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On non-uniform black branes

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Steven S. Gubser

High Energy Physics - Theory · Submitted 20 Oct 2001 · 11 references

Pith's verdict: The paper that actually built non-uniform black string solutions and pinned the leading coefficients η1 ≈ +1.45 and σ2 ≈ −0.59; the construction is solid, the jump from those signs to 'first order' is the soft part, and Gubser says so himself.

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  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDhep-th/0110193 (version 2)
Pith IDSAK4FGCV, Pith's permanent record of this paper and its review
CitationSteven S. Gubser. "On non-uniform black branes." arXiv:hep-th/0110193 (2001). https://pith.science/paper/SAK4FGCV
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The paper

The paper claims

The paper argues that an uncharged five-dimensional black string undergoes a first-order transition from a uniform horizon to a non-uniform one, based on the signs of the leading perturbative corrections to the critical compactification尺度.

The reading

The paper asks what happens to a black string—a black-hole horizon wrapped around a compact extra dimension—once the uniform string becomes unstable to bunching up along that dimension. The author constructs static non-uniform solutions as a power series in the horizon's fractional lumpiness $lambda$ around the critical compactification scale $eta_c$, solving the five-dimensional vacuum Einstein equations order by order with numerical shooting. The central result is a pair of signs: the critical scale shifts upward at order $lambda^2$ ($eta_1 approx 1.45$), and the non-uniform string's entropy lies below that of a uniform string of the same mass at order $lambda^4$ ($sigma_2 approx -0.59$). A sympathetic reader should care because these signs say the new branch does not immediately provide a smooth, entropically favored path away from the uniform solution; instead the transition is likely to be first order, with a finite jump in shape and an entropy gap. The paper presents this as perturbative evidence, not a proof at finite amplitude.

What carries the argument

The machinery is a perturbative expansion around the critical uniform black string. The metric functions are expanded as $A=sum_{n,p}lambda^{n+2p}A_{n,p}(y)cos(nKx)$ and similarly for $B$ and $C$, with $K=sum_q lambda^{2q}k_q$; at each order the Einstein equations reduce to linear ordinary differential equations in the radial variable, solved by shooting with regularity at the horizon and exponential falloff at infinity. The load-bearing outputs are the shift in the dimensionless compactification parameter $eta=R_{rm Schwarzschild}k_{min}$, encoded in $eta_1$, and the entropy difference at fixed mass, encoded in $sigma_2$; the first law is used to show the order-$lambda^2$ entropy difference vanishes, leaving the sign of $sigma_2$ as the diagnostic.

Extended reading notes

Core claim

The paper's central claim is that the continuous branch of non-uniform five-dimensional uncharged black-string solutions that meets the uniform string at the black-string instability threshold does not describe a continuous, second-order phase transition. Working to third order in the lumpiness amplitude, the author finds $eta_1 approx 1.45 > 0$ and, using the first law plus the perturbative expansion, $sigma_2 approx -0.59 < 0$: near the threshold the non-uniform solutions have a slightly higher mass density at fixed compactification scale and a slightly lower entropy than uniform strings of the same mass. The simplest completion of the moduli-space curve consistent with these signs bends down to finite $lambda$ and only there rises above the uniform entropy, so a uniform string crossing the threshold must evolve non-adiabatically to a finitely non-uniform endpoint. That is what the paper means by a first-order transition.

Load-bearing premise

The inference from the signs of the leading small-$lambda$ coefficients $eta_1$ and $sigma_2$ to a first-order transition at finite $lambda$ assumes that the non-uniform branch continues smoothly, bends downward in the $eta$-$lambda$ plane, and does not encounter a new branch or non-analytic behavior before reaching the entropically favored finite-amplitude solution; the paper itself notes that low-order perturbative results could mislead at finite $lambda$.

Editorial extensions

If this is right

  • For a five-dimensional uncharged black string, the transition from uniform to non-uniform horizons is discontinuous: as the mass density falls through the critical value, the horizon should jump to a finite lumpiness rather than grow lumps smoothly.
  • The non-uniform branch near threshold is entropically disfavored relative to a uniform string of the same mass, so the system cannot reach it by a slow adiabatic deformation; the jump involves a finite amount of non-adiabatic evolution and classical gravitational radiation.
  • The endpoint of the black-string instability should be a static non-uniform string at finite amplitude, consistent with the horizon's inability to pinch off, not an array of separate black holes.
  • The first-order behavior is tied to the signs of the leading corrections and need not be generic: higher-dimensional or charged branes could have continuous transitions if those signs reverse.
  • The result gives a concrete target for numerical relativity: locate the finite-$lambda$ turning point where the non-uniform branch's entropy overtakes the uniform string's.

