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REVIEW 5 major objections 5 minor 21 references

Black hole evaporation and semiclassicality at large D

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Pith's one-line read At large spacetime dimension, black holes must have entropies above $(D/4\pi)^{D+3}\log D$ to evaporate semiclassically.

desk verdict A new and clearly stated timescale comparison that likely fixes a puzzle, but it rests on an unproven scrambling-time extrapolation and the abstract overstates the bound. read the letter →

arxiv 1908.08083 v1 pith:RWLWIIJ5 submitted 2019-08-21 hep-th gr-qc

Frederik Holdt-S{o}rensen , David A. McGady , Nico Wintergerst This is my paper · ORCID
classification hep-thgr-qc MSC 83C5783C4583E15 PACS 04.70.Dy04.60.-m04.50.-h
keywords blackholeevaporationlarge-DlimitsemiclassicalgravityscramblingtimeHawkingradiationBekenstein-Hawkingentropyinformationparadoxhigher-dimensional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

At large $D$, the usual test of semiclassicality—that the black hole be parametrically larger than the Planck length so that curvatures are small—is not sufficient. The paper shows that the decisive constraints come from the rate of change of the geometry and, most strongly, from the requirement that a black hole scrambles information faster than it evaporates. Because $D$-dimensional Hawking radiation is enormously powerful, black holes with entropies between $(D/4\pi)^D$ and $(D/4\pi)^{D+3}\log D$ satisfy the conventional curvature and quasi-staticity bounds yet evaporate faster than they can scramble. The paper therefore proposes the new necessary condition $t_{\rm scr}(D/4\pi)^{D+3}\log D$, equivalently $R_H/\ell_P\gtrsim D^{3/2}$, for semiclassical black holes. If correct, this sharpens where semiclassical gravity applies and explains apparent entropy-bound tensions at large $D$, while leaving $D\leq 26$ physics unchanged.

What carries the argument

The load-bearing objects are the $D$-dimensional Schwarzschild metric, its Bekenstein–Hawking entropy $S_{\rm BH}$, and two timescales. The Hawking temperature is $T_H=(D-3)/(4\pi R_H)$, which makes $R_H T_H\sim D$; the luminosity is $P_{\rm BH}\sim N_D(D/4\pi)^{D+2}/R_H^2$, whose growth with $D$ reflects the enormous phase space of emitted modes. From these follow the evaporation time $t_{\rm evap}$ and the scrambling time $t_{\rm scr}=(M_P/(2\pi T_H))\log S_{\rm BH}$. The paper's decisive identity is the ratio $t_{\rm evap}/t_{\rm scr}\sim(4\pi/D)^D N_D\,S_{\rm BH}/\log S_{\rm BH}$; demanding $t_{\rm scr}<t_{\rm evap}$ turns this ratio into the entropy bound $S_{\rm BH}>(D/4\pi)^{D+3}\log D$. The analysis is organized around families of large-$D$ black holes whose entropy scaling is tracked by a parameter $k$, which lets the paper translate each physical condition into a bound on $k$; the scrambling condition is the one that eliminates the intermediate window.

What would settle it

Compute the actual scrambling time of a large-$D$ Schwarzschild black hole from a microscopic or holographic model, for example from the decay of out-of-time-ordered correlators. If the result scales with $D$ more mildly than $(M_P/(2\pi T_H))\log S_{\rm BH}$—say as $S_{\rm BH}^{1/(D-2)}$ times a power of $D$ rather than times $\log S_{\rm BH}$—then there would be no large-$D$ window with $t_{\rm evap}<t_{\rm scr}$, and the bound $S_{\rm BH}>(D/4\pi)^{D+3}\log D$ would not be necessary.

