_H)$ with $rho_H = Tr_{vis} rho_{init}$, and that this definition reproduces the Bekenstein-Hawking value. Working with a Schwarzschild black hole enclosed in a mirror cavity and a conformal massless scalar field, the authors split the field into UP modes (radiated outward from the horizon) and SIDE modes (radiated into the interior), build positive-frequency combinations that make the Hartle-Hawking state thermal, and trace over the visible partners. Each visible mode then contributes a thermal entropy $s(2pi nu_J)$, and the sum over escaping and trapped modes near the horizon gives $S_{H0} = gamma A_H/(4l_{Pl}^2)$ with $gamma = 1/(360pi alpha)$. The result is independent of the cavity radius and of which null surface is used to separate visible from invisible modes, and the authors argue it does not depend on the number of field species, because the horizon-fluctuation parameter grows with the species count just as the emission rate does.

Load-bearing premise

The entire numerical match rests on assuming that quantum jitter of the horizon smears it over a distance set by $alpha l_{Pl}^2/r_g^2$, where the constant $alpha$ is not derived from any theory of quantum gravity but is fixed by requiring agreement with the standard area law.

Editorial extensions

If this is right

Black hole entropy is a pure quantum effect: it arises from particle pairs created by the gravitational field near the horizon, one member of each pair escaping and the other falling in, and it disappears in the classical limit.
The entropy does not depend on how many species of fields exist, because the horizon-fluctuation parameter grows with the species count at the same rate as the particle emission rate, cancelling the additive contributions.
The result is independent of the radius of the enclosing mirror cavity and of the choice of null surface used to split visible from invisible modes, so the construction is not tied to the particular setup used to compute it.
The generalization to rotating and charged black holes and to other field types is claimed to be straightforward, with each field contributing additively and the total still scaling with horizon area.
In the wormhole thought experiment where one mouth falls into a black hole, the entropy drop is explained as invisible near-horizon modes becoming visible; when the second mouth crosses the horizon or the wormhole is destroyed, the system forgets the intervention and the entropy returns to its initial value.

Reading between the lines

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

The derivation reduces the area law to a single number: the coefficient $alpha$ of horizon jitter. A first-principles quantization of horizon fluctuations that fixes $alpha$ would therefore be a sharp, quantitative test of the whole picture rather than a loose consistency check.
If the entropy really is the entanglement entropy of modes hidden behind a null surface, the same counting should apply to Rindler, de Sitter, and cosmological horizons, which suggests the area law has one common origin rather than a special one for black holes.
The cutoff $lambda$ is effectively a ultraviolet regulator standing in for unknown short-distance physics; for the claim to be robust, the coefficient should turn out to be regulator-independent, which the paper does not show.
Footnote [22] concedes that it is unclear whether trapped modes can be altered without changing the black hole's external parameters, and whether they contribute extra entropy; resolving that ambiguity would sharpen the bookkeeping of which modes count as degrees of freedom.

Original abstract

Modes of physical fields which are located inside a horizon and which cannot be observed by a distant observer are identified with dynamical degrees of freedom of a black hole. A new invariant statistical mechanical definition of a black-hole's entropy is proposed. It is shown that the main contribution to the entropy is given by thermally excited `invisible' modes propagating in the close vicinity of the horizon. A calculation based on the proposed definition yields a value of the entropy which is in good agreement with the usually adopted value.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,543 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.25 · compares with prior results 0.89 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Modes of physical fields which are located inside a horizon and which cannot be observed by a distant observer are identified with dynamical degrees of freedom of a black hole. A new invariant statistical mechanical definition of a black-hole's entropy is proposed. It is shown that the main contribution to the entropy is given by thermally excited `invisible' modes propagating in the close vicinity of the horizon. A calculation based on the proposed definition yields a value of the entropy which is in good agreement with the usually adopted value.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Modes of physical fields which are located inside a horizon and which cannot be observed by a distant observer are identified with dynamical degrees of freedom of a black hole.	contribution 0.64	0.55	0.15	asserted 1.00
s1	A new invariant statistical mechanical definition of a black-hole's entropy is proposed.	contribution 0.99	0.89	0.17	asserted 0.85
s2	It is shown that the main contribution to the entropy is given by thermally excited `invisible' modes propagating in the close vicinity of the horizon.	result 0.92	0.89	0.53	asserted 1.00
s3	A calculation based on the proposed definition yields a value of the entropy which is in good agreement with the usually adopted value.	result 1.00	0.92	0.25	asserted 0.95

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 0.94.

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sources read

arXiv listing abstract v2

not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (23)

[22]

L.Bombelli, R.K.Koul, J.Lee, and R.Sorkin, Phys.Rev. D34 373 (1986)

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[1]

+ 1. For l ≫ 1 the potential Vl has maximum ≈ 16µ2l/27 at x =xm ≈ 3/2. We call ‘escaping’ (or brieﬂy E-modes) theUP -modes withν >4µl/ √ 27 which (in the absence of the boundary B) are propagating almost freely to inﬁnity. We call ‘trapped’ (or brieﬂy T-modes) the UP -modes with ν <4µl/ √ 27 which are mainly reﬂected by the potential barrier and returned ...

