REVIEW
Black Strings and p-Branes are Unstable
High Energy Physics - Theory · Submitted 13 Jan 1993 · 17 references
Pith's verdict: The instability is real and the derivation is clean; the numerical existence proof is thin, but the stress-test's claim that the zero/pole discriminator is inverted does not survive a careful read.
T0 · AI review
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Referee report
The paper
The paper claims
Black strings and p-branes — black holes stretched uniformly into extra flat dimensions — are classically unstable to long-wavelength perturbations that would pinch the horizon into separate lumps.
The reading
What carries the argument
The Lichnerowicz operator $Delta_L h_{ab} = -Box h_{ab} - 2 R^{c}{}_{a}{}^{d}{}_{b} h_{cd}$, applied to a Kaluza-Klein decomposition of the perturbation into scalar, vector and tensor pieces with respect to the $D$-dimensional Schwarzschild factor. The load-bearing step is the choice of initial data surface: instead of a surface through the neck of the Schwarzschild wormhole, the authors use one ending on the future event horizon, following Whitt's treatment of the mathematically identical fourth-order gravity problem. That choice avoids the fixed point of the isometries used in the mode decomposition and so avoids the mode-superposition obstruction that made an earlier analysis inconclusive. The instability is then read off as a zero of the ratio $R = A_-/A_+$ between the two allowed horizon behaviours, located numerically with an adaptive-step Runge-Kutta integrator.
Extended reading notes
Core claim
The paper claims that uncharged black strings and p-branes are classically unstable. Working with the metric $ds^2 = -V dt^2 + V^{-1} dr^2 + r^2 dOmega^2_{D-2} + dx^i dx_i$ with $V = 1-(r_+/r)^{D-3}$, the authors perturb the Einstein equations in transverse trace-free gauge and restrict to a $D$-dimensional s-wave of the form $h_{mu nu} = e^{Omega t} e^{i mu_i x^i} H_{mu nu}$ with all mixed and purely extra-dimensional components zero. The Lichnerowicz equation then reduces to $(Delta^D_L + sum_i mu_i^2) h_{mu nu} = 0$, and the gauge conditions collapse it to a single second-order ordinary differential equation for $H_{tr}$. Solutions behave as $e^{-sqrt{Omega^2+mu^2}, r}$ at infinity and as $(r-r_+)^{-1 pm r_+ Omega/(D-3)}$ at the horizon; the boundary conditions select the decaying solution at infinity and the positive root at the future horizon. Integrating inward and forming the ratio $R = A_-/A_+$ of the two horizon behaviours, the authors report zeros of $R$ — hence regular unstable solutions — for a range of $Omega$ and $mu$ for every $4 le D le 9$. Because $mu^2 neq 0$, the perturbation cannot be pure gauge, so the instability is physical rather than a coordinate artifact. The authors note that compactifying the extra dimensions on a scale below the largest unstable wavelength quantises $mu_i$ above the unstable range and removes the instability, so large four-dimensional astrophysical black holes are unaffected.
Load-bearing premise
The existence of unstable modes rests on a numerical zero of the horizon amplitude ratio in exactly the regime where the authors say an analytic argument is unavailable, and the horizon is a coordinate singularity there, so the sign change they track could in principle be a numerical artifact rather than a real growing mode.
Editorial extensions
If this is right
- Black strings and p-branes are not stable endpoints of gravitational collapse: a linearised perturbation grows in the exterior spacetime, so the static brane solution cannot persist.
- If the horizon fragments, the endpoint is a chain of separate black holes; periodic black hole solutions are already known, so a final state is available — and the intermediate configuration exposes a naked singularity, which would violate cosmic censorship.
- A less spectacular alternative is that the instability prevents such branes from ever forming from collapse in the first place.
- Compactifying the extra dimensions on a scale smaller than the largest unstable wavelength pushes the allowed $mu_i$ above the unstable band, so small 'black doughnuts' are stable and contemporary four-dimensional black holes are untouched.
- Charged branes are conjectured to be unstable as well, and the axionic case may end in a line of black holes threaded by a cosmic axion string.
Reading between the lines
- The threshold is set by a ratio, not an absolute scale: the unstable band in $mu$ scales as $1/r_+$, so for a fixed compactification radius $R_c$ there is a critical horizon size below which the brane is stable — a clean, checkable statement about which black objects can exist in a given compactification.
