Multi-Time Measurements in Hawking Radiation: Information at Higher-Order Correlations

Charis Anastopoulos; Ntina Savvidou

REVIEW

Multi-Time Measurements in Hawking Radiation: Information at Higher-Order Correlations

Charis Anastopoulos, Ntina Savvidou

General Relativity and Quantum Cosmology · Submitted 1 Sep 2019 · 67 references

Pith's verdict: Clear conceptual point—HW theorem doesn't constrain multi-time correlations—but the Schwarzschild quantitative demonstration leans on an approximation the authors admit fails at the Hawking peak.

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arXiv ID1909.00438 (version 2)

Pith IDLS264OO4, Pith's permanent record of this paper and its review

CitationCharis Anastopoulos, Ntina Savvidou. "Multi-Time Measurements in Hawking Radiation: Information at Higher-Order Correlations." arXiv:1909.00438 (2019). https://pith.science/paper/LS264OO4

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Editor's note

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Here's the short version: this paper has a clear, correct conceptual point that is worth taking seriously, and a quantitative claim that is not yet backed up in the regime that matters.

The conceptual point is Sec. 2: the Hawking-Wald theorem constrains the reduced single-time density matrix at I+; it says nothing about multi-time correlations. That is right, and the paper states it carefully, including the caveat that the split into Hawking modes and horizon modes is Cauchy-surface dependent. The UdW/QTP probability framework is their own, heavily self-cited, but it is a legitimate way to define two-time joint detection probabilities, and the Hawking calculation is a new evaluation against the standard Unruh vacuum two-point function with no free parameters. The Unruh detector-pair result (Sec. 4) is a clean re-derivation of their earlier non-thermal coincidence effect, and that part is exact within the model. Credit where due: they explicitly label the pre-collapse information storage as a conjecture, and they flag the approximation problem.

The soft spot is exactly where the stress-test note lands. The angular and scattering-history dependence in Sec. 5.4 comes from a geometric-optics / stationary-phase treatment of the radial modes. The authors themselves say it is unreliable at ω ∼ κ, and that is precisely where the Hawking spectrum peaks. So the headline result—that Schwarzschild multi-time correlations carry non-trivial angular and scattering information—is demonstrated only in a regime that is illustrative rather than physical. In addition, Eqs. (66) and (70) contain formal squared-delta-function expressions that need a regulator; without that, they are not well-defined. This is not fatal to the overall message, but it means the specific Hawking claim needs either a controlled low-energy evaluation or an honest downgrade to 'suggestive'.

The citation pattern is fine; the self-citations are to their own framework, but the actual new computation is independent of those results. No free parameters, no invented entities.

Bottom line: the paper deserves a serious referee. It advances the discussion by pointing out a genuinely overlooked class of observables, and the flaw is a gap between a plausible conceptual argument and a quantitative demonstration, not a mistake in the core logic. I would send it to peer review and ask for a major revision that either supplies a better approximation at low energies or explicitly recasts the Hawking results as a toy-model demonstration.

Referee report

Significance

If the Schwarzschild result held as stated, the paper would open a genuinely new channel for black-hole information studies: multi-time correlations would evade the HW theorem and provide observables not encoded in the single-time reduced density matrix. The paper is transparent and self-contained in its detector derivation, uses no free parameters, and provides an exact analytic treatment of the accelerated-detector case. The open-systems argument of Sec. 2 is a useful clarification of why a Gibbsian reduced state does not constrain temporal correlations. However, the quantitative Schwarzschild demonstration rests on an approximation the authors themselves flag as unreliable in the physically most relevant regime, and the displayed expressions contain formal singular angular-delta products. The abstract's central claim therefore goes beyond what the present evidence establishes.

Major objections (3)

[Sec. 5.4, text near Eq. (67)] The quantitative demonstration that Schwarzschild multi-time correlations depend on angular variables and scattering history uses the geometric-optics approximation to the radial mode functions. The paper states that this approximation "is unreliable at low energies, including the most interesting regime of ω ∼ κ." Since the Hawking flux is concentrated at energies of order κ, the claimed information-bearing structure is demonstrated only in a regime away from the dominant Hawking quanta. Because the exact low-energy radial mode functions are not analytically known, the possibility remains that the exact low-energy modes wash out the angle and scattering dependence. The authors need to provide an error estimate, a numerical evaluation of Eq. (64) in the ω ∼ κ regime, or a clearly stated reformulation of the claim that does not assert the physical-regime result on the basis of geometric optics.
[Secs. 5.3–5.4, Eqs. (66) and (70)] The coincidence function is expressed through formal products of angular delta distributions, δ²(Θ)δ²(0) in Eq. (66) and [δ²(Θ)]², δ²(Θ)S_E(Θ), and S_E(Θ)² in Eq. (70). These expressions are not well-defined distributions on the two-sphere without a regularization prescription. The paper calls the δ²(0) term a formal divergence arising from the effectively two-dimensional calculation, but it then uses these expressions to make physical statements such as the lightcone bunching of Hawking photons and the explicit angular dependence of correlations. Without a finite angular-size detector model or another regulator, the quantitative predictions of Secs. 5.3 and 5.4 are not evaluable, and the claim of angular dependence may be an artifact of the singular limit. The authors should present a regulated version of the coincidence function and show that the qualitative conclusions survive the regulator removal.
[Appendix B, Eqs. (86)–(91)] The stationary-phase evaluation in Appendix B uses a first-order expansion of the scattering phase and then discards rapidly oscillatory terms on the grounds that they are suppressed by energy coarse-graining. The condition for this suppression is that the relevant length scale L satisfy L ΔE ≫ 1, but the paper does not state when this holds for the cases studied, e.g., for the detector separations Δr* entering Eq. (86) or the transmission/reflection delay differences in Eqs. (88) and (91). If the condition is not met, the interference terms cannot be dropped and the resulting coincidence function, and hence the scattering-history interpretation, would change. A precise statement of the parameter regime for each of the three cases is needed before the Appendix B results can be regarded as supporting the central claim.