View on Pith
[2]

J.D.Bekenstein, Nuov.Cim.Lett. 4, 737 (1972)

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[3]

J.D.Bekenstien, Phys.Rev. D7, 2333 (1973)

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

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J.D.Bekenstien, Phys.Rev. D9, 3292 (1974)

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[6]

J.M.Bardeen, B.Carter, and S.W.Hawking, Comm.Math.Phys. 31, 181 (1973)

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[7]

J.D.Bekenstein, Phys.Today 33, no.1, 24

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J.W.York, Phys.Rev. D28, 2929 (1983)

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[9]

W.H.Zurek and K.S.Thorne, Phys.Rev.Lett. 54, 2171 (1985)

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[10]

K.S.Thorne, W.H.Zurek, and R.H.Price, in Black Holes: The Membrane Paradigm , edited by K.S.Thorne, R.H.Price, and D.A.MacDonald (Yale University Press, New Haven, 1986), p.280

View on Pith 1986
[11]

A.G.Doroshkevich and I.D.Novikov, Sov.Phys.JETF 74, 2 (1978)

View on Pith 1978
[12]

There is a fundamental problem connected with the information loss in black holes. Hawking (Phys.Rev. D14, 2460 (1976)) argued that an initial pure quantum state that underwent gravitational collapse into a black hole after the black hole evaporation might produce a mixed quantum state with non-zero entropy. This prob- lem known as the problem of informat...

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[13]

V.P.Frolov and I.D.Novikov. Phys.Rev.D48 1607 (1993)

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[14]

For the deﬁnition of U P-modes see e.g. Ref.[9]. Our deﬁ- nition of SIDE -modes correspond to the left-hand-sector modes in the thermoﬁeld-dynamical description of the thermal radiation of black holes in: W.Israel, Phys.Lett. 57A, 107 (1976)

View on Pith 1976
[15]

Strictly speaking s(z) = (1 + n) ln(1 +n) − n ln(n), where n = Γ(exp z − 1)−1, and Γ is the probability for the wave packet to reach Σ v. For Γ = 0 s(z) = 0, while for Γ = 1 one gets the expression ( 19). The approximation we use is analogous to the one proposed by DeWitt (Phys.Reports 19C, 295 (1975)). We put Γ = 1 for the modes which reach Σ v and Γ = 0...

View on Pith 1975
[16]

According to the Euler-Maclaurin formula ∑∞l=0 F (l) ≈ 1/2F (0) − 1/12F ′(0) + ∫ ∞0 F (x)dx. The omitted terms F (0) and F ′(0) are important for the calculation of the contribution of the scalar massless ﬁeld into quantum radiation at inﬁnity. (The authors are grateful to Andrei Zelnikov for this remark.) It will be shown later that the main contribution...

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[17]

J.M.York. In: Quantum Theory of Gravity. Essays in honor of the 60th birthday of Bryce S.DeWitt , Ed.S.M.Christensen, (Adam Hilger Ltd, Bristol) (1984)

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J.D.Bekenstein. In: Quantum Theory of Gravity. Es- says in honor of the 60th birthday of Bryce S.DeWitt , Ed.S.M.Christensen, (Adam Hilger Ltd, Bristol) (1984)

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[19]

Our case is a particular example of the situation when a physical quantity is measured several times each time with a diﬀerent accuracy. The formula below is a special case of general result for the accuracies of such obser- vations which can be found e.g. in: J.R.Taylor. An in- troduction to error analysis. (Univ.Science Books, Mill Valley, California) 1982

View on Pith 1982
[20]

R.S.Simkins. ‘Massive scalar particle emission from Schwarzschild black holes’, PhD Thesis, The Pennsylva- nia State Univ., (1986) (unpublished)

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[21]

M.Srednicki, Phys.Rev.Lett. 71 666 (1993)

View on Pith 1993
[23]

The SIDE -modes are propagating in the region lying to the future with respect to the horizon, while the U P- modes are (at least partly) propagating in the region lying to the past with respect to the horizon. Due to the evident causal asymmetry between U P and SIDE - modes it is not quite clear whether one can also change the states of T -modes without ...

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Cites it as support 1 paper

There is some evidence that quantum gravity should be digital, from the finiteness of black hole entropy in generalized thermodynamics [ref] (based on the idea that the black hole entropy should count the entropy of some set of microscopic degrees of freedom associated with the horizon [ref] .) Also, in loop quantum gravity, area and volume operators have discrete eigenvalues [ref] .

A discrete, unitary, causal theory of quantum gravity · Motivating Remarks · 1.00

Cites it as background 11 papers

Regarding the gravitational entropy, there is numerous literature [ref] on the identification of Bekenstein-Hawking entropy $S_BH$ with the vacuum entanglement entropy of quantum gravitational degrees of freedom.