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Extended reading notes

Core claim

The central claim is that in $D\gg1$ dimensions, small curvature is insufficient for a semiclassical description of evaporating black holes; the decisive requirement is that a black hole scramble information faster than it evaporates. The paper computes the evaporation time from the $D$-dimensional blackbody luminosity $P_{\rm BH}\sim N_D(D/4\pi)^{D+2}/R_H^2$ and takes the scrambling time to be the standard fast-scrambling expression $t_{\rm scr}=(M_P/(2\pi T_H))\log S_{\rm BH}$. At fixed entropy the ratio $t_{\rm evap}/t_{\rm scr}$ falls like $(4\pi/D)^D$, so for any fixed $S_{\rm BH}$ there is a dimension above which evaporation is faster than scrambling. The paper identifies the interval $(D/4\pi)^D\lesssim S_{\rm BH}\lesssim(D/4\pi)^{D+3}\log D$ as one in which the conventional bounds on curvature, softness of radiation, and quasi-staticity all hold, yet the black hole evaporates before it scrambles. Its conclusion is that semiclassical unitarity imposes $S_{\rm BH}>(D/4\pi)^{D+3}\log D$ (Eq. 22), corresponding to $R_H/\ell_P\gtrsim D^{3/2}$, and that this new large-$D$ constraint becomes operative only above $D\sim26$, leaving the familiar low-dimensional picture unchanged.

Load-bearing premise

The load-bearing premise is that the four-dimensional fast-scrambling formula $t_{\rm scr}=(M_P/(2\pi T_H))\log S_{\rm BH}$ continues to hold in every dimension, together with the judgement that a black hole which evaporates before it scrambles cannot be described semiclassically; if scrambling is genuinely faster at large $D$ than this formula predicts, or if $t_{\rm scr}<t_{\rm evap}$ is not required, the entropy bound collapses.

Editorial extensions

If this is right

  • For $D\gg1$, any black hole with $S_{\rm BH}\lesssim(D/4\pi)^{D+3}\log D$—even one with sub-Planckian horizon curvature—cannot be described by semiclassical evaporation, because it would radiate its information before scrambling it.
  • Semiclassical large-$D$ black holes have radii $R_H/\ell_P\gtrsim D^{3/2}$ and Hawking temperatures bounded by $T_H/M_P\lesssim1/(4\pi\sqrt{D})$, so even these enormous black holes radiate at temperatures well below the Planck scale.
  • The new bound removes an apparent large-$D$ puzzle about hyperentropic matter: configurations approaching Bekenstein-type entropy bounds sit outside the semiclassical regime, so no semiclassical tension arises.
  • In $D\leq26$, the scrambling condition does not invalidate any otherwise-semiclassical Schwarzschild black hole, so the standard large-radius semiclassical picture familiar from four and five dimensions remains intact.

Reading between the lines

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

  • If the criterion is accepted, it converts the large-$D$ entropy floor into an effective species bound: for fixed $S_{\rm BH}$, the number of gravitationally coupled modes in $D$ dimensions is constrained by $N_D\lesssim(4\pi/D)^D S_{\rm BH}/\log S_{\rm BH}$ up to order-one factors, a large-$D$ analogue of the large-$N$ species bounds.
  • In $D>26$, the paper's logic implies that quantum gravity must take over for black holes that by conventional geometric measures are large and weakly curved; the natural next step is to look for scrambling dynamics in the near-horizon stringy description of such black holes.
  • The same phase-space argument should apply to charged or rotating black holes and to other spacetime asymptotics: any setting in which the radiation phase space grows steeply with $D$ should exhibit a similar semiclassicality floor based on $t_{\rm scr}<t_{\rm evap}$, though the precise exponent will shift.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper studies D-dimensional Schwarzschild black holes at large D and compares their Hawking evaporation time with their scrambling time. The authors find that, at sufficiently large D, demanding that black holes scramble faster than they evaporate, t_scr < t_evap, imposes a lower bound on the entropy, S_BH > (D/4π)^{D+3} log D (Eq. 22), which is parametrically stronger than the quasistatic bound R_H/ℓ_P ≳ D and implies R_H/ℓ_P ≳ D^{3/2}. They conclude that small curvature is not sufficient for semiclassicality at large D, and they comment on implications for string theory and for hyperentropic matter.

Significance. If the central bound is correct, the paper identifies a new, concrete obstruction to semiclassical black hole evaporation at large D, with potential implications for large-D limits of quantum gravity and for previously proposed entropy bounds. The paper is careful to use manifestly dimensionless quantities and makes its main assumptions explicit. It also connects to the existing literature on fast scrambling and on large-N species bounds, which strengthens its conceptual interest. However, the result is conditional on two substantial assumptions: the validity of the Hayden-Preskill/Sekino-Susskind scrambling time in arbitrary dimension, and the physical necessity of t_scr < t_evap for semiclassicality. The paper also contains a major inconsistency between the abstract's D^{D+3} bound and the derived (D/4π)^{D+3} bound.

