REVIEW
Hydrodynamics of R-charged black holes
High Energy Physics - Theory · Submitted 20 Jan 2006 · 29 references
Pith's verdict: The η/s = 1/4π result at finite R-charge is correct and cross-checked; the reader's decoupling worry doesn't land, and the remaining soft spots are cosmetic.
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Referee report
The paper
The paper claims
The 1/4π shear-viscosity-to-entropy ratio survives nonzero R-charge density in the strongly coupled plasma.
The reading
What carries the argument
The load-bearing device is the reduction of the shear perturbation to a single scalar problem. Because the background is translationally invariant and the shear mode is off-diagonal, the component $h_{xy}$ is claimed to decouple from every other perturbation and to satisfy the same equation as a massless minimally coupled scalar in the charged black hole geometry, Eq. (3.3) of the paper. That reduction turns the Kubo formula into a one-dimensional boundary-value problem whose zero-frequency limit is read off from the near-horizon expansion. The second device is the holographic dictionary for retarded correlators, which converts the quadratic boundary action, including the counterterm $W = -(H^{1/3}/L) sum_i H_i^{-1}$ needed to regulate it, into the two-point functions; the position of the resulting hydrodynamic pole then supplies an independent measurement of $eta/s$ through $D = eta/(epsilon+P)$. The background itself is the Behrndt-Cvetic-Sabra solution, rescaled from a spherical to a translationally invariant horizon so that a hydrodynamic limit is meaningful.
Extended reading notes
Core claim
The central claim is that for $N=4$ $SU(N_c)$ super Yang-Mills at large $N_c$ and large 't Hooft coupling, with three independent R-charge chemical potentials turned on, the shear viscosity divided by the entropy density is exactly $1/4pi$ for every point inside the domain of thermodynamic stability. The authors obtain this twice. First, from the Kubo formula applied to the off-diagonal metric perturbation $h_{xy}$, which they argue obeys the equation of a minimally coupled massless scalar in the charged black hole background; the resulting viscosity, $eta = (pi/8) N^2 T_0^3 sqrt((1+kappa_1)(1+kappa_2)(1+kappa_3))$, divided by the Bekenstein-Hawking entropy density $s = (pi^2/2) N^2 T_0^3 sqrt((1+kappa_1)(1+kappa_2)(1+kappa_3))$, gives $1/4pi$ identically in the charges $kappa_i$. Second, for the technically simpler single-charge black hole, they solve the coupled shear-channel system for the metric and gauge perturbations, extract the retarded correlators of $T^{mu nu}$ and $J^mu$, and find a diffusion pole with $omega = -i D q^2$ and $D = eta/(epsilon+P)$; substituting the thermodynamic identity $epsilon + P = 2(1+kappa)/(2+kappa), T_H s$ reproduces the same ratio. A by-product is the thermal conductivity $kappa_T = pi N^2 T_H^2 (1+kappa)^2 / (kappa(kappa+2))$, which obeys $kappa_T mu^2/(eta T_H) = 8 pi^2$, and the observation that at the stability boundary $mu_c = pi T_H$ both transport coefficients stay finite while their first derivatives diverge with critical exponent $1/2$.
Load-bearing premise
The argument assumes that the off-diagonal metric perturbation $h_{xy}$ decouples from the gauge and scalar perturbations and obeys a single minimally coupled scalar equation even when all three R-charges are nonzero; the paper supports this by a symmetry argument and by the explicit coupled-system check in the single-charge case, but does not demonstrate it directly for the general three-charge background.
Editorial extensions
If this is right
- The universality of $eta/s = 1/4pi$ is not an artefact of zero charge density: it holds across the whole thermodynamically stable region of the three-chemical-potential phase diagram.
- The shear viscosity itself does depend on the charges, growing as $eta = (pi N^2 T_H^3/8)(1 + (1/2)sum_i kappa_i + ...)$ at small charge, so the statement of universality is genuinely about the ratio and not about the viscosity.
- Thermal conductivity for a single charge is fixed by the same data, and the combination $kappa_T mu^2/(eta T_H)$ is a pure number, $8 pi^2$, for all temperatures and chemical potentials.
- Near the boundary of thermodynamic stability at $mu = pi T_H$, both $eta$ and $kappa_T$ remain finite while their derivatives blow up as the inverse square root of the distance to the critical point.
- Because conformal invariance fixes the speed of sound at $1/sqrt{3}$ and the bulk viscosity at zero, the shear channel plus the thermal conductivity essentially completes the first-order transport description of the theory at finite chemical potential.
Reading between the lines
- The pattern suggests the ratio is protected by a symmetry of the gravitational action rather than by the details of the solution; a natural extension is to ask whether any black hole with an Einstein-frame horizon and a standard bifurcate Killing horizon inherits $1/4pi$ regardless of the matter sector, which would make the result a statement about horizon geometry rather than about R-charge.