Emergent Gravity in a Holographic Universe · Entanglement entropy of a causal diamond · 0.96
An interesting attempt to understand the Bekenstein-Hawking formula of black hole entropy $S_BH=A/4G$, with $A$ the black hole horizon area and $G$ the gravitational constant, is to relate it to the entanglement entropy ($S_E$) across the black hole horizon [ref] . $S_E$ quantifies the entanglement between the degrees of freedom inside and outside the black hole.

On the Renormalization of Entanglement Entropy · Introduction · 0.94
This construction was originally introduced in order to understand aspects of black hole entropy [ref] , although it has since found important applications in a variety of other areas of physics [ref] .

Investigations on entanglement entropy in gravity · Generalized entropy as entanglement entropy · 0.96

Discussion

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L.Bombelli, R.K.Koul, J.Lee, and R.Sorkin, Phys.Rev. D34 373 (1986)

View on Pith 1986

[2] [1]

+ 1. For l ≫ 1 the potential Vl has maximum ≈ 16µ2l/27 at x =xm ≈ 3/2. We call ‘escaping’ (or brieﬂy E-modes) theUP -modes withν >4µl/ √ 27 which (in the absence of the boundary B) are propagating almost freely to inﬁnity. We call ‘trapped’ (or brieﬂy T-modes) the UP -modes with ν <4µl/ √ 27 which are mainly reﬂected by the potential barrier and returned ...

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J.D.Bekenstein, Nuov.Cim.Lett. 4, 737 (1972)

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J.D.Bekenstien, Phys.Rev. D7, 2333 (1973)

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

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J.D.Bekenstien, Phys.Rev. D9, 3292 (1974)

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J.M.Bardeen, B.Carter, and S.W.Hawking, Comm.Math.Phys. 31, 181 (1973)

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J.D.Bekenstein, Phys.Today 33, no.1, 24

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J.W.York, Phys.Rev. D28, 2929 (1983)

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W.H.Zurek and K.S.Thorne, Phys.Rev.Lett. 54, 2171 (1985)

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K.S.Thorne, W.H.Zurek, and R.H.Price, in Black Holes: The Membrane Paradigm , edited by K.S.Thorne, R.H.Price, and D.A.MacDonald (Yale University Press, New Haven, 1986), p.280

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A.G.Doroshkevich and I.D.Novikov, Sov.Phys.JETF 74, 2 (1978)

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There is a fundamental problem connected with the information loss in black holes. Hawking (Phys.Rev. D14, 2460 (1976)) argued that an initial pure quantum state that underwent gravitational collapse into a black hole after the black hole evaporation might produce a mixed quantum state with non-zero entropy. This prob- lem known as the problem of informat...

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V.P.Frolov and I.D.Novikov. Phys.Rev.D48 1607 (1993)

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For the deﬁnition of U P-modes see e.g. Ref.[9]. Our deﬁ- nition of SIDE -modes correspond to the left-hand-sector modes in the thermoﬁeld-dynamical description of the thermal radiation of black holes in: W.Israel, Phys.Lett. 57A, 107 (1976)

View on Pith 1976

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Strictly speaking s(z) = (1 + n) ln(1 +n) − n ln(n), where n = Γ(exp z − 1)−1, and Γ is the probability for the wave packet to reach Σ v. For Γ = 0 s(z) = 0, while for Γ = 1 one gets the expression ( 19). The approximation we use is analogous to the one proposed by DeWitt (Phys.Reports 19C, 295 (1975)). We put Γ = 1 for the modes which reach Σ v and Γ = 0...

View on Pith 1975

[17] [16]

According to the Euler-Maclaurin formula ∑∞l=0 F (l) ≈ 1/2F (0) − 1/12F ′(0) + ∫ ∞0 F (x)dx. The omitted terms F (0) and F ′(0) are important for the calculation of the contribution of the scalar massless ﬁeld into quantum radiation at inﬁnity. (The authors are grateful to Andrei Zelnikov for this remark.) It will be shown later that the main contribution...

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J.M.York. In: Quantum Theory of Gravity. Essays in honor of the 60th birthday of Bryce S.DeWitt , Ed.S.M.Christensen, (Adam Hilger Ltd, Bristol) (1984)

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J.D.Bekenstein. In: Quantum Theory of Gravity. Es- says in honor of the 60th birthday of Bryce S.DeWitt , Ed.S.M.Christensen, (Adam Hilger Ltd, Bristol) (1984)

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Our case is a particular example of the situation when a physical quantity is measured several times each time with a diﬀerent accuracy. The formula below is a special case of general result for the accuracies of such obser- vations which can be found e.g. in: J.R.Taylor. An in- troduction to error analysis. (Univ.Science Books, Mill Valley, California) 1982

View on Pith 1982

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R.S.Simkins. ‘Massive scalar particle emission from Schwarzschild black holes’, PhD Thesis, The Pennsylva- nia State Univ., (1986) (unpublished)

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M.Srednicki, Phys.Rev.Lett. 71 666 (1993)

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The SIDE -modes are propagating in the region lying to the future with respect to the horizon, while the U P- modes are (at least partly) propagating in the region lying to the past with respect to the horizon. Due to the evident causal asymmetry between U P and SIDE - modes it is not quite clear whether one can also change the states of T -modes without ...

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