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High Energy Physics - Theory · arXiv:hep-th/0602017 v3 · submitted 2 Feb 2006 · 31 references

Rayleigh-Plateau and Gregory-Laflamme instabilities of black strings

Vitor Cardoso, Oscar J. C. Dias

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“Many and very general arguments indicate that the event horizon behaves as a stretched membrane. We explore this analogy by associating the Gregory-Laflamme instability of black strings with a classical membrane instability known as Rayleigh-Plateau instability.”

more of the abstract

We show that the key features of the black string instability can be reproduced using this viewpoint. In particular, we get good agreement for the threshold mode in all dimensions and exact agreement for large spacetime dimensionality. The instability timescale is also well described within this model, as well as the dimensionality dependence. We conjecture that general non-axisymmetric perturbations are stable. We further argue that the instability of ultra-spinning black holes follows from this model.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

It can be wrong. Challenge this review or reply to it.

Holds up, with conditions. The parameter-free threshold match R0 kc = sqrt(D-2) against Kol-Sorkin is real and worth citing; the timescale claim is softer than the abstract admits. This line is wrong

2 major objections · 7 minor

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The paper says, at Figure 2 and the paragraph beginning 'We compute the Rayleigh-Plateau instability timescale'

The review did not quote this passage.

Pith says: Objection 1 of 2 · major

The D-dimensional dispersion relation that underlies Figure 2 is never written down. Eq. (8) is the D = 3 result only, and the text states that 'we have generalized Rayleigh's procedure for higher dimensions' without giving the generalization or citing where it appears.

more of objection 1

As a result the reader cannot check the peak growth rates, the claimed dimensionality dependence, or the proposed redshift explanation for the roughly order-of-magnitude offset from the GL values. Since the abstract lists the timescale as one of the reproduced key features (abstract:s4), this is load-bearing. Either give the D-dimensional dispersion relation explicitly or cite a source where it is derived, and state clearly that the redshift reconciliation is a conjecture rather than a computed result.

The paper says, at Abstract, sentences s3 and s4

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

The abstract overstates the threshold agreement. 'Good agreement for the threshold mode in all dimensions' (abstract:s3) is not borne out at low D: Table I gives 1.41 versus 0.876 at D = 4 (about 61% high), 1.73 versus 1.27 at D = 5 (about 36% high), and 2.00 versus 1.58 at D = 6 (about 27% high).

more of objection 2

The body table is honest and the text correctly emphasizes exact asymptotic agreement, so the fix is to recalibrate the abstract to say that agreement improves with D and becomes exact asymptotically. The same applies to 'well described' in abstract:s4, which is stronger than the body's own 'qualitative behavior matches surprisingly well' and sits uneasily with the acknowledged order-of-magnitude discrepancy.

7 minor objections

The paper says, at Eqs. (1), (6), and (8)

Pith says: minor objection 1 of 7

The symbol T denotes the black-hole temperature in Eq. (1) and the surface tension in Eqs. (6)-(8). The identification T = T_H/4 is made in passing; it would help to state it explicitly at the point of use and to consider a distinct symbol for the surface tension to avoid confusion.

The paper says, at Eq. (8)

Pith says: minor objection 2 of 7

The argument of the Bessel functions is ikR0, which is imaginary. Rewriting the expression in terms of modified Bessel functions I0 and I1 would make the reality and sign of Omega-squared transparent and would let the reader see immediately where the instability sets in.

show 5 more

The paper says, at Table I

Pith says: minor objection 3 of 7

The table header says 'D spatial dimensions', while the GL literature and the cited Kol-Sorkin values are usually quoted in spacetime dimension d = D + 1. A short note in the caption fixing the convention would let readers match the entries to the source without ambiguity.

The paper says, at Paragraph following Eq. (2)

Pith says: minor objection 4 of 7

The sentence 'the threshold wavenumber increases with dimension number D and so does the maximum growth timescale' appears to mean the maximum growth rate; the timescale itself decreases with D, as the later discussion of the growth rate confirms. Please correct the wording.

The paper says, at Abstract and paragraph on non-axisymmetric modes

Pith says: minor objection 5 of 7

The abstract says 'We conjecture that general non-axisymmetric perturbations are stable', but the body notes that Kudoh has explicitly verified this for the GL instability. Please clarify that the conjecture concerns the fluid model, while the GL statement is supported by the cited work.

The paper says, at Ref. [19] and the redshift discussion

Pith says: minor objection 6 of 7

The redshift explanation for the order-of-magnitude discrepancy is attributed to a private communication. Since this is the only offered reconciliation of a quantitative mismatch, it should be labeled as a private communication in the text and framed explicitly as a speculation.