- Because the entropy comparison is thermodynamic rather than dynamical, the same reasoning should extend to any horizon with a translationally invariant flat direction, including boosted branes and branes embedded in curved backgrounds, which the paper does not examine.
- The paper reports isolated points in the $(Omega, mu)$ plane rather than the boundary of the unstable region; a reader could ask whether the unstable band closes at small $mu$ or runs all the way to $mu = 0$, which would change how the compactification escape clause is stated.
- The cosmic-censorship conclusion depends on the perturbed horizon shrinking rather than oscillating in place; a nonlinear evolution, not available here, is what would decide between fragmentation and a ringing response.
Original abstract
We investigate the evolution of small perturbations around black strings and branes which are low energy solutions of string theory. For simplicity we focus attention on the zero charge case and show that there are unstable modes for a range of time frequency and wavelength in the extra $10-D$ dimensions. These perturbations can be stabililized if the extra dimensions are compactified to a scale smaller than the minimum wavelength for which instability occurs and thus will not affect large astrophysical black holes in four dimensions. We comment on the implications of this result for the Cosmic Censorship Hypothesis.
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.
The abstract, sentence by sentence
We investigate the evolution of small perturbations around black strings and branes which are low energy solutions of string theory. For simplicity we focus attention on the zero charge case and show that there are unstable modes for a range of time frequency and wavelength in the extra $10-D$ dimensions. These perturbations can be stabililized if the extra dimensions are compactified to a scale smaller than the minimum wavelength for which instability occurs and thus will not affect large astrophysical black holes in four dimensions. We comment on the implications of this result for the Cosmic Censorship Hypothesis.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the evolution of small perturbations around black strings and branes which are low energy solutions of string theory. | contribution 0.95 | 0.90 | 0.04 | unclear 0.57 |
| s1 | For simplicity we focus attention on the zero charge case and show that there are unstable modes for a range of time frequency and wavelength in the extra $10-D$ dimensions. | result 0.85 | 0.92 | 0.45 | asserted 1.00 |
| s2 | These perturbations can be stabililized if the extra dimensions are compactified to a scale smaller than the minimum wavelength for which instability occurs and thus will not affect large astrophysical black holes in four dimensions. | interpretation 0.60 | 0.75 | 0.26 | hedged possibility 0.57 |
| s3 | We comment on the implications of this result for the Cosmic Censorship Hypothesis. | interpretation 0.99 | 0.35 | 0.04 | unclear 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
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- P abstract
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- T body sentences
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- R cross-paper relations
References
How later papers used it
360 later papers cite it
4 later papers by other authors are listed below: 2 used its method or tool and 2 compared their own results with it.
Uses its method or tool 2 of 5 papers with this label
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Followed this paper by expanding the functions A, B and C in a small parameter and considering a Fourier series in the z coordinate.
Einstein-Gauss-Bonnet black strings (2010) · 1.00 · paraphrased: the licence does not allow quoting
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Then, following [ref] we perform an expansion of the functions $A,B,C,ψ$ in terms of a small parameter $ϵ$ and consider a Fourier series in the $z$-coordinate with
Gregory-Laflamme-type instability of boson strings and related phases in D=5 Kaluza-Klein theory (2025) · The uniform strings and the zero mode · 0.98
Compares with it 2 of 38 papers with this label
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Said that its own solution corroborates with this paper for the specific values of Omega equal to 4/a and -4/a.
Effect of temperature gradient on heavy quark anti-quark potential using gravity dual model (2016) · 0.99 · paraphrased: the licence does not allow quoting
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Said that its own numerical values of the quantity decrease with d, which is also the behavior with d seen in this paper's black strings.
String Theory in a Pinch: Resolving the Gregory-Laflamme Singularity (2024) · 0.95 · paraphrased: the licence does not allow quoting
How the literature uses it
325 arXiv papers cite this in 500 sentences
Papers whose main result relates to this one's
States a conflicting result 1 paper
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No instability has been found.
