REVIEW
Classical and Thermodynamic Stability of Black Branes
High Energy Physics - Theory · Submitted 9 Apr 2001 · 36 references
Pith's verdict: The mechanism linking the Gregory-Laflamme threshold mode to a Euclidean negative mode is the real contribution here, and it holds up; the 'only if' half of the equivalence is the soft leg, and the paper says so itself.
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Referee report
The paper
The paper claims
A black brane's classical instability may be nothing but negative specific heat, decided by a single thermodynamic sign.
The reading
What carries the argument
Two pieces of machinery carry the argument. The first is a one-parameter family of off-shell Euclidean geometries, built by keeping the spatial sections of a black hole of charge $λ$ but arbitrary temperature while fixing the period $β = 1/T$ at the cavity wall. Along this family the action takes the form $I = βE(x) - S(x)$, so that stationarity reproduces the equilibrium condition $dE/dS = T$ and the second derivative is proportional to $dE/dT$; negative specific heat therefore means the black hole is not a local minimum of the Euclidean action and a negative mode must exist. The second is the mode correspondence itself: the longitudinal s-wave ansatz of Gregory and Laflamme, rewritten in a gauge where the perturbation vanishes along the worldvolume directions, produces eigenvalue equations of exactly the form governing quadratic fluctuations about the Euclidean black brane, with $λ = -μ^{2}$ negative. The paper checks the correspondence numerically against known Schwarzschild negative-mode eigenvalues, converting $λ = -0.19M^{-2}$ into $μ_* = 0.44$, $λ = -1.60r_+^{-2}$ into $μ_* = 0.63$ and $λ = -3.39r_+^{-2}$ into $μ_* = 0.92$, all matching the Gregory-Laflamme critical wavelengths.
Extended reading notes
Core claim
The paper's claim is an equivalence: a non-extremal black brane with a non-compact translational symmetry is classically Gregory-Laflamme unstable if and only if it is locally thermodynamically unstable, i.e. has negative specific heat at fixed charge. The bridge is a correspondence of modes. A classical instability requires a threshold mode that is time-independent, regular on the future horizon and decaying at infinity; dropping its $e^{iμ_i z_i}$ factor and Wick rotating turns it into a Euclidean negative mode of the dimensionally reduced black hole, with eigenvalue $λ = -μ^{2}$. The converse runs the argument backwards: negative specific heat forces a negative mode of the lower-dimensional black hole, which oxidises to a translationally invariant negative mode of the brane, and the equations of motion then force the extra conditions $Z = Y_μ = 0$ needed to reinterpret it as a classical threshold unstable mode. Applied to the black $p$-branes of type II supergravity, the equivalence predicts that the instability disappears near extremality for $p = 1, 2, 4$ but persists all the way down to extremality for $p = 5, 6$; the black D3-brane lies outside the class of solutions treated, and the near-extremal M2 and M5 branes are inferred to be stable by dimensional reduction.
Load-bearing premise
The equivalence is derived only for longitudinal s-wave perturbations with the form-field perturbation set to zero, so the 'only if' direction rests on the assumption that no other channel, higher partial wave or form-field mode, produces a growing mode on a brane whose specific heat has turned positive.
Original abstract
It is argued that many non-extremal black branes exhibit a classical Gregory-Laflamme instability if, and only if, they are locally thermodynamically unstable. For some black branes, the Gregory-Laflamme instability must therefore disappear near extremality. For the black $p$-branes of the type II supergravity theories, the Gregory-Laflamme instability disappears near extremality for $p=1,2,4$ but persists all the way down to extremality for $p=5,6$ (the black D3-brane is not covered by the analysis of this paper). This implies that the instability also vanishes for the near-extremal black M2 and M5-brane 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.