The paper says, at Endpoint discussion

Pith says: minor objection 7 of 7

The fluid model gives a critical dimension between D = 10 and D = 11, while Sorkin's gravitational result is between d = 13 and d = 14. The mismatch is noted but not addressed; a sentence explaining why the two critical dimensions need not coincide would strengthen the discussion.

What would settle it

Derive and write out the D-dimensional Rayleigh-Plateau dispersion relation (the generalization of Eq. 8 that the paper says it computed for Figure 2), then evaluate the peak growth rate with the stated T = T_H/4 and rho = energy density. Check whether a horizon redshift factor g_tt^{1/2} (or a first-order gravitational correction) reduces the ~10x discrepancy to O(1). If it does not, the abstract's claim that the timescale is 'well described' (abstract:s4) is unsupported and should be removed or labeled qualitative.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up, with conditions confidence high
  2. Skeptic agrees 4 min 26 s · 8,614 tokens read, 14,507 written
  3. Editor wrote the verdict above 36 s · 9,492 tokens read, 1,538 written
  4. Referee major revision: 2 major objections, 7 minor 2 min 55 s · 8,829 tokens read, 8,907 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

Short version: the threshold result is real and worth your attention; the timescale claim is softer than the abstract lets on. What's new here is small but genuine. The D-dimensional Rayleigh-Plateau threshold R0 kc = sqrt(D-2) falls out of a surface-area minimization (their Eqs. 5-7) with no black hole input at all — the surface tension cancels at threshold. Matching that against Kol and Sorkin's independently computed Gregory-Laflamme numbers is a real parameter-free comparison, not a fit. Table I is honest: 1.41 vs 0.876 at D=4, converging to 6.78 vs 6.72 at D=49 and 9.80 vs 9.75 at D=99. The large-D agreement is exact and is the strongest thing in the paper. The observation that the fluid analogue also gives stability for non-axisymmetric modes is a nice consistency check, though it is Hovdebo-Myers and Kudoh's result, cited rather than derived here. Where it's soft: the abstract's 'good agreement for the threshold mode in all dimensions' (abstract:s3) doesn't match its own table. At D=4 the fluid number is 61% high, at D=5 it's 36% high. The body is candid about this; the abstract should say agreement improves with dimension. The timescale claim (abstract:s4) is weaker. They assign the horizon a surface tension TH/4 and a density equal to the energy density by dimensional analogy, then quote a growth rate. Eq. 8 is the D=3 dispersion relation; the higher-dimensional one behind Figure 2 is never written down, so you can't check the peak or the claimed redshift fix. They do acknowledge the peak rate is about ten times the gravitational value and attribute it to neglected gravity outside the cylinder — a plausible story, but asserted, not computed. Read Figure 2 as qualitative. The ultra-spinning black hole section is explicitly a conjecture with no calculation; take it as a pointer. Who should read it: anyone working on black string or brane stability or the membrane paradigm. It's a four-page letter, cheap to read, and the threshold formula is a useful thing to have in your head. Recommendation: send it out. The load-bearing claim holds up; ask for the D-dimensional dispersion relation and a softened abstract.

The strongest objection

The threshold result (Eqs. 5-7, R0 kc = sqrt(D-2)) is a correct surface-area minimization and is robust to the surface-tension identification, since T cancels at threshold. The soft spot is the timescale claim (abstract:s4). The paper writes the D=3 dispersion relation (Eq. 8) and states it generalized Rayleigh's procedure to higher D (Figure 2), but the D-dimensional dispersion relation is never written down, so the reader cannot check the peak growth rate or the claimed redshift fix. The effective surface tension T = T_H/4 and density = energy density are assigned by dimensional analogy, not derived. The authors acknowledge the ~10x discrepancy and attribute it to neglected gravity/redshift, but this is asserted, not computed. Separately, the abstract's 'good agreement in all dimensions' (abstract:s3) overstates the low-D entries: at D=4 the fluid threshold is 1.41 vs. GL 0.876 (61% high), and the body table is honest but the abstract is not. The threshold claim survives; the timescale and 'all dimensions' language do not.