Papers citing it · 328 sentences without a confident label
Contests it 1 paper
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However, [ref] showed that for small $𝔪$ the dispersion relations under the assumption of vanishing shear modulus agree with numerical results.[We believe that the discrepancy between hydrodynamic and numerical results identified in [ref] is due to the fact that [ref] assumed a vanishing shear modulus in their hydrodynamic calculations.] According to the analysis of [ref] , it is expected that a Gregory-Laflamme like instability [ref] is present for specific values of the parameters, including when $ℳ=0$.
Viscoelastic hydrodynamics and holography · Thermal state in $D=4$ · contest 0.74
Builds on it 2 papers
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Considering the metric [ref], it was shown by Gregory [ref] that the instability for the flat black string initially discovered in [ref] simply generalizes to this warped case.
Quantum Effects in Black Holes from the Schwarzschild Black String? · GL instability and Hawking radiation · 0.95
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A corollary of our approach is the first data for the Gregory-Laflamme instability [ref] for rotating black branes.
Classical and thermodynamic stability of black holes · Introduction · 0.97
Uses its method or tool 4 papers
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Taking a transverse tracefree gauge, $h^b_b = 0$ and $∇_b h^b_c = 0$, the equation, $Δ_L(ΔH_bc) = 0$, reduces to [ref] : [equation] with $R_bdc^e$, being the Riemann curvature tensor.
Effect of temperature gradient on heavy quark anti-quark potential using gravity dual model · Perturbation of Black String · 0.82
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Then, following [ref] we perform an expansion of the functions $A,B,C,ψ$ in terms of a small parameter $ϵ$ and consider a Fourier series in the $z$-coordinate with
Gregory-Laflamme-type instability of boson strings and related phases in D=5 Kaluza-Klein theory · The uniform strings and the zero mode · 0.98
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Then, following [ref] we perform an expansion of the functions $A,B,C$ in terms of a small parameter $ϵ$ and consider a Fourier series in the $z$ coordinate.
Einstein-Gauss-Bonnet black strings · Perturbations of the uniform metric · 1.00
Uses its data or values 3 papers
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(Note that the uniform solutions have $M=Lm/G$.) The limit $μ_2→0$ corresponds to the Schwarzschild black string solution in a Kaluza-Klein theory, in which case the GL unstable mode occurs for $μ_1 ≃0.0649519$ [ref] .
The Gregory-Laflamme instability and non-uniform generalizations of NUT strings · The Gregory-Laflamme instability · 0.81
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Solving $H_tr$-equation, one found that the GL instability bound is given by $0<k_z<0.876/r_+$ which is a long wavelength instability [ref] .
Two instabilities of Schwarzschild-AdS black holes in Einstein-Weyl-scalar theory · Introduction · 0.92
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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 34 papers
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We find that, at the transition [ref] corresponding to the BH/BS transition [ref] , there is a topology change which strikingly resembles the counterpart on the gravity side [ref] .
On the shape of a D-brane bound state and its topology change · Matrix quantum mechanics and black hole/black string topology change · 0.94
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This is in accordance with the results of [ref] .
The Schwarzschild-Black String AdS Soliton: Instability and Holographic Heat Transport · Reduction to Gregory-Laflamme Instability · 0.85
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For these cases, our numerical results give values of $_*$ that decrease with $d$, namely, [equation] which is also the behavior with $d$ seen in black strings [ref] .
String Theory in a Pinch: Resolving the Gregory-Laflamme Singularity · Zero mode · 0.95
Cites it as support 1 paper
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These studies have been performed only in quadratic gravity [ref] and for higher-dimensional scenarios [ref] , confirming the fragmentation instability in these cases.
When Black Holes Can Wear Pants · Black holes do wear pants beyond GR · 0.99
Cites it as background 113 papers
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Almost 30 years ago, Gregory and Laflamme (GL) discovered that black strings (and black $p$-branes in general) can be unstable to long wavelength perturbations that break the translational symmetry along the compact extra dimensions [ref] .
The endpoint of the Gregory-Laflamme instability of black strings revisited · Introduction · 0.97
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It is known as the Gregory-Laflamme (GL) instability [ref] .
String Stars in Anti de Sitter Space · Thermodynamic instabilities · 0.82
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The two classes have different thermodynamic behaviors, and there is a first-order Gregory–Laflamme [ref] phase transition between them in the gravity dual.
Testing holography using lattice super-Yang--Mills on a 2-torus · Introduction · 0.92
Discussion
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