- The divergence of the derivative of $eta$ at $mu_c = pi T_H$ with exponent $1/2$ is the mean-field value; it would be worth checking whether the same exponent appears in the correlation length of the would-be critical mode, since that would tie the transport anomaly to an underlying second-order transition rather than to a mere edge of the grand canonical ensemble.
- The paper leaves open what phase replaces the unstable large-charge black hole, and it also leaves the three-charge thermal conductivity uncomputed; the second is a bookkeeping exercise in the same coupled system, while the first is a genuinely different question about the phase diagram of the dual gauge theory.
- If the thermal-conductivity relation generalises to all three charges with the same numerical constant, it would give a charge-independent statement analogous to a Wiedemann-Franz law for strongly coupled plasmas, testable in principle against any future lattice or hydrodynamic benchmark at finite density.
Original abstract
We consider hydrodynamics of N=4 supersymmetric SU(N_c) Yang-Mills plasma at a nonzero density of R-charge. In the regime of large N_c and large 't Hooft coupling the gravity dual description involves an asymptotically Anti- de Sitter five-dimensional charged black hole solution of Behrnd, Cvetic and Sabra. We compute the shear viscosity as a function of chemical potentials conjugated to the three U(1) \subset SO(6)_R charges. The ratio of the shear viscosity to entropy density is independent of the chemical potentials and is equal to 1/4\pi. For a single charge black hole we also compute the thermal conductivity, and investigate the critical behavior of the transport coefficients near the boundary of thermodynamic stability.
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 consider hydrodynamics of N=4 supersymmetric SU(N_c) Yang-Mills plasma at a nonzero density of R-charge. In the regime of large N_c and large 't Hooft coupling the gravity dual description involves an asymptotically Anti- de Sitter five-dimensional charged black hole solution of Behrnd, Cvetic and Sabra. We compute the shear viscosity as a function of chemical potentials conjugated to the three U(1) \subset SO(6)_R charges. The ratio of the shear viscosity to entropy density is independent of the chemical potentials and is equal to 1/4\pi. For a single charge black hole we also compute the thermal conductivity, and investigate the critical behavior of the transport coefficients near the boundary of thermodynamic stability.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider hydrodynamics of N=4 supersymmetric SU(N_c) Yang-Mills plasma at a nonzero density of R-charge. | contribution 0.66 | 0.56 | 0.04 | asserted 0.72 |
| s1 | In the regime of large N_c and large 't Hooft coupling the gravity dual description involves an asymptotically Anti- de Sitter five-dimensional charged black hole solution of Behrnd, Cvetic and Sabra. | method 0.75 | 0.11 | 0.03 | asserted 1.00 |
| s2 | We compute the shear viscosity as a function of chemical potentials conjugated to the three U(1) \subset SO(6)_R charges. | contribution 0.94 | 0.96 | 0.08 | asserted 0.99 |
| s3 | The ratio of the shear viscosity to entropy density is independent of the chemical potentials and is equal to 1/4\pi. | result 1.00 | 0.93 | 0.71 | asserted 1.00 |
| s4 | For a single charge black hole we also compute the thermal conductivity, and investigate the critical behavior of the transport coefficients near the boundary of thermodynamic stability. | contribution 0.98 | 0.96 | 0.06 | asserted 0.99 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-th/0601157 by the authors.
References
How later papers used it
90 later papers cite it
3 later papers by other authors are listed below: 2 used its method or tool and 1 compared its own results with it.
Uses its method or tool 2 of 4 papers with this label
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Constructed the Hessian matrix for variations of energy density with respect to entropy and R-charge densities following this paper, giving a necessary condition for local thermodynamic stability.
Superconducting instabilities of R-charged black branes (2012) · 0.97 · paraphrased: the licence does not allow quoting
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Followed this paper's analysis to determine the critical behaviour of transport coefficients at the boundary of thermodynamic instability.
Hydrodynamics of R-charged D1-branes (2010) · 0.99 · paraphrased: the licence does not allow quoting
Compares with it 1 of 11 papers with this label
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Showed its calculation in some detail to contrast its method with the well known work of this paper.
R-Charged Black Holes and Holographic Optics (2013) · 0.96 · paraphrased: the licence does not allow quoting
How the literature uses it
87 arXiv papers cite this in 141 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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The quotient of the shear viscosity over the entropy density exhibits no deviation from the well known result 1/4\pi.
Papers citing it · 83 sentences without a confident label
Builds on it 3 papers
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Though, for single R-charged black holes, thermal properties were discussed in [ref] and [ref] , we extended these results in several ways.
Notes on R-charged black holes near criticality and gauge theory · Conclusion · 0.98
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Although this was worked out in [ref] , we construct here the master variables which decouple the perturbation equations, and allow us to go beyond the hydrodynamic limit, in studying the optical properties.