Reading between the lines

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

  • The same sign test could be run for charged branes and for $p>1$: if the leading shift in the critical parameter is positive and the leading entropy difference negative, the transition should be first order; if either sign reverses, a continuous branch becomes plausible.
  • The entropy gap at the jump implies a calculable latent heat; scaling the compactification torus large while keeping curvature small should make the gravitational-wave burst from the non-adiabatic transition a controlled classical signal.
  • The crystallization analogy suggests a broader phase diagram: for $p>1$, phase locking among several unstable wavevectors could produce a lattice of maxima and first-order transitions between different numbers of maxima, even if the one-dimensional string transition were continuous.
  • A numerical search for the finite-$lambda$ crossing where non-uniform entropy equals uniform entropy would turn the schematic curve in the paper into a quantitative phase boundary, and would test whether the branch really bends down rather than terminating.
Original abstract

Certain black branes are unstable toward fluctuations that lead to non-uniform mass distributions. We study static, non-uniform solutions that differ only perturbatively from uniform ones. For uncharged black strings in five dimensions, we find evidence of a first order transition from uniform to non-uniform solutions.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,846 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.21 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Certain black branes are unstable toward fluctuations that lead to non-uniform mass distributions. We study static, non-uniform solutions that differ only perturbatively from uniform ones. For uncharged black strings in five dimensions, we find evidence of a first order transition from uniform to non-uniform solutions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Certain black branes are unstable toward fluctuations that lead to non-uniform mass distributions. background 1.00 0.10 0.10 asserted 0.95
s1 We study static, non-uniform solutions that differ only perturbatively from uniform ones. contribution 0.98 0.90 0.06 asserted 0.97
s2 For uncharged black strings in five dimensions, we find evidence of a first order transition from uniform to non-uniform solutions. result 1.00 0.96 0.79 asserted 0.67

What kind of paper the classifier reports this is: theoretical result 0.83 · evidence from simulation or computation 0.65.

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
not run
  • 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. (a) Possible curves, drawn schematically in the

Figures reproduced from arXiv:hep-th/0110193 by the authors.

References
How later papers used it

55 later papers cite it

Evidence: substantial. 55 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 106 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 6 of 9 papers with this label
Cites it as support 1 paper

  • Said that, given the higher-order perturbative analysis of this paper, it does not seem possible to have other branches than the known ones coming out of the Gregory-Laflamme point of the uniform black string.

    Multi-black hole configurations on the cylinder (2007) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

53 arXiv papers cite this in 101 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 · 63 sentences without a confident label

Builds on it 1 paper
  • We review, correct and extend the work of [ref] by constructing lumpy black holes in AdS$_5×S^5$ and AdS$_4×S^7$ with non-uniformity parameters (see Section [ref] for the definition) $λ∼25$ and $λ∼104$ (for $ℓ= 1$) respectively.[Following [ref] , from now on we will exclusively use $λ$ to denote the non-uniformity parameter, as is customary in the field.

    Critical lumpy black holes in AdS${}_p\times S^q$ · Introduction · 0.87

Uses its method or tool 9 papers
Uses its data or values 2 papers
  • Numerical computation gives the following data, [equation] which in principle specifies the solution to the levels $(1,1)$, $(2,0)$, $(0,2)$ and $(3,1)$ if one integrates these data out from the horizon to infinity (although one would require more precision than the two significant figures given here).[ Note that we have chosen the constant circle size `scheme' of [ref] so that the wavenumber in ([ref]) is unperturbed at higher orders in $λ$. ] For these levels the functions $a_n,m$ , $b_n,m$ and $c_n,m$ all exponentially decay asymptotically except for $(2,0)$ which has power law decay going as, [equation] The variation of the mass $M$, area $A$ and surface gravity $κ$ for a non-uniform string takes the form, [equation] where, [equation] and we have now rescaled our solution to have a circle size $L$ again.

    General black holes in Kaluza-Klein theory · Inhomogeneous black strings · 0.85

  • The values of $k_c$ for $d=4,...,14$, as obtained in [ref] , are listed in Table 1 of [ref] .

    Black Holes in Higher-Dimensional Gravity · Uniform black string and Gregory-Laflamme instability · 0.99

Compares with it 2 papers
Cites it as support 1 paper
  • Also, it does not seem possible that one can have other branches than the known ones coming out of the Gregory-Laflamme point (or its $k$-copies) of the uniform black string given the higher-order perturbative analysis of Ref. [ref] .

    Multi-black hole configurations on the cylinder · Discussion of the phase structure · 0.97

Cites it as background 19 papers

Discussion

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