The threshold claim is the load-bearing one and it holds up. I re-derived the surface-area minimization of Eqs. (5)-(7) for a hyper-cylinder with cross-section S^{D-2}: the second-order area coefficient is proportional to R0^2 k^2 - (D-2), giving R0 kc = sqrt(D-2) exactly. The surface tension T cancels at threshold, so the threshold is independent of the T = T_H/4 identification. The comparison to Kol-Sorkin's GL values is apples-to-apples (D spatial = spacetime - 1; D=4 corresponds to 5D GL = 0.876). The large-D agreement is asymptotic (both approach sqrt(D)), which the paper states correctly. So the central threshold claim is sound. The soft spot is the timescale (abstract:s4). The paper writes Eq. (8) for D=3 and says it generalized to higher D (Figure 2), but the D-dimensional dispersion relation is never displayed. The T and rho assignments are dimensional analogies, not derivations. The ~10x peak-rate discrepancy is acknowledged and attributed to neglected gravity/redshift, but the fix is asserted, not computed. This is a real gap in a claim the abstract makes. The reader's weakest_assumption identifies exactly this, so I agree. A secondary observation: the large-D 'exact agreement' is the regime where the membrane paradigm is expected to hold by construction (locally flat horizon, long-wavelength perturbation), so it is a consistency check rather than an independent test; the small-D regime, where the analogy is non-trivial, shows 20-60% disagreement. This supports the reader's call to soften 'good agreement in all dimensions' (abstract:s3) to 'agreement improving with dimension.' The non-axisymmetric stability statement is a conjecture/citation (Hovdebo-Myers, Kudoh), and the ultra-spinning argument is explicitly a conjecture with no calculation; both are appropriately labeled and not load-bearing. The reader's CONDITIONAL verdict is well-calibrated; I would not change it.

The referee's report

If the analogy holds, it supplies a simple and intuitive membrane picture of a robust gravitational instability, and it connects two literatures that are usually kept separate. The threshold result is the durable contribution: R0 kc = sqrt(D-2) follows from surface-area minimization alone, the surface tension cancels at threshold, and the comparison against the independently computed GL values is therefore a genuine parameter-free test rather than a fit. The agreement is asymptotic and becomes excellent at large D. The timescale comparison is weaker: it depends on an assumed identification of surface tension and density with black-hole quantities, and the body itself describes the match as qualitative. The paper is appropriately hedged in the body on the endpoint and ultra-spinning discussions, though the abstract is not similarly calibrated.

Pith's reading of the paper (a machine's summary, not the authors')

A black string is a black hole whose horizon is stretched into a cylinder, and it is known to be unstable: ripples along the long axis grow for wavelengths above a critical value. This paper argues that the instability is not a peculiarly gravitational effect at all, but the same physics as a thin column of water pinching into drops, the Rayleigh-Plateau instability. The authors minimise the surface area of a perturbed fluid hyper-cylinder and find a threshold wavelength $R_0 k_c = 2.5ex$

What carries the argument. The carrying object is the surface-area (potential-energy) functional of a perturbed cylinder. Writing the disturbed radius as $r(z) = R_0 + 2.5ex$

Core claim. The claim is that the Gregory-Laflamme instability of a black string and the Rayleigh-Plateau instability of a fluid cylinder are the same instability seen in two languages. Minimising the surface area of a perturbed hyper-cylinder with $D-1$ spatial directions gives a critical wavenumber $R_0 k_c = 2.5ex$

The premise it rests on. To predict how fast the instability grows, the paper treats the horizon as a fluid whose surface tension equals the black hole temperature and whose density equals its energy density, while ignoring the gravity outside the cylinder; the threshold wavelength does not need this identification, but the growth-rate claim does.

If the paper is right (Pith's inferences, not the authors')

The critical wavelength of a black string's instability can be read off from a surface-area calculation alone, with no gravitational perturbation theory, giving $R_0 k_c = 2.5ex$

Non-axisymmetric perturbations of a black string should be stable, since the fluid potential energy for angular mode $m$ picks up a term $+m^2$ that never lowers the energy.

Rotating black holes in more than six dimensions, which have no Kerr-like upper bound on spin, should be unstable before their surface flattens, by the same reasoning that destabilises a sufficiently rapidly spinning liquid drop.

The endpoint of the instability should depend on dimension: in the fluid model a sphere stops being the lower-area endpoint between $D=10$ and $D=11$, mirroring the gravitational finding that the black string stops being entropically unstable to a spherical black hole above a critical dimension.

If the correspondence is more than an analogy, the effective surface tension and density of a horizon should be derivable from horizon thermodynamics alone, which would turn the membrane picture into a calculational tool for dynamical processes rather than a bookkeeping device.

The fluid model predicts a family of non-uniform stationary cylinder shapes analogous to the non-uniform black string solutions; looking for that branch in the surface-area functional and testing its stability would be a concrete, gravity-free check of the correspondence.

The roughly factor-of-ten overshoot in the predicted peak growth rate is the natural place to look for the first quantitative failure: a careful gravitational calculation of the growth-rate spectrum at intermediate dimensions would show whether the mismatch is a constant normalisation or a genuine dimensional disagreement.

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The paper

Abstract

Many and very general arguments indicate that the event horizon behaves as a stretched membrane. We explore this analogy by associating the Gregory-Laflamme instability of black strings with a classical membrane instability known as Rayleigh-Plateau instability. We show that the key features of the black string instability can be reproduced using this viewpoint. In particular, we get good agreement for the threshold mode in all dimensions and exact agreement for large spacetime dimensionality. The instability timescale is also well described within this model, as well as the dimensionality dependence. We conjecture that general non-axisymmetric perturbations are stable. We further argue that the instability of ultra-spinning black holes follows from this model.