Minor objections (4)

[Sec. 3.1, Eq. (9)] The displayed Glauber formula contains a duplicated factor: after the field negative-frequency parts there are two identical factors Ê^{(-)}_{i1}(X1), which appears to be a typo; the repeated index should presumably be i2.
[Sec. 5.4, paragraph after Eq. (70)] The sentence "The first term in Eq. (66) refers to a pair of outcoming Hawking quanta" should refer to Eq. (70), not Eq. (66), since the enumerated terms are those of the Case II formula.
[Appendix A, Eq. (78)] The expression for the coincidence function has an unbalanced parenthesis in the final line; the intended argument of the second Gaussian appears to be (Δτ + q)/√2, but as typeset the formula is ambiguous.
[Sec. 5.2, Eq. (64)] The notation F_{ω}(Θ,r,r′) appears with positive and negative frequencies after the substitution ω = −E ± ξ/2; since F is defined for real ω through the extension R_{−ω,𝓁} = R^{*}_{ω,𝓁}, a brief reminder of this convention near Eq. (64) would improve readability.

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The paper

The paper claims

The paper shows that multi-time correlations in Hawking radiation can carry information that the Hawking-Wald theorem does not constrain, and derives a concrete two-detector coincidence formula for Schwarzschild.

The reading

The paper argues that the standard conclusion that Hawking radiation carries no information rests on a narrow measurement notion. It derives joint probabilities for detections at two different times, using generalized Unruh-DeWitt detectors, and applies them to a scalar field in the Schwarzschild geometry in the Unruh vacuum, the vacuum state that models a black hole formed by collapse. The two-time coincidence function turns out to depend on the angular separation of the detectors and on the scattering history of Hawking quanta in the potential barrier. Such dependence is absent from the single-time thermal density matrix established by the Hawking-Wald theorem, so the theorem does not rule out information being stored in multi-time correlations. The authors conjecture that backreaction transfers pre-collapse information into these higher-order correlations.

What carries the argument

The load-bearing object is the two-time coincidence function $C(E_1,E_2;\Delta t)$ built from a pair of generalized Unruh-DeWitt detectors, pointlike quantum systems that exchange field quanta with the field and record the proper time of each click. For stationary detector paths and $E\sigma\gg1$, the dominant term reduces to Eq. (64): an integral over frequency of products $F_\omega(\Theta,r,r')F_{\omega'}(\Theta,r,r')^*$, where $F_\omega$ is a sum over angular-momentum modes weighted by Legendre polynomials. This object does the work: it is where angular separation and scattering history enter, and it replaces the single function of energy, the grey-body factor, that fully characterizes single-time measurements.

Extended reading notes

Core claim

The central discovery is that the informational content of Hawking radiation is not exhausted by single-time measurements. For two static detectors in the Schwarzschild geometry with the field in the Unruh vacuum, the paper derives a two-time coincidence function $Z(E,\Delta t)$ whose integrand contains the full angular and radial mode structure of the field. In the geometric-optics/stationary-phase limit, the coincidence splits into pieces that track outgoing Hawking quanta, quanta reflected off the potential barrier, and cross terms oscillating with the distance to the reflection point; these pieces depend on the angular separation $\Theta$ and on the transmission and reflection amplitudes of the barrier. The authors conclude that multi-time correlations are generically non-thermal and can store non-trivial information about the history of the quanta, and that the Hawking-Wald theorem, being a statement only about asymptotic single-time observables, does not forbid such storage.

Load-bearing premise

The load-bearing premise is that using classical ray paths to approximate the quantum waves does not wash out the angular and scattering dependence; the paper itself says this approximation is unreliable at low energies, including the regime $\omega\sim\kappa$ where the most relevant effects would appear.