major comments (5)
  1. [Abstract and Section II.B, Eq. (22)] The abstract and the introduction state that asymptotically only black holes with entropies S ≥ D^{D+3} log D are semiclassical, but the derivation in Section II.B, Eq. (22), gives S_BH > (D/4π)^{D+3} log D. These two expressions differ by a factor of (4π)^{D+3}, which is exponentially large. The factor (D/4π)^D arises naturally from the luminosity formula (10), so the derived form appears to be the correct one; the abstract and introduction should be corrected to match Eq. (22).
  2. [Section II.B, Eqs. (21)-(22)] The threshold quoted in Eq. (22) does not by itself ensure t_scr < t_evap. Substituting S_BH = (D/4π)^{D+3} log D into the upper bound (21) gives tevap/tscr ≈ e^2/(64π^3) ≈ 3.7 × 10^{-3} < 1, so scrambling is still slower than evaporation at this entropy. The correct parametric threshold is larger by an order-one factor, roughly (64π^3/e^2) (D/4π)^{D+3} log D. This does not change the leading large-D scaling, but the numerical bound as stated should be corrected.
  3. [Section II.A, Eq. (17)] The condition t_scr < t_evap is presented as a requirement that 'forces us' to impose (Eq. 17), but it is not derived from the semiclassical approximation. Semiclassical Hawking radiation is not manifestly unitary, and importing unitarity as a criterion of semiclassicality is an additional physical assumption. The paper should clearly state Eq. (17) as an assumption or postulate, and discuss possible justification, rather than presenting it as an inevitable consequence.
  4. [Section IV and Eq. (12)] The central result depends on the validity of the scrambling time formula (12) in arbitrary D. The authors explicitly acknowledge in Section IV that this is an assumption. This is load-bearing: a D-dependent correction to the prefactor, t_scr ~ f(D) (M_P/(2π T_H)) log S_BH, would rescale the derived bound (22) to (D/4π)^{D+3}/f(D) times logarithms. If f(D) grows as a power of D, the claimed window of semiclassical black holes, and even the parametric scaling, could change. The expectation that scrambling is non-local and therefore D-insensitive is a conjecture, not a derivation; the result should be presented as conditional on this assumption, or the assumption should receive additional support.
  5. [Section I.A, Eq. (13)] The family scaling in Eq. (13) appears internally inconsistent. With S_BH = Ŝ0 D^{Dk/2}, the relation S_BH ∼ (R_H/ℓ_P)^{D-2} gives R_H/ℓ_P ∼ D^{k/2}, not D^{(k+1)/2} as stated. If the intended definition is S_BH = Ŝ0 D^{D(k+1)/2}, then the bounds derived in Eqs. (18) and (19) change; in particular, the quasistatic bound may be k ≥ 1 rather than k ≥ 2. This inconsistency affects the comparison with the 'conventional' bounds and should be fixed.
minor comments (5)
  1. [Section II.A, Eq. (14)] The expression for the curvature invariant is given as (√D/S_BH)^{4/D}, which for large D behaves as S_BH^{-4/D}, omitting the D^4 prefactor that is present in the exact expression (D-1)(D-2)^2(D-3)/(R_H/ℓ_P)^4. This makes the curvature bound appear much weaker than it is; the correct large-D scaling is D^4 S_BH^{-4/(D-2)}.
  2. [Section II.B, Eq. (21)] The constant factor e^2/(4π)^2 in Eq. (21) is not derived in the main text. It comes from the constants in Eqs. (10) and (11) and from the lower bound on N_D; a short derivation or a cross-reference would improve readability.
  3. [Section II.B, after Eq. (22)] The text switches from t_scr < t_evap in Eq. (17) to t_scr ≪ t_evap after Eq. (22). The distinction matters for the numerical threshold; the paper should use one condition consistently.
  4. [Section III, reference list] Reference [5] is spelled 'Battacharya' but should be 'Bhattacharyya'.
  5. [Throughout] The abstract and Eq. (3) use D^{D+3} log D while Eq. (22) uses (D/4π)^{D+3} log D. Please ensure that all instances of the final bound are consistent after the correction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the large-D semiclassicality bound follows from standard formulas and two explicitly stated assumptions, with no fitted inputs or self-citation chain.