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. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.99

“We show that the key features of the black string instability can be reproduced using this viewpoint.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.63 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v3). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
Many and very general arguments indicate that the event horizon behaves as a stretched membrane. background 1.00 0.04 0.03 hedged possibility 0.90
We explore this analogy by associating the Gregory-Laflamme instability of black strings with a classical membrane instability known as Rayleigh-Plateau instability. contribution 0.80 0.85 0.15 asserted 0.82
We show that the key features of the black string instability can be reproduced using this viewpoint. result 0.70 0.94 0.57 asserted 1.00
In particular, we get good agreement for the threshold mode in all dimensions and exact agreement for large spacetime dimensionality. result 1.00 0.94 0.25 asserted 0.99
The instability timescale is also well described within this model, as well as the dimensionality dependence. result 1.00 0.88 0.10 asserted 0.93
We conjecture that general non-axisymmetric perturbations are stable. interpretation 0.47 0.44 0.14 speculative 1.00
We further argue that the instability of ultra-spinning black holes follows from this model. interpretation 0.91 0.62 0.11 speculative 0.42

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from argument 0.53

What was examined, and what was not

Sources read

  • arXiv listing abstract v3

Not examined

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  • figures and tables
  • references and citing sentences
  • relations to other papers

Levels run

  • P abstract

Not run

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

How the labels were made

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,405 input tokens, $0.0002 reported.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

Figures (2)
Figure 1 of the paper
Figure 1. FIG. 1: Black strings and fluid cylinders are unstable to per
Figure 2 of the paper
Figure 2. FIG. 2: Rayleigh-Plateau instability of a hyper-cylinder i

Figures are the authors', reproduced from the paper.

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Cited by 20 Pith papers

Reviewed papers on Pith that cite this one. The full citation record

  1. Rings, Ripples, and Rotation: Connecting Black Holes to Black Rings hep-th · Feb 2014
  2. New Horizons for Black Holes and Branes hep-th · Dec 2009
  3. Plasmarings as dual black rings hep-th · May 2007
  4. Hydrodynamic properties in soliton field theory hep-th · Oct 2025
  5. Carrollian hydrodynamics and symplectic structure on stretched horizons gr-qc · Nov 2022
  6. Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass gr-qc · Apr 2013
  7. Black Brane Viscosity and the Gregory-Laflamme Instability hep-th · Mar 2010
  8. A new approach to static numerical relativity, and its application to Kaluza-Klein black holes gr-qc · May 2009
  9. Dyonic AdS black holes from magnetohydrodynamics hep-th · Dec 2008
  10. Black Holes as Lumps of Fluid hep-th · Nov 2008
  11. Numerical Methods for Finding Stationary Gravitational Solutions hep-th · Oct 2015
  12. Ultraspinning instability of rotating black holes hep-th · Jun 2010
  13. High and Low Dimensions in The Black Hole Negative Mode hep-th · Jun 2007
  14. The Large D Limit of Einstein's Equations hep-th · Mar 2020
  15. AdS/CFT Duality User Guide hep-th · Sep 2014
  16. Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence hep-th · May 2009
  17. Instabilities of Black Strings and Branes hep-th · Jan 2007
  18. Exploring New Physics Frontiers Through Numerical Relativity gr-qc · Aug 2014
  19. The fluid/gravity correspondence hep-th · Jul 2011
  20. Black Holes in Higher-Dimensional Gravity hep-th · Feb 2008
How the literature uses it

32 arXiv papers cite this in 54 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 · 40 sentences without a confident label

Compares with it 5 papers
  • One sees from Eq. ([ref]) that the wavenumber of the onset of instability [$k_c>0$, $ω(k_c) = 0$] is given by $k_c = √(n)/r_0$, irrespective of the viscosity, which generalizes the result of the inviscid case [ref] and is similar to the dimensional dependence of Gregory-Laflamme instability [ref] .

    One-Dimensional Approximation of Viscous Flows · Instability of viscous flows · 0.95

  • See Refs. [ref] for more on the similarities between the and instabilities.

    Black hole-black string phase transitions from hydrodynamics · The Equivalent Plateau Problem and Instability · 0.86

  • This allows us to compute thermodynamic quantities of the fluid that are identified with those for the black hole, something that was impossible in [ref] .

    Black Holes as Lumps of Fluid · Equivalent variational principles for spinning lumps · 0.94

Cites it as support 1 paper
  • This discussion at least strongly suggests that the end point of the Gregory-Laflamme instability consists of localized black holes [ref] .

    The fluid/gravity correspondence · Theories with a deconfinement transition · 0.99

Cites it as background 7 papers

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