R-Charged Black Holes and Holographic Optics · R charged Black Holes in Five Dimensions · 0.96
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From here, the natural generalization of the original Minkowskian AdS/CFT prescription [ref] is just[In [ref] the same authors also deal with a mixed operator situation.
Holographic Operator Mixing and Quasinormal Modes on the Brane · Holographic Green's functions · 0.91
Uses its method or tool 4 papers
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Following [ref] one can choose $u^μ$ and $ρ_i$'s so that [equation] Note that [equation] We also have [equation] Using which one gets [equation] Now $_̣μ(su^μ- ∑_i=1^mμ^i/ Tν^μ_i)$ can be interpreted as the divergence of the entropy current which implies right hand side is positive.
Holographic electrical and thermal conductivity in strongly coupled gauge theory with multiple chemical potentials · Thermal conductivity · 0.94
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Constructing the Hessian matrix for variations of the energy density $ε$, ℋ = ∂^2 ε/∂s∂ρ_I with respect to the entropy density $s$ and the R-charge densities $ρ_I$ following [ref] , a necessary condition for local thermodynamic stability is given by ℋ ∝2-q_1-q_2-q_3 + q_1 q_2 q_3 >0.
Superconducting instabilities of R-charged black branes · Multiple black brane branches · 0.97
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Using the Lorentzian AdS/CFT recipe of [ref] , for the retarded two-point functions of the form $⟨T_μνT_αβ⟩$ we find [equation]
Holography and the sound of criticality · Boundary action and retarded Green's functions · 0.95
Uses its data or values 4 papers
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The dependence of the charge density on the chemical potential was found in [ref] .
Holographic spectral functions and diffusion constants for fundamental matter · A simple example: spectral function of $R$ currents in $N=4$ SYM · 0.90
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Beginning with our model theory, we note that we will show in Sec. [ref] (see Eq. [ref]) that to linear order in $μ$ and to lowest order in $$ and $β$ the baryon number density is given (for a choice of normalization, see Section [ref]) by [equation] giving a susceptibility [equation] From Refs. [ref] we know that in $𝒩=4$ SYM theory we have $χ_∞/χ_0 = 1 / 2$ where $χ_∞$ is the susceptibility in the infinite coupling limit (corresponding to $=β=0$ in the above), while $χ_0$ is the susceptibility in the non-interacting limit.
Coupling Constant Corrections in a Holographic Model of Heavy Ion Collisions with Nonzero Baryon Number Density · The holographic model · 0.89
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The explicit calculation performed in Ref. [ref] leads to
Gauge/String Duality, Hot QCD and Heavy Ion Collisions · Holographic susceptibilities · 0.82
Compares with it 10 papers
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We know from the study of the thermal conductivity using hydrodynamics, in such a system without magnetic field, eq(5.1) of [ref] , that the thermal conductivity obey Re κ_xx=(sρ+μT)^2 T Reσ_xx, where $s, ρ$ are entropy density and charge density respectively.
RG flow of transport quantities · A Puzzle · 0.88
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It turns out that our expression for the current is precisely agree with that of Landau-Lifshitz [ref] .
Hydrodynamics with conserved current from the gravity dual · Introduction · 1.00
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Notice that the lowest order term above matches the two-derivative entropy computation of [ref] .
Black holes in five-dimensional gauged supergravity with higher derivatives · Thermodynamics · 0.93
Cites it as support 2 papers
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That such a breakdown can be expected when the characteristic thermal wave length $1/T$ is not small compared with the curvature length scales was noted already in [ref] , and in the AdS/CFT context it has been observed for example in [ref] .
Isotropic AdS/CFT fireball · Thermodynamics on static FRW · 0.98
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Meanwhile, it has been observed that some simpler types of matter do not modify the viscosity–entropy ratio (e.g., [ref] ): Our proposal indicates that this pattern is really a universal feature of black brane hydrodynamics.
The ratio of shear viscosity to entropy density in generalized theories of gravity · Introduction · 0.96
Cites it as background 28 papers
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The holographic study for the system with finite density in hydrodynamic regime was made in [ref] .
Photo-emission rate of sQGP at finite density · Introduction · 0.83
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In the above framework the low energy or long wavelength behavior of $N = 4$ supersymmetric $SU(N_c)$ Yang-Mills was studied in [ref] , in the presence of non-zero chemical potential.
Holographic electrical and thermal conductivity in strongly coupled gauge theory with multiple chemical potentials · Introduction · 0.89
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Thermodynamic instabilities in the STU model have been known for some time [ref] , and the hydrodynamic instability in the solution with a single chemical potential was identified in ref.
Instability in ${\cal N}=4$ supersymmetric Yang-Mills theory at finite density · Introduction · 0.86
Discussion
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