Editorial extensions

If this is right

Hawking radiation can, in principle, store information in temporal correlations without violating the Hawking-Wald theorem, since that theorem governs only single-time reduced density matrices.
Multi-time coincidence measurements near the horizon can distinguish outgoing from reflected Hawking quanta, yielding information about the potential barrier that is invisible in single-detector counting rates.
The thermal description of a black hole is a coarse-graining at the level of the two-point function; higher-order correlation functions are not thermal and should be included in any information accounting for evaporation.
A mechanism for storing pre-collapse information exists entirely within semiclassical gravity, through backreaction transferring information to higher-order correlations, so the information-loss paradox need not wait on quantum gravity for a storage channel.
The same reasoning transfers to the Unruh effect: uniformly accelerated detectors show multi-time coincidence correlations that deviate from thermal ones by a distance-dependent factor, so quantum-field correlations along accelerated worldlines are not Gibbsian.

Reading between the lines

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

Editorial extension: a direct numerical evaluation of the coincidence formula with exact Schwarzschild mode functions near $\omega\sim\kappa$ would turn the geometric-optics demonstration into a quantitative prediction; the paper itself flags this as the regime where its approximation is unreliable.
Editorial extension: if the near-horizon reflected-quantum cross terms survive at low energy, two detectors placed on opposite sides of the barrier could effectively measure the scattering phase shifts of the potential, making multi-time Hawking coincidences a probe of the barrier shape.
Editorial extension: the same generalized detector formalism could be applied to analogue black-hole systems, where the equivalent of the outgoing/reflected pair terms might be observed in a laboratory.

Original abstract

It is believed that no information can be stored in Hawking radiation, because correlations between quanta of different field modes vanish. However, such correlations have been defined only with reference to a single moment of time. In this article, we develop a method for the evaluation of {\em multi-time} correlations. We find that these correlations are highly non-trivial: for a scalar field in the Schwarzschild black hole, multi-time correlations have an explicit dependence on angular variables and on the scattering history of Hawking quanta. This result leads us to the conjecture that some pre-collapse information can be stored in multi-time correlations after backreaction effects have been incorporated in the physical description.

The compiled record

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,228 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.43 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.79 · 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.

It is believed that no information can be stored in Hawking radiation, because correlations between quanta of different field modes vanish. However, such correlations have been defined only with reference to a single moment of time. In this article, we develop a method for the evaluation of {\em multi-time} correlations. We find that these correlations are highly non-trivial: for a scalar field in the Schwarzschild black hole, multi-time correlations have an explicit dependence on angular variables and on the scattering history of Hawking quanta. This result leads us to the conjecture that some pre-collapse information can be stored in multi-time correlations after backreaction effects have been incorporated in the physical description.

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	It is believed that no information can be stored in Hawking radiation, because correlations between quanta of different field modes vanish.	background 1.00	0.03	0.02	hedged possibility 0.48
s1	However, such correlations have been defined only with reference to a single moment of time.	gap 0.98	0.06	0.02	asserted 0.99
s2	In this article, we develop a method for the evaluation of {\em multi-time} correlations.	method 0.51	0.93	0.04	asserted 0.98
s3	We find that these correlations are highly non-trivial: for a scalar field in the Schwarzschild black hole, multi-time correlations have an explicit dependence on angular variables and on the scattering history of Hawking quanta.	result 1.00	0.95	0.30	asserted 1.00
s4	This result leads us to the conjecture that some pre-collapse information can be stored in multi-time correlations after backreaction effects have been incorporated in the physical description.	interpretation 1.00	0.23	0.78	speculative 0.99

What kind of paper the classifier reports this is: method or tool 0.62 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 1.** Left: Penrose diagram of a collapsing black hole. Right: Sequential events of

**Figure 2.** Penrose diagram for black hole collapse and total evaporation.

**Figure 3.** The function M, defined by Eq. (81) is plotted as a function of x for different values of b. B Stationary phase evaluation of correlations in Hawking radiation In this section, we evaluate the coincidence function for the Unruh vacuum in the regime where the asymptotic expressions (53, 54) for r∗ → ±∞ apply. Since the integral (64) is strongly dominated by values of ω around −E, we employ the usual stationary phase approximation for wave-packet propagation in one-dimensional scattering. That is, we expand ln −→Aω,` and Bω,` as a series around ω = −E, keeping only the zero-th order…

Figures reproduced from arXiv:1909.00438 by the authors.

References

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sinϵu sinhau−gσ(v/ √

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sinϵv sinhav ) , (77) In Eq. (77), we approximated sinh( aτ) sinh(aq) ∼ sgn(q) 1 4e−a(|τ|+|q|), since aτ >> aσ >>1 and a|q|>>aσ >> 1. By Eq. (37), the coincidence function involves two separate contributions. The ﬁrst comes from oscillations with frequency ϵ1 +ϵ2 around τ =q, and the second from oscillations with same frequency around τ =−q. We evaluate e...

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Discussion

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[1] [1]

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sinϵv sinhav ) , (77) In Eq. (77), we approximated sinh( aτ) sinh(aq) ∼ sgn(q) 1 4e−a(|τ|+|q|), since aτ >> aσ >>1 and a|q|>>aσ >> 1. By Eq. (37), the coincidence function involves two separate contributions. The ﬁrst comes from oscillations with frequency ϵ1 +ϵ2 around τ =q, and the second from oscillations with same frequency around τ =−q. We evaluate e...

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