full rationale

The derivation is self-contained in the relevant sense. The paper starts from standard D-dimensional Schwarzschild relations (Eqs. 4-7), the D-dimensional blackbody luminosity (Eqs. 8-10), and published expressions for the evaporation time (Eq. 11) and the Hayden-Preskill/Sekino-Susskind scrambling time (Eq. 12). The semiclassicality conditions (14)-(16) are stated independently (sub-Planckian curvature, softness of radiation, quasistatic geometry), and the scrambling condition t_scr < t_evap is introduced as an additional explicit criterion (Eq. 17: 'forces us to impose'). Eq. (22) is obtained by algebraically solving t_scr < t_evap using Eqs. (11)-(12); no parameter is fitted to data, and no quantity is defined in terms of the final bound. The paper also flags its one genuinely fragile input, Eq. (12), as an assumption: 'One of the main assumptions underlying this work is the validity of expression (12) for the scrambling time in any D.' That is an acknowledged physical assumption, not a hidden circular step. There are no load-bearing self-citations by the present authors, and no uniqueness theorem from prior work is invoked to forbid alternatives. The abstract's D^{D+3} log D versus Eq. (22)'s (D/4π)^{D+3} log D is a numerical or simplification discrepancy, not a circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the only parameter is the order-one family normalization Ŝ0, which drops out of the leading scaling. The central bound depends on four main assumptions: the extrapolated scrambling time formula, the new t_scr < t_evap criterion, the blackbody luminosity form, and the minimal mode count. None of these are invented entities; they are standard or explicitly flagged assumptions.

free parameters (1)
  • Family normalization Ŝ0
    Order-one constant defining the large-D family S_BH(k,D) = Ŝ0 D^(Dk/2) (Eq. 13). It is not fitted; the central bound (22) is independent of its value at leading order.
assumptions (5)
  • domain assumption The scrambling time in arbitrary dimension is t_scr = (M_P/(2π T_H)) log S_BH (Eq. 12).
    Extrapolates the Hayden-Preskill/Sekino-Susskind result, established for low dimensions, to all D. The authors flag this as the main assumption in Sec. IV.
  • ad hoc to paper A semiclassical, unitary black hole must scramble information faster than it evaporates, t_scr < t_evap (Eq. 17).
    Introduced in Sec. II.A as a new necessary condition. It is motivated by unitary evolution but is not derived from a fundamental principle.
  • domain assumption The black hole radiates as a blackbody with greybody factor γ_D(R_H) approaching unity at large D (Eq. 10).
    Standard in the large-D limit; radiation is dominated by wavelengths small compared to R_H, suppressing greybody corrections. Cited to [12] and [6].
  • domain assumption The number of massless modes N_D is at least the number of graviton polarizations, D(D-3)/2, used for the upper bound in Eq. (21).
    In pure gravity N_D = D(D-3)/2. Using this minimal value yields the weakest bound on S_BH.
  • domain assumption The k-indexed family S_BH = Ŝ0 D^(Dk/2) (Eq. 13) covers the semiclassical black holes of interest.
    These are the large-D limits studied by Emparan et al. [4]; the bounds are evaluated on this family.

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Cite this review

Pith. "Pith review of Black hole evaporation and semiclassicality at large D." pith.science (2026). https://pith.science/paper/RWLWIIJ5

@misc{pith2026190808083,
  author       = {Pith},
  title        = {Pith review of: Black hole evaporation and semiclassicality at large D},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RWLWIIJ5}},
  note         = {Machine review of arXiv:1908.08083}
}
abstract

Black holes of sufficiently large initial radius are expected to be well described by a semiclassical analysis at least until half of their initial mass has evaporated away. For a small number of spacetime dimensions, this holds as long as the black hole is parametrically larger than the Planck length. In that case, curvatures are small and backreaction onto geometry is expected to be well described by a time-dependent classical metric. We point out that at large $D$, small curvature is insufficient to guarantee a valid semiclassical description of black holes. Instead, the strongest bounds come from demanding that the rate of change of the geometry is small and that black holes scramble information faster than they evaporate. This is a consequence of the enormous power of Hawking radiation in $D$-dimensions due to the large available phase space and the resulting minuscule evaporation times. Asymptotically, only black holes with entropies $S \geq D^{D+3} \log D$ are semiclassical. We comment on implications for realistic quantum gravity models in $D \leq 26$ as well as relations to bounds on theories with a large number of gravitationally interacting light species.

Discussion (0). Continue with ORCID to comment.

Reference graph

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