The abstract, sentence by sentence
It is argued that many non-extremal black branes exhibit a classical Gregory-Laflamme instability if, and only if, they are locally thermodynamically unstable. For some black branes, the Gregory-Laflamme instability must therefore disappear near extremality. For the black $p$-branes of the type II supergravity theories, the Gregory-Laflamme instability disappears near extremality for $p=1,2,4$ but persists all the way down to extremality for $p=5,6$ (the black D3-brane is not covered by the analysis of this paper). This implies that the instability also vanishes for the near-extremal black M2 and M5-brane solutions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It is argued that many non-extremal black branes exhibit a classical Gregory-Laflamme instability if, and only if, they are locally thermodynamically unstable. | result 0.92 | 0.49 | 0.86 | asserted 0.59 |
| s1 | For some black branes, the Gregory-Laflamme instability must therefore disappear near extremality. | interpretation 0.84 | 0.36 | 0.15 | asserted 0.79 |
| s2 | For the black $p$-branes of the type II supergravity theories, the Gregory-Laflamme instability disappears near extremality for $p=1,2,4$ but persists all the way down to extremality for $p=5,6$ (the black D3-brane is not covered by the analysis of this paper). | result 0.99 | 0.76 | 0.25 | asserted 1.00 |
| s3 | This implies that the instability also vanishes for the near-extremal black M2 and M5-brane solutions. | interpretation 0.95 | 0.63 | 0.12 | asserted 0.91 |
What was examined, and what was not
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References
How later papers used it
50 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool.
Uses its method or tool 2 papers
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Used the gauge used in this paper, obtaining the transverse condition as an equation of motion.
Negative mode of Schwarzschild black hole from the thermodynamic instability (2010) · 0.93 · paraphrased: the licence does not allow quoting
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Took from this paper the idea of considering a family of off-shell geometries for the Euclidean path integral for which the Euclidean action decreases in a certain direction if the black hole is locally thermodynamically unstable.
Classical and thermodynamic stability of black holes (2010) · 0.94 · paraphrased: the licence does not allow quoting
How the literature uses it
45 arXiv papers cite this in 68 sentences
Papers citing it · 53 sentences without a confident label
Uses its method or tool 2 papers
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The idea, taken from Ref. [ref] , is to consider a family of off-shell geometries for the Euclidean path integral for which one can show that the Euclidean action decreases in a certain direction if the black hole is locally thermodynamically unstable.
Classical and thermodynamic stability of black holes · Negative modes and local thermodynamic stability · 0.94
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(Notice that we use the gauge used in [ref] and the transverse condition is obtained as a equation of motion.) We also did not compute the eigenvalue.
Negative mode of Schwarzschild black hole from the thermodynamic instability · Summary · 0.93
Cites it as support 2 papers
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This means scalarized BHs cannot be a local minimum of the action and therefore must have a negative mode [ref] , despite being dynamically preferred.
A little hair can make a big difference: thermodynamic stability of quasi-bald asymptotically-flat black holes · Local thermodynamic stability · 0.98
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The dynamical modes are eventually equivalent to $10$ dimensional Euclidean negative modes, and both are related to the same negative mode of the ($d+1$ dimensional) small AdS black hole that we mentioned at the beginning of the section [ref] .
String Stars in Anti de Sitter Space · Thermodynamic instabilities · 0.97
Cites it as background 10 papers
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Gross, Perry and Yaffe had previously discovered that the Schwarzschild solution in asymptotically flat spacetime suffers from a Euclidean negative mode [ref] , later shown to be directly related to its negative specific heat capacity [ref] , and thus it can never be a dominant saddle point in the Euclidean path integral which computes the canonical partition function.
New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity · Introduction · 0.96
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The interesting relationship between dynamical and thermodynamic stability, which is well known as the correlated stability conjecture (or the Gubser–Mitra conjecture [ref] ), has been suggested and confirmed for some black objects (strings, branes and so on) by many authors [ref] .
Thermodynamic and dynamical stability of Freund-Rubin compactification · Introduction and summary · 0.89
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Non-supersymmetric black holes often exhibit dynamical instabilities, potentially describing transitions toward more stable phases ( [ref] ).
New analytic solutions of non-SUSY black branes in $10$-D heterotic supergravity and the phase transition · Introduction · 0.87
Discussion
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